# The GNU Emacs Calculator

*Calc* is an advanced desk calculator and mathematical tool
written by Dave Gillespie that runs as part of the GNU Emacs environment.

This manual, also written (mostly) by Dave Gillespie, is divided into three major parts: “Getting Started,” the “Calc Tutorial,” and the “Calc Reference.” The Tutorial introduces all the major aspects of Calculator use in an easy, hands-on way. The remainder of the manual is a complete reference to the features of the Calculator.

This file documents Calc, the GNU Emacs calculator, included with GNU Emacs 29.1.

Copyright © 1990–1991, 2001–2023 Free Software Foundation, Inc.

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.3 or any later version published by the Free Software Foundation; with the Invariant Sections being just “GNU GENERAL PUBLIC LICENSE”, with the Front-Cover Texts being “A GNU Manual,” and with the Back-Cover Texts as in (a) below. A copy of the license is included in the section entitled “GNU Free Documentation License.”

(a) The FSF’s Back-Cover Text is: “You have the freedom to copy and modify this GNU manual.”

## Table of Contents

- 1 Getting Started
- 2 Tutorial
- 2.1 Basic Tutorial
- 2.2 Arithmetic Tutorial
- 2.3 Vector/Matrix Tutorial
- 2.4 Types Tutorial
- 2.5 Algebra and Calculus Tutorial
- 2.6 Programming Tutorial
- 2.7 Answers to Exercises
- 2.7.1 RPN Tutorial Exercise 1
- 2.7.2 RPN Tutorial Exercise 2
- 2.7.3 RPN Tutorial Exercise 3
- 2.7.4 RPN Tutorial Exercise 4
- 2.7.5 Algebraic Entry Tutorial Exercise 1
- 2.7.6 Algebraic Entry Tutorial Exercise 2
- 2.7.7 Algebraic Entry Tutorial Exercise 3
- 2.7.8 Modes Tutorial Exercise 1
- 2.7.9 Modes Tutorial Exercise 2
- 2.7.10 Modes Tutorial Exercise 3
- 2.7.11 Modes Tutorial Exercise 4
- 2.7.12 Arithmetic Tutorial Exercise 1
- 2.7.13 Arithmetic Tutorial Exercise 2
- 2.7.14 Vector Tutorial Exercise 1
- 2.7.15 Vector Tutorial Exercise 2
- 2.7.16 Matrix Tutorial Exercise 1
- 2.7.17 Matrix Tutorial Exercise 2
- 2.7.18 Matrix Tutorial Exercise 3
- 2.7.19 List Tutorial Exercise 1
- 2.7.20 List Tutorial Exercise 2
- 2.7.21 List Tutorial Exercise 3
- 2.7.22 List Tutorial Exercise 4
- 2.7.23 List Tutorial Exercise 5
- 2.7.24 List Tutorial Exercise 6
- 2.7.25 List Tutorial Exercise 7
- 2.7.26 List Tutorial Exercise 8
- 2.7.27 List Tutorial Exercise 9
- 2.7.28 List Tutorial Exercise 10
- 2.7.29 List Tutorial Exercise 11
- 2.7.30 List Tutorial Exercise 12
- 2.7.31 List Tutorial Exercise 13
- 2.7.32 List Tutorial Exercise 14
- 2.7.33 Types Tutorial Exercise 1
- 2.7.34 Types Tutorial Exercise 2
- 2.7.35 Types Tutorial Exercise 3
- 2.7.36 Types Tutorial Exercise 4
- 2.7.37 Types Tutorial Exercise 5
- 2.7.38 Types Tutorial Exercise 6
- 2.7.39 Types Tutorial Exercise 7
- 2.7.40 Types Tutorial Exercise 8
- 2.7.41 Types Tutorial Exercise 9
- 2.7.42 Types Tutorial Exercise 10
- 2.7.43 Types Tutorial Exercise 11
- 2.7.44 Types Tutorial Exercise 12
- 2.7.45 Types Tutorial Exercise 13
- 2.7.46 Types Tutorial Exercise 14
- 2.7.47 Types Tutorial Exercise 15
- 2.7.48 Algebra Tutorial Exercise 1
- 2.7.49 Algebra Tutorial Exercise 2
- 2.7.50 Algebra Tutorial Exercise 3
- 2.7.51 Algebra Tutorial Exercise 4
- 2.7.52 Rewrites Tutorial Exercise 1
- 2.7.53 Rewrites Tutorial Exercise 2
- 2.7.54 Rewrites Tutorial Exercise 3
- 2.7.55 Rewrites Tutorial Exercise 4
- 2.7.56 Rewrites Tutorial Exercise 5
- 2.7.57 Rewrites Tutorial Exercise 6
- 2.7.58 Programming Tutorial Exercise 1
- 2.7.59 Programming Tutorial Exercise 2
- 2.7.60 Programming Tutorial Exercise 3
- 2.7.61 Programming Tutorial Exercise 4
- 2.7.62 Programming Tutorial Exercise 5
- 2.7.63 Programming Tutorial Exercise 6
- 2.7.64 Programming Tutorial Exercise 7
- 2.7.65 Programming Tutorial Exercise 8
- 2.7.66 Programming Tutorial Exercise 9
- 2.7.67 Programming Tutorial Exercise 10
- 2.7.68 Programming Tutorial Exercise 11
- 2.7.69 Programming Tutorial Exercise 12

- 3 Introduction
- 4 Data Types
- 5 Stack and Trail Commands
- 6 Mode Settings
- 6.1 General Mode Commands
- 6.2 Precision
- 6.3 Inverse and Hyperbolic Flags
- 6.4 Calculation Modes
- 6.5 Simplification Modes
- 6.6 Declarations
- 6.7 Display Modes
- 6.8 Language Modes
- 6.9 The
`Modes`

Variable - 6.10 The Calc Mode Line

- 7 Arithmetic Functions
- 8 Scientific Functions
- 9 Vector/Matrix Functions
- 10 Algebra
- 10.1 Selecting Sub-Formulas
- 10.2 Algebraic Manipulation
- 10.3 Simplifying Formulas
- 10.4 Polynomials
- 10.5 Calculus
- 10.6 Solving Equations
- 10.7 Numerical Solutions
- 10.8 Curve Fitting
- 10.9 Summations
- 10.10 Logical Operations
- 10.11 Rewrite Rules
- 10.11.1 Entering Rewrite Rules
- 10.11.2 Basic Rewrite Rules
- 10.11.3 Conditional Rewrite Rules
- 10.11.4 Algebraic Properties of Rewrite Rules
- 10.11.5 Other Features of Rewrite Rules
- 10.11.6 Composing Patterns in Rewrite Rules
- 10.11.7 Nested Formulas with Rewrite Rules
- 10.11.8 Multi-Phase Rewrite Rules
- 10.11.9 Selections with Rewrite Rules
- 10.11.10 Matching Commands
- 10.11.11 Automatic Rewrites
- 10.11.12 Debugging Rewrites
- 10.11.13 Examples of Rewrite Rules

- 11 Operating on Units
- 12 Storing and Recalling
- 13 Graphics
- 14 Kill and Yank Functions
- 15 Keypad Mode
- 16 Embedded Mode
- 17 Programming
- 17.1 Creating User Keys
- 17.2 Programming with Keyboard Macros
- 17.3 Invocation Macros
- 17.4 Programming with Formulas
- 17.5 Programming with Lisp

- Appendix A GNU GENERAL PUBLIC LICENSE
- Appendix B GNU Free Documentation License
- Appendix C Customizing Calc
- Appendix D Reporting Bugs
- Appendix E Calc Summary
- Index of Key Sequences
- Index of Calculator Commands
- Index of Algebraic Functions
- Concept Index
- Index of Variables
- Index of Lisp Math Functions

## Short Table of Contents

- 1 Getting Started
- 2 Tutorial
- 3 Introduction
- 4 Data Types
- 5 Stack and Trail Commands
- 6 Mode Settings
- 7 Arithmetic Functions
- 8 Scientific Functions
- 9 Vector/Matrix Functions
- 10 Algebra
- 11 Operating on Units
- 12 Storing and Recalling
- 13 Graphics
- 14 Kill and Yank Functions
- 15 Keypad Mode
- 16 Embedded Mode
- 17 Programming
- Appendix A GNU GENERAL PUBLIC LICENSE
- Appendix B GNU Free Documentation License
- Appendix C Customizing Calc
- Appendix D Reporting Bugs
- Appendix E Calc Summary
- Index of Key Sequences
- Index of Calculator Commands
- Index of Algebraic Functions
- Concept Index
- Index of Variables
- Index of Lisp Math Functions

## 1 Getting Started

This chapter provides a general overview of Calc, the GNU Emacs Calculator: What it is, how to start it and how to exit from it, and what are the various ways that it can be used.

- What is Calc?
- About This Manual
- Notations Used in This Manual
- A Demonstration of Calc
- Using Calc
- History and Acknowledgments

### 1.1 What is Calc?

*Calc* is an advanced calculator and mathematical tool that runs as
part of the GNU Emacs environment. Very roughly based on the HP-28/48
series of calculators, its many features include:

- Choice of algebraic or Reverse Polish notation (RPN), i.e. stack-based, entry of calculations.
- Arbitrary precision integers and floating-point numbers.
- Arithmetic on rational numbers, complex numbers (rectangular and polar), error forms with standard deviations, open and closed intervals, vectors and matrices, dates and times, infinities, sets, quantities with units, and algebraic formulas.
- Mathematical operations such as logarithms and trigonometric functions.
- Programmer’s features (bitwise operations, non-decimal numbers).
- Financial functions such as future value and internal rate of return.
- Number theoretical features such as prime factorization and arithmetic
modulo
`m`for any`m`. - Algebraic manipulation features, including symbolic calculus.
- Moving data to and from regular editing buffers.
- Embedded mode for manipulating Calc formulas and data directly inside any editing buffer.
- Graphics using GNUPLOT, a versatile (and free) plotting program.
- Easy programming using keyboard macros, algebraic formulas, algebraic rewrite rules, or extended Emacs Lisp.

Calc tries to include a little something for everyone; as a result it is large and might be intimidating to the first-time user. If you plan to use Calc only as a traditional desk calculator, all you really need to read is the “Getting Started” chapter of this manual and possibly the first few sections of the tutorial. As you become more comfortable with the program you can learn its additional features. Calc does not have the scope and depth of a fully-functional symbolic math package, but Calc has the advantages of convenience, portability, and freedom.

### 1.2 About This Manual

This document serves as a complete description of the GNU Emacs Calculator. It works both as an introduction for novices and as a reference for experienced users. While it helps to have some experience with GNU Emacs in order to get the most out of Calc, this manual ought to be readable even if you don’t know or use Emacs regularly.

This manual is divided into three major parts: the “Getting Started” chapter you are reading now, the Calc tutorial, and the Calc reference manual.

If you are in a hurry to use Calc, there is a brief “demonstration” below which illustrates the major features of Calc in just a couple of pages. If you don’t have time to go through the full tutorial, this will show you everything you need to know to begin. See A Demonstration of Calc.

The tutorial chapter walks you through the various parts of Calc with lots of hands-on examples and explanations. If you are new to Calc and you have some time, try going through at least the beginning of the tutorial. The tutorial includes about 70 exercises with answers. These exercises give you some guided practice with Calc, as well as pointing out some interesting and unusual ways to use its features.

The reference section discusses Calc in complete depth. You can read the reference from start to finish if you want to learn every aspect of Calc. Or, you can look in the table of contents or the Concept Index to find the parts of the manual that discuss the things you need to know.

Every Calc keyboard command is listed in the Calc Summary, and also
in the Key Index. Algebraic functions, `M-x` commands, and
variables also have their own indices.

You can access this manual on-line at any time within Calc by pressing
the `h i` key sequence. Outside of the Calc window, you can press
`C-x * i` to read the manual on-line. From within Calc the command
`h t` will jump directly to the Tutorial; from outside of Calc the
command `C-x * t` will jump to the Tutorial and start Calc if
necessary. Pressing `h s` or `C-x * s` will take you directly
to the Calc Summary. Within Calc, you can also go to the part of the
manual describing any Calc key, function, or variable using
`h k`, `h f`, or `h v`, respectively. See Help Commands.

The Calc manual can be printed, but because the manual is so large, you
should only make a printed copy if you really need it. To print the
manual, you will need the TeX typesetting program (this is a free
program by Donald Knuth at Stanford University) as well as the
`texindex` program and `texinfo.tex` file, both of which can
be obtained from the FSF as part of the `texinfo`

package.
To print the Calc manual in one huge tome, you will need the
Emacs source, which contains the source code to this manual,
`calc.texi`. Change to the `doc/misc` subdirectory of the
Emacs source distribution, which contains source code for this manual,
and type `make calc.pdf`. (Don’t worry if you get some “overfull
box” warnings while TeX runs.) The result will be this entire
manual as a pdf file.

### 1.3 Notations Used in This Manual

This section describes the various notations that are used throughout the Calc manual.

In keystroke sequences, uppercase letters mean you must hold down
the shift key while typing the letter. Keys pressed with Control
held down are shown as `C-x`. Keys pressed with Meta held down
are shown as `M-x`. Other notations are `RET` for the
Return key, `SPC` for the space bar, `TAB` for the Tab key,
`DEL` for the Delete key, and `LFD` for the Line-Feed key.
The `DEL` key is called Backspace on some keyboards, it is
whatever key you would use to correct a simple typing error when
regularly using Emacs.

(If you don’t have the `LFD` or `TAB` keys on your keyboard,
the `C-j` and `C-i` keys are equivalent to them, respectively.
If you don’t have a Meta key, look for Alt or Extend Char. You can
also press `ESC` or `C-[` first to get the same effect, so
that `M-x`, ` ESC x`, and

`C-[ x`are all equivalent.)

Sometimes the `RET` key is not shown when it is “obvious”
that you must press `RET` to proceed. For example, the `RET`
is usually omitted in key sequences like `M-x calc-keypad RET`.

Commands are generally shown like this: `p` (`calc-precision`

)
or `C-x * k` (`calc-keypad`

). This means that the command is
normally used by pressing the `p` key or `C-x * k` key sequence,
but it also has the full-name equivalent shown, e.g., `M-x calc-precision`.

Commands that correspond to functions in algebraic notation
are written: `C` (`calc-cos`

) [`cos`

]. This means
the `C` key is equivalent to `M-x calc-cos`, and that
the corresponding function in an algebraic-style formula would
be ‘`cos( x)`’.

A few commands don’t have key equivalents: `calc-sincos`

[`sincos`

].

### 1.4 A Demonstration of Calc

This section will show some typical small problems being solved with Calc. The focus is more on demonstration than explanation, but everything you see here will be covered more thoroughly in the Tutorial.

To begin, start Emacs if necessary (usually the command `emacs`

does this), and type `C-x * c` to start the
Calculator. (You can also use `M-x calc` if this doesn’t work.
See Starting Calc, for various ways of starting the Calculator.)

Be sure to type all the sample input exactly, especially noting the
difference between lower-case and upper-case letters. Remember,
`RET`, `TAB`, `DEL`, and `SPC` are the Return, Tab,
Delete, and Space keys.

**RPN calculation.** In Reverse Polish notation (RPN), you type
the input number(s) first, then the command to operate on the numbers.

Type `2 RET 3 + Q` to compute
the square root of 2+3, which is 2.2360679775.

Type `P 2 ^` to compute
the value of ‘`pi`’ squared, 9.86960440109.

Type `TAB` to exchange the order of these two results.

Type `- I H S` to subtract these results and compute the Inverse
Hyperbolic sine of the difference, 2.72996136574.

Type `DEL` to erase this result.

**Algebraic calculation.** You can also enter calculations using
conventional “algebraic” notation. To enter an algebraic formula,
use the apostrophe key.

Type `' sqrt(2+3) RET` to compute
the square root of 2+3.

Type `' pi^2 RET` to enter
‘

`pi`’ squared. To evaluate this symbolic formula as a number, type

`=`.

Type `' arcsinh($ - $$) RET` to subtract the second-most-recent
result from the most-recent and compute the Inverse Hyperbolic sine.

**Keypad mode.** If you are using the X window system, press
`C-x * k` to get Keypad mode. (If you don’t use X, skip to
the next section.)

Click on the `2`, `ENTER`, `3`, `+`, and `SQRT`
“buttons” using your left mouse button.

Click on `PI`, `2`, and `y^x`

.

Click on `INV`, then `ENTER` to swap the two results.

Click on `-`, `INV`, `HYP`, and `SIN`.

Click on `<-` to erase the result, then click `OFF` to turn
the Keypad Calculator off.

**Grabbing data.** Type `C-x * x` if necessary to exit Calc.
Now select the following numbers as an Emacs region: “Mark” the
front of the list by typing `C- SPC` or

`C-@`there, then move to the other end of the list. (Either get this list from the on-line copy of this manual, accessed by

`C-x * i`, or just type these numbers into a scratch file.) Now type

`C-x * g`to “grab” these numbers into Calc.

1.23 1.97 1.6 2 1.19 1.08

The result ‘`[1.23, 1.97, 1.6, 2, 1.19, 1.08]`’ is a Calc “vector.”
Type `V R +` to compute the sum of these numbers.

Type `U` to Undo this command, then type `V R *` to compute
the product of the numbers.

You can also grab data as a rectangular matrix. Place the cursor on
the upper-leftmost ‘`1`’ and set the mark, then move to just after
the lower-right ‘`8`’ and press `C-x * r`.

Type `v t` to transpose this
3x2
matrix into a
2x3
matrix. Type `v u` to unpack the rows into two separate
vectors. Now type `V R + TAB V R +` to compute the sums
of the two original columns. (There is also a special
grab-and-sum-columns command,

`C-x * :`.)

**Units conversion.** Units are entered algebraically.
Type `' 43 mi/hr RET` to enter the quantity 43 miles-per-hour.
Type

`u c km/hr`. Type

`RET``u c m/s`.

`RET`**Date arithmetic.** Type `t N` to get the current date and
time. Type `90 +` to find the date 90 days from now. Type
`' <25 dec 87> RET` to enter a date, then

`- 7 /`to see how many weeks have passed since then.

**Algebra.** Algebraic entries can also include formulas
or equations involving variables. Type `' [x + y = a, x y = 1] RET`
to enter a pair of equations involving three variables.
(Note the leading apostrophe in this example; also, note that the space
in ‘

`x y`’ is required.) Type

`a S x,y`to solve these equations for the variables ‘

`RET``x`’ and ‘

`y`’.

Type `d B` to view the solutions in more readable notation.
Type `d C` to view them in C language notation, `d T`
to view them in the notation for the TeX typesetting system,
and `d L` to view them in the notation for the LaTeX typesetting
system. Type `d N` to return to normal notation.

Type `7.5`, then `s l a RET` to let ‘

`a = 7.5`’ in these formulas. (That’s the letter

`l`, not the numeral

`1`.)

**Help functions.** You can read about any command in the on-line
manual. Type `C-x * c` to return to Calc after each of these
commands: `h k t N` to read about the `t N` command,
`h f sqrt RET` to read about the

`sqrt`

function, and
`h s`to read the Calc summary.

Press `DEL` repeatedly to remove any leftover results from the stack.
To exit from Calc, press `q` or `C-x * c` again.

### 1.5 Using Calc

Calc has several user interfaces that are specialized for different kinds of tasks. As well as Calc’s standard interface, there are Quick mode, Keypad mode, and Embedded mode.

- Starting Calc
- The Standard Calc Interface
- Quick Mode (Overview)
- Keypad Mode (Overview)
- Standalone Operation
- Embedded Mode (Overview)
- Other
`C-x *`Commands

#### 1.5.1 Starting Calc

On most systems, you can type `C-x *` to start the Calculator.
The key sequence `C-x *` is bound to the command `calc-dispatch`

,
which can be rebound if convenient (see Customizing Calc).

When you press `C-x *`, Emacs waits for you to press a second key to
complete the command. In this case, you will follow `C-x *` with a
letter (upper- or lower-case, it doesn’t matter for `C-x *`) that says
which Calc interface you want to use.

To get Calc’s standard interface, type `C-x * c`. To get
Keypad mode, type `C-x * k`. Type `C-x * ?` to get a brief
list of the available options, and type a second `?` to get
a complete list.

To ease typing, `C-x * *` also works to start Calc. It starts the
same interface (either `C-x * c` or `C-x * k`) that you last
used, selecting the `C-x * c` interface by default.

If `C-x *` doesn’t work for you, you can always type explicit
commands like `M-x calc` (for the standard user interface) or
`M-x calc-keypad` (for Keypad mode). First type `M-x`
(that’s Meta with the letter `x`), then, at the prompt,
type the full command (like `calc-keypad`) and press Return.

The same commands (like `C-x * c` or `C-x * *`) that start
the Calculator also turn it off if it is already on.

#### 1.5.2 The Standard Calc Interface

Calc’s standard interface acts like a traditional RPN calculator,
operated by the normal Emacs keyboard. When you type `C-x * c`
to start the Calculator, the Emacs screen splits into two windows
with the file you were editing on top and Calc on the bottom.

... --**-Emacs: myfile (Fundamental)----All---------------------- --- Emacs Calculator Mode --- |Emacs Calculator Trail 2: 17.3 | 17.3 1: -5 | 3 . | 2 | 4 | * 8 | ->-5 | --%*-Calc: 12 Deg (Calculator)----All----- --%*- *Calc Trail*

In this figure, the mode-line for `myfile` has moved up and the
“Calculator” window has appeared below it. As you can see, Calc
actually makes two windows side-by-side. The lefthand one is
called the *stack window* and the righthand one is called the
*trail window.* The stack holds the numbers involved in the
calculation you are currently performing. The trail holds a complete
record of all calculations you have done. In a desk calculator with
a printer, the trail corresponds to the paper tape that records what
you do.

In this case, the trail shows that four numbers (17.3, 3, 2, and 4)
were first entered into the Calculator, then the 2 and 4 were
multiplied to get 8, then the 3 and 8 were subtracted to get *-5*.
(The ‘`>`’ symbol shows that this was the most recent calculation.)
The net result is the two numbers 17.3 and *-5* sitting on the stack.

Most Calculator commands deal explicitly with the stack only, but there is a set of commands that allow you to search back through the trail and retrieve any previous result.

Calc commands use the digits, letters, and punctuation keys.
Shifted (i.e., upper-case) letters are different from lowercase
letters. Some letters are *prefix* keys that begin two-letter
commands. For example, `e` means “enter exponent” and shifted
`E` means ‘`e^x`’. With the `d` (“display modes”) prefix
the letter “e” takes on very different meanings: `d e` means
“engineering notation” and `d E` means “*eqn* language mode.”

There is nothing stopping you from switching out of the Calc
window and back into your editing window, say by using the Emacs
`C-x o` (`other-window`

) command. When the cursor is
inside a regular window, Emacs acts just like normal. When the
cursor is in the Calc stack or trail windows, keys are interpreted
as Calc commands.

When you quit by pressing `C-x * c` a second time, the Calculator
windows go away but the actual Stack and Trail are not gone, just
hidden. When you press `C-x * c` once again you will get the
same stack and trail contents you had when you last used the
Calculator.

The Calculator does not remember its state between Emacs sessions.
Thus if you quit Emacs and start it again, `C-x * c` will give you
a fresh stack and trail. There is a command (`m m`) that lets
you save your favorite mode settings between sessions, though.
One of the things it saves is which user interface (standard or
Keypad) you last used; otherwise, a freshly started Emacs will
always treat `C-x * *` the same as `C-x * c`.

The `q` key is another equivalent way to turn the Calculator off.

If you type `C-x * b` first and then `C-x * c`, you get a
full-screen version of Calc (`full-calc`

) in which the stack and
trail windows are still side-by-side but are now as tall as the whole
Emacs screen. When you press `q` or `C-x * c` again to quit,
the file you were editing before reappears. The `C-x * b` key
switches back and forth between “big” full-screen mode and the
normal partial-screen mode.

Finally, `C-x * o` (`calc-other-window`

) is like `C-x * c`
except that the Calc window is not selected. The buffer you were
editing before remains selected instead. If you are in a Calc window,
then `C-x * o` will switch you out of it, being careful not to
switch you to the Calc Trail window. So `C-x * o` is a handy
way to switch out of Calc momentarily to edit your file; you can then
type `C-x * c` to switch back into Calc when you are done.

#### 1.5.3 Quick Mode (Overview)

*Quick mode* is a quick way to use Calc when you don’t need the
full complexity of the stack and trail. To use it, type `C-x * q`
(`quick-calc`

) in any regular editing buffer.

Quick mode is very simple: It prompts you to type any formula in
standard algebraic notation (like ‘`4 - 2/3`’) and then displays
the result at the bottom of the Emacs screen (*3.33333333333*
in this case). You are then back in the same editing buffer you
were in before, ready to continue editing or to type `C-x * q`
again to do another quick calculation. The result of the calculation
will also be in the Emacs “kill ring” so that a `C-y` command
at this point will yank the result into your editing buffer.

Calc mode settings affect Quick mode, too, though you will have to
go into regular Calc (with `C-x * c`) to change the mode settings.

See “Quick Calculator” Mode, for further information.

#### 1.5.4 Keypad Mode (Overview)

*Keypad mode* is a mouse-based interface to the Calculator.
It is designed for use with terminals that support a mouse. If you
don’t have a mouse, you will have to operate Keypad mode with your
arrow keys (which is probably more trouble than it’s worth).

Type `C-x * k` to turn Keypad mode on or off. Once again you
get two new windows, this time on the righthand side of the screen
instead of at the bottom. The upper window is the familiar Calc
Stack; the lower window is a picture of a typical calculator keypad.

|--- Emacs Calculator Mode --- |2: 17.3 |1: -5 | . |--%*-Calc: 12 Deg (Calcul |----+----+--Calc---+----+----1 |FLR |CEIL|RND |TRNC|CLN2|FLT | |----+----+----+----+----+----| | LN |EXP | |ABS |IDIV|MOD | |----+----+----+----+----+----| |SIN |COS |TAN |SQRT|y^x |1/x | |----+----+----+----+----+----| | ENTER |+/- |EEX |UNDO| <- | |-----+---+-+--+--+-+---++----| | INV | 7 | 8 | 9 | / | |-----+-----+-----+-----+-----| | HYP | 4 | 5 | 6 | * | |-----+-----+-----+-----+-----| |EXEC | 1 | 2 | 3 | - | |-----+-----+-----+-----+-----| | OFF | 0 | . | PI | + | |-----+-----+-----+-----+-----+

Keypad mode is much easier for beginners to learn, because there is no need to memorize lots of obscure key sequences. But not all commands in regular Calc are available on the Keypad. You can always switch the cursor into the Calc stack window to use standard Calc commands if you need. Serious Calc users, though, often find they prefer the standard interface over Keypad mode.

To operate the Calculator, just click on the “buttons” of the
keypad using your left mouse button. To enter the two numbers
shown here you would click `1 7 . 3 ENTER 5 +/- ENTER`; to
add them together you would then click `+` (to get 12.3 on
the stack).

If you click the right mouse button, the top three rows of the keypad change to show other sets of commands, such as advanced math functions, vector operations, and operations on binary numbers.

Because Keypad mode doesn’t use the regular keyboard, Calc leaves the cursor in your original editing buffer. You can type in this buffer in the usual way while also clicking on the Calculator keypad. One advantage of Keypad mode is that you don’t need an explicit command to switch between editing and calculating.

If you press `C-x * b` first, you get a full-screen Keypad mode
(`full-calc-keypad`

) with three windows: The keypad in the lower
left, the stack in the lower right, and the trail on top.

See Keypad Mode, for further information.

#### 1.5.5 Standalone Operation

If you are not in Emacs at the moment but you wish to use Calc, you must start Emacs first. If all you want is to run Calc, you can give the commands:

emacs -f full-calc

or

emacs -f full-calc-keypad

which run a full-screen Calculator (as if by `C-x * b C-x * c`) or
a full-screen X-based Calculator (as if by `C-x * b C-x * k`).
In standalone operation, quitting the Calculator (by pressing
`q` or clicking on the keypad `EXIT` button) quits Emacs
itself.

#### 1.5.6 Embedded Mode (Overview)

*Embedded mode* is a way to use Calc directly from inside an
editing buffer. Suppose you have a formula written as part of a
document like this:

The derivative of ln(ln(x)) is

and you wish to have Calc compute and format the derivative for you and store this derivative in the buffer automatically. To do this with Embedded mode, first copy the formula down to where you want the result to be, leaving a blank line before and after the formula:

The derivative of ln(ln(x)) is ln(ln(x))

Now, move the cursor onto this new formula and press `C-x * e`.
Calc will read the formula (using the surrounding blank lines to tell
how much text to read), then push this formula (invisibly) onto the Calc
stack. The cursor will stay on the formula in the editing buffer, but
the line with the formula will now appear as it would on the Calc stack
(in this case, it will be left-aligned) and the buffer’s mode line will
change to look like the Calc mode line (with mode indicators like
‘`12 Deg`’ and so on). Even though you are still in your editing
buffer, the keyboard now acts like the Calc keyboard, and any new result
you get is copied from the stack back into the buffer. To take the
derivative, you would type `a d x RET`.

The derivative of ln(ln(x)) is 1 / x ln(x)

(Note that by default, Calc gives division lower precedence than multiplication,
so that ‘`1 / x ln(x)`’ is equivalent to ‘`1 / (x ln(x))`’.)

To make this look nicer, you might want to press `d =` to center
the formula, and even `d B` to use Big display mode.

The derivative of ln(ln(x)) is % [calc-mode: justify: center] % [calc-mode: language: big] 1 ------- x ln(x)

Calc has added annotations to the file to help it remember the modes that were used for this formula. They are formatted like comments in the TeX typesetting language, just in case you are using TeX or LaTeX. (In this example TeX is not being used, so you might want to move these comments up to the top of the file or otherwise put them out of the way.)

As an extra flourish, we can add an equation number using a
righthand label: Type `d } (1) RET`.

% [calc-mode: justify: center] % [calc-mode: language: big] % [calc-mode: right-label: " (1)"] 1 ------- (1) ln(x) x

To leave Embedded mode, type `C-x * e` again. The mode line
and keyboard will revert to the way they were before.

The related command `C-x * w` operates on a single word, which
generally means a single number, inside text. It searches for an
expression which “looks” like a number containing the point.
Here’s an example of its use (before you try this, remove the Calc
annotations or use a new buffer so that the extra settings in the
annotations don’t take effect):

A slope of one-third corresponds to an angle of 1 degrees.

Place the cursor on the ‘`1`’, then type `C-x * w` to enable
Embedded mode on that number. Now type `3 /` (to get one-third),
and `I T` (the Inverse Tangent converts a slope into an angle),
then `C-x * w` again to exit Embedded mode.

A slope of one-third corresponds to an angle of 18.4349488229 degrees.

See Embedded Mode, for full details.

#### 1.5.7 Other `C-x *` Commands

Two more Calc-related commands are `C-x * g` and `C-x * r`,
which “grab” data from a selected region of a buffer into the
Calculator. The region is defined in the usual Emacs way, by
a “mark” placed at one end of the region, and the Emacs
cursor or “point” placed at the other.

The `C-x * g` command reads the region in the usual left-to-right,
top-to-bottom order. The result is packaged into a Calc vector
of numbers and placed on the stack. Calc (in its standard
user interface) is then started. Type `v u` if you want
to unpack this vector into separate numbers on the stack. Also,
`C-u C-x * g` interprets the region as a single number or
formula.

The `C-x * r` command reads a rectangle, with the point and
mark defining opposite corners of the rectangle. The result
is a matrix of numbers on the Calculator stack.

Complementary to these is `C-x * y`, which “yanks” the
value at the top of the Calc stack back into an editing buffer.
If you type `C-x * y` while in such a buffer, the value is
yanked at the current position. If you type `C-x * y` while
in the Calc buffer, Calc makes an educated guess as to which
editing buffer you want to use. The Calc window does not have
to be visible in order to use this command, as long as there
is something on the Calc stack.

Here, for reference, is the complete list of `C-x *` commands.
The shift, control, and meta keys are ignored for the keystroke
following `C-x *`.

Commands for turning Calc on and off:

`*`Turn Calc on or off, employing the same user interface as last time.

`=, +, -, /, \, &, #`Alternatives for

`*`.`C`Turn Calc on or off using its standard bottom-of-the-screen interface. If Calc is already turned on but the cursor is not in the Calc window, move the cursor into the window.

`O`Same as

`C`, but don’t select the new Calc window. If Calc is already turned on and the cursor is in the Calc window, move it out of that window.`B`Control whether

`C-x * c`and`C-x * k`use the full screen.`Q`Use Quick mode for a single short calculation.

`K`Turn Calc Keypad mode on or off.

`E`Turn Calc Embedded mode on or off at the current formula.

`J`Turn Calc Embedded mode on or off, select the interesting part.

`W`Turn Calc Embedded mode on or off at the current word (number).

`Z`Turn Calc on in a user-defined way, as defined by a

`Z I`command.`X`Quit Calc; turn off standard, Keypad, or Embedded mode if on. (This is like

`q`or`OFF`inside of Calc.)

Commands for moving data into and out of the Calculator:

`G`Grab the region into the Calculator as a vector.

`R`Grab the rectangular region into the Calculator as a matrix.

`:`Grab the rectangular region and compute the sums of its columns.

`_`Grab the rectangular region and compute the sums of its rows.

`Y`Yank a value from the Calculator into the current editing buffer.

Commands for use with Embedded mode:

`A`“Activate” the current buffer. Locate all formulas that contain ‘

`:=`’ or ‘`=>`’ symbols and record their locations so that they can be updated automatically as variables are changed.`D`Duplicate the current formula immediately below and select the duplicate.

`F`Insert a new formula at the current point.

`N`Move the cursor to the next active formula in the buffer.

`P`Move the cursor to the previous active formula in the buffer.

`U`Update (i.e., as if by the

`=`key) the formula at the current point.```Edit (as if by

`calc-edit`

) the formula at the current point.

Miscellaneous commands:

`I`Run the Emacs Info system to read the Calc manual. (This is the same as

`h i`inside of Calc.)`T`Run the Emacs Info system to read the Calc Tutorial.

`S`Run the Emacs Info system to read the Calc Summary.

`L`Load Calc entirely into memory. (Normally the various parts are loaded only as they are needed.)

`M`Read a region of written keystroke names (like

`C-n a b c`) and record them as the current keyboard macro.`RET``0`(This is the “zero” digit key.) Reset the Calculator to its initial state: Empty stack, and initial mode settings.

### 1.6 History and Acknowledgments

Calc was originally started as a two-week project to occupy a lull
in the author’s schedule. Basically, a friend asked if I remembered
the value of
‘`2^32`’.
I didn’t offhand, but I said, “that’s easy, just call up an
`xcalc`

.” `Xcalc`

duly reported that the answer to our
question was ‘`4.294967e+09`’—with no way to see the full ten
digits even though we knew they were there in the program’s memory! I
was so annoyed, I vowed to write a calculator of my own, once and for
all.

I chose Emacs Lisp, a) because I had always been curious about it and b) because, being only a text editor extension language after all, Emacs Lisp would surely reach its limits long before the project got too far out of hand.

To make a long story short, Emacs Lisp turned out to be a distressingly solid implementation of Lisp, and the humble task of calculating turned out to be more open-ended than one might have expected.

Emacs Lisp didn’t have built-in floating point math (now it does), so this had to be simulated in software. In fact, Emacs integers would only comfortably fit six decimal digits or so (at the time)—not enough for a decent calculator. So I had to write my own high-precision integer code as well, and once I had this I figured that arbitrary-size integers were just as easy as large integers. Arbitrary floating-point precision was the logical next step. Also, since the large integer arithmetic was there anyway it seemed only fair to give the user direct access to it, which in turn made it practical to support fractions as well as floats. All these features inspired me to look around for other data types that might be worth having.

Around this time, my friend Rick Koshi showed me his nifty new HP-28 calculator. It allowed the user to manipulate formulas as well as numerical quantities, and it could also operate on matrices. I decided that these would be good for Calc to have, too. And once things had gone this far, I figured I might as well take a look at serious algebra systems for further ideas. Since these systems did far more than I could ever hope to implement, I decided to focus on rewrite rules and other programming features so that users could implement what they needed for themselves.

Rick complained that matrices were hard to read, so I put in code to format them in a 2D style. Once these routines were in place, Big mode was obligatory. Gee, what other language modes would be useful?

Scott Hemphill and Allen Knutson, two friends with a strong mathematical bent, contributed ideas and algorithms for a number of Calc features including modulo forms, primality testing, and float-to-fraction conversion.

Units were added at the eager insistence of Mass Sivilotti. Later, Ulrich Mueller at CERN and Przemek Klosowski at NIST provided invaluable expert assistance with the units table. As far as I can remember, the idea of using algebraic formulas and variables to represent units dates back to an ancient article in Byte magazine about muMath, an early algebra system for microcomputers.

Many people have contributed to Calc by reporting bugs and suggesting
features, large and small. A few deserve special mention: Tim Peters,
who helped develop the ideas that led to the selection commands, rewrite
rules, and many other algebra features; FranÃ§ois
Pinard, who contributed an early prototype of the Calc Summary appendix
as well as providing valuable suggestions in many other areas of Calc;
Carl Witty, whose eagle eyes discovered many typographical and factual
errors in the Calc manual; Tim Kay, who drove the development of
Embedded mode; Ove Ewerlid, who made many suggestions relating to the
algebra commands and contributed some code for polynomial operations;
Randal Schwartz, who suggested the `calc-eval`

function; Juha
Sarlin, who first worked out how to split Calc into quickly-loading
parts; Bob Weiner, who helped immensely with the Lucid Emacs port; and
Robert J. Chassell, who suggested the Calc Tutorial and exercises as
well as many other things.

Among the books used in the development of Calc were Knuth’s *Art
of Computer Programming* (especially volume II, *Seminumerical
Algorithms*); *Numerical Recipes* by Press, Flannery, Teukolsky,
and Vetterling; Bevington’s *Data Reduction and Error Analysis
for the Physical Sciences*; *Concrete Mathematics* by Graham,
Knuth, and Patashnik; Steele’s *Common Lisp, the Language*; the
*CRC Standard Math Tables* (William H. Beyer, ed.); and
Abramowitz and Stegun’s venerable *Handbook of Mathematical
Functions*. Also, of course, Calc could not have been written without
the excellent *GNU Emacs Lisp Reference Manual*, by Bil Lewis and
Dan LaLiberte.

Final thanks go to Richard Stallman, without whose fine implementations of the Emacs editor, language, and environment, Calc would have been finished in two weeks.

## 2 Tutorial

This chapter explains how to use Calc and its many features, in a step-by-step, tutorial way. You are encouraged to run Calc and work along with the examples as you read (see Starting Calc). If you are already familiar with advanced calculators, you may wish to skip on to the rest of this manual.

This tutorial describes the standard user interface of Calc only. The Quick mode and Keypad mode interfaces are fairly self-explanatory. See Embedded Mode, for a description of the Embedded mode interface.

The easiest way to read this tutorial on-line is to have two windows on
your Emacs screen, one with Calc and one with the Info system. Press
`C-x * t` to set this up; the on-line tutorial will be opened in the
current window and Calc will be started in another window. From the
Info window, the command `C-x * c` can be used to switch to the Calc
window and `C-x * o` can be used to switch back to the Info window.
(If you have a printed copy of the manual you can use that instead; in
that case you only need to press `C-x * c` to start Calc.)

This tutorial is designed to be done in sequence. But the rest of this manual does not assume you have gone through the tutorial. The tutorial does not cover everything in the Calculator, but it touches on most general areas.

You may wish to print out a copy of the Calc Summary and keep notes on it as you learn Calc. See About This Manual, to see how to make a printed summary. See Calc Summary.

- Basic Tutorial
- Arithmetic Tutorial
- Vector/Matrix Tutorial
- Types Tutorial
- Algebra and Calculus Tutorial
- Programming Tutorial
- Answers to Exercises

### 2.1 Basic Tutorial

In this section, we learn how RPN and algebraic-style calculations work, how to undo and redo an operation done by mistake, and how to control various modes of the Calculator.

#### 2.1.1 RPN Calculations and the Stack

Calc normally uses Reverse Polish notation (RPN). You may be familiar with the RPN system from Hewlett-Packard calculators, FORTH, or PostScript. (Reverse Polish Notation is named after the Polish mathematician Jan Lukasiewicz.)

The central component of an RPN calculator is the *stack*. A
calculator stack is like a stack of dishes. New dishes (numbers) are
added at the top of the stack, and numbers are normally only removed
from the top of the stack.

In an operation like ‘`2+3`’, the 2 and 3 are called the *operands*
and the ‘`+`’ is the *operator*. In an RPN calculator you always
enter the operands first, then the operator. Each time you type a
number, Calc adds or *pushes* it onto the top of the Stack.
When you press an operator key like `+`, Calc *pops* the appropriate
number of operands from the stack and pushes back the result.

Thus we could add the numbers 2 and 3 in an RPN calculator by typing:
`2 RET 3 RET +`. (The

`RET`key, Return, corresponds to the

`ENTER`key on traditional RPN calculators.) Try this now if you wish; type

`C-x * c`to switch into the Calc window (you can type

`C-x * c`again or

`C-x * o`to switch back to the Tutorial window). The first four keystrokes “push” the numbers 2 and 3 onto the stack. The

`+`key “pops” the top two numbers from the stack, adds them, and pushes the result (5) back onto the stack. Here’s how the stack will look at various points throughout the calculation:

. 1: 2 2: 2 1: 5 . . 1: 3 . . C-x * c 2RET3RET+DEL

The ‘`.`’ symbol is a marker that represents the top of the stack.
Note that the “top” of the stack is really shown at the bottom of
the Stack window. This may seem backwards, but it turns out to be
less distracting in regular use.

The numbers ‘`1:`’ and ‘`2:`’ on the left are *stack level
numbers*. Old RPN calculators always had four stack levels called
‘`x`’, ‘`y`’, ‘`z`’, and ‘`t`’. Calc’s stack can grow
as large as you like, so it uses numbers instead of letters. Some
stack-manipulation commands accept a numeric argument that says
which stack level to work on. Normal commands like `+` always
work on the top few levels of the stack.

The Stack buffer is just an Emacs buffer, and you can move around in
it using the regular Emacs motion commands. But no matter where the
cursor is, even if you have scrolled the ‘`.`’ marker out of
view, most Calc commands always move the cursor back down to level 1
before doing anything. It is possible to move the ‘`.`’ marker
upwards through the stack, temporarily “hiding” some numbers from
commands like `+`. This is called *stack truncation* and
we will not cover it in this tutorial; see Truncating the Stack,
if you are interested.

You don’t really need the second `RET` in `2 RET 3
RET +`. That’s because if you type any operator name or
other non-numeric key when you are entering a number, the Calculator
automatically enters that number and then does the requested command.
Thus

`2`will work just as well.

`RET`3 +Examples in this tutorial will often omit `RET` even when the
stack displays shown would only happen if you did press `RET`:

1: 2 2: 2 1: 5 . 1: 3 . . 2RET3 +

Here, after pressing `3` the stack would really show ‘`1: 2`’
with ‘`Calc: 3`’ in the minibuffer. In these situations, you can
press the optional `RET` to see the stack as the figure shows.

(•) **Exercise 1.** (This tutorial will include exercises
at various points. Try them if you wish. Answers to all the exercises
are located at the end of the Tutorial chapter. Each exercise will
include a cross-reference to its particular answer. If you are
reading with the Emacs Info system, press `f` and the
exercise number to go to the answer, then the letter `l` to
return to where you were.)

Here’s the first exercise: What will the keystrokes `1 RET 2
RET 3 RET 4 + * -` compute? (‘

`*`’ is the symbol for multiplication.) Figure it out by hand, then try it with Calc to see if you’re right. See 1. (•)

(•) **Exercise 2.** Compute
‘`2*4 + 7*9.5 + 5/4`’
using the stack. See 2. (•)

The `DEL` key is called Backspace on some keyboards. It is
whatever key you would use to correct a simple typing error when
regularly using Emacs. The `DEL` key pops and throws away the
top value on the stack. (You can still get that value back from
the Trail if you should need it later on.) There are many places
in this tutorial where we assume you have used `DEL` to erase the
results of the previous example at the beginning of a new example.
In the few places where it is really important to use `DEL` to
clear away old results, the text will remind you to do so.

(It won’t hurt to let things accumulate on the stack, except that whenever you give a display-mode-changing command Calc will have to spend a long time reformatting such a large stack.)

Since the `-` key is also an operator (it subtracts the top two
stack elements), how does one enter a negative number? Calc uses
the `_` (underscore) key to act like the minus sign in a number.
So, typing `-5 RET` won’t work because the

`-`key will try to do a subtraction, but

`_5`works just fine.

`RET`You can also press `n`, which means “change sign.” It changes
the number at the top of the stack (or the number being entered)
from positive to negative or vice-versa: `5 n RET`.

If you press `RET` when you’re not entering a number, the effect
is to duplicate the top number on the stack. Consider this calculation:

1: 3 2: 3 1: 9 2: 9 1: 81 . 1: 3 . 1: 9 . . . 3RETRET*RET*

(Of course, an easier way to do this would be `3 RET 4 ^`,
to raise 3 to the fourth power.)

The space-bar key (denoted `SPC` here) performs the same function
as `RET`; you could replace all three occurrences of `RET` in
the above example with `SPC` and the effect would be the same.

Another stack manipulation key is `TAB`. This exchanges the top
two stack entries. Suppose you have computed `2 RET 3 +`
to get 5, and then you realize what you really wanted to compute
was ‘

`20 / (2+3)`’.

1: 5 2: 5 2: 20 1: 4 . 1: 20 1: 5 . . . 2RET3 + 20TAB/

Planning ahead, the calculation would have gone like this:

1: 20 2: 20 3: 20 2: 20 1: 4 . 1: 2 2: 2 1: 5 . . 1: 3 . . 20RET2RET3 + /

A related stack command is `M- TAB` (hold

`META`and type

`TAB`). It rotates the top three elements of the stack upward, bringing the object in level 3 to the top.

1: 10 2: 10 3: 10 3: 20 3: 30 . 1: 20 2: 20 2: 30 2: 10 . 1: 30 1: 10 1: 20 . . . 10RET20RET30RETM-TABM-TAB

(•) **Exercise 3.** Suppose the numbers 10, 20, and 30 are
on the stack. Figure out how to add one to the number in level 2
without affecting the rest of the stack. Also figure out how to add
one to the number in level 3. See 3. (•)

Operations like `+`, `-`, `*`, `/`, and `^` pop two
arguments from the stack and push a result. Operations like `n` and
`Q` (square root) pop a single number and push the result. You can
think of them as simply operating on the top element of the stack.

1: 3 1: 9 2: 9 1: 25 1: 5 . . 1: 16 . . . 3RETRET* 4RETRET* + Q

(Note that capital `Q` means to hold down the Shift key while
typing `q`. Remember, plain unshifted `q` is the Quit command.)

Here we’ve used the Pythagorean Theorem to determine the hypotenuse of a
right triangle. Calc actually has a built-in command for that called
`f h`, but let’s suppose we can’t remember the necessary keystrokes.
We can still enter it by its full name using `M-x` notation:

1: 3 2: 3 1: 5 . 1: 4 . . 3RET4RETM-x calc-hypot

All Calculator commands begin with the word ‘`calc-`’. Since it
gets tiring to type this, Calc provides an `x` key which is just
like the regular Emacs `M-x` key except that it types the ‘`calc-`’
prefix for you:

1: 3 2: 3 1: 5 . 1: 4 . . 3RET4RETx hypot

What happens if you take the square root of a negative number?

1: 4 1: -4 1: (0, 2) . . . 4RETn Q

The notation ‘`(a, b)`’ represents a complex number.
Complex numbers are more traditionally written ‘`a + b i`’;
Calc can display in this format, too, but for now we’ll stick to the
‘`(a, b)`’ notation.

If you don’t know how complex numbers work, you can safely ignore this feature. Complex numbers only arise from operations that would be errors in a calculator that didn’t have complex numbers. (For example, taking the square root or logarithm of a negative number produces a complex result.)

Complex numbers are entered in the notation shown. The `(` and
`,` and `)` keys manipulate “incomplete complex numbers.”

1: ( ... 2: ( ... 1: (2, ... 1: (2, ... 1: (2, 3) . 1: 2 . 3 . . . ( 2 , 3 )

You can perform calculations while entering parts of incomplete objects. However, an incomplete object cannot actually participate in a calculation:

1: ( ... 2: ( ... 3: ( ... 1: ( ... 1: ( ... . 1: 2 2: 2 5 5 . 1: 3 . . . (error) ( 2RET3 + +

Adding 5 to an incomplete object makes no sense, so the last command produces an error message and leaves the stack the same.

Incomplete objects can’t participate in arithmetic, but they can be moved around by the regular stack commands.

2: 2 3: 2 3: 3 1: ( ... 1: (2, 3) 1: 3 2: 3 2: ( ... 2 . . 1: ( ... 1: 2 3 . . . 2RET3RET( M-TABM-TAB)

Note that the `,` (comma) key did not have to be used here.
When you press `)` all the stack entries between the incomplete
entry and the top are collected, so there’s never really a reason
to use the comma. It’s up to you.

(•) **Exercise 4.** To enter the complex number ‘`(2, 3)`’,
your friend Joe typed `( 2 , SPC 3 )`. What happened?
(Joe thought of a clever way to correct his mistake in only two
keystrokes, but it didn’t quite work. Try it to find out why.)
See 4. (•)

Vectors are entered the same way as complex numbers, but with square brackets in place of parentheses. We’ll meet vectors again later in the tutorial.

Any Emacs command can be given a *numeric prefix argument* by
typing a series of `META`-digits beforehand. If `META` is
awkward for you, you can instead type `C-u` followed by the
necessary digits. Numeric prefix arguments can be negative, as in
`M-- M-3 M-5` or `C-u - 3 5`. Calc commands use numeric
prefix arguments in a variety of ways. For example, a numeric prefix
on the `+` operator adds any number of stack entries at once:

1: 10 2: 10 3: 10 3: 10 1: 60 . 1: 20 2: 20 2: 20 . . 1: 30 1: 30 . . 10RET20RET30RETC-u 3 +

For stack manipulation commands like `RET`, a positive numeric
prefix argument operates on the top `n` stack entries at once. A
negative argument operates on the entry in level `n` only. An
argument of zero operates on the entire stack. In this example, we copy
the second-to-top element of the stack:

1: 10 2: 10 3: 10 3: 10 4: 10 . 1: 20 2: 20 2: 20 3: 20 . 1: 30 1: 30 2: 30 . . 1: 20 . 10RET20RET30RETC-u -2RET

Another common idiom is `M-0 DEL`, which clears the stack.
(The

`M-0`numeric prefix tells

`DEL`to operate on the entire stack.)

#### 2.1.2 Algebraic-Style Calculations

If you are not used to RPN notation, you may prefer to operate the
Calculator in Algebraic mode, which is closer to the way
non-RPN calculators work. In Algebraic mode, you enter formulas
in traditional ‘`2+3`’ notation.

**Notice:** Calc gives ‘`/`’ lower precedence than ‘`*`’, so
that ‘`a/b*c`’ is interpreted as ‘`a/(b*c)`’; this is not
standard across all computer languages. See below for details.

You don’t really need any special “mode” to enter algebraic formulas.
You can enter a formula at any time by pressing the apostrophe (`'`)
key. Answer the prompt with the desired formula, then press `RET`.
The formula is evaluated and the result is pushed onto the RPN stack.
If you don’t want to think in RPN at all, you can enter your whole
computation as a formula, read the result from the stack, then press
`DEL` to delete it from the stack.

Try pressing the apostrophe key, then `2+3+4`, then `RET`.
The result should be the number 9.

Algebraic formulas use the operators ‘`+`’, ‘`-`’, ‘`*`’,
‘`/`’, and ‘`^`’. You can use parentheses to make the order
of evaluation clear. In the absence of parentheses, ‘`^`’ is
evaluated first, then ‘`*`’, then ‘`/`’, then finally
‘`+`’ and ‘`-`’. For example, the expression

2 + 3*4*5 / 6*7^8 - 9

is equivalent to

2 + ((3*4*5) / (6*(7^8))) - 9

or, in large mathematical notation,

3 * 4 * 5 2 + --------- - 9 8 6 * 7

The result of this expression will be the number *-6.99999826533*.

Calc’s order of evaluation is the same as for most computer languages,
except that ‘`*`’ binds more strongly than ‘`/`’, as the above
example shows. As in normal mathematical notation, the ‘`*`’ symbol
can often be omitted: ‘`2 a`’ is the same as ‘`2*a`’.

Operators at the same level are evaluated from left to right, except
that ‘`^`’ is evaluated from right to left. Thus, ‘`2-3-4`’ is
equivalent to ‘`(2-3)-4`’ or *-5*, whereas ‘`2^3^4`’ is equivalent
to ‘`2^(3^4)`’ (a very large integer; try it!).

If you tire of typing the apostrophe all the time, there is
Algebraic mode, where Calc automatically senses
when you are about to type an algebraic expression. To enter this
mode, press the two letters `m a`. (An ‘`Alg`’ indicator
should appear in the Calc window’s mode line.)

Press `m a`, then `2+3+4` with no apostrophe, then `RET`.

In Algebraic mode, when you press any key that would normally begin
entering a number (such as a digit, a decimal point, or the `_`
key), or if you press `(` or `[`, Calc automatically begins
an algebraic entry.

Functions which do not have operator symbols like ‘`+`’ and ‘`*`’
must be entered in formulas using function-call notation. For example,
the function name corresponding to the square-root key `Q` is
`sqrt`

. To compute a square root in a formula, you would use
the notation ‘`sqrt( x)`’.

Press the apostrophe, then type `sqrt(5*2) - 3`. The result should
be ‘`0.16227766017`’.

Note that if the formula begins with a function name, you need to use
the apostrophe even if you are in Algebraic mode. If you type `arcsin`
out of the blue, the `a r` will be taken as an Algebraic Rewrite
command, and the `csin` will be taken as the name of the rewrite
rule to use!

Some people prefer to enter complex numbers and vectors in algebraic form because they find RPN entry with incomplete objects to be too distracting, even though they otherwise use Calc as an RPN calculator.

Still in Algebraic mode, type:

1: (2, 3) 2: (2, 3) 1: (8, -1) 2: (8, -1) 1: (9, -1) . 1: (1, -2) . 1: 1 . . . (2,3)RET(1,-2)RET* 1RET+

Algebraic mode allows us to enter complex numbers without pressing
an apostrophe first, but it also means we need to press `RET`
after every entry, even for a simple number like ‘`1`’.

(You can type `C-u m a` to enable a special Incomplete Algebraic
mode in which the `(` and `[` keys use algebraic entry even
though regular numeric keys still use RPN numeric entry. There is also
Total Algebraic mode, started by typing `m t`, in which all
normal keys begin algebraic entry. You must then use the `META` key
to type Calc commands: `M-m t` to get back out of Total Algebraic
mode, `M-q` to quit, etc.)

If you’re still in Algebraic mode, press `m a` again to turn it off.

Actual non-RPN calculators use a mixture of algebraic and RPN styles.
In general, operators of two numbers (like `+` and `*`)
use algebraic form, but operators of one number (like `n` and `Q`)
use RPN form. Also, a non-RPN calculator allows you to see the
intermediate results of a calculation as you go along. You can
accomplish this in Calc by performing your calculation as a series
of algebraic entries, using the `$` sign to tie them together.
In an algebraic formula, `$` represents the number on the top
of the stack. Here, we perform the calculation
‘`sqrt(2*4+1)`’,
which on a traditional calculator would be done by pressing
`2 * 4 + 1 =` and then the square-root key.

1: 8 1: 9 1: 3 . . . ' 2*4RET$+1RETQ

Notice that we didn’t need to press an apostrophe for the `$+1`,
because the dollar sign always begins an algebraic entry.

(•) **Exercise 1.** How could you get the same effect as
pressing `Q` but using an algebraic entry instead? How about
if the `Q` key on your keyboard were broken?
See 1. (•)

The notations `$$`, `$$$`, and so on stand for higher stack
entries. For example, `' $$+$ RET` is just like typing

`+`.

Algebraic formulas can include *variables*. To store in a
variable, press `s s`, then type the variable name, then press
`RET`. (There are actually two flavors of store command:
`s s` stores a number in a variable but also leaves the number
on the stack, while `s t` removes a number from the stack and
stores it in the variable.) A variable name should consist of one
or more letters or digits, beginning with a letter.

1: 17 . 1: a + a^2 1: 306 . . . 17 s t aRET' a+a^2RET=

The `=` key *evaluates* a formula by replacing all its
variables by the values that were stored in them.

For RPN calculations, you can recall a variable’s value on the
stack either by entering its name as a formula and pressing `=`,
or by using the `s r` command.

1: 17 2: 17 3: 17 2: 17 1: 306 . 1: 17 2: 17 1: 289 . . 1: 2 . . s r aRET' aRET= 2 ^ +

If you press a single digit for a variable name (as in `s t 3`, you
get one of ten *quick variables* `q0`

through `q9`

.
They are “quick” simply because you don’t have to type the letter
`q`

or the `RET` after their names. In fact, you can type
simply `s 3` as a shorthand for `s s 3`, and likewise for
`t 3` and `r 3`.

Any variables in an algebraic formula for which you have not stored values are left alone, even when you evaluate the formula.

1: 2 a + 2 b 1: 2 b + 34 . . ' 2a+2bRET=

Calls to function names which are undefined in Calc are also left alone, as are calls for which the value is undefined.

1: log10(0) + log10(x) + log10(5, 6) + foo(3) + 2 . ' log10(100) + log10(0) + log10(x) + log10(5,6) + foo(3)RET

In this example, the first call to `log10`

works, but the other
calls are not evaluated. In the second call, the logarithm is
undefined for that value of the argument; in the third, the argument
is symbolic, and in the fourth, there are too many arguments. In the
fifth case, there is no function called `foo`

. You will see a
“Wrong number of arguments” message referring to ‘`log10(5,6)`’.
Press the `w` (“why”) key to see any other messages that may
have arisen from the last calculation. In this case you will get
“logarithm of zero,” then “number expected: `x`

”. Calc
automatically displays the first message only if the message is
sufficiently important; for example, Calc considers “wrong number
of arguments” and “logarithm of zero” to be important enough to
report automatically, while a message like “number expected: `x`

”
will only show up if you explicitly press the `w` key.

(•) **Exercise 2.** Joe entered the formula ‘`2 x y`’,
stored 5 in `x`

, pressed `=`, and got the expected result,
‘`10 y`’. He then tried the same for the formula ‘`2 x (1+y)`’,
expecting ‘`10 (1+y)`’, but it didn’t work. Why not?
See 2. (•)

(•) **Exercise 3.** What result would you expect
`1 RET 0 /` to give? What if you then type

`0 *`? See 3. (•)

One interesting way to work with variables is to use the
*evaluates-to* (‘`=>`’) operator. It works like this:
Enter a formula algebraically in the usual way, but follow
the formula with an ‘`=>`’ symbol. (There is also an `s =`
command which builds an ‘`=>`’ formula using the stack.) On
the stack, you will see two copies of the formula with an ‘`=>`’
between them. The lefthand formula is exactly like you typed it;
the righthand formula has been evaluated as if by typing `=`.

2: 2 + 3 => 5 2: 2 + 3 => 5 1: 2 a + 2 b => 34 + 2 b 1: 2 a + 2 b => 20 + 2 b . . ' 2+3 =>RET' 2a+2bRETs = 10 s t aRET

Notice that the instant we stored a new value in `a`

, all
‘`=>`’ operators already on the stack that referred to ‘`a`’
were updated to use the new value. With ‘`=>`’, you can push a
set of formulas on the stack, then change the variables experimentally
to see the effects on the formulas’ values.

You can also “unstore” a variable when you are through with it:

2: 2 + 3 => 5 1: 2 a + 2 b => 2 a + 2 b . s u aRET

We will encounter formulas involving variables and functions again when we discuss the algebra and calculus features of the Calculator.

#### 2.1.3 Undo and Redo

If you make a mistake, you can usually correct it by pressing shift-`U`,
the “undo” command. First, clear the stack (`M-0 DEL`) and exit
and restart Calc (

`C-x * * C-x * *`) to make sure things start off with a clean slate. Now:

1: 2 2: 2 1: 8 2: 2 1: 6 . 1: 3 . 1: 3 . . . 2RET3 ^ U *

You can undo any number of times. Calc keeps a complete record of all you have done since you last opened the Calc window. After the above example, you could type:

1: 6 2: 2 1: 2 . . . 1: 3 . . (error) U U U U

You can also type `D` to “redo” a command that you have undone
mistakenly.

. 1: 2 2: 2 1: 6 1: 6 . 1: 3 . . . (error) D D D D

It was not possible to redo past the ‘`6`’, since that was placed there
by something other than an undo command.

You can think of undo and redo as a sort of “time machine.” Press
`U` to go backward in time, `D` to go forward. If you go
backward and do something (like `*`) then, as any science fiction
reader knows, you have changed your future and you cannot go forward
again. Thus, the inability to redo past the ‘`6`’ even though there
was an earlier undo command.

You can always recall an earlier result using the Trail. We’ve ignored
the trail so far, but it has been faithfully recording everything we
did since we loaded the Calculator. If the Trail is not displayed,
press `t d` now to turn it on.

Let’s try grabbing an earlier result. The ‘`8`’ we computed was
undone by a `U` command, and was lost even to Redo when we pressed
`*`, but it’s still there in the trail. There should be a little
‘`>`’ arrow (the *trail pointer*) resting on the last trail
entry. If there isn’t, press `t ]` to reset the trail pointer.
Now, press `t p` to move the arrow onto the line containing
‘`8`’, and press `t y` to “yank” that number back onto the
stack.

If you press `t ]` again, you will see that even our Yank command
went into the trail.

Let’s go further back in time. Earlier in the tutorial we computed
a huge integer using the formula ‘`2^3^4`’. We don’t remember
what it was, but the first digits were “241”. Press `t r`
(which stands for trail-search-reverse), then type `241`.
The trail cursor will jump back to the next previous occurrence of
the string “241” in the trail. This is just a regular Emacs
incremental search; you can now press `C-s` or `C-r` to
continue the search forwards or backwards as you like.

To finish the search, press `RET`. This halts the incremental
search and leaves the trail pointer at the thing we found. Now we
can type `t y` to yank that number onto the stack. If we hadn’t
remembered the “241”, we could simply have searched for `2^3^4`,
then pressed ` RET t n` to halt and then move to the next item.

You may have noticed that all the trail-related commands begin with
the letter `t`. (The store-and-recall commands, on the other hand,
all began with `s`.) Calc has so many commands that there aren’t
enough keys for all of them, so various commands are grouped into
two-letter sequences where the first letter is called the *prefix*
key. If you type a prefix key by accident, you can press `C-g`
to cancel it. (In fact, you can press `C-g` to cancel almost
anything in Emacs.) To get help on a prefix key, press that key
followed by `?`. Some prefixes have several lines of help,
so you need to press `?` repeatedly to see them all.
You can also type `h h` to see all the help at once.

Try pressing `t ?` now. You will see a line of the form,

trail/time: Display; Fwd, Back; Next, Prev, Here, [, ]; Yank: [MORE] t-

The word “trail” indicates that the `t` prefix key contains
trail-related commands. Each entry on the line shows one command,
with a single capital letter showing which letter you press to get
that command. We have used `t n`, `t p`, `t ]`, and
`t y` so far. The ‘`[MORE]`’ means you can press `?`
again to see more `t`-prefix commands. Notice that the commands
are roughly divided (by semicolons) into related groups.

When you are in the help display for a prefix key, the prefix is
still active. If you press another key, like `y` for example,
it will be interpreted as a `t y` command. If all you wanted
was to look at the help messages, press `C-g` afterwards to cancel
the prefix.

One more way to correct an error is by editing the stack entries.
The actual Stack buffer is marked read-only and must not be edited
directly, but you can press ``` (grave accent)
to edit a stack entry.

Try entering ‘`3.141439`’ now. If this is supposed to represent
‘`pi`’, it’s got several errors. Press ``` to edit this number.
Now use the normal Emacs cursor motion and editing keys to change
the second 4 to a 5, and to transpose the 3 and the 9. When you
press `RET`, the number on the stack will be replaced by your
new number. This works for formulas, vectors, and all other types
of values you can put on the stack. The ``` key also works
during entry of a number or algebraic formula.

#### 2.1.4 Mode-Setting Commands

Calc has many types of *modes* that affect the way it interprets
your commands or the way it displays data. We have already seen one
mode, namely Algebraic mode. There are many others, too; we’ll
try some of the most common ones here.

Perhaps the most fundamental mode in Calc is the current *precision*.
Notice the ‘`12`’ on the Calc window’s mode line:

--%*-Calc: 12 Deg (Calculator)----All------

Most of the symbols there are Emacs things you don’t need to worry
about, but the ‘`12`’ and the ‘`Deg`’ are mode indicators.
The ‘`12`’ means that calculations should always be carried to
12 significant figures. That is why, when we type `1 RET 7 /`,
we get ‘

`0.142857142857`’ with exactly 12 digits, not counting leading and trailing zeros.

You can set the precision to anything you like by pressing `p`,
then entering a suitable number. Try pressing `p 30 RET`,
then doing

`1`again:

`RET`7 /1: 0.142857142857 2: 0.142857142857142857142857142857 .

Although the precision can be set arbitrarily high, Calc always
has to have *some* value for the current precision. After
all, the true value ‘`1/7`’ is an infinitely repeating decimal;
Calc has to stop somewhere.

Of course, calculations are slower the more digits you request.
Press `p 12` now to set the precision back down to the default.

Calculations always use the current precision. For example, even
though we have a 30-digit value for ‘`1/7`’ on the stack, if
we use it in a calculation in 12-digit mode it will be rounded
down to 12 digits before it is used. Try it; press `RET` to
duplicate the number, then `1 +`. Notice that the `RET`
key didn’t round the number, because it doesn’t do any calculation.
But the instant we pressed `+`, the number was rounded down.

1: 0.142857142857 2: 0.142857142857142857142857142857 3: 1.14285714286 .

In fact, since we added a digit on the left, we had to lose one
digit on the right from even the 12-digit value of ‘`1/7`’.

How did we get more than 12 digits when we computed ‘`2^3^4`’? The
answer is that Calc makes a distinction between *integers* and
*floating-point* numbers, or *floats*. An integer is a number
that does not contain a decimal point. There is no such thing as an
“infinitely repeating fraction integer,” so Calc doesn’t have to limit
itself. If you asked for ‘`2^10000`’ (don’t try this!), you would
have to wait a long time but you would eventually get an exact answer.
If you ask for ‘`2.^10000`’, you will quickly get an answer which is
correct only to 12 places. The decimal point tells Calc that it should
use floating-point arithmetic to get the answer, not exact integer
arithmetic.

You can use the `F` (`calc-floor`

) command to convert a
floating-point value to an integer, and `c f` (`calc-float`

)
to convert an integer to floating-point form.

Let’s try entering that last calculation:

1: 2. 2: 2. 1: 1.99506311689e3010 . 1: 10000 . . 2.0RET10000RET^

Notice the letter ‘`e`’ in there. It represents “times ten to the
power of,” and is used by Calc automatically whenever writing the
number out fully would introduce more extra zeros than you probably
want to see. You can enter numbers in this notation, too.

1: 2. 2: 2. 1: 1.99506311678e3010 . 1: 10000. . . 2.0RET1e4RET^

Hey, the answer is different! Look closely at the middle columns
of the two examples. In the first, the stack contained the
exact integer ‘`10000`’, but in the second it contained
a floating-point value with a decimal point. When you raise a
number to an integer power, Calc uses repeated squaring and
multiplication to get the answer. When you use a floating-point
power, Calc uses logarithms and exponentials. As you can see,
a slight error crept in during one of these methods. Which
one should we trust? Let’s raise the precision a bit and find
out:

. 1: 2. 2: 2. 1: 1.995063116880828e3010 . 1: 10000. . . p 16RET2.RET1e4 ^ p 12RET

Presumably, it doesn’t matter whether we do this higher-precision calculation using an integer or floating-point power, since we have added enough “guard digits” to trust the first 12 digits no matter what. And the verdict is… Integer powers were more accurate; in fact, the result was only off by one unit in the last place.

Calc does many of its internal calculations to a slightly higher
precision, but it doesn’t always bump the precision up enough.
In each case, Calc added about two digits of precision during
its calculation and then rounded back down to 12 digits
afterward. In one case, it was enough; in the other, it
wasn’t. If you really need `x` digits of precision, it
never hurts to do the calculation with a few extra guard digits.

What if we want guard digits but don’t want to look at them?
We can set the *float format*. Calc supports four major
formats for floating-point numbers, called *normal*,
*fixed-point*, *scientific notation*, and *engineering
notation*. You get them by pressing `d n`, `d f`,
`d s`, and `d e`, respectively. In each case, you can
supply a numeric prefix argument which says how many digits
should be displayed. As an example, let’s put a few numbers
onto the stack and try some different display modes. First,
use `M-0 DEL` to clear the stack, then enter the four
numbers shown here:

4: 12345 4: 12345 4: 12345 4: 12345 4: 12345 3: 12345. 3: 12300. 3: 1.2345e4 3: 1.23e4 3: 12345.000 2: 123.45 2: 123. 2: 1.2345e2 2: 1.23e2 2: 123.450 1: 12.345 1: 12.3 1: 1.2345e1 1: 1.23e1 1: 12.345 . . . . . d n M-3 d n d s M-3 d s M-3 d f

Notice that when we typed `M-3 d n`, the numbers were rounded down
to three significant digits, but then when we typed `d s` all
five significant figures reappeared. The float format does not
affect how numbers are stored, it only affects how they are
displayed. Only the current precision governs the actual rounding
of numbers in the Calculator’s memory.

Engineering notation, not shown here, is like scientific notation except the exponent (the power-of-ten part) is always adjusted to be a multiple of three (as in “kilo,” “micro,” etc.). As a result there will be one, two, or three digits before the decimal point.

Whenever you change a display-related mode, Calc redraws everything
in the stack. This may be slow if there are many things on the stack,
so Calc allows you to type shift-`H` before any mode command to
prevent it from updating the stack. Anything Calc displays after the
mode-changing command will appear in the new format.

4: 12345 4: 12345 4: 12345 4: 12345 4: 12345 3: 12345.000 3: 12345.000 3: 12345.000 3: 1.2345e4 3: 12345. 2: 123.450 2: 123.450 2: 1.2345e1 2: 1.2345e1 2: 123.45 1: 12.345 1: 1.2345e1 1: 1.2345e2 1: 1.2345e2 1: 12.345 . . . . . H d sDELUTABdSPCd n

Here the `H d s` command changes to scientific notation but without
updating the screen. Deleting the top stack entry and undoing it back
causes it to show up in the new format; swapping the top two stack
entries reformats both entries. The `d SPC` command refreshes the
whole stack. The

`d n`command changes back to the normal float format; since it doesn’t have an

`H`prefix, it also updates all the stack entries to be in

`d n`format.

Notice that the integer ‘`12345`’ was not affected by any
of the float formats. Integers are integers, and are always
displayed exactly.

Large integers have their own problems. Let’s look back at
the result of `2^3^4`.

2417851639229258349412352

Quick—how many digits does this have? Try typing `d g`:

2,417,851,639,229,258,349,412,352

Now how many digits does this have? It’s much easier to tell!
We can actually group digits into clumps of any size. Some
people prefer `M-5 d g`:

24178,51639,22925,83494,12352

Let’s see what happens to floating-point numbers when they are grouped.
First, type `p 25 RET` to make sure we have enough precision
to get ourselves into trouble. Now, type

`1e13 /`:

24,17851,63922.9258349412352

The integer part is grouped but the fractional part isn’t. Now try
`M-- M-5 d g` (that’s meta-minus-sign, meta-five):

24,17851,63922.92583,49412,352

If you find it hard to tell the decimal point from the commas, try
changing the grouping character to a space with `d , SPC`:

24 17851 63922.92583 49412 352

Type `d , ,` to restore the normal grouping character, then
`d g` again to turn grouping off. Also, press `p 12` to
restore the default precision.

Press `U` enough times to get the original big integer back.
(Notice that `U` does not undo each mode-setting command; if
you want to undo a mode-setting command, you have to do it yourself.)
Now, type `d r 16 RET`:

16#200000000000000000000

The number is now displayed in *hexadecimal*, or “base-16” form.
Suddenly it looks pretty simple; this should be no surprise, since we
got this number by computing a power of two, and 16 is a power of 2.
In fact, we can use `d r 2 RET` to see it in actual binary
form:

2#1000000000000000000000000000000000000000000000000000000 …

We don’t have enough space here to show all the zeros! They won’t
fit on a typical screen, either, so you will have to use horizontal
scrolling to see them all. Press `<` and `>` to scroll the
stack window left and right by half its width. Another way to view
something large is to press ``` (grave accent) to edit the top of
stack in a separate window. (Press `C-c C-c` when you are done.)

You can enter non-decimal numbers using the `#` symbol, too.
Let’s see what the hexadecimal number ‘`5FE`’ looks like in
binary. Type `16#5FE` (the letters can be typed in upper or
lower case; they will always appear in upper case). It will also
help to turn grouping on with `d g`:

2#101,1111,1110

Notice that `d g` groups by fours by default if the display radix
is binary or hexadecimal, but by threes if it is decimal, octal, or any
other radix.

Now let’s see that number in decimal; type `d r 10`:

1,534

Numbers are not *stored* with any particular radix attached. They’re
just numbers; they can be entered in any radix, and are always displayed
in whatever radix you’ve chosen with `d r`. The current radix applies
to integers, fractions, and floats.

(•) **Exercise 1.** Your friend Joe tried to enter one-third
as ‘`3#0.1`’ in `d r 3` mode with a precision of 12. He got
‘`3#0.0222222...`’ (with 25 2’s) in the display. When he multiplied
that by three, he got ‘`3#0.222222...`’ instead of the expected
‘`3#1`’. Next, Joe entered ‘`3#0.2`’ and, to his great relief,
saw ‘`3#0.2`’ on the screen. But when he typed `2 /`, he got
‘`3#0.10000001`’ (some zeros omitted). What’s going on here?
See 1. (•)

(•) **Exercise 2.** Scientific notation works in non-decimal
modes in the natural way (the exponent is a power of the radix instead of
a power of ten, although the exponent itself is always written in decimal).
Thus ‘`8#1.23e3 = 8#1230.0`’. Suppose we have the hexadecimal number
‘`f.e8f`’ times 16 to the 15th power: We write ‘`16#f.e8fe15`’.
What is wrong with this picture? What could we write instead that would
work better? See 2. (•)

The `m` prefix key has another set of modes, relating to the way
Calc interprets your inputs and does computations. Whereas `d`-prefix
modes generally affect the way things look, `m`-prefix modes affect
the way they are actually computed.

The most popular `m`-prefix mode is the *angular mode*. Notice
the ‘`Deg`’ indicator in the mode line. This means that if you use
a command that interprets a number as an angle, it will assume the
angle is measured in degrees. For example,

1: 45 1: 0.707106781187 1: 0.500000000001 1: 0.5 . . . . 45 S 2 ^ c 1

The shift-`S` command computes the sine of an angle. The sine
of 45 degrees is
‘`sqrt(2)/2`’;
squaring this yields ‘`2/4 = 0.5`’. However, there has been a slight
roundoff error because the representation of
‘`sqrt(2)/2`’
wasn’t exact. The `c 1` command is a handy way to clean up numbers
in this case; it temporarily reduces the precision by one digit while it
re-rounds the number on the top of the stack.

(•) **Exercise 3.** Your friend Joe computed the sine
of 45 degrees as shown above, then, hoping to avoid an inexact
result, he increased the precision to 16 digits before squaring.
What happened? See 3. (•)

To do this calculation in radians, we would type `m r` first.
(The indicator changes to ‘`Rad`’.) 45 degrees corresponds to
‘`pi/4`’ radians. To get ‘`pi`’, press the `P` key. (Once
again, this is a shifted capital `P`. Remember, unshifted
`p` sets the precision.)

1: 3.14159265359 1: 0.785398163398 1: 0.707106781187 . . . P 4 / m r S

Likewise, inverse trigonometric functions generate results in either radians or degrees, depending on the current angular mode.

1: 0.707106781187 1: 0.785398163398 1: 45. . . . .5 Q m r I S m d U I S

Here we compute the Inverse Sine of
‘`sqrt(0.5)`’,
first in radians, then in degrees.

Use `c d` and `c r` to convert a number from radians to degrees
and vice-versa.

1: 45 1: 0.785398163397 1: 45. . . . 45 c r c d

Another interesting mode is *Fraction mode*. Normally,
dividing two integers produces a floating-point result if the
quotient can’t be expressed as an exact integer. Fraction mode
causes integer division to produce a fraction, i.e., a rational
number, instead.

2: 12 1: 1.33333333333 1: 4:3 1: 9 . . . 12RET9 / m f U / m f

In the first case, we get an approximate floating-point result. In the second case, we get an exact fractional result (four-thirds).

You can enter a fraction at any time using `:` notation.
(Calc uses `:` instead of `/` as the fraction separator
because `/` is already used to divide the top two stack
elements.) Calculations involving fractions will always
produce exact fractional results; Fraction mode only says
what to do when dividing two integers.

(•) **Exercise 4.** If fractional arithmetic is exact,
why would you ever use floating-point numbers instead?
See 4. (•)

Typing `m f` doesn’t change any existing values in the stack.
In the above example, we had to Undo the division and do it over
again when we changed to Fraction mode. But if you use the
evaluates-to operator you can get commands like `m f` to
recompute for you.

1: 12 / 9 => 1.33333333333 1: 12 / 9 => 1.333 1: 12 / 9 => 4:3 . . . ' 12/9 =>RETp 4RETm f

In this example, the righthand side of the ‘`=>`’ operator
on the stack is recomputed when we change the precision, then
again when we change to Fraction mode. All ‘`=>`’ expressions
on the stack are recomputed every time you change any mode that
might affect their values.

### 2.2 Arithmetic Tutorial

In this section, we explore the arithmetic and scientific functions available in the Calculator.

The standard arithmetic commands are `+`, `-`, `*`, `/`,
and `^`. Each normally takes two numbers from the top of the stack
and pushes back a result. The `n` and `&` keys perform
change-sign and reciprocal operations, respectively.

1: 5 1: 0.2 1: 5. 1: -5. 1: 5. . . . . . 5 & & n n

You can apply a “binary operator” like `+` across any number of
stack entries by giving it a numeric prefix. You can also apply it
pairwise to several stack elements along with the top one if you use
a negative prefix.

3: 2 1: 9 3: 2 4: 2 3: 12 2: 3 . 2: 3 3: 3 2: 13 1: 4 1: 4 2: 4 1: 14 . . 1: 10 . . 2RET3RET4 M-3 + U 10 M-- M-3 +

You can apply a “unary operator” like `&` to the top `n`
stack entries with a numeric prefix, too.

3: 2 3: 0.5 3: 0.5 2: 3 2: 0.333333333333 2: 3. 1: 4 1: 0.25 1: 4. . . . 2RET3RET4 M-3 & M-2 &

Notice that the results here are left in floating-point form.
We can convert them back to integers by pressing `F`, the
“floor” function. This function rounds down to the next lower
integer. There is also `R`, which rounds to the nearest
integer.

7: 2. 7: 2 7: 2 6: 2.4 6: 2 6: 2 5: 2.5 5: 2 5: 3 4: 2.6 4: 2 4: 3 3: -2. 3: -2 3: -2 2: -2.4 2: -3 2: -2 1: -2.6 1: -3 1: -3 . . . M-7 F U M-7 R

Since dividing-and-flooring (i.e., “integer quotient”) is such a
common operation, Calc provides a special command for that purpose, the
backslash `\`. Another common arithmetic operator is `%`, which
computes the remainder that would arise from a `\` operation, i.e.,
the “modulo” of two numbers. For example,

2: 1234 1: 12 2: 1234 1: 34 1: 100 . 1: 100 . . . 1234RET100 \ U %

These commands actually work for any real numbers, not just integers.

2: 3.1415 1: 3 2: 3.1415 1: 0.1415 1: 1 . 1: 1 . . . 3.1415RET1 \ U %

(•) **Exercise 1.** The `\` command would appear to be a
frill, since you could always do the same thing with `/ F`. Think
of a situation where this is not true—`/ F` would be inadequate.
Now think of a way you could get around the problem if Calc didn’t
provide a `\` command. See 1. (•)

We’ve already seen the `Q` (square root) and `S` (sine)
commands. Other commands along those lines are `C` (cosine),
`T` (tangent), `E` (‘`e^x`’) and `L` (natural
logarithm). These can be modified by the `I` (inverse) and
`H` (hyperbolic) prefix keys.

Let’s compute the sine and cosine of an angle, and verify the
identity
‘`sin(x)^2 + cos(x)^2 = 1`’.
We’ll arbitrarily pick *-64* degrees as a good value for ‘`x`’.
With the angular mode set to degrees (type `m d`), do:

2: -64 2: -64 2: -0.89879 2: -0.89879 1: 1. 1: -64 1: -0.89879 1: -64 1: 0.43837 . . . . . 64 nRETRETSTABC f h

(For brevity, we’re showing only five digits of the results here. You can of course do these calculations to any precision you like.)

Remember, `f h` is the `calc-hypot`

, or square-root of sum
of squares, command.

Another identity is
‘`tan(x) = sin(x) / cos(x)`’.

2: -0.89879 1: -2.0503 1: -64. 1: 0.43837 . . . U / I T

A physical interpretation of this calculation is that if you move
‘`0.89879`’ units downward and ‘`0.43837`’ units to the right,
your direction of motion is *-64* degrees from horizontal. Suppose
we move in the opposite direction, up and to the left:

2: -0.89879 2: 0.89879 1: -2.0503 1: -64. 1: 0.43837 1: -0.43837 . . . . U U M-2 n / I T

How can the angle be the same? The answer is that the `/` operation
loses information about the signs of its inputs. Because the quotient
is negative, we know exactly one of the inputs was negative, but we
can’t tell which one. There is an `f T` [`arctan2`

] function which
computes the inverse tangent of the quotient of a pair of numbers.
Since you feed it the two original numbers, it has enough information
to give you a full 360-degree answer.

2: 0.89879 1: 116. 3: 116. 2: 116. 1: 180. 1: -0.43837 . 2: -0.89879 1: -64. . . 1: 0.43837 . . U U f T M-RETM-2 n f T -

The resulting angles differ by 180 degrees; in other words, they point in opposite directions, just as we would expect.

The `META`-`RET` we used in the third step is the
“last-arguments” command. It is sort of like Undo, except that it
restores the arguments of the last command to the stack without removing
the command’s result. It is useful in situations like this one,
where we need to do several operations on the same inputs. We could
have accomplished the same thing by using `M-2 RET` to duplicate
the top two stack elements right after the

`U U`, then a pair of

`M-`commands to cycle the 116 up around the duplicates.

`TAB`A similar identity is supposed to hold for hyperbolic sines and cosines,
except that it is the *difference*
‘`cosh(x)^2 - sinh(x)^2`’
that always equals one. Let’s try to verify this identity.

2: -64 2: -64 2: -64 2: 9.7192e54 2: 9.7192e54 1: -64 1: 3.1175e27 1: 9.7192e54 1: -64 1: 9.7192e54 . . . . . 64 nRETRETH C 2 ^TABH S 2 ^

Something’s obviously wrong, because when we subtract these numbers
the answer will clearly be zero! But if you think about it, if these
numbers *did* differ by one, it would be in the 55th decimal
place. The difference we seek has been lost entirely to roundoff
error.

We could verify this hypothesis by doing the actual calculation with, say, 60 decimal places of precision. This will be slow, but not enormously so. Try it if you wish; sure enough, the answer is 0.99999, reasonably close to 1.

Of course, a more reasonable way to verify the identity is to use
a more reasonable value for ‘`x`’!

Some Calculator commands use the Hyperbolic prefix for other purposes.
The logarithm and exponential functions, for example, work to the base
‘`e`’ normally but use base-10 instead if you use the Hyperbolic
prefix.

1: 1000 1: 6.9077 1: 1000 1: 3 . . . . 1000 L U H L

First, we mistakenly compute a natural logarithm. Then we undo and compute a common logarithm instead.

The `B` key computes a general base-`b` logarithm for any
value of `b`.

2: 1000 1: 3 1: 1000. 2: 1000. 1: 6.9077 1: 10 . . 1: 2.71828 . . . 1000RET10 B H E H P B

Here we first use `B` to compute the base-10 logarithm, then use
the “hyperbolic” exponential as a cheap hack to recover the number
1000, then use `B` again to compute the natural logarithm. Note
that `P` with the hyperbolic prefix pushes the constant ‘`e`’
onto the stack.

You may have noticed that both times we took the base-10 logarithm
of 1000, we got an exact integer result. Calc always tries to give
an exact rational result for calculations involving rational numbers
where possible. But when we used `H E`, the result was a
floating-point number for no apparent reason. In fact, if we had
computed `10 RET 3 ^` we

*would*have gotten an exact integer 1000. But the

`H E`command is rigged to generate a floating-point result all of the time so that

`1000 H E`will not waste time computing a thousand-digit integer when all you probably wanted was ‘

`1e1000`’.

(•) **Exercise 2.** Find a pair of integer inputs to
the `B` command for which Calc could find an exact rational
result but doesn’t. See 2. (•)

The Calculator also has a set of functions relating to combinatorics
and statistics. You may be familiar with the *factorial* function,
which computes the product of all the integers up to a given number.

1: 100 1: 93326215443... 1: 100. 1: 9.3326e157 . . . . 100 ! U c f !

Recall, the `c f` command converts the integer or fraction at the
top of the stack to floating-point format. If you take the factorial
of a floating-point number, you get a floating-point result
accurate to the current precision. But if you give `!` an
exact integer, you get an exact integer result (158 digits long
in this case).

If you take the factorial of a non-integer, Calc uses a generalized
factorial function defined in terms of Euler’s Gamma function
‘`gamma(n)`’
(which is itself available as the `f g` command).

3: 4. 3: 24. 1: 5.5 1: 52.342777847 2: 4.5 2: 52.3427777847 . . 1: 5. 1: 120. . . M-3 ! M-0DEL5.5 f g

Here we verify the identity
‘` n! = gamma(n+1)`’.

The binomial coefficient `n`-choose-`m`
is defined by
‘`n! / m! (n-m)!`’
for all reals ‘`n`’ and ‘`m`’. The intermediate results in this
formula can become quite large even if the final result is small; the
`k c` command computes a binomial coefficient in a way that avoids
large intermediate values.

The `k` prefix key defines several common functions out of
combinatorics and number theory. Here we compute the binomial
coefficient 30-choose-20, then determine its prime factorization.

2: 30 1: 30045015 1: [3, 3, 5, 7, 11, 13, 23, 29] 1: 20 . . . 30RET20 k c k f

You can verify these prime factors by using `V R *` to multiply
together the elements of this vector. The result is the original
number, 30045015.

Suppose a program you are writing needs a hash table with at least 10000 entries. It’s best to use a prime number as the actual size of a hash table. Calc can compute the next prime number after 10000:

1: 10000 1: 10007 1: 9973 . . . 10000 k n I k n

Just for kicks we’ve also computed the next prime *less* than
10000.

See Financial Functions, for a description of the Calculator
commands that deal with business and financial calculations (functions
like `pv`

, `rate`

, and `sln`

).

See Binary Number Functions, to read about the commands for operating
on binary numbers (like `and`

, `xor`

, and `lsh`

).

### 2.3 Vector/Matrix Tutorial

A *vector* is a list of numbers or other Calc data objects.
Calc provides a large set of commands that operate on vectors. Some
are familiar operations from vector analysis. Others simply treat
a vector as a list of objects.

#### 2.3.1 Vector Analysis

If you add two vectors, the result is a vector of the sums of the elements, taken pairwise.

1: [1, 2, 3] 2: [1, 2, 3] 1: [8, 8, 3] . 1: [7, 6, 0] . . [1,2,3] s 1 [7 6 0] s 2 +

Note that we can separate the vector elements with either commas or
spaces. This is true whether we are using incomplete vectors or
algebraic entry. The `s 1` and `s 2` commands save these
vectors so we can easily reuse them later.

If you multiply two vectors, the result is the sum of the products
of the elements taken pairwise. This is called the *dot product*
of the vectors.

2: [1, 2, 3] 1: 19 1: [7, 6, 0] . . r 1 r 2 *

The dot product of two vectors is equal to the product of their
lengths times the cosine of the angle between them. (Here the vector
is interpreted as a line from the origin ‘`(0,0,0)`’ to the
specified point in three-dimensional space.) The `A`
(absolute value) command can be used to compute the length of a
vector.

3: 19 3: 19 1: 0.550782 1: 56.579 2: [1, 2, 3] 2: 3.741657 . . 1: [7, 6, 0] 1: 9.219544 . . M-RETM-2 A * / I C

First we recall the arguments to the dot product command, then we compute the absolute values of the top two stack entries to obtain the lengths of the vectors, then we divide the dot product by the product of the lengths to get the cosine of the angle. The inverse cosine finds that the angle between the vectors is about 56 degrees.

The *cross product* of two vectors is a vector whose length
is the product of the lengths of the inputs times the sine of the
angle between them, and whose direction is perpendicular to both
input vectors. Unlike the dot product, the cross product is
defined only for three-dimensional vectors. Let’s double-check
our computation of the angle using the cross product.

2: [1, 2, 3] 3: [-18, 21, -8] 1: [-0.52, 0.61, -0.23] 1: 56.579 1: [7, 6, 0] 2: [1, 2, 3] . . . 1: [7, 6, 0] . r 1 r 2 V C s 3 M-RETM-2 A * / A I S

First we recall the original vectors and compute their cross product, which we also store for later reference. Now we divide the vector by the product of the lengths of the original vectors. The length of this vector should be the sine of the angle; sure enough, it is!

Vector-related commands generally begin with the `v` prefix key.
Some are uppercase letters and some are lowercase. To make it easier
to type these commands, the shift-`V` prefix key acts the same as
the `v` key. (See General Mode Commands, for a way to make all
prefix keys have this property.)

If we take the dot product of two perpendicular vectors we expect to get zero, since the cosine of 90 degrees is zero. Let’s check that the cross product is indeed perpendicular to both inputs:

2: [1, 2, 3] 1: 0 2: [7, 6, 0] 1: 0 1: [-18, 21, -8] . 1: [-18, 21, -8] . . . r 1 r 3 *DELr 2 r 3 *

(•) **Exercise 1.** Given a vector on the top of the
stack, what keystrokes would you use to *normalize* the
vector, i.e., to reduce its length to one without changing its
direction? See 1. (•)

(•) **Exercise 2.** Suppose a certain particle can be
at any of several positions along a ruler. You have a list of
those positions in the form of a vector, and another list of the
probabilities for the particle to be at the corresponding positions.
Find the average position of the particle.
See 2. (•)

#### 2.3.2 Matrices

A *matrix* is just a vector of vectors, all the same length.
This means you can enter a matrix using nested brackets. You can
also use the semicolon character to enter a matrix. We’ll show
both methods here:

1: [ [ 1, 2, 3 ] 1: [ [ 1, 2, 3 ] [ 4, 5, 6 ] ] [ 4, 5, 6 ] ] . . [[1 2 3] [4 5 6]] ' [1 2 3; 4 5 6]RET

We’ll be using this matrix again, so type `s 4` to save it now.

Note that semicolons work with incomplete vectors, but they work better in algebraic entry. That’s why we use the apostrophe in the second example.

When two matrices are multiplied, the lefthand matrix must have
the same number of columns as the righthand matrix has rows.
Row ‘`i`’, column ‘`j`’ of the result is effectively the
dot product of row ‘`i`’ of the left matrix by column ‘`j`’
of the right matrix.

If we try to duplicate this matrix and multiply it by itself, the dimensions are wrong and the multiplication cannot take place:

1: [ [ 1, 2, 3 ] * [ [ 1, 2, 3 ] [ 4, 5, 6 ] ] [ 4, 5, 6 ] ] .RET*

Though rather hard to read, this is a formula which shows the product
of two matrices. The ‘`*`’ function, having invalid arguments, has
been left in symbolic form.

We can multiply the matrices if we *transpose* one of them first.

2: [ [ 1, 2, 3 ] 1: [ [ 14, 32 ] 1: [ [ 17, 22, 27 ] [ 4, 5, 6 ] ] [ 32, 77 ] ] [ 22, 29, 36 ] 1: [ [ 1, 4 ] . [ 27, 36, 45 ] ] [ 2, 5 ] . [ 3, 6 ] ] . U v t * UTAB*

Matrix multiplication is not commutative; indeed, switching the order of the operands can even change the dimensions of the result matrix, as happened here!

If you multiply a plain vector by a matrix, it is treated as a single row or column depending on which side of the matrix it is on. The result is a plain vector which should also be interpreted as a row or column as appropriate.

2: [ [ 1, 2, 3 ] 1: [14, 32] [ 4, 5, 6 ] ] . 1: [1, 2, 3] . r 4 r 1 *

Multiplying in the other order wouldn’t work because the number of rows in the matrix is different from the number of elements in the vector.

(•) **Exercise 1.** Use ‘`*`’ to sum along the rows
of the above
2x3
matrix to get ‘`[6, 15]`’. Now use ‘`*`’ to sum along the columns
to get ‘`[5, 7, 9]`’.
See 1. (•)

An *identity matrix* is a square matrix with ones along the
diagonal and zeros elsewhere. It has the property that multiplication
by an identity matrix, on the left or on the right, always produces
the original matrix.

1: [ [ 1, 2, 3 ] 2: [ [ 1, 2, 3 ] 1: [ [ 1, 2, 3 ] [ 4, 5, 6 ] ] [ 4, 5, 6 ] ] [ 4, 5, 6 ] ] . 1: [ [ 1, 0, 0 ] . [ 0, 1, 0 ] [ 0, 0, 1 ] ] . r 4 v i 3RET*

If a matrix is square, it is often possible to find its *inverse*,
that is, a matrix which, when multiplied by the original matrix, yields
an identity matrix. The `&` (reciprocal) key also computes the
inverse of a matrix.

1: [ [ 1, 2, 3 ] 1: [ [ -2.4, 1.2, -0.2 ] [ 4, 5, 6 ] [ 2.8, -1.4, 0.4 ] [ 7, 6, 0 ] ] [ -0.73333, 0.53333, -0.2 ] ] . . r 4 r 2 | s 5 &

The vertical bar `|` *concatenates* numbers, vectors, and
matrices together. Here we have used it to add a new row onto
our matrix to make it square.

We can multiply these two matrices in either order to get an identity.

1: [ [ 1., 0., 0. ] 1: [ [ 1., 0., 0. ] [ 0., 1., 0. ] [ 0., 1., 0. ] [ 0., 0., 1. ] ] [ 0., 0., 1. ] ] . . M-RET* UTAB*

Matrix inverses are related to systems of linear equations in algebra. Suppose we had the following set of equations:

a + 2b + 3c = 6 4a + 5b + 6c = 2 7a + 6b = 3

This can be cast into the matrix equation,

[ [ 1, 2, 3 ] [ [ a ] [ [ 6 ] [ 4, 5, 6 ] * [ b ] = [ 2 ] [ 7, 6, 0 ] ] [ c ] ] [ 3 ] ]

We can solve this system of equations by multiplying both sides by the inverse of the matrix. Calc can do this all in one step:

2: [6, 2, 3] 1: [-12.6, 15.2, -3.93333] 1: [ [ 1, 2, 3 ] . [ 4, 5, 6 ] [ 7, 6, 0 ] ] . [6,2,3] r 5 /

The result is the ‘`[a, b, c]`’ vector that solves the equations.
(Dividing by a square matrix is equivalent to multiplying by its
inverse.)

Let’s verify this solution:

2: [ [ 1, 2, 3 ] 1: [6., 2., 3.] [ 4, 5, 6 ] . [ 7, 6, 0 ] ] 1: [-12.6, 15.2, -3.93333] . r 5TAB*

Note that we had to be careful about the order in which we multiplied
the matrix and vector. If we multiplied in the other order, Calc would
assume the vector was a row vector in order to make the dimensions
come out right, and the answer would be incorrect. If you
don’t feel safe letting Calc take either interpretation of your
vectors, use explicit
Nx1
or
1xN
matrices instead. In this case, you would enter the original column
vector as ‘`[[6], [2], [3]]`’ or ‘`[6; 2; 3]`’.

(•) **Exercise 2.** Algebraic entry allows you to make
vectors and matrices that include variables. Solve the following
system of equations to get expressions for ‘`x`’ and ‘`y`’
in terms of ‘`a`’ and ‘`b`’.

x + a y = 6 x + b y = 10

See 2. (•)

(•) **Exercise 3.** A system of equations is “over-determined”
if it has more equations than variables. It is often the case that
there are no values for the variables that will satisfy all the
equations at once, but it is still useful to find a set of values
which “nearly” satisfy all the equations. In terms of matrix equations,
you can’t solve ‘`A X = B`’ directly because the matrix ‘`A`’
is not square for an over-determined system. Matrix inversion works
only for square matrices. One common trick is to multiply both sides
on the left by the transpose of ‘`A`’:
‘`trn(A)*A*X = trn(A)*B`’.
Now
‘`trn(A)*A`’
is a square matrix so a solution is possible. It turns out that the
‘`X`’ vector you compute in this way will be a “least-squares”
solution, which can be regarded as the “closest” solution to the set
of equations. Use Calc to solve the following over-determined
system:

a + 2b + 3c = 6 4a + 5b + 6c = 2 7a + 6b = 3 2a + 4b + 6c = 11

See 3. (•)

#### 2.3.3 Vectors as Lists

Although Calc has a number of features for manipulating vectors and
matrices as mathematical objects, you can also treat vectors as
simple lists of values. For example, we saw that the `k f`
command returns a vector which is a list of the prime factors of a
number.

You can pack and unpack stack entries into vectors:

3: 10 1: [10, 20, 30] 3: 10 2: 20 . 2: 20 1: 30 1: 30 . . M-3 v p v u

You can also build vectors out of consecutive integers, or out of many copies of a given value:

1: [1, 2, 3, 4] 2: [1, 2, 3, 4] 2: [1, 2, 3, 4] . 1: 17 1: [17, 17, 17, 17] . . v x 4RET17 v b 4RET

You can apply an operator to every element of a vector using the
*map* command.

1: [17, 34, 51, 68] 1: [289, 1156, 2601, 4624] 1: [17, 34, 51, 68] . . . V M * 2 V M ^ V M Q

In the first step, we multiply the vector of integers by the vector of 17’s elementwise. In the second step, we raise each element to the power two. (The general rule is that both operands must be vectors of the same length, or else one must be a vector and the other a plain number.) In the final step, we take the square root of each element.

(•) **Exercise 1.** Compute a vector of powers of two
from
‘`2^-4`’
to ‘`2^4`’. See 1. (•)

You can also *reduce* a binary operator across a vector.
For example, reducing ‘`*`’ computes the product of all the
elements in the vector:

1: 123123 1: [3, 7, 11, 13, 41] 1: 123123 . . . 123123 k f V R *

In this example, we decompose 123123 into its prime factors, then multiply those factors together again to yield the original number.

We could compute a dot product “by hand” using mapping and reduction:

2: [1, 2, 3] 1: [7, 12, 0] 1: 19 1: [7, 6, 0] . . . r 1 r 2 V M * V R +

Recalling two vectors from the previous section, we compute the sum of pairwise products of the elements to get the same answer for the dot product as before.

A slight variant of vector reduction is the *accumulate* operation,
`V U`. This produces a vector of the intermediate results from
a corresponding reduction. Here we compute a table of factorials:

1: [1, 2, 3, 4, 5, 6] 1: [1, 2, 6, 24, 120, 720] . . v x 6RETV U *

Calc allows vectors to grow as large as you like, although it gets rather slow if vectors have more than about a hundred elements. Actually, most of the time is spent formatting these large vectors for display, not calculating on them. Try the following experiment (if your computer is very fast you may need to substitute a larger vector size).

1: [1, 2, 3, 4, ... 1: [2, 3, 4, 5, ... . . v x 500RET1 V M +

Now press `v .` (the letter `v`, then a period) and try the
experiment again. In `v .` mode, long vectors are displayed
“abbreviated” like this:

1: [1, 2, 3, ..., 500] 1: [2, 3, 4, ..., 501] . . v x 500RET1 V M +

(where now the ‘`...`’ is actually part of the Calc display).
You will find both operations are now much faster. But notice that
even in `v .` mode, the full vectors are still shown in the Trail.
Type `t .` to cause the trail to abbreviate as well, and try the
experiment one more time. Operations on long vectors are now quite
fast! (But of course if you use `t .` you will lose the ability
to get old vectors back using the `t y` command.)

An easy way to view a full vector when `v .` mode is active is
to press ``` (grave accent) to edit the vector; editing always works
with the full, unabbreviated value.

As a larger example, let’s try to fit a straight line to some data, using the method of least squares. (Calc has a built-in command for least-squares curve fitting, but we’ll do it by hand here just to practice working with vectors.) Suppose we have the following list of values in a file we have loaded into Emacs:

x y --- --- 1.34 0.234 1.41 0.298 1.49 0.402 1.56 0.412 1.64 0.466 1.73 0.473 1.82 0.601 1.91 0.519 2.01 0.603 2.11 0.637 2.22 0.645 2.33 0.705 2.45 0.917 2.58 1.009 2.71 0.971 2.85 1.062 3.00 1.148 3.15 1.157 3.32 1.354

If you are reading this tutorial in printed form, you will find it
easiest to press `C-x * i` to enter the on-line Info version of
the manual and find this table there. (Press `g`, then type
`List Tutorial`, to jump straight to this section.)

Position the cursor at the upper-left corner of this table, just
to the left of the ‘`1.34`’. Press `C-@` to set the mark.
(On your system this may be `C-2`, `C- SPC`, or

`NUL`.) Now position the cursor to the lower-right, just after the ‘

`1.354`’. You have now defined this region as an Emacs “rectangle.” Still in the Info buffer, type

`C-x * r`. This command (

`calc-grab-rectangle`

) will pop you back into the Calculator, with
the contents of the rectangle you specified in the form of a matrix.
1: [ [ 1.34, 0.234 ] [ 1.41, 0.298 ] …

(You may wish to use `v .` mode to abbreviate the display of this
large matrix.)

We want to treat this as a pair of lists. The first step is to transpose this matrix into a pair of rows. Remember, a matrix is just a vector of vectors. So we can unpack the matrix into a pair of row vectors on the stack.

1: [ [ 1.34, 1.41, 1.49, ... ] 2: [1.34, 1.41, 1.49, ... ] [ 0.234, 0.298, 0.402, ... ] ] 1: [0.234, 0.298, 0.402, ... ] . . v t v u

Let’s store these in quick variables 1 and 2, respectively.

1: [1.34, 1.41, 1.49, ... ] . . t 2 t 1

(Recall that `t 2` is a variant of `s 2` that removes the
stored value from the stack.)

In a least squares fit, the slope ‘`m`’ is given by the formula

m = (N sum(x y) - sum(x) sum(y)) / (N sum(x^2) - sum(x)^2)

where
‘`sum(x)`’
represents the sum of all the values of ‘`x`’. While there is an
actual `sum`

function in Calc, it’s easier to sum a vector using a
simple reduction. First, let’s compute the four different sums that
this formula uses.

1: 41.63 1: 98.0003 . . r 1 V R + t 3 r 1 2 V M ^ V R + t 4

1: 13.613 1: 33.36554 . . r 2 V R + t 5 r 1 r 2 V M * V R + t 6

These are ‘`sum(x)`’, ‘`sum(x^2)`’, ‘`sum(y)`’, and ‘`sum(x y)`’,
respectively. (We could have used `*` to compute ‘`sum(x^2)`’ and
‘`sum(x y)`’.)

Finally, we also need ‘`N`’, the number of data points. This is just
the length of either of our lists.

1: 19 . r 1 v l t 7

(That’s `v` followed by a lower-case `l`.)

Now we grind through the formula:

1: 633.94526 2: 633.94526 1: 67.23607 . 1: 566.70919 . . r 7 r 6 * r 3 r 5 * -

2: 67.23607 3: 67.23607 2: 67.23607 1: 0.52141679 1: 1862.0057 2: 1862.0057 1: 128.9488 . . 1: 1733.0569 . . r 7 r 4 * r 3 2 ^ - / t 8

That gives us the slope ‘`m`’. The y-intercept ‘`b`’ can now
be found with the simple formula,

b = (sum(y) - m sum(x)) / N

1: 13.613 2: 13.613 1: -8.09358 1: -0.425978 . 1: 21.70658 . . . r 5 r 8 r 3 * - r 7 / t 9

Let’s “plot” this straight line approximation,
‘`m x + b`’,
and compare it with the original data.

1: [0.699, 0.735, ... ] 1: [0.273, 0.309, ... ] . . r 1 r 8 * r 9 + s 0

Notice that multiplying a vector by a constant, and adding a constant to a vector, can be done without mapping commands since these are common operations from vector algebra. As far as Calc is concerned, we’ve just been doing geometry in 19-dimensional space!

We can subtract this vector from our original ‘`y`’ vector to get
a feel for the error of our fit. Let’s find the maximum error:

1: [0.0387, 0.0112, ... ] 1: [0.0387, 0.0112, ... ] 1: 0.0897 . . . r 2 - V M A V R X

First we compute a vector of differences, then we take the absolute
values of these differences, then we reduce the `max`

function
across the vector. (The `max`

function is on the two-key sequence
`f x`; because it is so common to use `max`

in a vector
operation, the letters `X` and `N` are also accepted for
`max`

and `min`

in this context. In general, you answer
the `V M` or `V R` prompt with the actual key sequence that
invokes the function you want. You could have typed `V R f x` or
even `V R x max RET` if you had preferred.)

If your system has the GNUPLOT program, you can see graphs of your data and your straight line to see how well they match. (If you have GNUPLOT 3.0 or higher, the following instructions will work regardless of the kind of display you have. Some GNUPLOT 2.0, non-X-windows systems may require additional steps to view the graphs.)

Let’s start by plotting the original data. Recall the “`x`” and “`y`”
vectors onto the stack and press `g f`. This “fast” graphing
command does everything you need to do for simple, straightforward
plotting of data.

2: [1.34, 1.41, 1.49, ... ] 1: [0.234, 0.298, 0.402, ... ] . r 1 r 2 g f

If all goes well, you will shortly get a new window containing a graph
of the data. (If not, contact your GNUPLOT or Calc installer to find
out what went wrong.) In the X window system, this will be a separate
graphics window. For other kinds of displays, the default is to
display the graph in Emacs itself using rough character graphics.
Press `q` when you are done viewing the character graphics.

Next, let’s add the line we got from our least-squares fit.

2: [1.34, 1.41, 1.49, ... ] 1: [0.273, 0.309, 0.351, ... ] .DELr 0 g a g p

It’s not very useful to get symbols to mark the data points on this
second curve; you can type `g S g p` to remove them. Type `g q`
when you are done to remove the X graphics window and terminate GNUPLOT.

(•) **Exercise 2.** An earlier exercise showed how to do
least squares fitting to a general system of equations. Our 19 data
points are really 19 equations of the form ‘`y_i = m x_i + b`’ for
different pairs of ‘`(x_i,y_i)`’. Use the matrix-transpose method
to solve for ‘`m`’ and ‘`b`’, duplicating the above result.
See 2. (•)

(•) **Exercise 3.** If the input data do not form a
rectangle, you can use `C-x * g` (`calc-grab-region`

)
to grab the data the way Emacs normally works with regions—it reads
left-to-right, top-to-bottom, treating line breaks the same as spaces.
Use this command to find the geometric mean of the following numbers.
(The geometric mean is the `n`th root of the product of `n` numbers.)

2.3 6 22 15.1 7 15 14 7.5 2.5

The `C-x * g` command accepts numbers separated by spaces or commas,
with or without surrounding vector brackets.
See 3. (•)

As another example, a theorem about binomial coefficients tells
us that the alternating sum of binomial coefficients
`n`-choose-0 minus `n`-choose-1 plus `n`-choose-2, and so
on up to `n`-choose-`n`,
always comes out to zero. Let’s verify this
for ‘`n=6`’.

1: [1, 2, 3, 4, 5, 6, 7] 1: [0, 1, 2, 3, 4, 5, 6] . . v x 7RET1 -

1: [1, -6, 15, -20, 15, -6, 1] 1: 0 . . V M ' (-1)^$ choose(6,$)RETV R +

The `V M '` command prompts you to enter any algebraic expression
to define the function to map over the vector. The symbol ‘`$`’
inside this expression represents the argument to the function.
The Calculator applies this formula to each element of the vector,
substituting each element’s value for the ‘`$`’ sign(s) in turn.

To define a two-argument function, use ‘`$$`’ for the first
argument and ‘`$`’ for the second: `V M ' $$-$ RET` is
equivalent to

`V M -`. This is analogous to regular algebraic entry, where ‘

`$$`’ would refer to the next-to-top stack entry and ‘

`$`’ would refer to the top stack entry, and

`' $$-$`would act exactly like

`RET``-`.

Notice that the `V M '` command has recorded two things in the
trail: The result, as usual, and also a funny-looking thing marked
‘`oper`’ that represents the operator function you typed in.
The function is enclosed in ‘`< >`’ brackets, and the argument is
denoted by a ‘`#`’ sign. If there were several arguments, they
would be shown as ‘`#1`’, ‘`#2`’, and so on. (For example,
`V M ' $$-$` will put the function ‘`<#1 - #2>`’ on the
trail.) This object is a “nameless function”; you can use nameless
‘`< >`’ notation to answer the `V M '` prompt if you like.
Nameless function notation has the interesting, occasionally useful
property that a nameless function is not actually evaluated until
it is used. For example, `V M ' $+random(2.0)` evaluates
‘`random(2.0)`’ once and adds that random number to all elements
of the vector, but `V M ' <#+random(2.0)>` evaluates the
‘`random(2.0)`’ separately for each vector element.

Another group of operators that are often useful with `V M` are
the relational operators: `a =`, for example, compares two numbers
and gives the result 1 if they are equal, or 0 if not. Similarly,
`a <` checks for one number being less than another.

Other useful vector operations include `v v`, to reverse a
vector end-for-end; `V S`, to sort the elements of a vector
into increasing order; and `v r` and `v c`, to extract
one row or column of a matrix, or (in both cases) to extract one
element of a plain vector. With a negative argument, `v r`
and `v c` instead delete one row, column, or vector element.

(•) **Exercise 4.** The ‘`k`’th *divisor function*
is the sum of the ‘`k`’th powers of all the divisors of an
integer ‘`n`’. Figure out a method for computing the divisor
function for reasonably small values of ‘`n`’. As a test,
the 0th and 1st divisor functions of 30 are 8 and 72, respectively.
See 4. (•)

(•) **Exercise 5.** The `k f` command produces a
list of prime factors for a number. Sometimes it is important to
know that a number is *square-free*, i.e., that no prime occurs
more than once in its list of prime factors. Find a sequence of
keystrokes to tell if a number is square-free; your method should
leave 1 on the stack if it is, or 0 if it isn’t.
See 5. (•)

(•) **Exercise 6.** Build a list of lists that looks
like the following diagram. (You may wish to use the `v /`
command to enable multi-line display of vectors.)

1: [ [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 6] ]

See 6. (•)

(•) **Exercise 7.** Build the following list of lists.

1: [ [0], [1, 2], [3, 4, 5], [6, 7, 8, 9], [10, 11, 12, 13, 14], [15, 16, 17, 18, 19, 20] ]

See 7. (•)

(•) **Exercise 8.** Compute a list of values of Bessel’s
‘`J1`’
function ‘`besJ(1,x)`’ for ‘`x`’ from 0 to 5 in steps of 0.25.
Find the value of ‘`x`’ (from among the above set of values) for
which ‘`besJ(1,x)`’ is a maximum. Use an “automatic” method,
i.e., just reading along the list by hand to find the largest value
is not allowed! (There is an `a X` command which does this kind
of thing automatically; see Numerical Solutions.)
See 8. (•)

(•) **Exercise 9.** You are given an integer in the range
‘`0 <= N < 10^m`’
for ‘`m=12`’ (i.e., an integer of less than
twelve digits). Convert this integer into a vector of ‘`m`’
digits, each in the range from 0 to 9. In vector-of-digits notation,
add one to this integer to produce a vector of ‘`m+1`’ digits
(since there could be a carry out of the most significant digit).
Convert this vector back into a regular integer. A good integer
to try is 25129925999. See 9. (•)

(•) **Exercise 10.** Your friend Joe tried to use
`V R a =` to test if all numbers in a list were equal. What
happened? How would you do this test? See 10. (•)

(•) **Exercise 11.** The area of a circle of radius one
is ‘`pi`’. The area of the
2x2
square that encloses that circle is 4. So if we throw `n` darts at
random points in the square, about ‘`pi/4`’ of them will land inside
the circle. This gives us an entertaining way to estimate the value of
‘`pi`’. The `k r`
command picks a random number between zero and the value on the stack.
We could get a random floating-point number between *-1* and 1 by typing
`2.0 k r 1 -`. Build a vector of 100 random ‘`(x,y)`’ points in
this square, then use vector mapping and reduction to count how many
points lie inside the unit circle. Hint: Use the `v b` command.
See 11. (•)

(•) **Exercise 12.** The *matchstick problem* provides
another way to calculate ‘`pi`’. Say you have an infinite field
of vertical lines with a spacing of one inch. Toss a one-inch matchstick
onto the field. The probability that the matchstick will land crossing
a line turns out to be
‘`2/pi`’.
Toss 100 matchsticks to estimate ‘`pi`’. (If you want still more fun,
the probability that the GCD (`k g`) of two large integers is
one turns out to be
‘`6/pi^2`’.
That provides yet another way to estimate ‘`pi`’.)
See 12. (•)

(•) **Exercise 13.** An algebraic entry of a string in
double-quote marks, ‘`"hello"`’, creates a vector of the numerical
(ASCII) codes of the characters (here, ‘`[104, 101, 108, 108, 111]`’).
Sometimes it is convenient to compute a *hash code* of a string,
which is just an integer that represents the value of that string.
Two equal strings have the same hash code; two different strings
*probably* have different hash codes. (For example, Calc has
over 400 function names, but Emacs can quickly find the definition for
any given name because it has sorted the functions into “buckets” by
their hash codes. Sometimes a few names will hash into the same bucket,
but it is easier to search among a few names than among all the names.)
One popular hash function is computed as follows: First set ‘`h = 0`’.
Then, for each character from the string in turn, set ‘`h = 3h + c_i`’
where ‘`c_i`’ is the character’s ASCII code. If we have 511 buckets,
we then take the hash code modulo 511 to get the bucket number. Develop a
simple command or commands for converting string vectors into hash codes.
The hash code for ‘`"Testing, 1, 2, 3"`’ is 1960915098, which modulo
511 is 121. See 13. (•)

(•) **Exercise 14.** The `H V R` and `H V U`
commands do nested function evaluations. `H V U` takes a starting
value and a number of steps `n` from the stack; it then applies the
function you give to the starting value 0, 1, 2, up to `n` times
and returns a vector of the results. Use this command to create a
“random walk” of 50 steps. Start with the two-dimensional point
‘`(0,0)`’; then take one step a random distance between *-1* and 1
in both ‘`x`’ and ‘`y`’; then take another step, and so on. Use the
`g f` command to display this random walk. Now modify your random
walk to walk a unit distance, but in a random direction, at each step.
(Hint: The `sincos`

function returns a vector of the cosine and
sine of an angle.) See 14. (•)

### 2.4 Types Tutorial

Calc understands a variety of data types as well as simple numbers. In this section, we’ll experiment with each of these types in turn.

The numbers we’ve been using so far have mainly been either *integers*
or *floats*. We saw that floats are usually a good approximation to
the mathematical concept of real numbers, but they are only approximations
and are susceptible to roundoff error. Calc also supports *fractions*,
which can exactly represent any rational number.

1: 3628800 2: 3628800 1: 518400:7 1: 518414:7 1: 7:518414 . 1: 49 . . . . 10 ! 49RET: 2 + &

The `:` command divides two integers to get a fraction; `/`
would normally divide integers to get a floating-point result.
Notice we had to type `RET` between the `49` and the `:`
since the `:` would otherwise be interpreted as part of a
fraction beginning with 49.

You can convert between floating-point and fractional format using
`c f` and `c F`:

1: 1.35027217629e-5 1: 7:518414 . . c f c F

The `c F` command replaces a floating-point number with the
“simplest” fraction whose floating-point representation is the
same, to within the current precision.

1: 3.14159265359 1: 1146408:364913 1: 3.1416 1: 355:113 . . . . P c FDELp 5RETP c F

(•) **Exercise 1.** A calculation has produced the
result 1.26508260337. You suspect it is the square root of the
product of ‘`pi`’ and some rational number. Is it? (Be sure
to allow for roundoff error!) See 1. (•)

*Complex numbers* can be stored in both rectangular and polar form.

1: -9 1: (0, 3) 1: (3; 90.) 1: (6; 90.) 1: (2.4495; 45.) . . . . . 9 n Q c p 2 * Q

The square root of *-9* is by default rendered in rectangular form
(‘`0 + 3i`’), but we can convert it to polar form (3 with a
phase angle of 90 degrees). All the usual arithmetic and scientific
operations are defined on both types of complex numbers.

Another generalized kind of number is *infinity*. Infinity
isn’t really a number, but it can sometimes be treated like one.
Calc uses the symbol `inf`

to represent positive infinity,
i.e., a value greater than any real number. Naturally, you can
also write ‘`-inf`’ for minus infinity, a value less than any
real number. The word `inf`

can only be input using
algebraic entry.

2: inf 2: -inf 2: -inf 2: -inf 1: nan 1: -17 1: -inf 1: -inf 1: inf . . . . . ' infRET17 n *RET72 + A +

Since infinity is infinitely large, multiplying it by any finite
number (like *-17*) has no effect, except that since *-17*
is negative, it changes a plus infinity to a minus infinity.
(“A huge positive number, multiplied by *-17*, yields a huge
negative number.”) Adding any finite number to infinity also
leaves it unchanged. Taking an absolute value gives us plus
infinity again. Finally, we add this plus infinity to the minus
infinity we had earlier. If you work it out, you might expect
the answer to be *-72* for this. But the 72 has been completely
lost next to the infinities; by the time we compute ‘`inf - inf`’
the finite difference between them, if any, is undetectable.
So we say the result is *indeterminate*, which Calc writes
with the symbol `nan`

(for Not A Number).

Dividing by zero is normally treated as an error, but you can get
Calc to write an answer in terms of infinity by pressing `m i`
to turn on Infinite mode.

3: nan 2: nan 2: nan 2: nan 1: nan 2: 1 1: 1 / 0 1: uinf 1: uinf . 1: 0 . . . . 1RET0 / m i U / 17 n * +

Dividing by zero normally is left unevaluated, but after `m i`
it instead gives an infinite result. The answer is actually
`uinf`

, “undirected infinity.” If you look at a graph of
‘`1 / x`’ around ‘`x = 0`’, you’ll see that it goes toward
plus infinity as you approach zero from above, but toward minus
infinity as you approach from below. Since we said only ‘`1 / 0`’,
Calc knows that the answer is infinite but not in which direction.
That’s what `uinf`

means. Notice that multiplying `uinf`

by a negative number still leaves plain `uinf`

; there’s no
point in saying ‘`-uinf`’ because the sign of `uinf`

is
unknown anyway. Finally, we add `uinf`

to our `nan`

,
yielding `nan`

again. It’s easy to see that, because
`nan`

means “totally unknown” while `uinf`

means
“unknown sign but known to be infinite,” the more mysterious
`nan`

wins out when it is combined with `uinf`

, or, for
that matter, with anything else.

(•) **Exercise 2.** Predict what Calc will answer
for each of these formulas: ‘`inf / inf`’, ‘`exp(inf)`’,
‘`exp(-inf)`’, ‘`sqrt(-inf)`’, ‘`sqrt(uinf)`’,
‘`abs(uinf)`’, ‘`ln(0)`’.
See 2. (•)

(•) **Exercise 3.** We saw that ‘`inf - inf = nan`’,
which stands for an unknown value. Can `nan`

stand for
a complex number? Can it stand for infinity?
See 3. (•)

*HMS forms* represent a value in terms of hours, minutes, and
seconds.

1: 2@ 30' 0" 1: 3@ 30' 0" 2: 3@ 30' 0" 1: 2. . . 1: 1@ 45' 0." . . 2@ 30'RET1 +RET2 / /

HMS forms can also be used to hold angles in degrees, minutes, and seconds.

1: 0.5 1: 26.56505 1: 26@ 33' 54.18" 1: 0.44721 . . . . 0.5 I T c h S

First we convert the inverse tangent of 0.5 to degrees-minutes-seconds form, then we take the sine of that angle. Note that the trigonometric functions will accept HMS forms directly as input.

(•) **Exercise 4.** The Beatles’ *Abbey Road* is
47 minutes and 26 seconds long, and contains 17 songs. What is the
average length of a song on *Abbey Road*? If the Extended Disco
Version of *Abbey Road* added 20 seconds to the length of each
song, how long would the album be? See 4. (•)

A *date form* represents a date, or a date and time. Dates must
be entered using algebraic entry. Date forms are surrounded by
‘`< >`’ symbols; most standard formats for dates are recognized.

2: <Sun Jan 13, 1991> 1: 2.25 1: <6:00pm Thu Jan 10, 1991> . . ' <13 Jan 1991>, <1/10/91, 6pm>RET-

In this example, we enter two dates, then subtract to find the number of days between them. It is also possible to add an HMS form or a number (of days) to a date form to get another date form.

1: <4:45:59pm Mon Jan 14, 1991> 1: <2:50:59am Thu Jan 17, 1991> . . t N 2 + 10@ 5' +

The `t N` (“now”) command pushes the current date and time on the
stack; then we add two days, ten hours and five minutes to the date and
time. Other date-and-time related commands include `t J`, which
does Julian day conversions, `t W`, which finds the beginning of
the week in which a date form lies, and `t I`, which increments a
date by one or several months. See Date Arithmetic, for more.

(•) **Exercise 5.** How many days until the next
Friday the 13th? See 5. (•)

(•) **Exercise 6.** How many leap years will there be
between now and the year 10001 AD? See 6. (•)

An *error form* represents a mean value with an attached standard
deviation, or error estimate. Suppose our measurements indicate that
a certain telephone pole is about 30 meters away, with an estimated
error of 1 meter, and 8 meters tall, with an estimated error of 0.2
meters. What is the slope of a line from here to the top of the
pole, and what is the equivalent angle in degrees?

1: 8 +/- 0.2 2: 8 +/- 0.2 1: 0.266 +/- 0.011 1: 14.93 +/- 0.594 . 1: 30 +/- 1 . . . 8 p .2RET30 p 1 / I T

This means that the angle is about 15 degrees, and, assuming our original error estimates were valid standard deviations, there is about a 60% chance that the result is correct within 0.59 degrees.

(•) **Exercise 7.** The volume of a torus (a donut shape) is
‘`2 pi^2 R r^2`’
where ‘`R`’ is the radius of the circle that
defines the center of the tube and ‘`r`’ is the radius of the tube
itself. Suppose ‘`R`’ is 20 cm and ‘`r`’ is 4 cm, each known to
within 5 percent. What is the volume and the relative uncertainty of
the volume? See 7. (•)

An *interval form* represents a range of values. While an
error form is best for making statistical estimates, intervals give
you exact bounds on an answer. Suppose we additionally know that
our telephone pole is definitely between 28 and 31 meters away,
and that it is between 7.7 and 8.1 meters tall.

1: [7.7 .. 8.1] 2: [7.7 .. 8.1] 1: [0.24 .. 0.28] 1: [13.9 .. 16.1] . 1: [28 .. 31] . . . [ 7.7 .. 8.1 ] [ 28 .. 31 ] / I T

If our bounds were correct, then the angle to the top of the pole is sure to lie in the range shown.

The square brackets around these intervals indicate that the endpoints
themselves are allowable values. In other words, the distance to the
telephone pole is between 28 and 31, *inclusive*. You can also
make an interval that is exclusive of its endpoints by writing
parentheses instead of square brackets. You can even make an interval
which is inclusive (“closed”) on one end and exclusive (“open”) on
the other.

1: [1 .. 10) 1: (0.1 .. 1] 2: (0.1 .. 1] 1: (0.2 .. 3) . . 1: [2 .. 3) . . [ 1 .. 10 ) & [ 2 .. 3 ) *

The Calculator automatically keeps track of which end values should
be open and which should be closed. You can also make infinite or
semi-infinite intervals by using ‘`-inf`’ or ‘`inf`’ for one
or both endpoints.

(•) **Exercise 8.** What answer would you expect from
‘`1 / (0 .. 10)`’? What about ‘`1 / (-10 .. 0)`’? What
about ‘`1 / [0 .. 10]`’ (where the interval actually includes
zero)? What about ‘`1 / (-10 .. 10)`’?
See 8. (•)

(•) **Exercise 9.** Two easy ways of squaring a number
are ` RET *` and

`2 ^`. Normally these produce the same answer. Would you expect this still to hold true for interval forms? If not, which of these will result in a larger interval? See 9. (•)

A *modulo form* is used for performing arithmetic modulo `m`.
For example, arithmetic involving time is generally done modulo 12
or 24 hours.

1: 17 mod 24 1: 3 mod 24 1: 21 mod 24 1: 9 mod 24 . . . . 17 M 24RET10 + n 5 /

In this last step, Calc has divided by 5 modulo 24; i.e., it has found a
new number which, when multiplied by 5 modulo 24, produces the original
number, 21. If `m` is prime and the divisor is not a multiple of
`m`, it is always possible to find such a number. For non-prime
`m` like 24, it is only sometimes possible.

1: 10 mod 24 1: 16 mod 24 1: 1000000... 1: 16 . . . . 10 M 24RET100 ^ 10RET100 ^ 24 %

These two calculations get the same answer, but the first one is much more efficient because it avoids the huge intermediate value that arises in the second one.

(•) **Exercise 10.** A theorem of Pierre de Fermat
says that
‘`x^(n-1) mod n = 1`’
if ‘`n`’ is a prime number and ‘`x`’ is an integer less than
‘`n`’. If ‘`n`’ is *not* a prime number, this will
*not* be true for most values of ‘`x`’. Thus we can test
informally if a number is prime by trying this formula for several
values of ‘`x`’. Use this test to tell whether the following numbers
are prime: 811749613, 15485863. See 10. (•)

It is possible to use HMS forms as parts of error forms, intervals,
modulo forms, or as the phase part of a polar complex number.
For example, the `calc-time`

command pushes the current time
of day on the stack as an HMS/modulo form.

1: 17@ 34' 45" mod 24@ 0' 0" 1: 6@ 22' 15" mod 24@ 0' 0" . . x timeRETn

This calculation tells me it is six hours and 22 minutes until midnight.

(•) **Exercise 11.** A rule of thumb is that one year
is about
‘`pi * 10^7`’
seconds. What time will it be that many seconds from right now?
See 11. (•)

(•) **Exercise 12.** You are preparing to order packaging
for the CD release of the Extended Disco Version of *Abbey Road*.
You are told that the songs will actually be anywhere from 20 to 60
seconds longer than the originals. One CD can hold about 75 minutes
of music. Should you order single or double packages?
See 12. (•)

Another kind of data the Calculator can manipulate is numbers with
*units*. This isn’t strictly a new data type; it’s simply an
application of algebraic expressions, where we use variables with
suggestive names like ‘`cm`’ and ‘`in`’ to represent units
like centimeters and inches.

1: 2 in 1: 5.08 cm 1: 0.027778 fath 1: 0.0508 m . . . . ' 2inRETu c cmRETu c fathRETu b

We enter the quantity “2 inches” (actually an algebraic expression
which means two times the variable ‘`in`’), then we convert it
first to centimeters, then to fathoms, then finally to “base” units,
which in this case means meters.

1: 9 acre 1: 3 sqrt(acre) 1: 190.84 m 1: 190.84 m + 30 cm . . . . ' 9 acreRETQ u s ' $+30 cmRET

1: 191.14 m 1: 36536.3046 m^2 1: 365363046 cm^2 . . . u s 2 ^ u c cgs

Since units expressions are really just formulas, taking the square
root of ‘`acre`’ is undefined. After all, `acre`

might be an
algebraic variable that you will someday assign a value. We use the
“units-simplify” command to simplify the expression with variables
being interpreted as unit names.

In the final step, we have converted not to a particular unit, but to a units system. The “cgs” system uses centimeters instead of meters as its standard unit of length.

There is a wide variety of units defined in the Calculator.

1: 55 mph 1: 88.5139 kph 1: 88.5139 km / hr 1: 8.201407e-8 c . . . . ' 55 mphRETu c kphRETu c km/hrRETu c cRET

We express a speed first in miles per hour, then in kilometers per hour, then again using a slightly more explicit notation, then finally in terms of fractions of the speed of light.

Temperature conversions are a bit more tricky. There are two ways to interpret “20 degrees Fahrenheit”—it could mean an actual temperature, or it could mean a change in temperature. For normal units there is no difference, but temperature units have an offset as well as a scale factor and so there must be two explicit commands for them.

1: 20 degF 1: 11.1111 degC 1: -6.666 degC . . . . ' 20 degFRETu c degCRETU u t degCRET

First we convert a change of 20 degrees Fahrenheit into an equivalent change in degrees Celsius (or Centigrade). Then, we convert the absolute temperature 20 degrees Fahrenheit into Celsius.

For simple unit conversions, you can put a plain number on the stack.
Then `u c` and `u t` will prompt for both old and new units.
When you use this method, you’re responsible for remembering which
numbers are in which units:

1: 55 1: 88.5139 1: 8.201407e-8 . . . 55 u c mphRETkphRETu c km/hrRETcRET

To see a complete list of built-in units, type `u v`. Press
`C-x * c` again to re-enter the Calculator when you’re done looking
at the units table.

(•) **Exercise 13.** How many seconds are there really
in a year? See 13. (•)

(•) **Exercise 14.** Supercomputer designs are limited by
the speed of light (and of electricity, which is nearly as fast).
Suppose a computer has a 4.1 ns (nanosecond) clock cycle, and its
cabinet is one meter across. Is speed of light going to be a
significant factor in its design? See 14. (•)

(•) **Exercise 15.** Sam the Slug normally travels about
five yards in an hour. He has obtained a supply of Power Pills; each
Power Pill he eats doubles his speed. How many Power Pills can he
swallow and still travel legally on most US highways?
See 15. (•)

### 2.5 Algebra and Calculus Tutorial

This section shows how to use Calc’s algebra facilities to solve equations, do simple calculus problems, and manipulate algebraic formulas.

#### 2.5.1 Basic Algebra

If you enter a formula in Algebraic mode that refers to variables, the formula itself is pushed onto the stack. You can manipulate formulas as regular data objects.

1: 2 x^2 - 6 1: 6 - 2 x^2 1: (3 x^2 + y) (6 - 2 x^2) . . . ' 2x^2-6RETn ' 3x^2+yRET*

(•) **Exercise 1.** Do `' x RET Q 2 ^` and

`' x`both wind up with the same result (‘

`RET`2 ^ Q`x`’)? Why or why not? See 1. (•)

There are also commands for doing common algebraic operations on formulas. Continuing with the formula from the last example,

1: 18 x^2 - 6 x^4 + 6 y - 2 y x^2 1: (18 - 2 y) x^2 - 6 x^4 + 6 y . . a x a c xRET

First we “expand” using the distributive law, then we “collect”
terms involving like powers of ‘`x`’.

Let’s find the value of this expression when ‘`x`’ is 2 and ‘`y`’
is one-half.

1: 17 x^2 - 6 x^4 + 3 1: -25 . . 1:2 s l yRET2 s l xRET

The `s l` command means “let”; it takes a number from the top of
the stack and temporarily assigns it as the value of the variable
you specify. It then evaluates (as if by the `=` key) the
next expression on the stack. After this command, the variable goes
back to its original value, if any.

(An earlier exercise in this tutorial involved storing a value in the
variable `x`

; if this value is still there, you will have to
unstore it with `s u x RET` before the above example will work
properly.)

Let’s find the maximum value of our original expression when ‘`y`’
is one-half and ‘`x`’ ranges over all possible values. We can
do this by taking the derivative with respect to ‘`x`’ and examining
values of ‘`x`’ for which the derivative is zero. If the second
derivative of the function at that value of ‘`x`’ is negative,
the function has a local maximum there.

1: 17 x^2 - 6 x^4 + 3 1: 34 x - 24 x^3 . . UDELs 1 a d xRETs 2

Well, the derivative is clearly zero when ‘`x`’ is zero. To find
the other root(s), let’s divide through by ‘`x`’ and then solve:

1: (34 x - 24 x^3) / x 1: 34 - 24 x^2 . . ' xRET/ a x

1: 0.70588 x^2 = 1 1: x = 1.19023 . . 0 a = s 3 a S xRET

Now we compute the second derivative and plug in our values of ‘`x`’:

1: 1.19023 2: 1.19023 2: 1.19023 . 1: 34 x - 24 x^3 1: 34 - 72 x^2 . . a . r 2 a d xRETs 4

(The `a .` command extracts just the righthand side of an equation.
Another method would have been to use `v u` to unpack the equation
‘`x = 1.19`’ to ‘`x`’ and ‘`1.19`’, then use `M-- M-2 DEL`
to delete the ‘

`x`’.)

2: 34 - 72 x^2 1: -68. 2: 34 - 72 x^2 1: 34 1: 1.19023 . 1: 0 . . .TABs l xRETUDEL0 s l xRET

The first of these second derivatives is negative, so we know the function
has a maximum value at ‘`x = 1.19023`’. (The function also has a
local *minimum* at ‘`x = 0`’.)

When we solved for ‘`x`’, we got only one value even though
‘`0.70588 x^2 = 1`’ is a quadratic equation that ought to have
two solutions. The reason is that `a S` normally returns a
single “principal” solution. If it needs to come up with an
arbitrary sign (as occurs in the quadratic formula) it picks ‘`+`’.
If it needs an arbitrary integer, it picks zero. We can get a full
solution by pressing `H` (the Hyperbolic flag) before `a S`.

1: 0.70588 x^2 = 1 1: x = 1.19023 s1 1: x = -1.19023 . . . r 3 H a S xRETs 5 1 n s l s1RET

Calc has invented the variable ‘`s1`’ to represent an unknown sign;
it is supposed to be either *+1* or *-1*. Here we have used
the “let” command to evaluate the expression when the sign is negative.
If we plugged this into our second derivative we would get the same,
negative, answer, so ‘`x = -1.19023`’ is also a maximum.

To find the actual maximum value, we must plug our two values of ‘`x`’
into the original formula.

2: 17 x^2 - 6 x^4 + 3 1: 24.08333 s1^2 - 12.04166 s1^4 + 3 1: x = 1.19023 s1 . . r 1 r 5 s lRET

(Here we see another way to use `s l`; if its input is an equation
with a variable on the lefthand side, then `s l` treats the equation
like an assignment to that variable if you don’t give a variable name.)

It’s clear that this will have the same value for either sign of
`s1`

, but let’s work it out anyway, just for the exercise:

2: [-1, 1] 1: [15.04166, 15.04166] 1: 24.08333 s1^2 ... . . [ 1 n , 1 ]TABV M $RET

Here we have used a vector mapping operation to evaluate the function
at several values of ‘`s1`’ at once. `V M $` is like `V M '`
except that it takes the formula from the top of the stack. The
formula is interpreted as a function to apply across the vector at the
next-to-top stack level. Since a formula on the stack can’t contain
‘`$`’ signs, Calc assumes the variables in the formula stand for
different arguments. It prompts you for an *argument list*, giving
the list of all variables in the formula in alphabetical order as the
default list. In this case the default is ‘`(s1)`’, which is just
what we want so we simply press `RET` at the prompt.

If there had been several different values, we could have used
`V R X` to find the global maximum.

Calc has a built-in `a P` command that solves an equation using
`H a S` and returns a vector of all the solutions. It simply
automates the job we just did by hand. Applied to our original
cubic polynomial, it would produce the vector of solutions
‘`[1.19023, -1.19023, 0]`’. (There is also an `a X` command
which finds a local maximum of a function. It uses a numerical search
method rather than examining the derivatives, and thus requires you
to provide some kind of initial guess to show it where to look.)

(•) **Exercise 2.** Given a vector of the roots of a
polynomial (such as the output of an `a P` command), what
sequence of commands would you use to reconstruct the original
polynomial? (The answer will be unique to within a constant
multiple; choose the solution where the leading coefficient is one.)
See 2. (•)

The `m s` command enables Symbolic mode, in which formulas
like ‘`sqrt(5)`’ that can’t be evaluated exactly are left in
symbolic form rather than giving a floating-point approximate answer.
Fraction mode (`m f`) is also useful when doing algebra.

2: 34 x - 24 x^3 2: 34 x - 24 x^3 1: 34 x - 24 x^3 1: [sqrt(51) / 6, sqrt(51) / -6, 0] . . r 2RETm s m f a P xRET

One more mode that makes reading formulas easier is Big mode.

3 2: 34 x - 24 x ____ ____ V 51 V 51 1: [-----, -----, 0] 6 -6 . d B

Here things like powers, square roots, and quotients and fractions are displayed in a two-dimensional pictorial form. Calc has other language modes as well, such as C mode, FORTRAN mode, TeX mode and LaTeX mode.

2: 34*x - 24*pow(x, 3) 2: 34*x - 24*x**3 1: {sqrt(51) / 6, sqrt(51) / -6, 0} 1: /sqrt(51) / 6, sqrt(51) / -6, 0/ . . d C d F

3: 34 x - 24 x^3 2: [{\sqrt{51} \over 6}, {\sqrt{51} \over -6}, 0] 1: {2 \over 3} \sqrt{5} . d T ' 2 \sqrt{5} \over 3RET

As you can see, language modes affect both entry and display of formulas. They affect such things as the names used for built-in functions, the set of arithmetic operators and their precedences, and notations for vectors and matrices.

Notice that ‘`sqrt(51)`’ may cause problems with older
implementations of C and FORTRAN, which would require something more
like ‘`sqrt(51.0)`’. It is always wise to check over the formulas
produced by the various language modes to make sure they are fully
correct.

Type `m s`, `m f`, and `d N` to reset these modes. (You
may prefer to remain in Big mode, but all the examples in the tutorial
are shown in normal mode.)

What is the area under the portion of this curve from ‘`x = 1`’ to ‘`2`’?
This is simply the integral of the function:

1: 17 x^2 - 6 x^4 + 3 1: 5.6666 x^3 - 1.2 x^5 + 3 x . . r 1 a i x

We want to evaluate this at our two values for ‘`x`’ and subtract.
One way to do it is again with vector mapping and reduction:

2: [2, 1] 1: [12.93333, 7.46666] 1: 5.46666 1: 5.6666 x^3 ... . . [ 2 , 1 ]TABV M $RETV R -

(•) **Exercise 3.** Find the integral from 1 to ‘`y`’
of
‘`x sin(pi x)`’
(where the sine is calculated in radians). Find the values of the
integral for integers ‘`y`’ from 1 to 5. See 3. (•)

Calc’s integrator can do many simple integrals symbolically, but many
others are beyond its capabilities. Suppose we wish to find the area
under the curve
‘`sin(x) ln(x)`’
over the same range of ‘`x`’. If you entered this formula and typed
`a i x RET` (don’t bother to try this), Calc would work for a
long time but would be unable to find a solution. In fact, there is no
closed-form solution to this integral. Now what do we do?

One approach would be to do the integral numerically. It is not hard to do this by hand using vector mapping and reduction. It is rather slow, though, since the sine and logarithm functions take a long time. We can save some time by reducing the working precision.

3: 10 1: [1, 1.1, 1.2, ... , 1.8, 1.9] 2: 1 . 1: 0.1 . 10RET1RET.1RETC-u v x

(Note that we have used the extended version of `v x`; we could
also have used plain `v x` as follows: `v x 10 RET 9 + .1 *`.)

2: [1, 1.1, ... ] 1: [0., 0.084941, 0.16993, ... ] 1: ln(x) sin(x) . . ' sin(x) ln(x)RETs 1 m r p 5RETV M $RET

1: 3.4195 0.34195 . . V R + 0.1 *

(If you got wildly different results, did you remember to switch to Radians mode?)

Here we have divided the curve into ten segments of equal width; approximating these segments as rectangular boxes (i.e., assuming the curve is nearly flat at that resolution), we compute the areas of the boxes (height times width), then sum the areas. (It is faster to sum first, then multiply by the width, since the width is the same for every box.)

The true value of this integral turns out to be about 0.374, so we’re not doing too well. Let’s try another approach.

1: ln(x) sin(x) 1: 0.84147 x + 0.11957 (x - 1)^2 - ... . . r 1 a t x=1RET4RET

Here we have computed the Taylor series expansion of the function
about the point ‘`x=1`’. We can now integrate this polynomial
approximation, since polynomials are easy to integrate.

1: 0.42074 x^2 + ... 1: [-0.0446, -0.42073] 1: 0.3761 . . . a i xRET[ 2 , 1 ]TABV M $RETV R -

Better! By increasing the precision and/or asking for more terms
in the Taylor series, we can get a result as accurate as we like.
(Taylor series converge better away from singularities in the
function such as the one at `ln(0)`

, so it would also help to
expand the series about the points ‘`x=2`’ or ‘`x=1.5`’ instead
of ‘`x=1`’.)

(•) **Exercise 4.** Our first method approximated the
curve by stairsteps of width 0.1; the total area was then the sum
of the areas of the rectangles under these stairsteps. Our second
method approximated the function by a polynomial, which turned out
to be a better approximation than stairsteps. A third method is
*Simpson’s rule*, which is like the stairstep method except
that the steps are not required to be flat. Simpson’s rule boils
down to the formula,

(h/3) * (f(a) + 4 f(a+h) + 2 f(a+2h) + 4 f(a+3h) + ... + 2 f(a+(n-2)*h) + 4 f(a+(n-1)*h) + f(a+n*h))

where ‘`n`’ (which must be even) is the number of slices and ‘`h`’
is the width of each slice. These are 10 and 0.1 in our example.
For reference, here is the corresponding formula for the stairstep
method:

h * (f(a) + f(a+h) + f(a+2h) + f(a+3h) + ... + f(a+(n-2)*h) + f(a+(n-1)*h))

Compute the integral from 1 to 2 of
‘`sin(x) ln(x)`’
using Simpson’s rule with 10 slices.
See 4. (•)

Calc has a built-in `a I` command for doing numerical integration.
It uses *Romberg’s method*, which is a more sophisticated cousin
of Simpson’s rule. In particular, it knows how to keep refining the
result until the current precision is satisfied.

Aside from the commands we’ve seen so far, Calc also provides a
large set of commands for operating on parts of formulas. You
indicate the desired sub-formula by placing the cursor on any part
of the formula before giving a *selection* command. Selections won’t
be covered in the tutorial; see Selecting Sub-Formulas, for
details and examples.

#### 2.5.2 Rewrite Rules

No matter how many built-in commands Calc provided for doing algebra,
there would always be something you wanted to do that Calc didn’t have
in its repertoire. So Calc also provides a *rewrite rule* system
that you can use to define your own algebraic manipulations.

Suppose we want to simplify this trigonometric formula:

1: 2 sec(x)^2 / tan(x)^2 - 2 / tan(x)^2 . ' 2sec(x)^2/tan(x)^2 - 2/tan(x)^2RETs 1

If we were simplifying this by hand, we’d probably combine over the common
denominator. The `a n` algebra command will do this, but we’ll do
it with a rewrite rule just for practice.

Rewrite rules are written with the ‘`:=`’ symbol.

1: (2 sec(x)^2 - 2) / tan(x)^2 . a r a/x + b/x := (a+b)/xRET

(The “assignment operator” ‘`:=`’ has several uses in Calc. All
by itself the formula ‘`a/x + b/x := (a+b)/x`’ doesn’t do anything,
but when it is given to the `a r` command, that command interprets
it as a rewrite rule.)

The lefthand side, ‘`a/x + b/x`’, is called the *pattern* of the
rewrite rule. Calc searches the formula on the stack for parts that
match the pattern. Variables in a rewrite pattern are called
*meta-variables*, and when matching the pattern each meta-variable
can match any sub-formula. Here, the meta-variable ‘`a`’ matched
the expression ‘`2 sec(x)^2`’, the meta-variable ‘`b`’ matched
the constant ‘`-2`’ and the meta-variable ‘`x`’ matched
the expression ‘`tan(x)^2`’.

This rule points out several interesting features of rewrite patterns.
First, if a meta-variable appears several times in a pattern, it must
match the same thing everywhere. This rule detects common denominators
because the same meta-variable ‘`x`’ is used in both of the
denominators.

Second, meta-variable names are independent from variables in the
target formula. Notice that the meta-variable ‘`x`’ here matches
the subformula ‘`tan(x)^2`’; Calc never confuses the two meanings of
‘`x`’.

And third, rewrite patterns know a little bit about the algebraic
properties of formulas. The pattern called for a sum of two quotients;
Calc was able to match a difference of two quotients by matching
‘`a = 2 sec(x)^2`’, ‘`b = -2`’, and ‘`x = tan(x)^2`’.

When the pattern part of a rewrite rule matches a part of the formula,
that part is replaced by the righthand side with all the meta-variables
substituted with the things they matched. So the result is
‘`(2 sec(x)^2 - 2) / tan(x)^2`’.

We could just as easily have written ‘`a/x - b/x := (a-b)/x`’ for
the rule. It would have worked just the same in all cases. (If we
really wanted the rule to apply only to ‘`+`’ or only to ‘`-`’,
we could have used the `plain`

symbol. See Algebraic Properties of Rewrite Rules, for some examples of this.)

One more rewrite will complete the job. We want to use the identity
‘`tan(x)^2 + 1 = sec(x)^2`’, but of course we must first rearrange
the identity in a way that matches our formula. The obvious rule
would be ‘`2 sec(x)^2 - 2 := 2 tan(x)^2`’, but a little thought shows
that the rule ‘`sec(x)^2 := 1 + tan(x)^2`’ will also work. The
latter rule has a more general pattern so it will work in many other
situations, too.

1: 2 . a r sec(x)^2 := 1 + tan(x)^2RET

You may ask, what’s the point of using the most general rule if you
have to type it in every time anyway? The answer is that Calc allows
you to store a rewrite rule in a variable, then give the variable
name in the `a r` command. In fact, this is the preferred way to
use rewrites. For one, if you need a rule once you’ll most likely
need it again later. Also, if the rule doesn’t work quite right you
can simply Undo, edit the variable, and run the rule again without
having to retype it.

' a/x + b/x := (a+b)/xRETs t mergeRET' sec(x)^2 := 1 + tan(x)^2RETs t secsqrRET1: 2 sec(x)^2 / tan(x)^2 - 2 / tan(x)^2 1: 2 . . r 1 a r mergeRETa r secsqrRET

To edit a variable, type `s e` and the variable name, use regular
Emacs editing commands as necessary, then type `C-c C-c` to store
the edited value back into the variable.
You can also use `s e` to create a new variable if you wish.

Notice that the first time you use each rule, Calc puts up a “compiling”
message briefly. The pattern matcher converts rules into a special
optimized pattern-matching language rather than using them directly.
This allows `a r` to apply even rather complicated rules very
efficiently. If the rule is stored in a variable, Calc compiles it
only once and stores the compiled form along with the variable. That’s
another good reason to store your rules in variables rather than
entering them on the fly.

(•) **Exercise 1.** Type `m s` to get Symbolic
mode, then enter the formula ‘`(2 + sqrt(2)) / (1 + sqrt(2))`’.
Using a rewrite rule, simplify this formula by multiplying the top and
bottom by the conjugate ‘`1 - sqrt(2)`’. The result will have
to be expanded by the distributive law; do this with another
rewrite. See 1. (•)

The `a r` command can also accept a vector of rewrite rules, or
a variable containing a vector of rules.

1: [merge, secsqr] 1: [a/x + b/x := (a + b)/x, ... ] . . ' [merge,secsqr]RET=

1: 2 sec(x)^2 / tan(x)^2 - 2 / tan(x)^2 1: 2 . . s t trigRETr 1 a r trigRET

Calc tries all the rules you give against all parts of the formula, repeating until no further change is possible. (The exact order in which things are tried is rather complex, but for simple rules like the ones we’ve used here the order doesn’t really matter. See Nested Formulas with Rewrite Rules.)

Calc actually repeats only up to 100 times, just in case your rule set
has gotten into an infinite loop. You can give a numeric prefix argument
to `a r` to specify any limit. In particular, `M-1 a r` does
only one rewrite at a time.

1: (2 sec(x)^2 - 2) / tan(x)^2 1: 2 . . r 1 M-1 a r trigRETM-1 a r trigRET

You can type `M-0 a r` if you want no limit at all on the number
of rewrites that occur.

Rewrite rules can also be *conditional*. Simply follow the rule
with a ‘`::`’ symbol and the desired condition. For example,

1: sin(x + 2 pi) + sin(x + 3 pi) + sin(x + 4 pi) . ' sin(x+2pi) + sin(x+3pi) + sin(x+4pi)RET

1: sin(x + 3 pi) + 2 sin(x) . a r sin(a + k pi) := sin(a) :: k % 2 = 0RET

(Recall, ‘`k % 2`’ is the remainder from dividing ‘`k`’ by 2,
which will be zero only when ‘`k`’ is an even integer.)

An interesting point is that the variable ‘`pi`’ was matched
literally rather than acting as a meta-variable.
This is because it is a special-constant variable. The special
constants ‘`e`’, ‘`i`’, ‘`phi`’, and so on also match literally.
A common error with rewrite
rules is to write, say, ‘`f(a,b,c,d,e) := g(a+b+c+d+e)`’, expecting
to match any ‘`f`’ with five arguments but in fact matching
only when the fifth argument is literally ‘`e`’!

Rewrite rules provide an interesting way to define your own functions.
Suppose we want to define ‘`fib(n)`’ to produce the `n`th
Fibonacci number. The first two Fibonacci numbers are each 1;
later numbers are formed by summing the two preceding numbers in
the sequence. This is easy to express in a set of three rules:

' [fib(1) := 1, fib(2) := 1, fib(n) := fib(n-1) + fib(n-2)]RETs t fib 1: fib(7) 1: 13 . . ' fib(7)RETa r fibRET

One thing that is guaranteed about the order that rewrites are tried
is that, for any given subformula, earlier rules in the rule set will
be tried for that subformula before later ones. So even though the
first and third rules both match ‘`fib(1)`’, we know the first will
be used preferentially.

This rule set has one dangerous bug: Suppose we apply it to the
formula ‘`fib(x)`’? (Don’t actually try this.) The third rule
will match ‘`fib(x)`’ and replace it with ‘`fib(x-1) + fib(x-2)`’.
Each of these will then be replaced to get ‘`fib(x-2) + 2 fib(x-3) +
fib(x-4)`’, and so on, expanding forever. What we really want is to apply
the third rule only when ‘`n`’ is an integer greater than two. Type
`s e fib RET`, then edit the third rule to:

fib(n) := fib(n-1) + fib(n-2) :: integer(n) :: n > 2

Now:

1: fib(6) + fib(x) + fib(0) 1: fib(x) + fib(0) + 8 . . ' fib(6)+fib(x)+fib(0)RETa r fibRET

We’ve created a new function, `fib`

, and a new command,
`a r fib RET`, which means “evaluate all

`fib`

calls in
this formula.” To make things easier still, we can tell Calc to
apply these rules automatically by storing them in the special
variable `EvalRules`

.
1: [fib(1) := ...] . 1: [8, 13] . . s r fibRETs t EvalRulesRET' [fib(6), fib(7)]RET

It turns out that this rule set has the problem that it does far
more work than it needs to when ‘`n`’ is large. Consider the
first few steps of the computation of ‘`fib(6)`’:

fib(6) = fib(5) + fib(4) = fib(4) + fib(3) + fib(3) + fib(2) = fib(3) + fib(2) + fib(2) + fib(1) + fib(2) + fib(1) + 1 = ...

Note that ‘`fib(3)`’ appears three times here. Unless Calc’s
algebraic simplifier notices the multiple ‘`fib(3)`’s and combines
them (and, as it happens, it doesn’t), this rule set does lots of
needless recomputation. To cure the problem, type `s e EvalRules`

to edit the rules (or just `s E`, a shorthand command for editing
`EvalRules`

) and add another condition:

fib(n) := fib(n-1) + fib(n-2) :: integer(n) :: n > 2 :: remember

If a ‘`:: remember`’ condition appears anywhere in a rule, then if
that rule succeeds Calc will add another rule that describes that match
to the front of the rule set. (Remembering works in any rule set, but
for technical reasons it is most effective in `EvalRules`

.) For
example, if the rule rewrites ‘`fib(7)`’ to something that evaluates
to 13, then the rule ‘`fib(7) := 13`’ will be added to the rule set.

Type `' fib(8) RET` to compute the eighth Fibonacci number, then
type

`s E`again to see what has happened to the rule set.

With the `remember`

feature, our rule set can now compute
‘`fib( n)`’ in just

`n`steps. In the process it builds up a table of all Fibonacci numbers up to

`n`. After we have computed the result for a particular

`n`, we can get it back (and the results for all smaller

`n`) later in just one step.

All Calc operations will run somewhat slower whenever `EvalRules`

contains any rules. You should type `s u EvalRules RET` now to
un-store the variable.

(•) **Exercise 2.** Sometimes it is possible to reformulate
a problem to reduce the amount of recursion necessary to solve it.
Create a rule that, in about `n` simple steps and without recourse
to the `remember`

option, replaces ‘`fib( n, 1, 1)`’ with
‘

`fib(1,`’ where

`x`,`y`)`x`and

`y`are the

`n`th and

`n+1`st Fibonacci numbers, respectively. This rule is rather clunky to use, so add a couple more rules to make the “user interface” the same as for our first version: enter ‘

`fib(`’, get back a plain number. See 2. (•)

`n`)There are many more things that rewrites can do. For example, there
are ‘`&&&`’ and ‘`|||`’ pattern operators that create “and”
and “or” combinations of rules. As one really simple example, we
could combine our first two Fibonacci rules thusly:

[fib(1 ||| 2) := 1, fib(n) := ... ]

That means “`fib`

of something matching either 1 or 2 rewrites
to 1.”

You can also make meta-variables optional by enclosing them in `opt`

.
For example, the pattern ‘`a + b x`’ matches ‘`2 + 3 x`’ but not
‘`2 + x`’ or ‘`3 x`’ or ‘`x`’. The pattern ‘`opt(a) + opt(b) x`’
matches all of these forms, filling in a default of zero for ‘`a`’
and one for ‘`b`’.

(•) **Exercise 3.** Your friend Joe had ‘`2 + 3 x`’
on the stack and tried to use the rule
‘`opt(a) + opt(b) x := f(a, b, x)`’. What happened?
See 3. (•)

(•) **Exercise 4.** Starting with a positive integer ‘`a`’,
divide ‘`a`’ by two if it is even, otherwise compute ‘`3 a + 1`’.
Now repeat this step over and over. A famous unproved conjecture
is that for any starting ‘`a`’, the sequence always eventually
reaches 1. Given the formula ‘`seq( a, 0)`’, write a set of
rules that convert this into ‘

`seq(1,`’ where

`n`)`n`is the number of steps it took the sequence to reach the value 1. Now enhance the rules to accept ‘

`seq(`’ as a starting configuration, and to stop with just the number

`a`)`n`by itself. Now make the result be a vector of values in the sequence, from

`a`to 1. (The formula ‘

`’ appends the vectors`

`x`|`y``x`and

`y`.) For example, rewriting ‘

`seq(6)`’ should yield the vector ‘

`[6, 3, 10, 5, 16, 8, 4, 2, 1]`’. See 4. (•)

(•) **Exercise 5.** Define, using rewrite rules, a function
‘`nterms( x)`’ that returns the number of terms in the sum

`x`, or 1 if

`x`is not a sum. (A

*sum*for our purposes is one or more non-sum terms separated by ‘

`+`’ or ‘

`-`’ signs, so that ‘

`2 - 3 (x + y) + x y`’ is a sum of three terms.) See 5. (•)

(•) **Exercise 6.** A Taylor series for a function is an
infinite series that exactly equals the value of that function at
values of ‘`x`’ near zero.

cos(x) = 1 - x^2 / 2! + x^4 / 4! - x^6 / 6! + ...

The `a t` command produces a *truncated Taylor series* which
is obtained by dropping all the terms higher than, say, ‘`x^2`’.
Calc represents the truncated Taylor series as a polynomial in ‘`x`’.
Mathematicians often write a truncated series using a “big-O” notation
that records what was the lowest term that was truncated.

cos(x) = 1 - x^2 / 2! + O(x^3)

The meaning of ‘`O(x^3)`’ is “a quantity which is negligibly small
if ‘`x^3`’ is considered negligibly small as ‘`x`’ goes to zero.”

The exercise is to create rewrite rules that simplify sums and products of
power series represented as ‘` polynomial + O(var^n)`’.
For example, given ‘

`1 - x^2 / 2 + O(x^3)`’ and ‘

`x - x^3 / 6 + O(x^4)`’ on the stack, we want to be able to type

`*`and get the result ‘

`x - 2:3 x^3 + O(x^4)`’. Don’t worry if the terms of the sum are rearranged. (This one is rather tricky; the solution at the end of this chapter uses 6 rewrite rules. Hint: The ‘

`constant(x)`’ condition tests whether ‘

`x`’ is a number.) See 6. (•)

Just for kicks, try adding the rule `2+3 := 6`

to `EvalRules`

.
What happens? (Be sure to remove this rule afterward, or you might get
a nasty surprise when you use Calc to balance your checkbook!)

See Rewrite Rules, for the whole story on rewrite rules.

### 2.6 Programming Tutorial

The Calculator is written entirely in Emacs Lisp, a highly extensible language. If you know Lisp, you can program the Calculator to do anything you like. Rewrite rules also work as a powerful programming system. But Lisp and rewrite rules take a while to master, and often all you want to do is define a new function or repeat a command a few times. Calc has features that allow you to do these things easily.

One very limited form of programming is defining your own functions.
Calc’s `Z F` command allows you to define a function name and
key sequence to correspond to any formula. Programming commands use
the shift-`Z` prefix; the user commands they create use the lower
case `z` prefix.

1: x + x^2 / 2 + x^3 / 6 + 1 1: x + x^2 / 2 + x^3 / 6 + 1 . . ' 1 + x + x^2/2! + x^3/3!RETZ F e myexpRETRETRETy

This polynomial is a Taylor series approximation to ‘`exp(x)`’.
The `Z F` command asks a number of questions. The above answers
say that the key sequence for our function should be `z e`; the
`M-x` equivalent should be `calc-myexp`

; the name of the
function in algebraic formulas should also be `myexp`

; the
default argument list ‘`(x)`’ is acceptable; and finally `y`
answers the question “leave it in symbolic form for non-constant
arguments?”

1: 1.3495 2: 1.3495 3: 1.3495 . 1: 1.34986 2: 1.34986 . 1: myexp(a + 1) . .3 z e .3 E ' a+1RETz e

First we call our new `exp`

approximation with 0.3 as an
argument, and compare it with the true `exp`

function. Then
we note that, as requested, if we try to give `z e` an
argument that isn’t a plain number, it leaves the `myexp`

function call in symbolic form. If we had answered `n` to the
final question, ‘`myexp(a + 1)`’ would have evaluated by plugging
in ‘`a + 1`’ for ‘`x`’ in the defining formula.

(•) **Exercise 1.** The “sine integral” function
‘`Si(x)`’
is defined as the integral of ‘`sin(t)/t`’ for
‘`t = 0`’ to ‘`x`’ in radians. (It was invented because this
integral has no solution in terms of basic functions; if you give it
to Calc’s `a i` command, it will ponder it for a long time and then
give up.) We can use the numerical integration command, however,
which in algebraic notation is written like ‘`ninteg(f(t), t, 0, x)`’
with any integrand ‘`f(t)`’. Define a `z s` command and
`Si`

function that implement this. You will need to edit the
default argument list a bit. As a test, ‘`Si(1)`’ should return
0.946083. (If you don’t get this answer, you might want to check that
Calc is in Radians mode. Also, `ninteg`

will run a lot faster if
you reduce the precision to, say, six digits beforehand.)
See 1. (•)

The simplest way to do real “programming” of Emacs is to define a
*keyboard macro*. A keyboard macro is simply a sequence of
keystrokes which Emacs has stored away and can play back on demand.
For example, if you find yourself typing `H a S x RET` often,
you may wish to program a keyboard macro to type this for you.

1: y = sqrt(x) 1: x = y^2 . . ' y=sqrt(x)RETC-x ( H a S xRETC-x ) 1: y = cos(x) 1: x = s1 arccos(y) + 2 n1 pi . . ' y=cos(x)RETX

When you type `C-x (`, Emacs begins recording. But it is also
still ready to execute your keystrokes, so you’re really “training”
Emacs by walking it through the procedure once. When you type
`C-x )`, the macro is recorded. You can now type `X` to
re-execute the same keystrokes.

You can give a name to your macro by typing `Z K`.

1: . 1: y = x^4 1: x = s2 sqrt(s1 sqrt(y)) . . Z K xRET' y=x^4RETz x

Notice that we use shift-`Z` to define the command, and lower-case
`z` to call it up.

Keyboard macros can call other macros.

1: abs(x) 1: x = s1 y 1: 2 / x 1: x = 2 / y . . . . ' abs(x)RETC-x ( ' yRETa = z x C-x ) ' 2/xRETX

(•) **Exercise 2.** Define a keyboard macro to negate
the item in level 3 of the stack, without disturbing the rest of
the stack. See 2. (•)

(•) **Exercise 3.** Define keyboard macros to compute
the following functions:

- Compute
‘
`sin(x) / x`’, where ‘`x`’ is the number on the top of the stack. - Compute the base-‘
`b`’ logarithm, just like the`B`key except the arguments are taken in the opposite order. - Produce a vector of integers from 1 to the integer on the top of the stack.

See 3. (•)

(•) **Exercise 4.** Define a keyboard macro to compute
the average (mean) value of a list of numbers.
See 4. (•)

In many programs, some of the steps must execute several times.
Calc has *looping* commands that allow this. Loops are useful
inside keyboard macros, but actually work at any time.

1: x^6 2: x^6 1: 360 x^2 . 1: 4 . . ' x^6RET4 Z < a d xRETZ >

Here we have computed the fourth derivative of ‘`x^6`’ by
enclosing a derivative command in a “repeat loop” structure.
This structure pops a repeat count from the stack, then
executes the body of the loop that many times.

If you make a mistake while entering the body of the loop,
type `Z C-g` to cancel the loop command.

Here’s another example:

3: 1 2: 10946 2: 1 1: 17711 1: 20 . . 1RETRET20 Z <TABC-j + Z >

The numbers in levels 2 and 1 should be the 21st and 22nd Fibonacci
numbers, respectively. (To see what’s going on, try a few repetitions
of the loop body by hand; `C-j`, also on the Line-Feed or `LFD`
key if you have one, makes a copy of the number in level 2.)

A fascinating property of the Fibonacci numbers is that the ‘`n`’th
Fibonacci number can be found directly by computing
‘`phi^n / sqrt(5)`’
and then rounding to the nearest integer, where
‘`phi`’,
the “golden ratio,” is
‘`(1 + sqrt(5)) / 2`’.
(For convenience, this constant is available from the `phi`

variable, or the `I H P` command.)

1: 1.61803 1: 24476.0000409 1: 10945.9999817 1: 10946 . . . . I H P 21 ^ 5 Q / R

(•) **Exercise 5.** The *continued fraction*
representation of
‘`phi`’
is
‘`1 + 1/(1 + 1/(1 + 1/( ... )))`’.
We can compute an approximate value by carrying this however far
and then replacing the innermost
‘`1/( ... )`’
by 1. Approximate
‘`phi`’
using a twenty-term continued fraction.
See 5. (•)

(•) **Exercise 6.** Linear recurrences like the one for
Fibonacci numbers can be expressed in terms of matrices. Given a
vector ‘`[a, b]`’ determine a matrix which, when multiplied by this
vector, produces the vector ‘`[b, c]`’, where ‘`a`’, ‘`b`’ and
‘`c`’ are three successive Fibonacci numbers. Now write a program
that, given an integer ‘`n`’, computes the ‘`n`’th Fibonacci number
using matrix arithmetic. See 6. (•)

A more sophisticated kind of loop is the *for* loop. Suppose
we wish to compute the 20th “harmonic” number, which is equal to
the sum of the reciprocals of the integers from 1 to 20.

3: 0 1: 3.597739 2: 1 . 1: 20 . 0RET1RET20 Z ( & + 1 Z )

The “for” loop pops two numbers, the lower and upper limits, then repeats the body of the loop as an internal counter increases from the lower limit to the upper one. Just before executing the loop body, it pushes the current loop counter. When the loop body finishes, it pops the “step,” i.e., the amount by which to increment the loop counter. As you can see, our loop always uses a step of one.

This harmonic number function uses the stack to hold the running total as well as for the various loop housekeeping functions. If you find this disorienting, you can sum in a variable instead:

1: 0 2: 1 . 1: 3.597739 . 1: 20 . . 0 t 7 1RET20 Z ( & s + 7 1 Z ) r 7

The `s +` command adds the top-of-stack into the value in a
variable (and removes that value from the stack).

It’s worth noting that many jobs that call for a “for” loop can
also be done more easily by Calc’s high-level operations. Two
other ways to compute harmonic numbers are to use vector mapping
and reduction (`v x 20`, then `V M &`, then `V R +`),
or to use the summation command `a +`. Both of these are
probably easier than using loops. However, there are some
situations where loops really are the way to go:

(•) **Exercise 7.** Use a “for” loop to find the first
harmonic number which is greater than 4.0.
See 7. (•)

Of course, if we’re going to be using variables in our programs, we have to worry about the programs clobbering values that the caller was keeping in those same variables. This is easy to fix, though:

. 1: 0.6667 1: 0.6667 3: 0.6667 . . 2: 3.597739 1: 0.6667 . Z ` p 4RET2RET3 / s 7 s s aRETZ ' r 7 s r aRET

When we type `Z `` (that’s a grave accent), Calc saves
its mode settings and the contents of the ten “quick variables”
for later reference. When we type `Z '` (that’s an apostrophe
now), Calc restores those saved values. Thus the `p 4` and
`s 7` commands have no effect outside this sequence. Wrapping
this around the body of a keyboard macro ensures that it doesn’t
interfere with what the user of the macro was doing. Notice that
the contents of the stack, and the values of named variables,
survive past the `Z '` command.

The *Bernoulli numbers* are a sequence with the interesting
property that all of the odd Bernoulli numbers are zero, and the
even ones, while difficult to compute, can be roughly approximated
by the formula
‘`2 n! / (2 pi)^n`’.
Let’s write a keyboard macro to compute (approximate) Bernoulli numbers.
(Calc has a command, `k b`, to compute exact Bernoulli numbers, but
this command is very slow for large ‘`n`’ since the higher Bernoulli
numbers are very large fractions.)

1: 10 1: 0.0756823 . . 10 C-x (RET2 % Z [DEL0 Z : ' 2 $! / (2 pi)^$RET= Z ] C-x )

You can read `Z [` as “then,” `Z :` as “else,” and
`Z ]` as “end-if.” There is no need for an explicit “if”
command. For the purposes of `Z [`, the condition is “true”
if the value it pops from the stack is a nonzero number, or “false”
if it pops zero or something that is not a number (like a formula).
Here we take our integer argument modulo 2; this will be nonzero
if we’re asking for an odd Bernoulli number.

The actual tenth Bernoulli number is ‘`5/66`’.

3: 0.0756823 1: 0 1: 0.25305 1: 0 1: 1.16659 2: 5:66 . . . . 1: 0.0757575 . 10 k bRETc f M-0DEL11 XDEL12 XDEL13 XDEL14 X

Just to exercise loops a bit more, let’s compute a table of even Bernoulli numbers.

3: [] 1: [0.10132, 0.03079, 0.02340, 0.033197, ...] 2: 2 . 1: 30 . [ ] 2RET30 Z ( X | 2 Z )

The vertical-bar `|` is the vector-concatenation command. When
we execute it, the list we are building will be in stack level 2
(initially this is an empty list), and the next Bernoulli number
will be in level 1. The effect is to append the Bernoulli number
onto the end of the list. (To create a table of exact fractional
Bernoulli numbers, just replace `X` with `k b` in the above
sequence of keystrokes.)

With loops and conditionals, you can program essentially anything
in Calc. One other command that makes looping easier is `Z /`,
which takes a condition from the stack and breaks out of the enclosing
loop if the condition is true (non-zero). You can use this to make
“while” and “until” style loops.

If you make a mistake when entering a keyboard macro, you can edit
it using `Z E`. First, you must attach it to a key with `Z K`.
One technique is to enter a throwaway dummy definition for the macro,
then enter the real one in the edit command.

1: 3 1: 3 Calc Macro Edit Mode. . . Original keys: 1 <return> 2 + 1 ;; calc digits RET ;; calc-enter 2 ;; calc digits + ;; calc-plus C-x ( 1RET2 + C-x ) Z K hRETZ E h

A keyboard macro is stored as a pure keystroke sequence. The
`edmacro` package (invoked by `Z E`) scans along the
macro and tries to decode it back into human-readable steps.
Descriptions of the keystrokes are given as comments, which begin with
‘`;;`’, and which are ignored when the edited macro is saved.
Spaces and line breaks are also ignored when the edited macro is saved.
To enter a space into the macro, type `SPC`

. All the special
characters `RET`

, `LFD`

, `TAB`

, `SPC`

, `DEL`

,
and `NUL`

must be written in all uppercase, as must the prefixes
`C-`

and `M-`

.

Let’s edit in a new definition, for computing harmonic numbers.
First, erase the four lines of the old definition. Then, type
in the new definition (or use Emacs `M-w` and `C-y` commands
to copy it from this page of the Info file; you can of course skip
typing the comments, which begin with ‘`;;`’).

Z` ;; calc-kbd-push (Save local values) 0 ;; calc digits (Push a zero onto the stack) st ;; calc-store-into (Store it in the following variable) 1 ;; calc quick variable (Quick variable q1) 1 ;; calc digits (Initial value for the loop) TAB ;; calc-roll-down (Swap initial and final) Z( ;; calc-kbd-for (Begin the "for" loop) & ;; calc-inv (Take the reciprocal) s+ ;; calc-store-plus (Add to the following variable) 1 ;; calc quick variable (Quick variable q1) 1 ;; calc digits (The loop step is 1) Z) ;; calc-kbd-end-for (End the "for" loop) sr ;; calc-recall (Recall the final accumulated value) 1 ;; calc quick variable (Quick variable q1) Z' ;; calc-kbd-pop (Restore values)

Press `C-c C-c` to finish editing and return to the Calculator.

1: 20 1: 3.597739 . . 20 z h

The `edmacro` package defines a handy `read-kbd-macro`

command
which reads the current region of the current buffer as a sequence of
keystroke names, and defines that sequence on the `X`
(and `C-x e`) key. Because this is so useful, Calc puts this
command on the `C-x * m` key. Try reading in this macro in the
following form: Press `C-@` (or `C- SPC`) at
one end of the text below, then type

`C-x * m`at the other.

Z ` 0 t 1 1 TAB Z ( & s + 1 1 Z ) r 1 Z '

(•) **Exercise 8.** A general algorithm for solving
equations numerically is *Newton’s Method*. Given the equation
‘`f(x) = 0`’ for any function ‘`f`’, and an initial guess
‘`x_0`’ which is reasonably close to the desired solution, apply
this formula over and over:

new_x = x - f(x)/f'(x)

where ‘`f'(x)`’ is the derivative of ‘`f`’. The ‘`x`’
values will quickly converge to a solution, i.e., eventually
‘`new_x`’
and ‘`x`’ will be equal to within the limits
of the current precision. Write a program which takes a formula
involving the variable ‘`x`’, and an initial guess ‘`x_0`’,
on the stack, and produces a value of ‘`x`’ for which the formula
is zero. Use it to find a solution of
‘`sin(cos(x)) = 0.5`’
near ‘`x = 4.5`’. (Use angles measured in radians.) Note that
the built-in `a R` (`calc-find-root`

) command uses Newton’s
method when it is able. See 8. (•)

(•) **Exercise 9.** The *digamma* function
‘`psi(z)`’
is defined as the derivative of
‘`ln(gamma(z))`’.
For large values of ‘`z`’, it can be approximated by the infinite sum

psi(z) ~= ln(z) - 1/2z - sum(bern(2 n) / 2 n z^(2 n), n, 1, inf)

where
‘`sum`’
represents the sum over ‘`n`’ from 1 to infinity
(or to some limit high enough to give the desired accuracy), and
the `bern`

function produces (exact) Bernoulli numbers.
While this sum is not guaranteed to converge, in practice it is safe.
An interesting mathematical constant is Euler’s gamma, which is equal
to about 0.5772. One way to compute it is by the formula,
‘`gamma = -psi(1)`’.
Unfortunately, 1 isn’t a large enough argument
for the above formula to work (5 is a much safer value for ‘`z`’).
Fortunately, we can compute
‘`psi(1)`’
from
‘`psi(5)`’
using the recurrence
‘`psi(z+1) = psi(z) + 1/z`’.
Your task: Develop a program to compute
‘`psi(z)`’;
it should “pump up” ‘`z`’
if necessary to be greater than 5, then use the above summation
formula. Use looping commands to compute the sum. Use your function
to compute
‘`gamma`’
to twelve decimal places. (Calc has a built-in command
for Euler’s constant, `I P`, which you can use to check your answer.)
See 9. (•)

(•) **Exercise 10.** Given a polynomial in ‘`x`’ and
a number ‘`m`’ on the stack, where the polynomial is of degree
‘`m`’ or less (i.e., does not have any terms higher than ‘`x^m`’),
write a program to convert the polynomial into a list-of-coefficients
notation. For example, ‘`5 x^4 + (x + 1)^2`’ with ‘`m = 6`’
should produce the list ‘`[1, 2, 1, 0, 5, 0, 0]`’. Also develop
a way to convert from this form back to the standard algebraic form.
See 10. (•)

(•) **Exercise 11.** The *Stirling numbers of the
first kind* are defined by the recurrences,

s(n,n) = 1 for n >= 0, s(n,0) = 0 for n > 0, s(n+1,m) = s(n,m-1) - n s(n,m) for n >= m >= 1.

This can be implemented using a *recursive* program in Calc; the
program must invoke itself in order to calculate the two righthand
terms in the general formula. Since it always invokes itself with
“simpler” arguments, it’s easy to see that it must eventually finish
the computation. Recursion is a little difficult with Emacs keyboard
macros since the macro is executed before its definition is complete.
So here’s the recommended strategy: Create a “dummy macro” and assign
it to a key with, e.g., `Z K s`. Now enter the true definition,
using the `z s` command to call itself recursively, then assign it
to the same key with `Z K s`. Now the `z s` command will run
the complete recursive program. (Another way is to use `Z E`
or `C-x * m` (`read-kbd-macro`

) to read the whole macro at once,
thus avoiding the “training” phase.) The task: Write a program
that computes Stirling numbers of the first kind, given ‘`n`’ and
‘`m`’ on the stack. Test it with *small* inputs like
‘`s(4,2)`’. (There is a built-in command for Stirling numbers,
`k s`, which you can use to check your answers.)
See 11. (•)

The programming commands we’ve seen in this part of the tutorial are low-level, general-purpose operations. Often you will find that a higher-level function, such as vector mapping or rewrite rules, will do the job much more easily than a detailed, step-by-step program can:

(•) **Exercise 12.** Write another program for
computing Stirling numbers of the first kind, this time using
rewrite rules. Once again, ‘`n`’ and ‘`m`’ should be taken
from the stack. See 12. (•)

This ends the tutorial section of the Calc manual. Now you know enough about Calc to use it effectively for many kinds of calculations. But Calc has many features that were not even touched upon in this tutorial. The rest of this manual tells the whole story.

### 2.7 Answers to Exercises

This section includes answers to all the exercises in the Calc tutorial.

- RPN Tutorial Exercise 1
- RPN Tutorial Exercise 2
- RPN Tutorial Exercise 3
- RPN Tutorial Exercise 4
- Algebraic Entry Tutorial Exercise 1
- Algebraic Entry Tutorial Exercise 2
- Algebraic Entry Tutorial Exercise 3
- Modes Tutorial Exercise 1
- Modes Tutorial Exercise 2
- Modes Tutorial Exercise 3
- Modes Tutorial Exercise 4
- Arithmetic Tutorial Exercise 1
- Arithmetic Tutorial Exercise 2
- Vector Tutorial Exercise 1
- Vector Tutorial Exercise 2
- Matrix Tutorial Exercise 1
- Matrix Tutorial Exercise 2
- Matrix Tutorial Exercise 3
- List Tutorial Exercise 1
- List Tutorial Exercise 2
- List Tutorial Exercise 3
- List Tutorial Exercise 4
- List Tutorial Exercise 5
- List Tutorial Exercise 6
- List Tutorial Exercise 7
- List Tutorial Exercise 8
- List Tutorial Exercise 9
- List Tutorial Exercise 10
- List Tutorial Exercise 11
- List Tutorial Exercise 12
- List Tutorial Exercise 13
- List Tutorial Exercise 14
- Types Tutorial Exercise 1
- Types Tutorial Exercise 2
- Types Tutorial Exercise 3
- Types Tutorial Exercise 4
- Types Tutorial Exercise 5
- Types Tutorial Exercise 6
- Types Tutorial Exercise 7
- Types Tutorial Exercise 8
- Types Tutorial Exercise 9
- Types Tutorial Exercise 10
- Types Tutorial Exercise 11
- Types Tutorial Exercise 12
- Types Tutorial Exercise 13
- Types Tutorial Exercise 14
- Types Tutorial Exercise 15
- Algebra Tutorial Exercise 1
- Algebra Tutorial Exercise 2
- Algebra Tutorial Exercise 3
- Algebra Tutorial Exercise 4
- Rewrites Tutorial Exercise 1
- Rewrites Tutorial Exercise 2
- Rewrites Tutorial Exercise 3
- Rewrites Tutorial Exercise 4
- Rewrites Tutorial Exercise 5
- Rewrites Tutorial Exercise 6
- Programming Tutorial Exercise 1
- Programming Tutorial Exercise 2
- Programming Tutorial Exercise 3
- Programming Tutorial Exercise 4
- Programming Tutorial Exercise 5
- Programming Tutorial Exercise 6
- Programming Tutorial Exercise 7
- Programming Tutorial Exercise 8
- Programming Tutorial Exercise 9
- Programming Tutorial Exercise 10
- Programming Tutorial Exercise 11
- Programming Tutorial Exercise 12

#### 2.7.1 RPN Tutorial Exercise 1

`1 RET 2 RET 3 RET 4 + * -`

The result is
‘`1 - (2 * (3 + 4)) = -13`’.

#### 2.7.2 RPN Tutorial Exercise 2

‘`2*4 + 7*9.5 + 5/4 = 75.75`’

After computing the intermediate term
‘`2*4 = 8`’,
you can leave that result on the stack while you compute the second
term. With both of these results waiting on the stack you can then
compute the final term, then press `+ +` to add everything up.

2: 2 1: 8 3: 8 2: 8 1: 4 . 2: 7 1: 66.5 . 1: 9.5 . . 2RET4 * 7RET9.5 *

4: 8 3: 8 2: 8 1: 75.75 3: 66.5 2: 66.5 1: 67.75 . 2: 5 1: 1.25 . 1: 4 . . 5RET4 / + +

Alternatively, you could add the first two terms before going on with the third term.

2: 8 1: 74.5 3: 74.5 2: 74.5 1: 75.75 1: 66.5 . 2: 5 1: 1.25 . . 1: 4 . . ... + 5RET4 / +

On an old-style RPN calculator this second method would have the advantage of using only three stack levels. But since Calc’s stack can grow arbitrarily large this isn’t really an issue. Which method you choose is purely a matter of taste.

#### 2.7.3 RPN Tutorial Exercise 3

The `TAB` key provides a way to operate on the number in level 2.

3: 10 3: 10 4: 10 3: 10 3: 10 2: 20 2: 30 3: 30 2: 30 2: 21 1: 30 1: 20 2: 20 1: 21 1: 30 . . 1: 1 . . .TAB1 +TAB

Similarly, `M- TAB` gives you access to the number in level 3.

3: 10 3: 21 3: 21 3: 30 3: 11 2: 21 2: 30 2: 30 2: 11 2: 21 1: 30 1: 10 1: 11 1: 21 1: 30 . . . . . M-TAB1 + M-TABM-TAB

#### 2.7.4 RPN Tutorial Exercise 4

Either `( 2 , 3 )` or `( 2 SPC 3 )` would have worked,
but using both the comma and the space at once yields:

1: ( ... 2: ( ... 1: (2, ... 2: (2, ... 2: (2, ... . 1: 2 . 1: (2, ... 1: (2, 3) . . . ( 2 ,SPC3 )

Joe probably tried to type ` TAB DEL` to swap the
extra incomplete object to the top of the stack and delete it.
But a feature of Calc is that

`DEL`on an incomplete object deletes just one component out of that object, so he had to press

`DEL`twice to finish the job.

2: (2, ... 2: (2, 3) 2: (2, 3) 1: (2, 3) 1: (2, 3) 1: (2, ... 1: ( ... . . . .TABDELDEL

(As it turns out, deleting the second-to-top stack entry happens often
enough that Calc provides a special key, `M- DEL`, to do just that.

`M-`is just like

`DEL``, except that it doesn’t exhibit the “feature” that tripped poor Joe.)`

`TAB``DEL`#### 2.7.5 Algebraic Entry Tutorial Exercise 1

Type `' sqrt($) RET`.

If the `Q` key is broken, you could use `' $^0.5 RET`.
Or, RPN style,

`0.5 ^`.

(Actually, ‘`$^1:2`’, using the fraction one-half as the power, is
a closer equivalent, since ‘`9^0.5`’ yields ‘`3.0`’ whereas
‘`sqrt(9)`’ and ‘`9^1:2`’ yield the exact integer ‘`3`’.)

#### 2.7.6 Algebraic Entry Tutorial Exercise 2

In the formula ‘`2 x (1+y)`’, ‘`x`’ was interpreted as a function
name with ‘`1+y`’ as its argument. Assigning a value to a variable
has no relation to a function by the same name. Joe needed to use an
explicit ‘`*`’ symbol here: ‘`2 x*(1+y)`’.

#### 2.7.7 Algebraic Entry Tutorial Exercise 3

The result from `1 RET 0 /` will be the formula ‘

`1 / 0`’. The “function” ‘

`/`’ cannot be evaluated when its second argument is zero, so it is left in symbolic form. When you now type

`0 *`, the result will be zero because Calc uses the general rule that “zero times anything is zero.”

The `m i` command enables an *Infinite mode* in which ‘`1 / 0`’
results in a special symbol that represents “infinity.” If you
multiply infinity by zero, Calc uses another special new symbol to
show that the answer is “indeterminate.” See Infinities, for
further discussion of infinite and indeterminate values.

#### 2.7.8 Modes Tutorial Exercise 1

Calc always stores its floating-point numbers in decimal,
so even though one-third has
an exact base-3 representation (‘`3#0.1`’), it is still stored as
0.3333333 (chopped off after 12 or however many decimal digits) inside
the calculator’s memory. When this inexact number is converted back
to base 3 for display, it may still be slightly inexact. When we
multiply this number by 3, we get 0.999999, also an inexact value.

When Calc displays a number in base 3, it has to decide how many digits
to show. If the current precision is 12 (decimal) digits, that corresponds
to ‘`12 / log10(3) = 25.15`’ base-3 digits. Because 25.15 is not an
exact integer, Calc shows only 25 digits, with the result that stored
numbers carry a little bit of extra information that may not show up on
the screen. When Joe entered ‘`3#0.2`’, the stored number 0.666666
happened to round to a pleasing value when it lost that last 0.15 of a
digit, but it was still inexact in Calc’s memory. When he divided by 2,
he still got the dreaded inexact value 0.333333. (Actually, he divided
0.666667 by 2 to get 0.333334, which is why he got something a little
higher than `3#0.1`

instead of a little lower.)

If Joe didn’t want to be bothered with all this, he could have typed
`M-24 d n` to display with one less digit than the default. (If
you give `d n` a negative argument, it uses default-minus-that,
so `M-- d n` would be an easier way to get the same effect.) Those
inexact results would still be lurking there, but they would now be
rounded to nice, natural-looking values for display purposes. (Remember,
‘`0.022222`’ in base 3 is like ‘`0.099999`’ in base 10; rounding
off one digit will round the number up to ‘`0.1`’.) Depending on the
nature of your work, this hiding of the inexactness may be a benefit or
a danger. With the `d n` command, Calc gives you the choice.

Incidentally, another consequence of all this is that if you type
`M-30 d n` to display more digits than are “really there,”
you’ll see garbage digits at the end of the number. (In decimal
display mode, with decimally-stored numbers, these garbage digits are
always zero so they vanish and you don’t notice them.) Because Calc
rounds off that 0.15 digit, there is the danger that two numbers could
be slightly different internally but still look the same. If you feel
uneasy about this, set the `d n` precision to be a little higher
than normal; you’ll get ugly garbage digits, but you’ll always be able
to tell two distinct numbers apart.

An interesting side note is that most computers store their
floating-point numbers in binary, and convert to decimal for display.
Thus everyday programs have the same problem: Decimal 0.1 cannot be
represented exactly in binary (try it: `0.1 d 2`), so ‘`0.1 * 10`’
comes out as an inexact approximation to 1 on some machines (though
they generally arrange to hide it from you by rounding off one digit as
we did above). Because Calc works in decimal instead of binary, you can
be sure that numbers that look exact *are* exact as long as you stay
in decimal display mode.

It’s not hard to show that any number that can be represented exactly in binary, octal, or hexadecimal is also exact in decimal, so the kinds of problems we saw in this exercise are likely to be severe only when you use a relatively unusual radix like 3.

#### 2.7.9 Modes Tutorial Exercise 2

If the radix is 15 or higher, we can’t use the letter ‘`e`’ to mark
the exponent because ‘`e`’ is interpreted as a digit. When Calc
needs to display scientific notation in a high radix, it writes
‘`16#F.E8F*16.^15`’. You can enter a number like this as an
algebraic entry. Also, pressing `e` without any digits before it
normally types `1e`, but in a high radix it types `16.^` and
puts you in algebraic entry: `16#f.e8f RET e 15 RET *` is another
way to enter this number.

The reason Calc puts a decimal point in the ‘`16.^`’ is to prevent
huge integers from being generated if the exponent is large (consider
‘`16#1.23*16^1000`’, where we compute ‘`16^1000`’ as a giant
exact integer and then throw away most of the digits when we multiply
it by the floating-point ‘`16#1.23`’). While this wouldn’t normally
matter for display purposes, it could give you a nasty surprise if you
copied that number into a file and later moved it back into Calc.

#### 2.7.10 Modes Tutorial Exercise 3

The answer he got was ‘`0.5000000000006399`’.

The problem is not that the square operation is inexact, but that the sine of 45 that was already on the stack was accurate to only 12 places. Arbitrary-precision calculations still only give answers as good as their inputs.

The real problem is that there is no 12-digit number which, when
squared, comes out to 0.5 exactly. The `f [` and `f ]`
commands decrease or increase a number by one unit in the last
place (according to the current precision). They are useful for
determining facts like this.

1: 0.707106781187 1: 0.500000000001 . . 45 S 2 ^

1: 0.707106781187 1: 0.707106781186 1: 0.499999999999 . . . UDELf [ 2 ^

A high-precision calculation must be carried out in high precision all the way. The only number in the original problem which was known exactly was the quantity 45 degrees, so the precision must be raised before anything is done after the number 45 has been entered in order for the higher precision to be meaningful.

#### 2.7.11 Modes Tutorial Exercise 4

Many calculations involve real-world quantities, like the width and height of a piece of wood or the volume of a jar. Such quantities can’t be measured exactly anyway, and if the data that is input to a calculation is inexact, doing exact arithmetic on it is a waste of time.

Fractions become unwieldy after too many calculations have been done with them. For example, the sum of the reciprocals of the integers from 1 to 10 is 7381:2520. The sum from 1 to 30 is 9304682830147:2329089562800. After a point it will take a long time to add even one more term to this sum, but a floating-point calculation of the sum will not have this problem.

Also, rational numbers cannot express the results of all calculations.
There is no fractional form for the square root of two, so if you type
`2 Q`, Calc has no choice but to give you a floating-point answer.

#### 2.7.12 Arithmetic Tutorial Exercise 1

Dividing two integers that are larger than the current precision may
give a floating-point result that is inaccurate even when rounded
down to an integer. Consider ‘`123456789 / 2`’ when the current
precision is 6 digits. The true answer is ‘`61728394.5`’, but
with a precision of 6 this will be rounded to
‘`12345700. / 2. = 61728500.`’.
The result, when converted to an integer, will be off by 106.

Here are two solutions: Raise the precision enough that the
floating-point round-off error is strictly to the right of the
decimal point. Or, convert to Fraction mode so that ‘`123456789 / 2`’
produces the exact fraction ‘`123456789:2`’, which can be rounded
down by the `F` command without ever switching to floating-point
format.

#### 2.7.13 Arithmetic Tutorial Exercise 2

`27 RET 9 B` could give the exact result ‘

`3:2`’, but it does a floating-point calculation instead and produces ‘

`1.5`’.

Calc will find an exact result for a logarithm if the result is an integer or (when in Fraction mode) the reciprocal of an integer. But there is no efficient way to search the space of all possible rational numbers for an exact answer, so Calc doesn’t try.

#### 2.7.14 Vector Tutorial Exercise 1

Duplicate the vector, compute its length, then divide the vector
by its length: ` RET A /`.

1: [1, 2, 3] 2: [1, 2, 3] 1: [0.27, 0.53, 0.80] 1: 1. . 1: 3.74165738677 . . . r 1RETA / A

The final `A` command shows that the normalized vector does
indeed have unit length.

#### 2.7.15 Vector Tutorial Exercise 2

The average position is equal to the sum of the products of the
positions times their corresponding probabilities. This is the
definition of the dot product operation. So all you need to do
is to put the two vectors on the stack and press `*`.

#### 2.7.16 Matrix Tutorial Exercise 1

The trick is to multiply by a vector of ones. Use `r 4 [1 1 1] *` to
get the row sum. Similarly, use `[1 1] r 4 *` to get the column sum.

#### 2.7.17 Matrix Tutorial Exercise 2

x + a y = 6 x + b y = 10

Just enter the righthand side vector, then divide by the lefthand side matrix as usual.

1: [6, 10] 2: [6, 10] 1: [4 a / (a - b) + 6, 4 / (b - a) ] . 1: [ [ 1, a ] . [ 1, b ] ] . ' [6 10]RET' [1 a; 1 b]RET/

This can be made more readable using `d B` to enable Big display
mode:

4 a 4 1: [----- + 6, -----] a - b b - a

Type `d N` to return to Normal display mode afterwards.

#### 2.7.18 Matrix Tutorial Exercise 3

To solve
‘`trn(A) * A * X = trn(A) * B`’,
first we compute
‘`A2 = trn(A) * A`’
and
‘`B2 = trn(A) * B`’;
now, we have a system
‘`A2 * X = B2`’
which we can solve using Calc’s ‘`/`’ command.

a + 2b + 3c = 6 4a + 5b + 6c = 2 7a + 6b = 3 2a + 4b + 6c = 11

The first step is to enter the coefficient matrix. We’ll store it in
quick variable number 7 for later reference. Next, we compute the
‘`B2`’
vector.

1: [ [ 1, 2, 3 ] 2: [ [ 1, 4, 7, 2 ] 1: [57, 84, 96] [ 4, 5, 6 ] [ 2, 5, 6, 4 ] . [ 7, 6, 0 ] [ 3, 6, 0, 6 ] ] [ 2, 4, 6 ] ] 1: [6, 2, 3, 11] . . ' [1 2 3; 4 5 6; 7 6 0; 2 4 6]RETs 7 v t [6 2 3 11] *

Now we compute the matrix
‘`A2`’
and divide.

2: [57, 84, 96] 1: [-11.64, 14.08, -3.64] 1: [ [ 70, 72, 39 ] . [ 72, 81, 60 ] [ 39, 60, 81 ] ] . r 7 v t r 7 * /

(The actual computed answer will be slightly inexact due to round-off error.)

Notice that the answers are similar to those for the 3x3 system solved in the text. That’s because the fourth equation that was added to the system is almost identical to the first one multiplied by two. (If it were identical, we would have gotten the exact same answer since the 4x3 system would be equivalent to the original 3x3 system.)

Since the first and fourth equations aren’t quite equivalent, they can’t both be satisfied at once. Let’s plug our answers back into the original system of equations to see how well they match.

2: [-11.64, 14.08, -3.64] 1: [5.6, 2., 3., 11.2] 1: [ [ 1, 2, 3 ] . [ 4, 5, 6 ] [ 7, 6, 0 ] [ 2, 4, 6 ] ] . r 7TAB*

This is reasonably close to our original ‘`B`’ vector,
‘`[6, 2, 3, 11]`’.

#### 2.7.19 List Tutorial Exercise 1

We can use `v x` to build a vector of integers. This needs to be
adjusted to get the range of integers we desire. Mapping ‘`-`’
across the vector will accomplish this, although it turns out the
plain ‘`-`’ key will work just as well.

2: 2 2: 2 1: [1, 2, 3, 4, 5, 6, 7, 8, 9] 1: [-4, -3, -2, -1, 0, 1, 2, 3, 4] . . 2 v x 9RET5 V M - or 5 -

Now we use `V M ^` to map the exponentiation operator across the
vector.

1: [0.0625, 0.125, 0.25, 0.5, 1, 2, 4, 8, 16] . V M ^

#### 2.7.20 List Tutorial Exercise 2

Given ‘`x`’ and ‘`y`’ vectors in quick variables 1 and 2 as before,
the first job is to form the matrix that describes the problem.

m*x + b*1 = y

Thus we want a
19x2
matrix with our ‘`x`’ vector as one column and
ones as the other column. So, first we build the column of ones, then
we combine the two columns to form our ‘`A`’ matrix.

2: [1.34, 1.41, 1.49, ... ] 1: [ [ 1.34, 1 ] 1: [1, 1, 1, ...] [ 1.41, 1 ] . [ 1.49, 1 ] … r 1 1 v b 19RETM-2 v p v t s 3

Now we compute
‘`trn(A) * y`’
and
‘`trn(A) * A`’
and divide.

1: [33.36554, 13.613] 2: [33.36554, 13.613] . 1: [ [ 98.0003, 41.63 ] [ 41.63, 19 ] ] . v t r 2 * r 3 v t r 3 *

(Hey, those numbers look familiar!)

1: [0.52141679, -0.425978] . /

Since we were solving equations of the form
‘`m*x + b*1 = y`’,
these numbers should be ‘`m`’ and ‘`b`’, respectively. Sure
enough, they agree exactly with the result computed using `V M` and
`V R`!

The moral of this story: `V M` and `V R` will probably solve
your problem, but there is often an easier way using the higher-level
arithmetic functions!

In fact, there is a built-in `a F` command that does least-squares
fits. See Curve Fitting.

#### 2.7.21 List Tutorial Exercise 3

Move to one end of the list and press `C-@` (or `C- SPC` or
whatever) to set the mark, then move to the other end of the list
and type

`C-x * g`.

1: [2.3, 6, 22, 15.1, 7, 15, 14, 7.5, 2.5] .

To make things interesting, let’s assume we don’t know at a glance how many numbers are in this list. Then we could type:

2: [2.3, 6, 22, ... ] 2: [2.3, 6, 22, ... ] 1: [2.3, 6, 22, ... ] 1: 126356422.5 . .RETV R *

2: 126356422.5 2: 126356422.5 1: 7.94652913734 1: [2.3, 6, 22, ... ] 1: 9 . . .TABv l I ^

(The `I ^` command computes the `n`th root of a number.
You could also type `& ^` to take the reciprocal of 9 and
then raise the number to that power.)

#### 2.7.22 List Tutorial Exercise 4

A number ‘`j`’ is a divisor of ‘`n`’ if
‘`n % j = 0`’.
The first step is to get a vector that identifies the divisors.

2: 30 2: [0, 0, 0, 2, ...] 1: [1, 1, 1, 0, ...] 1: [1, 2, 3, 4, ...] 1: 0 . . . 30RETv x 30RETs 1 V M % 0 V M a = s 2

This vector has 1’s marking divisors of 30 and 0’s marking non-divisors.

The zeroth divisor function is just the total number of divisors. The first divisor function is the sum of the divisors.

1: 8 3: 8 2: 8 2: 8 2: [1, 2, 3, 4, ...] 1: [1, 2, 3, 0, ...] 1: 72 1: [1, 1, 1, 0, ...] . . . V R + r 1 r 2 V M * V R +

Once again, the last two steps just compute a dot product for which
a simple `*` would have worked equally well.

#### 2.7.23 List Tutorial Exercise 5

The obvious first step is to obtain the list of factors with `k f`.
This list will always be in sorted order, so if there are duplicates
they will be right next to each other. A suitable method is to compare
the list with a copy of itself shifted over by one.

1: [3, 7, 7, 7, 19] 2: [3, 7, 7, 7, 19] 2: [3, 7, 7, 7, 19, 0] . 1: [3, 7, 7, 7, 19, 0] 1: [0, 3, 7, 7, 7, 19] . . 19551 k fRET0 |TAB0TAB|

1: [0, 0, 1, 1, 0, 0] 1: 2 1: 0 . . . V M a = V R + 0 a =

Note that we have to arrange for both vectors to have the same length so that the mapping operation works; no prime factor will ever be zero, so adding zeros on the left and right is safe. From then on the job is pretty straightforward.

Incidentally, Calc provides the *MÃ¶bius Î¼*
function which is zero if and only if its argument is square-free. It
would be a much more convenient way to do the above test in practice.

#### 2.7.24 List Tutorial Exercise 6

First use `v x 6 RET` to get a list of integers, then

`V M v x`to get a list of lists of integers!

#### 2.7.25 List Tutorial Exercise 7

Here’s one solution. First, compute the triangular list from the previous
exercise and type `1 -` to subtract one from all the elements.

1: [ [0], [0, 1], [0, 1, 2], … 1 -

The numbers down the lefthand edge of the list we desire are called
the “triangular numbers” (now you know why!). The ‘`n`’th
triangular number is the sum of the integers from 1 to ‘`n`’, and
can be computed directly by the formula
‘`n * (n+1) / 2`’.

2: [ [0], [0, 1], ... ] 2: [ [0], [0, 1], ... ] 1: [0, 1, 2, 3, 4, 5] 1: [0, 1, 3, 6, 10, 15] . . v x 6RET1 - V M ' $ ($+1)/2RET

Adding this list to the above list of lists produces the desired result:

1: [ [0], [1, 2], [3, 4, 5], [6, 7, 8, 9], [10, 11, 12, 13, 14], [15, 16, 17, 18, 19, 20] ] . V M +

If we did not know the formula for triangular numbers, we could have
computed them using a `V U +` command. We could also have
gotten them the hard way by mapping a reduction across the original
triangular list.

2: [ [0], [0, 1], ... ] 2: [ [0], [0, 1], ... ] 1: [ [0], [0, 1], ... ] 1: [0, 1, 3, 6, 10, 15] . .RETV M V R +

(This means “map a `V R +` command across the vector,” and
since each element of the main vector is itself a small vector,
`V R +` computes the sum of its elements.)

#### 2.7.26 List Tutorial Exercise 8

The first step is to build a list of values of ‘`x`’.

1: [1, 2, 3, ..., 21] 1: [0, 1, 2, ..., 20] 1: [0, 0.25, 0.5, ..., 5] . . . v x 21RET1 - 4 / s 1

Next, we compute the Bessel function values.

1: [0., 0.124, 0.242, ..., -0.328] . V M ' besJ(1,$)RET

(Another way to do this would be `1 TAB V M f j`.)

A way to isolate the maximum value is to compute the maximum using
`V R X`, then compare all the Bessel values with that maximum.

2: [0., 0.124, 0.242, ... ] 1: [0, 0, 0, ... ] 2: [0, 0, 0, ... ] 1: 0.5801562 . 1: 1 . .RETV R X V M a =RETV R +DEL

It’s a good idea to verify, as in the last step above, that only
one value is equal to the maximum. (After all, a plot of
‘`sin(x)`’
might have many points all equal to the maximum value, 1.)

The vector we have now has a single 1 in the position that indicates
the maximum value of ‘`x`’. Now it is a simple matter to convert
this back into the corresponding value itself.

2: [0, 0, 0, ... ] 1: [0, 0., 0., ... ] 1: 1.75 1: [0, 0.25, 0.5, ... ] . . . r 1 V M * V R +

If `a =` had produced more than one ‘`1`’ value, this method
would have given the sum of all maximum ‘`x`’ values; not very
useful! In this case we could have used `v m` (`calc-mask-vector`

)
instead. This command deletes all elements of a “data” vector that
correspond to zeros in a “mask” vector, leaving us with, in this
example, a vector of maximum ‘`x`’ values.

The built-in `a X` command maximizes a function using more
efficient methods. Just for illustration, let’s use `a X`
to maximize ‘`besJ(1,x)`’ over this same interval.

2: besJ(1, x) 1: [1.84115, 0.581865] 1: [0 .. 5] . . ' besJ(1,x), [0..5]RETa X xRET

The output from `a X` is a vector containing the value of ‘`x`’
that maximizes the function, and the function’s value at that maximum.
As you can see, our simple search got quite close to the right answer.

#### 2.7.27 List Tutorial Exercise 9

Step one is to convert our integer into vector notation.

1: 25129925999 3: 25129925999 . 2: 10 1: [11, 10, 9, ..., 1, 0] . 25129925999RET10RET12RETv x 12RET-

1: 25129925999 1: [0, 2, 25, 251, 2512, ... ] 2: [100000000000, ... ] . . V M ^ s 1 V M \

(Recall, the `\` command computes an integer quotient.)

1: [0, 2, 5, 1, 2, 9, 9, 2, 5, 9, 9, 9] . 10 V M % s 2

Next we must increment this number. This involves adding one to the last digit, plus handling carries. There is a carry to the left out of a digit if that digit is a nine and all the digits to the right of it are nines.

1: [0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1] 1: [1, 1, 1, 0, 0, 1, ... ] . . 9 V M a = v v

1: [1, 1, 1, 0, 0, 0, ... ] 1: [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1] . . V U * v v 1 |

Accumulating `*` across a vector of ones and zeros will preserve
only the initial run of ones. These are the carries into all digits
except the rightmost digit. Concatenating a one on the right takes
care of aligning the carries properly, and also adding one to the
rightmost digit.

2: [0, 0, 0, 0, ... ] 1: [0, 0, 2, 5, 1, 2, 9, 9, 2, 6, 0, 0, 0] 1: [0, 0, 2, 5, ... ] . . 0 r 2 | V M + 10 V M %

Here we have concatenated 0 to the *left* of the original number;
this takes care of shifting the carries by one with respect to the
digits that generated them.

Finally, we must convert this list back into an integer.

3: [0, 0, 2, 5, ... ] 2: [0, 0, 2, 5, ... ] 2: 1000000000000 1: [1000000000000, 100000000000, ... ] 1: [100000000000, ... ] . . 10RET12 ^ r 1 |

1: [0, 0, 20000000000, 5000000000, ... ] 1: 25129926000 . . V M * V R +

Another way to do this final step would be to reduce the formula
‘`10 $$ + $`’ across the vector of digits.

1: [0, 0, 2, 5, ... ] 1: 25129926000 . . V R ' 10 $$ + $RET

#### 2.7.28 List Tutorial Exercise 10

For the list ‘`[a, b, c, d]`’, the result is ‘`((a = b) = c) = d`’,
which will compare ‘`a`’ and ‘`b`’ to produce a 1 or 0, which is
then compared with ‘`c`’ to produce another 1 or 0, which is then
compared with ‘`d`’. This is not at all what Joe wanted.

Here’s a more correct method:

1: [7, 7, 7, 8, 7] 2: [7, 7, 7, 8, 7] . 1: 7 . ' [7,7,7,8,7]RETRETv r 1RET

1: [1, 1, 1, 0, 1] 1: 0 . . V M a = V R *

#### 2.7.29 List Tutorial Exercise 11

The circle of unit radius consists of those points ‘`(x,y)`’ for which
‘`x^2 + y^2 < 1`’. We start by generating a vector of ‘`x^2`’
and a vector of ‘`y^2`’.

We can make this go a bit faster by using the `v .` and `t .`
commands.

2: [2., 2., ..., 2.] 2: [2., 2., ..., 2.] 1: [2., 2., ..., 2.] 1: [1.16, 1.98, ..., 0.81] . . v . t . 2. v b 100RETRETV M k r

2: [2., 2., ..., 2.] 1: [0.026, 0.96, ..., 0.036] 1: [0.026, 0.96, ..., 0.036] 2: [0.53, 0.81, ..., 0.094] . . 1 - 2 V M ^TABV M k r 1 - 2 V M ^

Now we sum the ‘`x^2`’ and ‘`y^2`’ values, compare with 1 to
get a vector of 1/0 truth values, then sum the truth values.

1: [0.56, 1.78, ..., 0.13] 1: [1, 0, ..., 1] 1: 84 . . . + 1 V M a < V R +

The ratio ‘`84/100`’ should approximate the ratio ‘`pi/4`’.

1: 0.84 1: 3.36 2: 3.36 1: 1.0695 . . 1: 3.14159 . 100 / 4 * P /

Our estimate, 3.36, is off by about 7%. We could get a better estimate by taking more points (say, 1000), but it’s clear that this method is not very efficient!

(Naturally, since this example uses random numbers your own answer will be slightly different from the one shown here!)

If you typed `v .` and `t .` before, type them again to
return to full-sized display of vectors.

#### 2.7.30 List Tutorial Exercise 12

This problem can be made a lot easier by taking advantage of some
symmetries. First of all, after some thought it’s clear that the
‘`y`’ axis can be ignored altogether. Just pick a random ‘`x`’
component for one end of the match, pick a random direction
‘`theta`’,
and see if ‘`x`’ and
‘`x + cos(theta)`’
(which is the ‘`x`’ coordinate of the other endpoint) cross a line.
The lines are at integer coordinates, so this happens when the two
numbers surround an integer.

Since the two endpoints are equivalent, we may as well choose the leftmost
of the two endpoints as ‘`x`’. Then ‘`theta`’ is an angle pointing
to the right, in the range -90 to 90 degrees. (We could use radians, but
it would feel like cheating to refer to ‘`pi/2`’ radians while trying
to estimate ‘`pi`’!)

In fact, since the field of lines is infinite we can choose the
coordinates 0 and 1 for the lines on either side of the leftmost
endpoint. The rightmost endpoint will be between 0 and 1 if the
match does not cross a line, or between 1 and 2 if it does. So:
Pick random ‘`x`’ and
‘`theta`’,
compute
‘`x + cos(theta)`’,
and count how many of the results are greater than one. Simple!

We can make this go a bit faster by using the `v .` and `t .`
commands.

1: [0.52, 0.71, ..., 0.72] 2: [0.52, 0.71, ..., 0.72] . 1: [78.4, 64.5, ..., -42.9] . v . t . 1. v b 100RETV M k r 180. v b 100RETV M k r 90 -

(The next step may be slow, depending on the speed of your computer.)

2: [0.52, 0.71, ..., 0.72] 1: [0.72, 1.14, ..., 1.45] 1: [0.20, 0.43, ..., 0.73] . . m d V M C +

1: [0, 1, ..., 1] 1: 0.64 1: 3.125 . . . 1 V M a > V R + 100 / 2TAB/

Let’s try the third method, too. We’ll use random integers up to
one million. The `k r` command with an integer argument picks
a random integer.

2: [1000000, 1000000, ..., 1000000] 2: [78489, 527587, ..., 814975] 1: [1000000, 1000000, ..., 1000000] 1: [324014, 358783, ..., 955450] . . 1000000 v b 100RETRETV M k rTABV M k r

1: [1, 1, ..., 25] 1: [1, 1, ..., 0] 1: 0.56 . . . V M k g 1 V M a = V R + 100 /

1: 10.714 1: 3.273 . . 6TAB/ Q

For a proof of this property of the GCD function, see section 4.5.2,
exercise 10, of Knuth’s *Art of Computer Programming*, volume II.

If you typed `v .` and `t .` before, type them again to
return to full-sized display of vectors.

#### 2.7.31 List Tutorial Exercise 13

First, we put the string on the stack as a vector of ASCII codes.

1: [84, 101, 115, ..., 51] . "Testing, 1, 2, 3RET

Note that the `"` key, like `$`, initiates algebraic entry so
there was no need to type an apostrophe. Also, Calc didn’t mind that
we omitted the closing `"`. (The same goes for all closing delimiters
like `)` and `]` at the end of a formula.

We’ll show two different approaches here. In the first, we note that
if the input vector is ‘`[a, b, c, d]`’, then the hash code is
‘`3 (3 (3a + b) + c) + d = 27a + 9b + 3c + d`’. In other words,
it’s a sum of descending powers of three times the ASCII codes.

2: [84, 101, 115, ..., 51] 2: [84, 101, 115, ..., 51] 1: 16 1: [15, 14, 13, ..., 0] . .RETv l v x 16RET-

2: [84, 101, 115, ..., 51] 1: 1960915098 1: 121 1: [14348907, ..., 1] . . . 3TABV M ^ * 511 %

Once again, `*` elegantly summarizes most of the computation.
But there’s an even more elegant approach: Reduce the formula
`3 $$ + $` across the vector. Recall that this represents a
function of two arguments that computes its first argument times three
plus its second argument.

1: [84, 101, 115, ..., 51] 1: 1960915098 . . "Testing, 1, 2, 3RETV R ' 3$$+$RET

If you did the decimal arithmetic exercise, this will be familiar. Basically, we’re turning a base-3 vector of digits into an integer, except that our “digits” are much larger than real digits.

Instead of typing `511 %` again to reduce the result, we can be
cleverer still and notice that rather than computing a huge integer
and taking the modulo at the end, we can take the modulo at each step
without affecting the result. While this means there are more
arithmetic operations, the numbers we operate on remain small so
the operations are faster.

1: [84, 101, 115, ..., 51] 1: 121 . . "Testing, 1, 2, 3RETV R ' (3$$+$)%511RET

Why does this work? Think about a two-step computation:
‘`3 (3a + b) + c`’. Taking a result modulo 511 basically means
subtracting off enough 511’s to put the result in the desired range.
So the result when we take the modulo after every step is,

3 (3 a + b - 511 m) + c - 511 n

for some suitable integers ‘`m`’ and ‘`n`’. Expanding out by
the distributive law yields

9 a + 3 b + c - 511*3 m - 511 n

The ‘`m`’ term in the latter formula is redundant because any
contribution it makes could just as easily be made by the ‘`n`’
term. So we can take it out to get an equivalent formula with
‘`n' = 3m + n`’,

9 a + 3 b + c - 511 n'

which is just the formula for taking the modulo only at the end of the calculation. Therefore the two methods are essentially the same.

Later in the tutorial we will encounter *modulo forms*, which
basically automate the idea of reducing every intermediate result
modulo some value `m`.

#### 2.7.32 List Tutorial Exercise 14

We want to use `H V U` to nest a function which adds a random
step to an ‘`(x,y)`’ coordinate. The function is a bit long, but
otherwise the problem is quite straightforward.

2: [0, 0] 1: [ [ 0, 0 ] 1: 50 [ 0.4288, -0.1695 ] . [ -0.4787, -0.9027 ] ... [0,0] 50 H V U ' <# + [random(2.0)-1, random(2.0)-1]>RET

Just as the text recommended, we used ‘`< >`’ nameless function
notation to keep the two `random`

calls from being evaluated
before nesting even begins.

We now have a vector of ‘`[x, y]`’ sub-vectors, which by Calc’s
rules acts like a matrix. We can transpose this matrix and unpack
to get a pair of vectors, ‘`x`’ and ‘`y`’, suitable for graphing.

2: [ 0, 0.4288, -0.4787, ... ] 1: [ 0, -0.1696, -0.9027, ... ] . v t v u g f

Incidentally, because the ‘`x`’ and ‘`y`’ are completely
independent in this case, we could have done two separate commands
to create our ‘`x`’ and ‘`y`’ vectors of numbers directly.

To make a random walk of unit steps, we note that `sincos`

of
a random direction exactly gives us an ‘`[x, y]`’ step of unit
length; in fact, the new nesting function is even briefer, though
we might want to lower the precision a bit for it.

2: [0, 0] 1: [ [ 0, 0 ] 1: 50 [ 0.1318, 0.9912 ] . [ -0.5965, 0.3061 ] ... [0,0] 50 m d p 6RETH V U ' <# + sincos(random(360.0))>RET

Another `v t v u g f` sequence will graph this new random walk.

An interesting twist on these random walk functions would be to use
complex numbers instead of 2-vectors to represent points on the plane.
In the first example, we’d use something like ‘`random + random*(0,1)`’,
and in the second we could use polar complex numbers with random phase
angles. (This exercise was first suggested in this form by Randal
Schwartz.)

#### 2.7.33 Types Tutorial Exercise 1

If the number is the square root of ‘`pi`’ times a rational number,
then its square, divided by ‘`pi`’, should be a rational number.

1: 1.26508260337 1: 0.509433962268 1: 2486645810:4881193627 . . . 2 ^ P / c F

Technically speaking this is a rational number, but not one that is likely to have arisen in the original problem. More likely, it just happens to be the fraction which most closely represents some irrational number to within 12 digits.

But perhaps our result was not quite exact. Let’s reduce the precision slightly and try again:

1: 0.509433962268 1: 27:53 . . U p 10RETc F

Aha! It’s unlikely that an irrational number would equal a fraction
this simple to within ten digits, so our original number was probably
‘`sqrt(27 pi / 53)`’.

Notice that we didn’t need to re-round the number when we reduced the precision. Remember, arithmetic operations always round their inputs to the current precision before they begin.

#### 2.7.34 Types Tutorial Exercise 2

‘`inf / inf = nan`’. Perhaps ‘`1`’ is the “obvious” answer.
But if ‘`17 inf = inf`’, then ‘`17 inf / inf = inf / inf = 17`’, too.

‘`exp(inf) = inf`’. It’s tempting to say that the exponential
of infinity must be “bigger” than “regular” infinity, but as
far as Calc is concerned all infinities are the same size.
In other words, as ‘`x`’ goes to infinity, ‘`e^x`’ also goes
to infinity, but the fact the ‘`e^x`’ grows much faster than
‘`x`’ is not relevant here.

‘`exp(-inf) = 0`’. Here we have a finite answer even though
the input is infinite.

‘`sqrt(-inf) = (0, 1) inf`’. Remember that ‘`(0, 1)`’
represents the imaginary number ‘`i`’. Here’s a derivation:
‘`sqrt(-inf) = sqrt((-1) * inf) = sqrt(-1) * sqrt(inf)`’.
The first part is, by definition, ‘`i`’; the second is `inf`

because, once again, all infinities are the same size.

‘`sqrt(uinf) = uinf`’. In fact, we do know something about the
direction because `sqrt`

is defined to return a value in the
right half of the complex plane. But Calc has no notation for this,
so it settles for the conservative answer `uinf`

.

‘`abs(uinf) = inf`’. No matter which direction ‘`x`’ points,
‘`abs(x)`’ always points along the positive real axis.

‘`ln(0) = -inf`’. Here we have an infinite answer to a finite
input. As in the ‘`1 / 0`’ case, Calc will only use infinities
here if you have turned on Infinite mode. Otherwise, it will
treat ‘`ln(0)`’ as an error.

#### 2.7.35 Types Tutorial Exercise 3

We can make ‘`inf - inf`’ be any real number we like, say,
‘`a`’, just by claiming that we added ‘`a`’ to the first
infinity but not to the second. This is just as true for complex
values of ‘`a`’, so `nan`

can stand for a complex number.
(And, similarly, `uinf`

can stand for an infinity that points
in any direction in the complex plane, such as ‘`(0, 1) inf`’).

In fact, we can multiply the first `inf`

by two. Surely
‘`2 inf - inf = inf`’, but also ‘`2 inf - inf = inf - inf = nan`’.
So `nan`

can even stand for infinity. Obviously it’s just
as easy to make it stand for minus infinity as for plus infinity.

The moral of this story is that “infinity” is a slippery fish
indeed, and Calc tries to handle it by having a very simple model
for infinities (only the direction counts, not the “size”); but
Calc is careful to write `nan`

any time this simple model is
unable to tell what the true answer is.

#### 2.7.36 Types Tutorial Exercise 4

2: 0@ 47' 26" 1: 0@ 2' 47.411765" 1: 17 . . 0@ 47' 26"RET17 /

The average song length is two minutes and 47.4 seconds.

2: 0@ 2' 47.411765" 1: 0@ 3' 7.411765" 1: 0@ 53' 6.000005" 1: 0@ 0' 20" . . . 20" + 17 *

The album would be 53 minutes and 6 seconds long.

#### 2.7.37 Types Tutorial Exercise 5

Let’s suppose it’s January 14, 1991. The easiest thing to do is
to keep trying 13ths of months until Calc reports a Friday.
We can do this by manually entering dates, or by using `t I`:

1: <Wed Feb 13, 1991> 1: <Wed Mar 13, 1991> 1: <Sat Apr 13, 1991> . . . ' <2/13>RETDEL' <3/13>RETt I

(Calc assumes the current year if you don’t say otherwise.)

This is getting tedious—we can keep advancing the date by typing
`t I` over and over again, but let’s automate the job by using
vector mapping. The `t I` command actually takes a second
“how-many-months” argument, which defaults to one. This
argument is exactly what we want to map over:

2: <Sat Apr 13, 1991> 1: [<Mon May 13, 1991>, <Thu Jun 13, 1991>, 1: [1, 2, 3, 4, 5, 6] <Sat Jul 13, 1991>, <Tue Aug 13, 1991>, . <Fri Sep 13, 1991>, <Sun Oct 13, 1991>] . v x 6RETV M t I

Et voilÃ , September 13, 1991 is a Friday.

1: 242 . ' <sep 13> - <jan 14>RET

And the answer to our original question: 242 days to go.

#### 2.7.38 Types Tutorial Exercise 6

The full rule for leap years is that they occur in every year divisible
by four, except that they don’t occur in years divisible by 100, except
that they *do* in years divisible by 400. We could work out the
answer by carefully counting the years divisible by four and the
exceptions, but there is a much simpler way that works even if we
don’t know the leap year rule.

Let’s assume the present year is 1991. Years have 365 days, except that leap years (whenever they occur) have 366 days. So let’s count the number of days between now and then, and compare that to the number of years times 365. The number of extra days we find must be equal to the number of leap years there were.

1: <Mon Jan 1, 10001> 2: <Mon Jan 1, 10001> 1: 2925593 . 1: <Tue Jan 1, 1991> . . ' <jan 1 10001>RET' <jan 1 1991>RET-

3: 2925593 2: 2925593 2: 2925593 1: 1943 2: 10001 1: 8010 1: 2923650 . 1: 1991 . . . 10001RET1991 - 365 * -

There will be 1943 leap years before the year 10001. (Assuming, of course, that the algorithm for computing leap years remains unchanged for that long. See Date Forms, for some interesting background information in that regard.)

#### 2.7.39 Types Tutorial Exercise 7

The relative errors must be converted to absolute errors so that
‘`+/-`’ notation may be used.

1: 1. 2: 1. . 1: 0.2 . 20RET.05 * 4RET.05 *

Now we simply chug through the formula.

1: 19.7392088022 1: 394.78 +/- 19.739 1: 6316.5 +/- 706.21 . . . 2 P 2 ^ * 20 p 1 * 4 p .2RET2 ^ *

It turns out the `v u` command will unpack an error form as
well as a vector. This saves us some retyping of numbers.

3: 6316.5 +/- 706.21 2: 6316.5 +/- 706.21 2: 6316.5 1: 0.1118 1: 706.21 . .RETv uTAB/

Thus the volume is 6316 cubic centimeters, within about 11 percent.

#### 2.7.40 Types Tutorial Exercise 8

The first answer is pretty simple: ‘`1 / (0 .. 10) = (0.1 .. inf)`’.
Since a number in the interval ‘`(0 .. 10)`’ can get arbitrarily
close to zero, its reciprocal can get arbitrarily large, so the answer
is an interval that effectively means, “any number greater than 0.1”
but with no upper bound.

The second answer, similarly, is ‘`1 / (-10 .. 0) = (-inf .. -0.1)`’.

Calc normally treats division by zero as an error, so that the formula
‘`1 / 0`’ is left unsimplified. Our third problem,
‘`1 / [0 .. 10]`’, also (potentially) divides by zero because zero
is now a member of the interval. So Calc leaves this one unevaluated, too.

If you turn on Infinite mode by pressing `m i`, you will
instead get the answer ‘`[0.1 .. inf]`’, which includes infinity
as a possible value.

The fourth calculation, ‘`1 / (-10 .. 10)`’, has the same problem.
Zero is buried inside the interval, but it’s still a possible value.
It’s not hard to see that the actual result of ‘`1 / (-10 .. 10)`’
will be either greater than *0.1*, or less than *-0.1*. Thus
the interval goes from minus infinity to plus infinity, with a “hole”
in it from *-0.1* to *0.1*. Calc doesn’t have any way to
represent this, so it just reports ‘`[-inf .. inf]`’ as the answer.
It may be disappointing to hear “the answer lies somewhere between
minus infinity and plus infinity, inclusive,” but that’s the best
that interval arithmetic can do in this case.

#### 2.7.41 Types Tutorial Exercise 9

1: [-3 .. 3] 2: [-3 .. 3] 2: [0 .. 9] . 1: [0 .. 9] 1: [-9 .. 9] . . [ 3 n .. 3 ]RET2 ^TABRET*

In the first case the result says, “if a number is between *-3* and
3, its square is between 0 and 9.” The second case says, “the product
of two numbers each between *-3* and 3 is between *-9* and 9.”

An interval form is not a number; it is a symbol that can stand for many different numbers. Two identical-looking interval forms can stand for different numbers.

The same issue arises when you try to square an error form.

#### 2.7.42 Types Tutorial Exercise 10

Testing the first number, we might arbitrarily choose 17 for ‘`x`’.

1: 17 mod 811749613 2: 17 mod 811749613 1: 533694123 mod 811749613 . 811749612 . . 17 M 811749613RET811749612 ^

Since 533694123 is (considerably) different from 1, the number 811749613 must not be prime.

It’s awkward to type the number in twice as we did above. There are various ways to avoid this, and algebraic entry is one. In fact, using a vector mapping operation we can perform several tests at once. Let’s use this method to test the second number.

2: [17, 42, 100000] 1: [1 mod 15485863, 1 mod ... ] 1: 15485863 . . [17 42 100000] 15485863RETV M ' ($$ mod $)^($-1)RET

The result is three ones (modulo ‘`n`’), so it’s very probable that
15485863 is prime. (In fact, this number is the millionth prime.)

Note that the functions ‘`($$^($-1)) mod $`’ or ‘`$$^($-1) % $`’
would have been hopelessly inefficient, since they would have calculated
the power using full integer arithmetic.

Calc has a `k p` command that does primality testing. For small
numbers it does an exact test; for large numbers it uses a variant
of the Fermat test we used here. You can use `k p` repeatedly
to prove that a large integer is prime with any desired probability.

#### 2.7.43 Types Tutorial Exercise 11

There are several ways to insert a calculated number into an HMS form. One way to convert a number of seconds to an HMS form is simply to multiply the number by an HMS form representing one second:

1: 31415926.5359 2: 31415926.5359 1: 8726@ 38' 46.5359" . 1: 0@ 0' 1" . . P 1e7 * 0@ 0' 1" *

2: 8726@ 38' 46.5359" 1: 6@ 6' 2.5359" mod 24@ 0' 0" 1: 15@ 27' 16" mod 24@ 0' 0" . . x timeRET+

It will be just after six in the morning.

The algebraic `hms`

function can also be used to build an
HMS form:

1: hms(0, 0, 10000000. pi) 1: 8726@ 38' 46.5359" . . ' hms(0, 0, 1e7 pi)RET=

The `=` key is necessary to evaluate the symbol ‘`pi`’ to
the actual number 3.14159...

#### 2.7.44 Types Tutorial Exercise 12

As we recall, there are 17 songs of about 2 minutes and 47 seconds each.

2: 0@ 2' 47" 1: [0@ 3' 7" .. 0@ 3' 47"] 1: [0@ 0' 20" .. 0@ 1' 0"] . . [ 0@ 20" .. 0@ 1' ] +

1: [0@ 52' 59." .. 1@ 4' 19."] . 17 *

No matter how long it is, the album will fit nicely on one CD.

#### 2.7.45 Types Tutorial Exercise 13

Type `' 1 yr RET u c s RET`. The answer is 31557600 seconds.

#### 2.7.46 Types Tutorial Exercise 14

How long will it take for a signal to get from one end of the computer to the other?

1: m / c 1: 3.3356 ns . . ' 1 m / cRETu c nsRET

(Recall, ‘`c`’ is a “unit” corresponding to the speed of light.)

1: 3.3356 ns 1: 0.81356 2: 4.1 ns . . ' 4.1 nsRET/

Thus a signal could take up to 81 percent of a clock cycle just to go from one place to another inside the computer, assuming the signal could actually attain the full speed of light. Pretty tight!

#### 2.7.47 Types Tutorial Exercise 15

The speed limit is 55 miles per hour on most highways. We want to find the ratio of Sam’s speed to the US speed limit.

1: 55 mph 2: 55 mph 3: 11 hr mph / yd . 1: 5 yd / hr . . ' 55 mphRET' 5 yd/hrRET/

The `u s` command cancels out these units to get a plain
number. Now we take the logarithm base two to find the final
answer, assuming that each successive pill doubles his speed.

1: 19360. 2: 19360. 1: 14.24 . 1: 2 . . u s 2 B

Thus Sam can take up to 14 pills without a worry.

#### 2.7.48 Algebra Tutorial Exercise 1

The result ‘`sqrt(x)^2`’ is simplified back to ‘`x`’ by the
Calculator, but ‘`sqrt(x^2)`’ is not. (Consider what happens
if ‘`x = -4`’.) If ‘`x`’ is real, this formula could be
simplified to ‘`abs(x)`’, but for general complex arguments even
that is not safe. (See Declarations, for a way to tell Calc
that ‘`x`’ is known to be real.)

#### 2.7.49 Algebra Tutorial Exercise 2

Suppose our roots are ‘`[a, b, c]`’. We want a polynomial which
is zero when ‘`x`’ is any of these values. The trivial polynomial
‘`x-a`’ is zero when ‘`x=a`’, so the product ‘`(x-a)(x-b)(x-c)`’
will do the job. We can use `a c x` to write this in a more
familiar form.

1: 34 x - 24 x^3 1: [1.19023, -1.19023, 0] . . r 2 a P xRET

1: [x - 1.19023, x + 1.19023, x] 1: x*(x + 1.19023) (x - 1.19023) . . V M ' x-$RETV R *

1: x^3 - 1.41666 x 1: 34 x - 24 x^3 . . a c xRET24 n * a x

Sure enough, our answer (multiplied by a suitable constant) is the same as the original polynomial.

#### 2.7.50 Algebra Tutorial Exercise 3

1: x sin(pi x) 1: sin(pi x) / pi^2 - x cos(pi x) / pi . . ' x sin(pi x)RETm r a i xRET

1: [y, 1] 2: sin(pi x) / pi^2 - x cos(pi x) / pi . ' [y,1]RETTAB

1: [sin(pi y) / pi^2 - y cos(pi y) / pi, 1 / pi] . V M $RET

1: sin(pi y) / pi^2 - y cos(pi y) / pi - 1 / pi . V R -

1: sin(3.14159 y) / 9.8696 - y cos(3.14159 y) / 3.14159 - 0.3183 . =

1: [0., -0.95493, 0.63662, -1.5915, 1.2732] . v x 5RETTABV M $RET

#### 2.7.51 Algebra Tutorial Exercise 4

The hard part is that `V R +` is no longer sufficient to add up all
the contributions from the slices, since the slices have varying
coefficients. So first we must come up with a vector of these
coefficients. Here’s one way:

2: -1 2: 3 1: [4, 2, ..., 4] 1: [1, 2, ..., 9] 1: [-1, 1, ..., -1] . . . 1 n v x 9RETV M ^ 3TAB-

1: [4, 2, ..., 4, 1] 1: [1, 4, 2, ..., 4, 1] . . 1 | 1TAB|

Now we compute the function values. Note that for this method we need eleven values, including both endpoints of the desired interval.

2: [1, 4, 2, ..., 4, 1] 1: [1, 1.1, 1.2, ... , 1.8, 1.9, 2.] . 11RET1RET.1RETC-u v x

2: [1, 4, 2, ..., 4, 1] 1: [0., 0.084941, 0.16993, ... ] . ' sin(x) ln(x)RETm r p 5RETV M $RET

Once again this calls for `V M * V R +`; a simple `*` does the
same thing.

1: 11.22 1: 1.122 1: 0.374 . . . * .1 * 3 /

Wow! That’s even better than the result from the Taylor series method.

#### 2.7.52 Rewrites Tutorial Exercise 1

We’ll use Big mode to make the formulas more readable.

___ V 2 + 2 1: (2 + sqrt(2)) / (1 + sqrt(2)) 1: --------- . ___ V 2 + 1 . ' (2+sqrt(2)) / (1+sqrt(2))RETd B

Multiplying by the conjugate helps because ‘`(a+b) (a-b) = a^2 - b^2`’.

___ ___ 1: (2 + V 2 ) (V 2 - 1) . a r a/(b+c) := a*(b-c) / (b^2-c^2)RET

___ 1: V 2 . a r a*(b+c) := a*b + a*c

(We could have used `a x` instead of a rewrite rule for the
second step.)

The multiply-by-conjugate rule turns out to be useful in many
different circumstances, such as when the denominator involves
sines and cosines or the imaginary constant `i`

.

#### 2.7.53 Rewrites Tutorial Exercise 2

Here is the rule set:

[ fib(n) := fib(n, 1, 1) :: integer(n) :: n >= 1, fib(1, x, y) := x, fib(n, x, y) := fib(n-1, y, x+y) ]

The first rule turns a one-argument `fib`

that people like to write
into a three-argument `fib`

that makes computation easier. The
second rule converts back from three-argument form once the computation
is done. The third rule does the computation itself. It basically
says that if ‘`x`’ and ‘`y`’ are two consecutive Fibonacci numbers,
then ‘`y`’ and ‘`x+y`’ are the next (overlapping) pair of Fibonacci
numbers.

Notice that because the number ‘`n`’ was “validated” by the
conditions on the first rule, there is no need to put conditions on
the other rules because the rule set would never get that far unless
the input were valid. That further speeds computation, since no
extra conditions need to be checked at every step.

Actually, a user with a nasty sense of humor could enter a bad
three-argument `fib`

call directly, say, ‘`fib(0, 1, 1)`’,
which would get the rules into an infinite loop. One thing that would
help keep this from happening by accident would be to use something like
‘`ZzFib`’ instead of `fib`

as the name of the three-argument
function.

#### 2.7.54 Rewrites Tutorial Exercise 3

He got an infinite loop. First, Calc did as expected and rewrote
‘`2 + 3 x`’ to ‘`f(2, 3, x)`’. Then it looked for ways to
apply the rule again, and found that ‘`f(2, 3, x)`’ looks like
‘`a + b x`’ with ‘`a = 0`’ and ‘`b = 1`’, so it rewrote to
‘`f(0, 1, f(2, 3, x))`’. It then wrapped another ‘`f(0, 1, ...)`’
around that, and so on, ad infinitum. Joe should have used `M-1 a r`
to make sure the rule applied only once.

(Actually, even the first step didn’t work as he expected. What Calc
really gives for `M-1 a r` in this situation is ‘`f(3 x, 1, 2)`’,
treating 2 as the “variable,” and ‘`3 x`’ as a constant being added
to it. While this may seem odd, it’s just as valid a solution as the
“obvious” one. One way to fix this would be to add the condition
‘`:: variable(x)`’ to the rule, to make sure the thing that matches
‘`x`’ is indeed a variable, or to change ‘`x`’ to ‘`quote(x)`’
on the lefthand side, so that the rule matches the actual variable
‘`x`’ rather than letting ‘`x`’ stand for something else.)

#### 2.7.55 Rewrites Tutorial Exercise 4

Here is a suitable set of rules to solve the first part of the problem:

[ seq(n, c) := seq(n/2, c+1) :: n%2 = 0, seq(n, c) := seq(3n+1, c+1) :: n%2 = 1 :: n > 1 ]

Given the initial formula ‘`seq(6, 0)`’, application of these
rules produces the following sequence of formulas:

seq( 3, 1) seq(10, 2) seq( 5, 3) seq(16, 4) seq( 8, 5) seq( 4, 6) seq( 2, 7) seq( 1, 8)

whereupon neither of the rules match, and rewriting stops.

We can pretty this up a bit with a couple more rules:

[ seq(n) := seq(n, 0), seq(1, c) := c, ... ]

Now, given ‘`seq(6)`’ as the starting configuration, we get 8
as the result.

The change to return a vector is quite simple:

[ seq(n) := seq(n, []) :: integer(n) :: n > 0, seq(1, v) := v | 1, seq(n, v) := seq(n/2, v | n) :: n%2 = 0, seq(n, v) := seq(3n+1, v | n) :: n%2 = 1 ]

Given ‘`seq(6)`’, the result is ‘`[6, 3, 10, 5, 16, 8, 4, 2, 1]`’.

Notice that the ‘`n > 1`’ guard is no longer necessary on the last
rule since the ‘`n = 1`’ case is now detected by another rule.
But a guard has been added to the initial rule to make sure the
initial value is suitable before the computation begins.

While still a good idea, this guard is not as vitally important as it
was for the `fib`

function, since calling, say, ‘`seq(x, [])`’
will not get into an infinite loop. Calc will not be able to prove
the symbol ‘`x`’ is either even or odd, so none of the rules will
apply and the rewrites will stop right away.

#### 2.7.56 Rewrites Tutorial Exercise 5

If ‘`x`’ is the sum ‘`a + b`’, then ‘`nterms(`

`x``)`

’ must
be ‘`nterms(`

`a``)`

’ plus ‘`nterms(`

`b``)`

’. If ‘`x`’
is not a sum, then ‘`nterms(`

`x``)`

’ = 1.

[ nterms(a + b) := nterms(a) + nterms(b), nterms(x) := 1 ]

Here we have taken advantage of the fact that earlier rules always
match before later rules; ‘`nterms(x)`’ will only be tried if we
already know that ‘`x`’ is not a sum.

#### 2.7.57 Rewrites Tutorial Exercise 6

Here is a rule set that will do the job:

[ a*(b + c) := a*b + a*c, opt(a) O(x^n) + opt(b) O(x^m) := O(x^n) :: n <= m :: constant(a) :: constant(b), opt(a) O(x^n) + opt(b) x^m := O(x^n) :: n <= m :: constant(a) :: constant(b), a O(x^n) := O(x^n) :: constant(a), x^opt(m) O(x^n) := O(x^(n+m)), O(x^n) O(x^m) := O(x^(n+m)) ]

If we really want the `+` and `*` keys to operate naturally
on power series, we should put these rules in `EvalRules`

. For
testing purposes, it is better to put them in a different variable,
say, `O`

, first.

The first rule just expands products of sums so that the rest of the
rules can assume they have an expanded-out polynomial to work with.
Note that this rule does not mention ‘`O`’ at all, so it will
apply to any product-of-sum it encounters—this rule may surprise
you if you put it into `EvalRules`

!

In the second rule, the sum of two O’s is changed to the smaller O.
The optional constant coefficients are there mostly so that
‘`O(x^2) - O(x^3)`’ and ‘`O(x^3) - O(x^2)`’ are handled
as well as ‘`O(x^2) + O(x^3)`’.

The third rule absorbs higher powers of ‘`x`’ into O’s.

The fourth rule says that a constant times a negligible quantity
is still negligible. (This rule will also match ‘`O(x^3) / 4`’,
with ‘`a = 1/4`’.)

The fifth rule rewrites, for example, ‘`x^2 O(x^3)`’ to ‘`O(x^5)`’.
(It is easy to see that if one of these forms is negligible, the other
is, too.) Notice the ‘`x^opt(m)`’ to pick up terms like
‘`x O(x^3)`’. Optional powers will match ‘`x`’ as ‘`x^1`’
but not 1 as ‘`x^0`’. This turns out to be exactly what we want here.

The sixth rule is the corresponding rule for products of two O’s.

Another way to solve this problem would be to create a new “data type”
that represents truncated power series. We might represent these as
function calls ‘`series( coefs, x)`’ where

`coefs`is a vector of coefficients for ‘

`x^0`’, ‘

`x^1`’, ‘

`x^2`’, and so on. Rules would exist for sums and products of such

`series`

objects, and as an optional convenience could also know how to combine a
`series`

object with a normal polynomial. (With this, and with a
rule that rewrites ‘`O(x^n)`’ to the equivalent

`series`

form,
you could still enter power series in exactly the same notation as
before.) Operations on such objects would probably be more efficient,
although the objects would be a bit harder to read.
Some other symbolic math programs provide a power series data type
similar to this. Mathematica, for example, has an object that looks
like ‘`PowerSeries[ x, x0, coefs, nmin,
nmax, den]`’, where

`x0`is the point about which the power series is taken (we’ve been assuming this was always zero), and

`nmin`,

`nmax`, and

`den`allow pseudo-power-series with fractional or negative powers. Also, the

`PowerSeries`

objects have a special display format that makes them look like
‘`2 x^2 + O(x^4)`’ when they are printed out. (See Compositions, for a way to do this in Calc, although for something as involved as this it would probably be better to write the formatting routine in Lisp.)

#### 2.7.58 Programming Tutorial Exercise 1

Just enter the formula ‘`ninteg(sin(t)/t, t, 0, x)`’, type
`Z F`, and answer the questions. Since this formula contains two
variables, the default argument list will be ‘`(t x)`’. We want to
change this to ‘`(x)`’ since ‘`t`’ is really a dummy variable
to be used within `ninteg`

.

The exact keystrokes are `Z F s Si RET RET C-b C-b DEL DEL RET y`.
(The

`C-b C-b`are what fix the argument list.)

`DEL``DEL`#### 2.7.59 Programming Tutorial Exercise 2

One way is to move the number to the top of the stack, operate on
it, then move it back: `C-x ( M- TAB n M-TAB M-TAB C-x )`.

Another way is to negate the top three stack entries, then negate
again the top two stack entries: `C-x ( M-3 n M-2 n C-x )`.

Finally, it turns out that a negative prefix argument causes a
command like `n` to operate on the specified stack entry only,
which is just what we want: `C-x ( M-- 3 n C-x )`.

Just for kicks, let’s also do it algebraically:
`C-x ( ' -$$$, $$, $ RET C-x )`.

#### 2.7.60 Programming Tutorial Exercise 3

Each of these functions can be computed using the stack, or using algebraic entry, whichever way you prefer:

Computing
‘`sin(x) / x`’:

Using the stack: `C-x ( RET S TAB / C-x )`.

Using algebraic entry: `C-x ( ' sin($)/$ RET C-x )`.

Computing the logarithm:

Using the stack: `C-x ( TAB B C-x )`

Using algebraic entry: `C-x ( ' log($,$$) RET C-x )`.

Computing the vector of integers:

Using the stack: `C-x ( 1 RET 1 C-u v x C-x )`. (Recall that

`C-u v x`takes the vector size, starting value, and increment from the stack.)

Alternatively: `C-x ( ~ v x C-x )`. (The `~` key pops a
number from the stack and uses it as the prefix argument for the
next command.)

Using algebraic entry: `C-x ( ' index($) RET C-x )`.

#### 2.7.61 Programming Tutorial Exercise 4

Here’s one way: `C-x ( RET V R + TAB v l / C-x )`.

#### 2.7.62 Programming Tutorial Exercise 5

2: 1 1: 1.61803398502 2: 1.61803398502 1: 20 . 1: 1.61803398875 . . 1RET20 Z < & 1 + Z > I H P

This answer is quite accurate.

#### 2.7.63 Programming Tutorial Exercise 6

Here is the matrix:

[ [ 0, 1 ] * [a, b] = [b, a + b] [ 1, 1 ] ]

Thus ‘`[0, 1; 1, 1]^n * [1, 1]`’ computes Fibonacci numbers ‘`n+1`’
and ‘`n+2`’. Here’s one program that does the job:

C-x ( ' [0, 1; 1, 1] ^ ($-1) * [1, 1]RETv uDELC-x )

This program is quite efficient because Calc knows how to raise a
matrix (or other value) to the power ‘`n`’ in only
‘`log(n,2)`’
steps. For example, this program can compute the 1000th Fibonacci
number (a 209-digit integer!) in about 10 steps; even though the
`Z < ... Z >` solution had much simpler steps, it would have
required so many steps that it would not have been practical.

#### 2.7.64 Programming Tutorial Exercise 7

The trick here is to compute the harmonic numbers differently, so that the loop counter itself accumulates the sum of reciprocals. We use a separate variable to hold the integer counter.

1: 1 2: 1 1: . . 1: 4 . 1 t 1 1RET4 Z ( t 2 r 1 1 + s 1 & Z )

The body of the loop goes as follows: First save the harmonic sum so far in variable 2. Then delete it from the stack; the for loop itself will take care of remembering it for us. Next, recall the count from variable 1, add one to it, and feed its reciprocal to the for loop to use as the step value. The for loop will increase the “loop counter” by that amount and keep going until the loop counter exceeds 4.

2: 31 3: 31 1: 3.99498713092 2: 3.99498713092 . 1: 4.02724519544 . r 1 r 2RET31 & +

Thus we find that the 30th harmonic number is 3.99, and the 31st harmonic number is 4.02.

#### 2.7.65 Programming Tutorial Exercise 8

The first step is to compute the derivative ‘`f'(x)`’ and thus
the formula
‘`x - f(x)/f'(x)`’.

(Because this definition is long, it will be repeated in concise form
below. You can use `C-x * m` to load it from there. While you are
entering a `Z ` Z '` body in a macro, Calc simply collects
keystrokes without executing them. In the following diagrams we’ll
pretend Calc actually executed the keystrokes as you typed them,
just for purposes of illustration.)

2: sin(cos(x)) - 0.5 3: 4.5 1: 4.5 2: sin(cos(x)) - 0.5 . 1: -(sin(x) cos(cos(x))) . ' sin(cos(x))-0.5RET4.5 m r C-x ( Z `TABRETa d xRET

2: 4.5 1: x + (sin(cos(x)) - 0.5) / sin(x) cos(cos(x)) . / ' xRETTAB- t 1

Now, we enter the loop. We’ll use a repeat loop with a 20-repetition
limit just in case the method fails to converge for some reason.
(Normally, the `Z /` command will stop the loop before all 20
repetitions are done.)

1: 4.5 3: 4.5 2: 4.5 . 2: x + (sin(cos(x)) ... 1: 5.24196456928 1: 4.5 . . 20 Z <RETr 1TABs l xRET

This is the new guess for ‘`x`’. Now we compare it with the
old one to see if we’ve converged.

3: 5.24196 2: 5.24196 1: 5.24196 1: 5.26345856348 2: 5.24196 1: 0 . . 1: 4.5 . .RETM-TABa = Z / Z > Z ' C-x )

The loop converges in just a few steps to this value. To check the result, we can simply substitute it back into the equation.

2: 5.26345856348 1: 0.499999999997 .RET' sin(cos($))RET

Let’s test the new definition again:

2: x^2 - 9 1: 3. 1: 1 . . ' x^2-9RET1 X

Once again, here’s the full Newton’s Method definition:

C-x ( Z `TABRETa d xRET/ ' xRETTAB- t 1 20 Z <RETr 1TABs l xRETRETM-TABa = Z / Z > Z ' C-x )

It turns out that Calc has a built-in command for applying a formula repeatedly until it converges to a number. See Nesting and Fixed Points, to see how to use it.

Also, of course, `a R` is a built-in command that uses Newton’s
method (among others) to look for numerical solutions to any equation.
See Root Finding.

#### 2.7.66 Programming Tutorial Exercise 9

The first step is to adjust ‘`z`’ to be greater than 5. A simple
“for” loop will do the job here. If ‘`z`’ is less than 5, we
reduce the problem using
‘`psi(z) = psi(z+1) - 1/z`’. We go
on to compute
‘`psi(z+1)`’,
and remember to add back a factor of ‘`-1/z`’ when we’re done. This
step is repeated until ‘`z > 5`’.

(Because this definition is long, it will be repeated in concise form
below. You can use `C-x * m` to load it from there. While you are
entering a `Z ` Z '` body in a macro, Calc simply collects
keystrokes without executing them. In the following diagrams we’ll
pretend Calc actually executed the keystrokes as you typed them,
just for purposes of illustration.)

1: 1. 1: 1. . . 1.0RETC-x ( Z ` s 1 0 t 2

Here, variable 1 holds ‘`z`’ and variable 2 holds the adjustment
factor. If ‘`z < 5`’, we use a loop to increase it.

(By the way, we started with ‘`1.0`’ instead of the integer 1 because
otherwise the calculation below will try to do exact fractional arithmetic,
and will never converge because fractions compare equal only if they
are exactly equal, not just equal to within the current precision.)

3: 1. 2: 1. 1: 6. 2: 1. 1: 1 . 1: 5 . .RET5 a < Z [ 5 Z ( & s + 2 1 s + 1 1 Z ) r 1 Z ]

Now we compute the initial part of the sum:
‘`ln(z) - 1/2z`’
minus the adjustment factor.

2: 1.79175946923 2: 1.7084261359 1: -0.57490719743 1: 0.0833333333333 1: 2.28333333333 . . . L r 1 2 * & - r 2 -

Now we evaluate the series. We’ll use another “for” loop counting
up the value of ‘`2 n`’. (Calc does have a summation command,
`a +`, but we’ll use loops just to get more practice with them.)

3: -0.5749 3: -0.5749 4: -0.5749 2: -0.5749 2: 2 2: 1:6 3: 1:6 1: 2.3148e-3 1: 40 1: 2 2: 2 . . . 1: 36. . 2RET40 Z (RETk bTABRETr 1TAB^ * /

3: -0.5749 3: -0.5772 2: -0.5772 1: -0.577215664892 2: -0.5749 2: -0.5772 1: 0 . 1: 2.3148e-3 1: -0.5749 . . .TABRETM-TAB-RETM-TABa = Z / 2 Z ) Z ' C-x )

This is the value of
‘`- gamma`’,
with a slight bit of roundoff error. To get a full 12 digits, let’s use
a higher precision:

2: -0.577215664892 2: -0.577215664892 1: 1. 1: -0.577215664901532 1.RETp 16RETX

Here’s the complete sequence of keystrokes:

C-x ( Z ` s 1 0 t 2RET5 a < Z [ 5 Z ( & s + 2 1 s + 1 1 Z ) r 1 Z ] L r 1 2 * & - r 2 - 2RET40 Z (RETk bTABRETr 1TAB^ * /TABRETM-TAB-RETM-TABa = Z / 2 Z ) Z ' C-x )

#### 2.7.67 Programming Tutorial Exercise 10

Taking the derivative of a term of the form ‘`x^n`’ will produce
a term like
‘`n x^(n-1)`’.
Taking the derivative of a constant
produces zero. From this it is easy to see that the ‘`n`’th
derivative of a polynomial, evaluated at ‘`x = 0`’, will equal the
coefficient on the ‘`x^n`’ term times ‘`n!`’.

(Because this definition is long, it will be repeated in concise form
below. You can use `C-x * m` to load it from there. While you are
entering a `Z ` Z '` body in a macro, Calc simply collects
keystrokes without executing them. In the following diagrams we’ll
pretend Calc actually executed the keystrokes as you typed them,
just for purposes of illustration.)

2: 5 x^4 + (x + 1)^2 3: 5 x^4 + (x + 1)^2 1: 6 2: 0 . 1: 6 . ' 5 x^4 + (x+1)^2RET6 C-x ( Z ` [ ] t 1 0TAB

Variable 1 will accumulate the vector of coefficients.

2: 0 3: 0 2: 5 x^4 + ... 1: 5 x^4 + ... 2: 5 x^4 + ... 1: 1 . 1: 1 . . Z (TABRET0 s l xRETM-TAB! / s | 1

Note that `s | 1` appends the top-of-stack value to the vector
in a variable; it is completely analogous to `s + 1`. We could
have written instead, `r 1 TAB | t 1`.

1: 20 x^3 + 2 x + 2 1: 0 1: [1, 2, 1, 0, 5, 0, 0] . . . a d xRET1 Z )DELr 1 Z ' C-x )

To convert back, a simple method is just to map the coefficients
against a table of powers of ‘`x`’.

2: [1, 2, 1, 0, 5, 0, 0] 2: [1, 2, 1, 0, 5, 0, 0] 1: 6 1: [0, 1, 2, 3, 4, 5, 6] . . 6RET1 + 0RET1 C-u v x

2: [1, 2, 1, 0, 5, 0, 0] 2: 1 + 2 x + x^2 + 5 x^4 1: [1, x, x^2, x^3, ... ] . . ' xRETTABV M ^ *

Once again, here are the whole polynomial to/from vector programs:

C-x ( Z ` [ ] t 1 0TABZ (TABRET0 s l xRETM-TAB! / s | 1 a d xRET1 Z ) r 1 Z ' C-x ) C-x ( 1 + 0RET1 C-u v x ' xRETTABV M ^ * C-x )

#### 2.7.68 Programming Tutorial Exercise 11

First we define a dummy program to go on the `z s` key. The true
`z s` key is supposed to take two numbers from the stack and
return one number, so `DEL` as a dummy definition will make
sure the stack comes out right.

2: 4 1: 4 2: 4 1: 2 . 1: 2 . . 4RET2 C-x (DELC-x ) Z K sRET2

The last step replaces the 2 that was eaten during the creation
of the dummy `z s` command. Now we move on to the real
definition. The recurrence needs to be rewritten slightly,
to the form ‘`s(n,m) = s(n-1,m-1) - (n-1) s(n-1,m)`’.

(Because this definition is long, it will be repeated in concise form
below. You can use `C-x * m` to load it from there.)

2: 4 4: 4 3: 4 2: 4 1: 2 3: 2 2: 2 1: 2 . 2: 4 1: 0 . 1: 2 . . C-x ( M-2RETa = Z [DELDEL1 Z :

4: 4 2: 4 2: 3 4: 3 4: 3 3: 3 3: 2 1: 2 1: 2 3: 2 3: 2 2: 2 2: 2 . . 2: 3 2: 3 1: 3 1: 0 1: 2 1: 1 . . . .RET0 a = Z [DELDEL0 Z :TAB1 -TABM-2RET1 - z s

(Note that the value 3 that our dummy `z s` produces is not correct;
it is merely a placeholder that will do just as well for now.)

3: 3 4: 3 3: 3 2: 3 1: -6 2: 3 3: 3 2: 3 1: 9 . 1: 2 2: 3 1: 3 . . 1: 2 . . M-TABM-TABTABRETM-TABz s * -

1: -6 2: 4 1: 11 2: 11 . 1: 2 . 1: 11 . . Z ] Z ] C-x ) Z K sRETDEL4RET2 z s M-RETk s

Even though the result that we got during the definition was highly
bogus, once the definition is complete the `z s` command gets
the right answers.

Here’s the full program once again:

C-x ( M-2RETa = Z [DELDEL1 Z :RET0 a = Z [DELDEL0 Z :TAB1 -TABM-2RET1 - z s M-TABM-TABTABRETM-TABz s * - Z ] Z ] C-x )

You can read this definition using `C-x * m` (`read-kbd-macro`

)
followed by `Z K s`, without having to make a dummy definition
first, because `read-kbd-macro`

doesn’t need to execute the
definition as it reads it in. For this reason, `C-x * m`

is often
the easiest way to create recursive programs in Calc.

#### 2.7.69 Programming Tutorial Exercise 12

This turns out to be a much easier way to solve the problem. Let’s
denote Stirling numbers as calls of the function ‘`s`’.

First, we store the rewrite rules corresponding to the definition of Stirling numbers in a convenient variable:

s e StirlingRulesRET[ s(n,n) := 1 :: n >= 0, s(n,0) := 0 :: n > 0, s(n,m) := s(n-1,m-1) - (n-1) s(n-1,m) :: n >= m :: m >= 1 ] C-c C-c

Now, it’s just a matter of applying the rules:

2: 4 1: s(4, 2) 1: 11 1: 2 . . . 4RET2 C-x ( ' s($$,$)RETa r StirlingRulesRETC-x )

As in the case of the `fib`

rules, it would be useful to put these
rules in `EvalRules`

and to add a ‘`:: remember`’ condition to
the last rule.

## 3 Introduction

This chapter is the beginning of the Calc reference manual. It covers basic concepts such as the stack, algebraic and numeric entry, undo, numeric prefix arguments, etc.

- Basic Commands
- Help Commands
- Stack Basics
- Numeric Entry
- Algebraic Entry
- “Quick Calculator” Mode
- Numeric Prefix Arguments
- Undoing Mistakes
- Error Messages
- Multiple Calculators
- Troubleshooting Commands

### 3.1 Basic Commands

To start the Calculator in its standard interface, type `M-x calc`.
By default this creates a pair of small windows, `*Calculator*`
and `*Calc Trail*`. The former displays the contents of the
Calculator stack and is manipulated exclusively through Calc commands.
It is possible (though not usually necessary) to create several Calc
mode buffers each of which has an independent stack, undo list, and
mode settings. There is exactly one Calc Trail buffer; it records a
list of the results of all calculations that have been done. The
Calc Trail buffer uses a variant of Calc mode, so Calculator commands
still work when the trail buffer’s window is selected. It is possible
to turn the trail window off, but the `*Calc Trail*` buffer itself
still exists and is updated silently. See Trail Commands.

In most installations, the `C-x * c` key sequence is a more
convenient way to start the Calculator. Also, `C-x * *`
is a synonym for `C-x * c` unless you last used Calc
in its Keypad mode.

Most Calc commands use one or two keystrokes. Lower- and upper-case
letters are distinct. Commands may also be entered in full `M-x` form;
for some commands this is the only form. As a convenience, the `x`
key (`calc-execute-extended-command`

)
is like `M-x` except that it enters the initial string ‘`calc-`’
for you. For example, the following key sequences are equivalent:
`S`, `M-x calc-sin RET`,

`x sin`.

`RET`Although Calc is designed to be used from the keyboard, some of Calc’s more common commands are available from a menu. In the menu, the arguments to the functions are given by referring to their stack level numbers.

The Calculator exists in many parts. When you type `C-x * c`, the
Emacs “auto-load” mechanism will bring in only the first part, which
contains the basic arithmetic functions. The other parts will be
auto-loaded the first time you use the more advanced commands like trig
functions or matrix operations. This is done to improve the response time
of the Calculator in the common case when all you need to do is a
little arithmetic. If for some reason the Calculator fails to load an
extension module automatically, you can force it to load all the
extensions by using the `C-x * L` (`calc-load-everything`

)
command. See Mode Settings.

If you type `M-x calc` or `C-x * c` with any numeric prefix argument,
the Calculator is loaded if necessary, but it is not actually started.
If the argument is positive, the `calc-ext` extensions are also
loaded if necessary. User-written Lisp code that wishes to make use
of Calc’s arithmetic routines can use ‘`(calc 0)`’ or ‘`(calc 1)`’
to auto-load the Calculator.

If you type `C-x * b`, then next time you use `C-x * c` you
will get a Calculator that uses the full height of the Emacs screen.
When full-screen mode is on, `C-x * c` runs the `full-calc`

command instead of `calc`

. From the Unix shell you can type
‘`emacs -f full-calc`’ to start a new Emacs specifically for use
as a calculator. When Calc is started from the Emacs command line
like this, Calc’s normal “quit” commands actually quit Emacs itself.

The `C-x * o` command is like `C-x * c` except that the Calc
window is not actually selected. If you are already in the Calc
window, `C-x * o` switches you out of it. (The regular Emacs
`C-x o` command would also work for this, but it has a
tendency to drop you into the Calc Trail window instead, which
`C-x * o` takes care not to do.)

For one quick calculation, you can type `C-x * q` (`quick-calc`

)
which prompts you for a formula (like ‘`2+3/4`’). The result is
displayed at the bottom of the Emacs screen without ever creating
any special Calculator windows. See “Quick Calculator” Mode.

Finally, if you are using the X window system you may want to try
`C-x * k` (`calc-keypad`

) which runs Calc with a
“calculator keypad” picture as well as a stack display. Click on
the keys with the mouse to operate the calculator. See Keypad Mode.

The `q` key (`calc-quit`

) exits Calc mode and closes the
Calculator’s window(s). It does not delete the Calculator buffers.
If you type `M-x calc` again, the Calculator will reappear with the
contents of the stack intact. Typing `C-x * c` or `C-x * *`
again from inside the Calculator buffer is equivalent to executing
`calc-quit`

; you can think of `C-x * *` as toggling the
Calculator on and off.

The `C-x * x` command also turns the Calculator off, no matter which
user interface (standard, Keypad, or Embedded) is currently active.
It also cancels `calc-edit`

mode if used from there.

The `d SPC` key sequence (

`calc-refresh`

) redraws the contents
of the Calculator buffer from memory. Use this if the contents of the
buffer have been damaged somehow.
The `o` key (`calc-realign`

) moves the cursor back to its
“home” position at the bottom of the Calculator buffer.

The `<` and `>` keys are bound to `calc-scroll-left`

and
`calc-scroll-right`

. These are just like the normal horizontal
scrolling commands except that they scroll one half-screen at a time by
default. (Calc formats its output to fit within the bounds of the
window whenever it can.)

The `{` and `}` keys are bound to `calc-scroll-down`

and `calc-scroll-up`

. They scroll up or down by one-half the
height of the Calc window.

The `C-x * 0` command (`calc-reset`

; that’s `C-x *` followed
by a zero) resets the Calculator to its initial state. This clears
the stack, resets all the modes to their initial values (the values
that were saved with `m m` (`calc-save-modes`

)), clears the
caches (see Caches), and so on. (It does *not* erase the
values of any variables.) With an argument of 0, Calc will be reset to
its default state; namely, the modes will be given their default values.
With a positive prefix argument, `C-x * 0` preserves the contents of
the stack but resets everything else to its initial state; with a
negative prefix argument, `C-x * 0` preserves the contents of the
stack but resets everything else to its default state.

### 3.2 Help Commands

The `?` key (`calc-help`

) displays a series of brief help messages.
Some keys (such as `b` and `d`) are prefix keys, like Emacs’s
`ESC` and `C-x` prefixes. You can type
`?` after a prefix to see a list of commands beginning with that
prefix. (If the message includes ‘`[MORE]`’, press `?` again
to see additional commands for that prefix.)

The `h h` (`calc-full-help`

) command displays all the `?`
responses at once. When printed, this makes a nice, compact (three pages)
summary of Calc keystrokes.

In general, the `h` key prefix introduces various commands that
provide help within Calc. Many of the `h` key functions are
Calc-specific analogues to the `C-h` functions for Emacs help.

The `h i` (`calc-info`

) command runs the Emacs Info system
to read this manual on-line. This is basically the same as typing
`C-h i` (the regular way to run the Info system), then, if Info
is not already in the Calc manual, selecting the beginning of the
manual. The `C-x * i` command is another way to read the Calc
manual; it is different from `h i` in that it works any time,
not just inside Calc. The plain `i` key is also equivalent to
`h i`, though this key is obsolete and may be replaced with a
different command in a future version of Calc.

The `h t` (`calc-tutorial`

) command runs the Info system on
the Tutorial section of the Calc manual. It is like `h i`,
except that it selects the starting node of the tutorial rather
than the beginning of the whole manual. (It actually selects the
node “Interactive Tutorial” which tells a few things about
using the Info system before going on to the actual tutorial.)
The `C-x * t` key is equivalent to `h t` (but it works at
all times).

The `h s` (`calc-info-summary`

) command runs the Info system
on the Summary node of the Calc manual. See Calc Summary. The `C-x * s`
key is equivalent to `h s`.

The `h k` (`calc-describe-key`

) command looks up a key
sequence in the Calc manual. For example, `h k H a S` looks
up the documentation on the `H a S` (`calc-solve-for`

)
command. This works by looking up the textual description of
the key(s) in the Key Index of the manual, then jumping to the
node indicated by the index.

Most Calc commands do not have traditional Emacs documentation
strings, since the `h k` command is both more convenient and
more instructive. This means the regular Emacs `C-h k`
(`describe-key`

) command will not be useful for Calc keystrokes.

The `h c` (`calc-describe-key-briefly`

) command reads a
key sequence and displays a brief one-line description of it at
the bottom of the screen. It looks for the key sequence in the
Summary node of the Calc manual; if it doesn’t find the sequence
there, it acts just like its regular Emacs counterpart `C-h c`
(`describe-key-briefly`

). For example, `h c H a S`
gives the description:

H a S runs calc-solve-for: a `H a S' v => fsolve(a,v) (?=notes)

which means the command `H a S` or `H M-x calc-solve-for`
takes a value ‘`a`’ from the stack, prompts for a value ‘`v`’,
then applies the algebraic function `fsolve`

to these values.
The ‘`?=notes`’ message means you can now type `?` to see
additional notes from the summary that apply to this command.

The `h f` (`calc-describe-function`

) command looks up an
algebraic function or a command name in the Calc manual. Enter an
algebraic function name to look up that function in the Function
Index or enter a command name beginning with ‘`calc-`’ to look it
up in the Command Index. This command will also look up operator
symbols that can appear in algebraic formulas, like ‘`%`’ and
‘`=>`’.

The `h v` (`calc-describe-variable`

) command looks up a
variable in the Calc manual. Enter a variable name like `pi`

or
`PlotRejects`

.

The `h b` (`calc-describe-bindings`

) command is just like
`C-h b`, except that only local (Calc-related) key bindings are
listed.

The `h n` or `h C-n` (`calc-view-news`

) command displays
the “news” or change history of Emacs, and jumps to the most recent
portion concerning Calc (if present). For older history, see the file
`etc/CALC-NEWS` in the Emacs distribution.

The `h C-c`, `h C-d`, and `h C-w` keys display copying,
distribution, and warranty information about Calc. These work by
pulling up the appropriate parts of the “Copying” or “Reporting
Bugs” sections of the manual.

### 3.3 Stack Basics

Calc uses RPN notation. If you are not familiar with RPN, see RPN Calculations and the Stack.

To add the numbers 1 and 2 in Calc you would type the keys:
`1 RET 2 +`.
(

`RET`corresponds to the

`ENTER`key on most calculators.) The first three keystrokes “push” the numbers 1 and 2 onto the stack. The

`+`key always “pops” the top two numbers from the stack, adds them, and pushes the result (3) back onto the stack. This number is ready for further calculations:

`5 -`pushes 5 onto the stack, then pops the 3 and 5, subtracts them, and pushes the result (

*-2*).

Note that the “top” of the stack actually appears at the *bottom*
of the buffer. A line containing a single ‘`.`’ character signifies
the end of the buffer; Calculator commands operate on the number(s)
directly above this line. The `d t` (`calc-truncate-stack`

)
command allows you to move the ‘`.`’ marker up and down in the stack;
see Truncating the Stack.

Stack elements are numbered consecutively, with number 1 being the top of
the stack. These line numbers are ordinarily displayed on the lefthand side
of the window. The `d l` (`calc-line-numbering`

) command controls
whether these numbers appear. (Line numbers may be turned off since they
slow the Calculator down a bit and also clutter the display.)

The unshifted letter `o` (`calc-realign`

) command repositions
the cursor to its top-of-stack “home” position. It also undoes any
horizontal scrolling in the window. If you give it a numeric prefix
argument, it instead moves the cursor to the specified stack element.

The `RET` (or equivalent `SPC`) key is only required to separate
two consecutive numbers.
(After all, if you typed `1 2` by themselves the Calculator
would enter the number 12.) If you press `RET` or `SPC` *not*
right after typing a number, the key duplicates the number on the top of
the stack. ` RET *` is thus a handy way to square a number.

The `DEL` key pops and throws away the top number on the stack.
The `TAB` key swaps the top two objects on the stack.
See Stack and Trail Commands, for descriptions of these and other stack-related
commands.

### 3.4 Numeric Entry

Pressing a digit or other numeric key begins numeric entry using the
minibuffer. The number is pushed on the stack when you press the `RET`
or `SPC` keys. If you press any other non-numeric key, the number is
pushed onto the stack and the appropriate operation is performed. If
you press a numeric key which is not valid, the key is ignored.

There are three different concepts corresponding to the word “minus,”
typified by ‘`a-b`’ (subtraction), ‘`-x`’
(change-sign), and ‘`-5`’ (negative number). Calc uses three
different keys for these operations, respectively:
`-`, `n`, and `_` (the underscore). The `-` key subtracts
the two numbers on the top of the stack. The `n` key changes the sign
of the number on the top of the stack or the number currently being entered.
The `_` key begins entry of a negative number or changes the sign of
the number currently being entered. The following sequences all enter the
number *-5* onto the stack: `0 RET 5 -`,

`5 n`,

`RET``5`,

`RET`n`_ 5`,

`RET``5 _`.

`RET`Some other keys are active during numeric entry, such as `#` for
non-decimal numbers, `:` for fractions, and `@` for HMS forms.
These notations are described later in this manual with the corresponding
data types. See Data Types.

During numeric entry, the only editing key available is `DEL`.

### 3.5 Algebraic Entry

The `'` (`calc-algebraic-entry`

) command can be used to enter
calculations in algebraic form. This is accomplished by typing the
apostrophe key, ’, followed by the expression in standard format:

' 2+3*4RET.

This will compute
‘`2+(3*4) = 14`’
and push it on the stack. If you wish you can
ignore the RPN aspect of Calc altogether and simply enter algebraic
expressions in this way. You may want to use `DEL` every so often to
clear previous results off the stack.

You can press the apostrophe key during normal numeric entry to switch the half-entered number into Algebraic entry mode. One reason to do this would be to fix a typo, as the full Emacs cursor motion and editing keys are available during algebraic entry but not during numeric entry.

In the same vein, during either numeric or algebraic entry you can
press ``` (grave accent) to switch to `calc-edit`

mode, where
you complete your half-finished entry in a separate buffer.
See Editing Stack Entries.

If you prefer algebraic entry, you can use the command `m a`
(`calc-algebraic-mode`

) to set Algebraic mode. In this mode,
digits and other keys that would normally start numeric entry instead
start full algebraic entry; as long as your formula begins with a digit
you can omit the apostrophe. Open parentheses and square brackets also
begin algebraic entry. You can still do RPN calculations in this mode,
but you will have to press `RET` to terminate every number:
`2 RET 3 RET * 4 RET +` would accomplish the same
thing as

`2*3+4`.

`RET`If you give a numeric prefix argument like `C-u` to the `m a`
command, it enables Incomplete Algebraic mode; this is like regular
Algebraic mode except that it applies to the `(` and `[` keys
only. Numeric keys still begin a numeric entry in this mode.

The `m t` (`calc-total-algebraic-mode`

) gives you an even
stronger algebraic-entry mode, in which *all* regular letter and
punctuation keys begin algebraic entry. Use this if you prefer typing
`sqrt( )` instead of `Q`, `factor( )` instead of
`a f`, and so on. To type regular Calc commands when you are in
Total Algebraic mode, hold down the `META` key. Thus `M-q`
is the command to quit Calc, `M-p` sets the precision, and
`M-m t` (or `M-m M-t`, if you prefer) turns Total Algebraic
mode back off again. Meta keys also terminate algebraic entry, so
that `2+3 M-S` is equivalent to `2+3 RET M-S`. The symbol
‘

`Alg*`’ will appear in the mode line whenever you are in this mode.

Pressing `'` (the apostrophe) a second time re-enters the previous
algebraic formula. You can then use the normal Emacs editing keys to
modify this formula to your liking before pressing `RET`.

Within a formula entered from the keyboard, the symbol `$`
represents the number on the top of the stack. If an entered formula
contains any `$` characters, the Calculator replaces the top of
stack with that formula rather than simply pushing the formula onto the
stack. Thus, `' 1+2 RET` pushes 3 on the stack, and

`$*2`replaces it with 6. Note that the

`RET``$`key always initiates algebraic entry; the

`'`is unnecessary if

`$`is the first character in the new formula.

Higher stack elements can be accessed from an entered formula with the
symbols `$$`, `$$$`, and so on. The number of stack elements
removed (to be replaced by the entered values) equals the number of dollar
signs in the longest such symbol in the formula. For example, ‘`$$+$$$`’
adds the second and third stack elements, replacing the top three elements
with the answer. (All information about the top stack element is thus lost
since no single ‘`$`’ appears in this formula.)

A slightly different way to refer to stack elements is with a dollar
sign followed by a number: ‘`$1`’, ‘`$2`’, and so on are much
like ‘`$`’, ‘`$$`’, etc., except that stack entries referred
to numerically are not replaced by the algebraic entry. That is, while
‘`$+1`’ replaces 5 on the stack with 6, ‘`$1+1`’ leaves the 5
on the stack and pushes an additional 6.

If a sequence of formulas are entered separated by commas, each formula
is pushed onto the stack in turn. For example, ‘`1,2,3`’ pushes
those three numbers onto the stack (leaving the 3 at the top), and
‘`$+1,$-1`’ replaces a 5 on the stack with 4 followed by 6. Also,
‘`$,$$`’ exchanges the top two elements of the stack, just like the
`TAB` key.

You can finish an algebraic entry with `M-=` or `M- RET` instead
of

`RET`. This uses

`=`to evaluate the variables in each formula that goes onto the stack. (Thus

`' pi`pushes the variable ‘

`RET``pi`’, but

`' pi M-`pushes 3.1415.)

`RET`If you finish your algebraic entry by pressing `LFD` (or `C-j`)
instead of `RET`, Calc disables simplification
(as if by `m O`; see Simplification Modes) while the entry
is being pushed on the stack. Thus `' 1+2 RET` pushes 3
on the stack, but

`' 1+2`pushes the formula ‘

`LFD``1+2`’; you might then press

`=`when it is time to evaluate this formula.

### 3.6 “Quick Calculator” Mode

There is another way to invoke the Calculator if all you need to do
is make one or two quick calculations. Type `C-x * q` (or
`M-x quick-calc`), then type any formula as an algebraic entry.
The Calculator will compute the result and display it in the echo
area, without ever actually putting up a Calc window.

You can use the `$` character in a Quick Calculator formula to
refer to the previous Quick Calculator result. Older results are
not retained; the Quick Calculator has no effect on the full
Calculator’s stack or trail. If you compute a result and then
forget what it was, just run `C-x * q`

again and enter
‘`$`’ as the formula.

If this is the first time you have used the Calculator in this Emacs
session, the `C-x * q` command will create the `*Calculator*`
buffer and perform all the usual initializations; it simply will
refrain from putting that buffer up in a new window. The Quick
Calculator refers to the `*Calculator*` buffer for all mode
settings. Thus, for example, to set the precision that the Quick
Calculator uses, simply run the full Calculator momentarily and use
the regular `p` command.

If you use `C-x * q`

from inside the Calculator buffer, the
effect is the same as pressing the apostrophe key (algebraic entry).

The result of a Quick calculation is placed in the Emacs “kill ring”
as well as being displayed. A subsequent `C-y` command will
yank the result into the editing buffer. You can also use this
to yank the result into the next `C-x * q` input line as a more
explicit alternative to `$` notation, or to yank the result
into the Calculator stack after typing `C-x * c`.

If you give a prefix argument to `C-x * q` or finish your formula
by typing `LFD` (or `C-j`) instead of `RET`, the result is
inserted immediately into the current buffer rather than going into
the kill ring.

Quick Calculator results are actually evaluated as if by the `=`
key (which replaces variable names by their stored values, if any).
If the formula you enter is an assignment to a variable using the
‘`:=`’ operator, say, ‘`foo := 2 + 3`’ or ‘`foo := foo + 1`’,
then the result of the evaluation is stored in that Calc variable.
See Storing and Recalling.

If the result is an integer and the current display radix is decimal, the number will also be displayed in hex, octal and binary formats. If the integer is in the range from 1 to 126, it will also be displayed as an ASCII character.

For example, the quoted character ‘`"x"`’ produces the vector
result ‘`[120]`’ (because 120 is the ASCII code of the lower-case
“x”; see Strings). Since this is a vector, not an integer, it
is displayed only according to the current mode settings. But
running Quick Calc again and entering ‘`120`’ will produce the
result ‘`120 (16#78, 8#170, x)`’ which shows the number in its
decimal, hexadecimal, octal, and ASCII forms.

Please note that the Quick Calculator is not any faster at loading or computing the answer than the full Calculator; the name “quick” merely refers to the fact that it’s much less hassle to use for small calculations.

### 3.7 Numeric Prefix Arguments

Many Calculator commands use numeric prefix arguments. Some, such as
`d s` (`calc-sci-notation`

), set a parameter to the value of
the prefix argument or use a default if you don’t use a prefix.
Others (like `d f` (`calc-fix-notation`

)) require an argument
and prompt for a number if you don’t give one as a prefix.

As a rule, stack-manipulation commands accept a numeric prefix argument
which is interpreted as an index into the stack. A positive argument
operates on the top `n` stack entries; a negative argument operates
on the `n`th stack entry in isolation; and a zero argument operates
on the entire stack.

Most commands that perform computations (such as the arithmetic and
scientific functions) accept a numeric prefix argument that allows the
operation to be applied across many stack elements. For unary operations
(that is, functions of one argument like absolute value or complex
conjugate), a positive prefix argument applies that function to the top
`n` stack entries simultaneously, and a negative argument applies it
to the `n`th stack entry only. For binary operations (functions of
two arguments like addition, GCD, and vector concatenation), a positive
prefix argument “reduces” the function across the top `n`
stack elements (for example, `C-u 5 +` sums the top 5 stack entries;
see Reducing and Mapping Vectors), and a negative argument maps the next-to-top
`n` stack elements with the top stack element as a second argument
(for example, `7 c-u -5 +` adds 7 to the top 5 stack elements).
This feature is not available for operations which use the numeric prefix
argument for some other purpose.

Numeric prefixes are specified the same way as always in Emacs: Press
a sequence of `META`-digits, or press `ESC` followed by digits,
or press `C-u` followed by digits. Some commands treat plain
`C-u` (without any actual digits) specially.

You can type `~` (`calc-num-prefix`

) to pop an integer from the
top of the stack and enter it as the numeric prefix for the next command.
For example, `C-u 16 p` sets the precision to 16 digits; an alternate
(silly) way to do this would be `2 RET 4 ^ ~ p`, i.e., compute 2
to the fourth power and set the precision to that value.

Conversely, if you have typed a numeric prefix argument the `~` key
pushes it onto the stack in the form of an integer.

### 3.8 Undoing Mistakes

The shift-`U` key (`calc-undo`

) undoes the most recent operation.
If that operation added or dropped objects from the stack, those objects
are removed or restored. If it was a “store” operation, you are
queried whether or not to restore the variable to its original value.
The `U` key may be pressed any number of times to undo successively
farther back in time; with a numeric prefix argument it undoes a
specified number of operations. When the Calculator is quit, as with
the `q` (`calc-quit`

) command, the undo history will be
truncated to the length of the customizable variable
`calc-undo-length`

(see Customizing Calc), which by default
is ‘`100`’. (Recall that `C-x * c` is synonymous with
`calc-quit`

while inside the Calculator; this also truncates the
undo history.)

Currently the mode-setting commands (like `calc-precision`

) are not
undoable. You can undo past a point where you changed a mode, but you
will need to reset the mode yourself.

The shift-`D` key (`calc-redo`

) redoes an operation that was
mistakenly undone. Pressing `U` with a negative prefix argument is
equivalent to executing `calc-redo`

. You can redo any number of
times, up to the number of recent consecutive undo commands. Redo
information is cleared whenever you give any command that adds new undo
information, i.e., if you undo, then enter a number on the stack or make
any other change, then it will be too late to redo.

The `M- RET` key (

`calc-last-args`

) is like undo in that
it restores the arguments of the most recent command onto the stack;
however, it does not remove the result of that command. Given a numeric
prefix argument, this command applies to the ‘`n`’th most recent command which removed items from the stack; it pushes those items back onto the stack.

The `K` (`calc-keep-args`

) command provides a related function
to `M- RET`. See Stack and Trail Commands.

It is also possible to recall previous results or inputs using the trail. See Trail Commands.

The standard Emacs `C-_` undo key is recognized as a synonym for `U`.

### 3.9 Error Messages

Many situations that would produce an error message in other calculators
simply create unsimplified formulas in the Emacs Calculator. For example,
`1 RET 0 /` pushes the formula ‘

`1 / 0`’;

`0 L`pushes the formula ‘

`ln(0)`’. Floating-point overflow and underflow are also reasons for this to happen.

When a function call must be left in symbolic form, Calc usually
produces a message explaining why. Messages that are probably
surprising or indicative of user errors are displayed automatically.
Other messages are simply kept in Calc’s memory and are displayed only
if you type `w` (`calc-why`

). You can also press `w` if
the same computation results in several messages. (The first message
will end with ‘`[w=more]`’ in this case.)

The `d w` (`calc-auto-why`

) command controls when error messages
are displayed automatically. (Calc effectively presses `w` for you
after your computation finishes.) By default, this occurs only for
“important” messages. The other possible modes are to report
*all* messages automatically, or to report none automatically (so
that you must always press `w` yourself to see the messages).

### 3.10 Multiple Calculators

It is possible to have any number of Calc mode buffers at once.
Usually this is done by executing `M-x another-calc`, which
is similar to `C-x * c` except that if a `*Calculator*`
buffer already exists, a new, independent one with a name of the
form `*Calculator*< n>` is created. You can also use the
command

`calc-mode`

to put any buffer into Calculator mode, but
this would ordinarily never be done.
The `q` (`calc-quit`

) command does not destroy a Calculator buffer;
it only closes its window. Use `M-x kill-buffer` to destroy a
Calculator buffer.

Each Calculator buffer keeps its own stack, undo list, and mode settings
such as precision, angular mode, and display formats. In Emacs terms,
variables such as `calc-stack`

are buffer-local variables. The
global default values of these variables are used only when a new
Calculator buffer is created. The `calc-quit`

command saves
the stack and mode settings of the buffer being quit as the new defaults.

There is only one trail buffer, `*Calc Trail*`, used by all
Calculator buffers.

### 3.11 Troubleshooting Commands

This section describes commands you can use in case a computation incorrectly fails or gives the wrong answer.

See Reporting Bugs, if you find a problem that appears to be due to a bug or deficiency in Calc.

#### 3.11.1 Autoloading Problems

The Calc program is split into many component files; components are loaded automatically as you use various commands that require them. Occasionally Calc may lose track of when a certain component is necessary; typically this means you will type a command and it won’t work because some function you’ve never heard of was undefined.

If this happens, the easiest workaround is to type `C-x * L`
(`calc-load-everything`

) to force all the parts of Calc to be
loaded right away. This will cause Emacs to take up a lot more
memory than it would otherwise, but it’s guaranteed to fix the problem.

#### 3.11.2 Recursion Depth

Calc uses recursion in many of its calculations. Emacs Lisp keeps a
variable `max-lisp-eval-depth`

which limits the amount of recursion
possible in an attempt to recover from program bugs. If a calculation
ever halts incorrectly with the message “Computation got stuck or
ran too long,” use the `M` command (`calc-more-recursion-depth`

)
to increase this limit. (Of course, this will not help if the
calculation really did get stuck due to some problem inside Calc.)

The limit is always increased (multiplied) by a factor of two. There
is also an `I M` (`calc-less-recursion-depth`

) command which
decreases this limit by a factor of two, down to a minimum value of 200.
The default value is 1000.

#### 3.11.3 Caches

Calc saves certain values after they have been computed once. For
example, the `P` (`calc-pi`

) command initially “knows” the
constant ‘`pi`’ to about 20 decimal places; if the current precision
is greater than this, it will recompute ‘`pi`’ using a series
approximation. This value will not need to be recomputed ever again
unless you raise the precision still further. Many operations such as
logarithms and sines make use of similarly cached values such as
‘`pi/4`’ and
‘`ln(2)`’.
The visible effect of caching is that
high-precision computations may seem to do extra work the first time.
Other things cached include powers of two (for the binary arithmetic
functions), matrix inverses and determinants, symbolic integrals, and
data points computed by the graphing commands.

If you suspect a Calculator cache has become corrupt, you can use the
`calc-flush-caches`

command to reset all caches to the empty state.
(This should only be necessary in the event of bugs in the Calculator.)
The `C-x * 0` (with the zero key) command also resets caches along
with all other aspects of the Calculator’s state.

#### 3.11.4 Debugging Calc

A few commands exist to help in the debugging of Calc commands. See Programming, to see the various ways that you can write your own Calc commands.

The `Z T` (`calc-timing`

) command turns on and off a mode
in which the timing of slow commands is reported in the Trail.
Any Calc command that takes two seconds or longer writes a line
to the Trail showing how many seconds it took. This value is
accurate only to within one second.

All steps of executing a command are included; in particular, time
taken to format the result for display in the stack and trail is
counted. Some prompts also count time taken waiting for them to
be answered, while others do not; this depends on the exact
implementation of the command. For best results, if you are timing
a sequence that includes prompts or multiple commands, define a
keyboard macro to run the whole sequence at once. Calc’s `X`
command (see Programming with Keyboard Macros) will then report the time taken
to execute the whole macro.

Another advantage of the `X` command is that while it is
executing, the stack and trail are not updated from step to step.
So if you expect the output of your test sequence to leave a result
that may take a long time to format and you don’t wish to count
this formatting time, end your sequence with a `DEL` keystroke
to clear the result from the stack. When you run the sequence with
`X`, Calc will never bother to format the large result.

Another thing `Z T` does is to increase the Emacs variable
`gc-cons-threshold`

to a much higher value (two million; the
usual default in Calc is 250,000) for the duration of each command.
This generally prevents garbage collection during the timing of
the command, though it may cause your Emacs process to grow
abnormally large. (Garbage collection time is a major unpredictable
factor in the timing of Emacs operations.)

Another command that is useful when debugging your own Lisp
extensions to Calc is `M-x calc-pass-errors`, which disables
the error handler that changes the “`max-lisp-eval-depth`

exceeded” message to the much more friendly “Computation got
stuck or ran too long.” This handler interferes with the Emacs
Lisp debugger’s `debug-on-error`

mode. Errors are reported
in the handler itself rather than at the true location of the
error. After you have executed `calc-pass-errors`

, Lisp
errors will be reported correctly but the user-friendly message
will be lost.

## 4 Data Types

This chapter discusses the various types of objects that can be placed on the Calculator stack, how they are displayed, and how they are entered. (See Data Type Formats, for information on how these data types are represented as underlying Lisp objects.)

Integers, fractions, and floats are various ways of describing real numbers. HMS forms also for many purposes act as real numbers. These types can be combined to form complex numbers, modulo forms, error forms, or interval forms. (But these last four types cannot be combined arbitrarily: error forms may not contain modulo forms, for example.) Finally, all these types of numbers may be combined into vectors, matrices, or algebraic formulas.

- Integers
- Fractions
- Floats
- Complex Numbers
- Infinities
- Vectors and Matrices
- Strings
- HMS Forms
- Date Forms
- Modulo Forms
- Error Forms
- Interval Forms
- Incomplete Objects
- Variables
- Formulas

### 4.1 Integers

The Calculator stores integers to arbitrary precision. Addition, subtraction, and multiplication of integers always yields an exact integer result. (If the result of a division or exponentiation of integers is not an integer, it is expressed in fractional or floating-point form according to the current Fraction mode. See Fraction Mode.)

A decimal integer is represented as an optional sign followed by a sequence of digits. Grouping (see Grouping Digits) can be used to insert a comma at every third digit for display purposes, but you must not type commas during the entry of numbers.

A non-decimal integer is represented as an optional sign, a radix
between 2 and 36, a ‘`#`’ symbol, and one or more digits. For radix 11
and above, the letters A through Z (upper- or lower-case) count as
digits and do not terminate numeric entry mode. See Radix Modes, for how
to set the default radix for display of integers. Numbers of any radix
may be entered at any time. If you press `#` at the beginning of a
number, the current display radix is used.

### 4.2 Fractions

A *fraction* is a ratio of two integers. Fractions are traditionally
written “2/3” but Calc uses the notation ‘`2:3`’. (The `/` key
performs RPN division; the following two sequences push the number
‘`2:3`’ on the stack: `2 : 3 RET`, or

`2`assuming Fraction mode has been enabled.) When the Calculator produces a fractional result it always reduces it to simplest form, which may in fact be an integer.

`RET`3 /Fractions may also be entered in a three-part form, where ‘`2:3:4`’
represents two-and-three-quarters. See Fraction Formats, for fraction
display formats.

Non-decimal fractions are entered and displayed as
‘` radix#num:denom`’ (or in the analogous three-part
form). The numerator and denominator always use the same radix.

### 4.3 Floats

A floating-point number or *float* is a number stored in scientific
notation. The number of significant digits in the fractional part is
governed by the current floating precision (see Precision). The
range of acceptable values is from
‘`10^-3999999`’
(inclusive) to
‘`10^4000000`’
(exclusive), plus the corresponding negative values and zero.

Calculations that would exceed the allowable range of values (such
as ‘`exp(exp(20))`’) are left in symbolic form by Calc. The
messages “floating-point overflow” or “floating-point underflow”
indicate that during the calculation a number would have been produced
that was too large or too close to zero, respectively, to be represented
by Calc. This does not necessarily mean the final result would have
overflowed, just that an overflow occurred while computing the result.
(In fact, it could report an underflow even though the final result
would have overflowed!)

If a rational number and a float are mixed in a calculation, the result
will in general be expressed as a float. Commands that require an integer
value (such as `k g` [`gcd`

]) will also accept integer-valued
floats, i.e., floating-point numbers with nothing after the decimal point.

Floats are identified by the presence of a decimal point and/or an
exponent. In general a float consists of an optional sign, digits
including an optional decimal point, and an optional exponent consisting
of an ‘`e`’, an optional sign, and up to seven exponent digits.
For example, ‘`23.5e-2`’ is 23.5 times ten to the minus-second power,
or 0.235.

Floating-point numbers are normally displayed in decimal notation with all significant figures shown. Exceedingly large or small numbers are displayed in scientific notation. Various other display options are available. See Float Formats.

Floating-point numbers are stored in decimal, not binary. The result of each operation is rounded to the nearest value representable in the number of significant digits specified by the current precision, rounding away from zero in the case of a tie. Thus (in the default display mode) what you see is exactly what you get. Some operations such as square roots and transcendental functions are performed with several digits of extra precision and then rounded down, in an effort to make the final result accurate to the full requested precision. However, accuracy is not rigorously guaranteed. If you suspect the validity of a result, try doing the same calculation in a higher precision. The Calculator’s arithmetic is not intended to be IEEE-conformant in any way.

While floats are always *stored* in decimal, they can be entered
and displayed in any radix just like integers and fractions. Since a
float that is entered in a radix other that 10 will be converted to
decimal, the number that Calc stores may not be exactly the number that
was entered, it will be the closest decimal approximation given the
current precision. The notation ‘` radix#ddd.ddd`’
is a floating-point number whose digits are in the specified radix.
Note that the ‘

`.`’ is more aptly referred to as a “radix point” than as a decimal point in this case. The number ‘

`8#123.4567`’ is defined as ‘

`8#1234567 * 8^-4`’. If the radix is 14 or less, you can use ‘

`e`’ notation to write a non-decimal number in scientific notation. The exponent is written in decimal, and is considered to be a power of the radix: ‘

`8#1234567e-4`’. If the radix is 15 or above, the letter ‘

`e`’ is a digit, so scientific notation must be written out, e.g., ‘

`16#123.4567*16^2`’. The first two exercises of the Modes Tutorial explore some of the properties of non-decimal floats.

### 4.4 Complex Numbers

There are two supported formats for complex numbers: rectangular and
polar. The default format is rectangular, displayed in the form
‘`( real,imag)`’ where

`real`is the real part and

`imag`is the imaginary part, each of which may be any real number. Rectangular complex numbers can also be displayed in ‘

`’ notation; see Complex Formats.`

`a`+`b`iPolar complex numbers are displayed in the form
‘`(`

`r``;`

`theta``)`

’
where `r` is the nonnegative magnitude and
`theta`
is the argument or phase angle. The range of
`theta`
depends on the current angular mode (see Angular Modes); it is
generally between *-180* and *+180* degrees or the equivalent range
in radians.

Complex numbers are entered in stages using incomplete objects. See Incomplete Objects.

Operations on rectangular complex numbers yield rectangular complex
results, and similarly for polar complex numbers. Where the two types
are mixed, or where new complex numbers arise (as for the square root of
a negative real), the current *Polar mode* is used to determine the
type. See Polar Mode.

A complex result in which the imaginary part is zero (or the phase angle
is 0 or 180 degrees or ‘`pi`’ radians) is automatically converted to a real
number.

### 4.5 Infinities

The word `inf`

represents the mathematical concept of *infinity*.
Calc actually has three slightly different infinity-like values:
`inf`

, `uinf`

, and `nan`

. These are just regular
variable names (see Variables); you should avoid using these
names for your own variables because Calc gives them special
treatment. Infinities, like all variable names, are normally
entered using algebraic entry.

Mathematically speaking, it is not rigorously correct to treat
“infinity” as if it were a number, but mathematicians often do
so informally. When they say that ‘`1 / inf = 0`’, what they
really mean is that ‘`1 / x`’, as ‘`x`’ becomes larger and
larger, becomes arbitrarily close to zero. So you can imagine
that if ‘`x`’ got “all the way to infinity,” then ‘`1 / x`’
would go all the way to zero. Similarly, when they say that
‘`exp(inf) = inf`’, they mean that
‘`exp(x)`’
grows without bound as ‘`x`’ grows. The symbol ‘`-inf`’ likewise
stands for an infinitely negative real value; for example, we say that
‘`exp(-inf) = 0`’. You can have an infinity pointing in any
direction on the complex plane: ‘`sqrt(-inf) = i inf`’.

The same concept of limits can be used to define ‘`1 / 0`’. We
really want the value that ‘`1 / x`’ approaches as ‘`x`’
approaches zero. But if all we have is ‘`1 / 0`’, we can’t
tell which direction ‘`x`’ was coming from. If ‘`x`’ was
positive and decreasing toward zero, then we should say that
‘`1 / 0 = inf`’. But if ‘`x`’ was negative and increasing
toward zero, the answer is ‘`1 / 0 = -inf`’. In fact, ‘`x`’
could be an imaginary number, giving the answer ‘`i inf`’ or
‘`-i inf`’. Calc uses the special symbol ‘`uinf`’ to mean
*undirected infinity*, i.e., a value which is infinitely
large but with an unknown sign (or direction on the complex plane).

Calc actually has three modes that say how infinities are handled.
Normally, infinities never arise from calculations that didn’t
already have them. Thus, ‘`1 / 0`’ is treated simply as an
error and left unevaluated. The `m i` (`calc-infinite-mode`

)
command (see Infinite Mode) enables a mode in which
‘`1 / 0`’ evaluates to `uinf`

instead. There is also
an alternative type of infinite mode which says to treat zeros
as if they were positive, so that ‘`1 / 0 = inf`’. While this
is less mathematically correct, it may be the answer you want in
some cases.

Since all infinities are “as large” as all others, Calc simplifies,
e.g., ‘`5 inf`’ to ‘`inf`’. Another example is
‘`5 - inf = -inf`’, where the ‘`-inf`’ is so large that
adding a finite number like five to it does not affect it.
Note that ‘`a - inf`’ also results in ‘`-inf`’; Calc assumes
that variables like `a`

always stand for finite quantities.
Just to show that infinities really are all the same size,
note that ‘`sqrt(inf) = inf^2 = exp(inf) = inf`’ in Calc’s
notation.

It’s not so easy to define certain formulas like ‘`0 * inf`’ and
‘`inf / inf`’. Depending on where these zeros and infinities
came from, the answer could be literally anything. The latter
formula could be the limit of ‘`x / x`’ (giving a result of one),
or ‘`2 x / x`’ (giving two), or ‘`x^2 / x`’ (giving `inf`

),
or ‘`x / x^2`’ (giving zero). Calc uses the symbol `nan`

to represent such an *indeterminate* value. (The name “nan”
comes from analogy with the “NAN” concept of IEEE standard
arithmetic; it stands for “Not A Number.” This is somewhat of a
misnomer, since `nan`

*does* stand for some number or
infinity, it’s just that *which* number it stands for
cannot be determined.) In Calc’s notation, ‘`0 * inf = nan`’
and ‘`inf / inf = nan`’. A few other common indeterminate
expressions are ‘`inf - inf`’ and ‘`inf ^ 0`’. Also,
‘`0 / 0 = nan`’ if you have turned on Infinite mode
(as described above).

Infinities are especially useful as parts of *intervals*.
See Interval Forms.

### 4.6 Vectors and Matrices

The *vector* data type is flexible and general. A vector is simply a
list of zero or more data objects. When these objects are numbers, the
whole is a vector in the mathematical sense. When these objects are
themselves vectors of equal (nonzero) length, the whole is a *matrix*.
A vector which is not a matrix is referred to here as a *plain vector*.

A vector is displayed as a list of values separated by commas and enclosed
in square brackets: ‘`[1, 2, 3]`’. Thus the following is a 2 row by
3 column matrix: ‘`[[1, 2, 3], [4, 5, 6]]`’. Vectors, like complex
numbers, are entered as incomplete objects. See Incomplete Objects.
During algebraic entry, vectors are entered all at once in the usual
brackets-and-commas form. Matrices may be entered algebraically as nested
vectors, or using the shortcut notation ‘`[1, 2, 3; 4, 5, 6]`’,
with rows separated by semicolons. The commas may usually be omitted
when entering vectors: ‘`[1 2 3]`’. Curly braces may be used in
place of brackets: ‘`{1, 2, 3}`’, but the commas are required in
this case.

Traditional vector and matrix arithmetic is also supported; see Basic Arithmetic and see Vector/Matrix Functions. Many other operations are applied to vectors element-wise. For example, the complex conjugate of a vector is a vector of the complex conjugates of its elements.

Algebraic functions for building vectors include ‘`vec(a, b, c)`’
to build ‘`[a, b, c]`’, ‘`cvec(a, n, m)`’ to build an
`n`x`m`
matrix of ‘`a`’s, and ‘`index(n)`’ to build a vector of integers
from 1 to ‘`n`’.

### 4.7 Strings

Character strings are not a special data type in the Calculator.
Rather, a string is represented simply as a vector all of whose
elements are integers in the range 0 to 255 (ASCII codes). You can
enter a string at any time by pressing the `"` key. Quotation
marks and backslashes are written ‘`\"`’ and ‘`\\`’, respectively,
inside strings. Other notations introduced by backslashes are:

\a 7 \^@ 0 \b 8 \^a-z 1-26 \e 27 \^[ 27 \f 12 \^\\ 28 \n 10 \^] 29 \r 13 \^^ 30 \t 9 \^_ 31 \^? 127

Finally, a backslash followed by three octal digits produces any character from its ASCII code.

Strings are normally displayed in vector-of-integers form. The
`d "` (`calc-display-strings`

) command toggles a mode in
which any vectors of small integers are displayed as quoted strings
instead.

The backslash notations shown above are also used for displaying
strings. Characters 128 and above are not translated by Calc; unless
you have an Emacs modified for 8-bit fonts, these will show up in
backslash-octal-digits notation. For characters below 32, and
for character 127, Calc uses the backslash-letter combination if
there is one, or otherwise uses a ‘`\^`’ sequence.

The only Calc feature that uses strings is *compositions*;
see Compositions. Strings also provide a convenient
way to do conversions between ASCII characters and integers.

There is a `string`

function which provides a different display
format for strings. Basically, ‘`string( s)`’, where

`s`is a vector of integers in the proper range, is displayed as the corresponding string of characters with no surrounding quotation marks or other modifications. Thus ‘

`string("ABC")`’ (or ‘

`string([65 66 67])`’) will look like ‘

`ABC`’ on the stack. This happens regardless of whether

`d "`has been used. The only way to turn it off is to use

`d U`(unformatted language mode) which will display ‘

`string("ABC")`’ instead.

Control characters are displayed somewhat differently by `string`

.
Characters below 32, and character 127, are shown using ‘`^`’ notation
(same as shown above, but without the backslash). The quote and
backslash characters are left alone, as are characters 128 and above.

The `bstring`

function is just like `string`

except that
the resulting string is breakable across multiple lines if it doesn’t
fit all on one line. Potential break points occur at every space
character in the string.

### 4.8 HMS Forms

*HMS* stands for Hours-Minutes-Seconds; when used as an angular
argument, the interpretation is Degrees-Minutes-Seconds. All functions
that operate on angles accept HMS forms. These are interpreted as
degrees regardless of the current angular mode. It is also possible to
use HMS as the angular mode so that calculated angles are expressed in
degrees, minutes, and seconds.

The default format for HMS values is
‘` hours@ mins' secs"`’. During entry, the letters
‘

`h`’ (for “hours”) or ‘

`o`’ (approximating the “degrees” symbol) are accepted as well as ‘

`@`’, ‘

`m`’ is accepted in place of ‘

`'`’, and ‘

`s`’ is accepted in place of ‘

`"`’. The

`hours`value is an integer (or integer-valued float). The

`mins`value is an integer or integer-valued float between 0 and 59. The

`secs`value is a real number between 0 (inclusive) and 60 (exclusive). A positive HMS form is interpreted as

`hours`+

`mins`/60 +

`secs`/3600. A negative HMS form is interpreted as

*-*

`hours`*-*

`mins`/60

*-*

`secs`/3600. Display format for HMS forms is quite flexible. See HMS Formats.

HMS forms can be added and subtracted. When they are added to numbers, the numbers are interpreted according to the current angular mode. HMS forms can also be multiplied and divided by real numbers. Dividing two HMS forms produces a real-valued ratio of the two angles.

Just for kicks, `M-x calc-time` pushes the current time of day on
the stack as an HMS form.

### 4.9 Date Forms

A *date form* represents a date and possibly an associated time.
Simple date arithmetic is supported: Adding a number to a date
produces a new date shifted by that many days; adding an HMS form to
a date shifts it by that many hours. Subtracting two date forms
computes the number of days between them (represented as a simple
number). Many other operations, such as multiplying two date forms,
are nonsensical and are not allowed by Calc.

Date forms are entered and displayed enclosed in ‘`< >`’ brackets.
The default format is, e.g., ‘`<Wed Jan 9, 1991>`’ for dates,
or ‘`<3:32:20pm Wed Jan 9, 1991>`’ for dates with times.
Input is flexible; date forms can be entered in any of the usual
notations for dates and times. See Date Formats.

Date forms are stored internally as numbers, specifically the number
of days since midnight on the morning of December 31 of the year 1 BC.
If the internal number is an integer, the form represents a date only;
if the internal number is a fraction or float, the form represents
a date and time. For example, ‘`<6:00am Thu Jan 10, 1991>`’
is represented by the number 726842.25. The standard precision of
12 decimal digits is enough to ensure that a (reasonable) date and
time can be stored without roundoff error.

If the current precision is greater than 12, date forms will keep additional digits in the seconds position. For example, if the precision is 15, the seconds will keep three digits after the decimal point. Decreasing the precision below 12 may cause the time part of a date form to become inaccurate. This can also happen if astronomically high years are used, though this will not be an issue in everyday (or even everymillennium) use. Note that date forms without times are stored as exact integers, so roundoff is never an issue for them.

You can use the `v p` (`calc-pack`

) and `v u`
(`calc-unpack`

) commands to get at the numerical representation
of a date form. See Packing and Unpacking.

Date forms can go arbitrarily far into the future or past. Negative
year numbers represent years BC. There is no “year 0”; the day
before ‘`<Mon Jan 1, +1>`’ is ‘`<Sun Dec 31, -1>`’. These are
days 1 and 0 respectively in Calc’s internal numbering scheme. The
Gregorian calendar is used for all dates, including dates before the
Gregorian calendar was invented (although that can be configured; see
below). Thus Calc’s use of the day number *-10000* to
represent August 15, 28 BC should be taken with a grain of salt.

Some historical background: The Julian calendar was created by Julius Caesar in the year 46 BC as an attempt to fix the confusion caused by the irregular Roman calendar that was used before that time. The Julian calendar introduced an extra day in all years divisible by four. After some initial confusion, the calendar was adopted around the year we call 8 AD. Some centuries later it became apparent that the Julian year of 365.25 days was itself not quite right. In 1582 Pope Gregory XIII introduced the Gregorian calendar, which added the new rule that years divisible by 100, but not by 400, were not to be considered leap years despite being divisible by four. Many countries delayed adoption of the Gregorian calendar because of religious differences. For example, Great Britain and the British colonies switched to the Gregorian calendar in September 1752, when the Julian calendar was eleven days behind the Gregorian calendar. That year in Britain, the day after September 2 was September 14. To take another example, Russia did not adopt the Gregorian calendar until 1918, and that year in Russia the day after January 31 was February 14. Calc’s reckoning therefore matches English practice starting in 1752 and Russian practice starting in 1918, but disagrees with earlier dates in both countries.

When the Julian calendar was introduced, it had January 1 as the first day of the year. By the Middle Ages, many European countries had changed the beginning of a new year to a different date, often to a religious festival. Almost all countries reverted to using January 1 as the beginning of the year by the time they adopted the Gregorian calendar.

Some calendars attempt to mimic the historical situation by using the
Gregorian calendar for recent dates and the Julian calendar for older
dates. The `cal`

program in most Unix implementations does this,
for example. While January 1 wasn’t always the beginning of a calendar
year, these hybrid calendars still use January 1 as the beginning of
the year even for older dates. The customizable variable
`calc-gregorian-switch`

(see Customizing Calc) can be set to
have Calc’s date forms switch from the Julian to Gregorian calendar at
any specified date.

Today’s timekeepers introduce an occasional “leap second”.
These do not occur regularly and Calc does not take these minor
effects into account. (If it did, it would have to report a
non-integer number of days between, say,
‘`<12:00am Mon Jan 1, 1900>`’ and
‘`<12:00am Sat Jan 1, 2000>`’.)

Another day counting system in common use is, confusingly, also called
“Julian.” Julian days go from noon to noon. The Julian day number
is the numbers of days since 12:00 noon (GMT) on November 24, 4714 BC
in the Gregorian calendar (i.e., January 1, 4713 BC in the Julian
calendar). In Calc’s scheme (in GMT) the Julian day origin is
*-1721424.5*, because Calc starts at midnight instead of noon.
Thus to convert a Calc date code obtained by unpacking a
date form into a Julian day number, simply add 1721424.5 after
compensating for the time zone difference. The built-in `t J`
command performs this conversion for you.

The Julian day number is based on the Julian cycle, which was invented in 1583 by Joseph Justus Scaliger. Scaliger named it the Julian cycle since it involves the Julian calendar, but some have suggested that Scaliger named it in honor of his father, Julius Caesar Scaliger. The Julian cycle is based on three other cycles: the indiction cycle, the Metonic cycle, and the solar cycle. The indiction cycle is a 15 year cycle originally used by the Romans for tax purposes but later used to date medieval documents. The Metonic cycle is a 19 year cycle; 19 years is close to being a common multiple of a solar year and a lunar month, and so every 19 years the phases of the moon will occur on the same days of the year. The solar cycle is a 28 year cycle; the Julian calendar repeats itself every 28 years. The smallest time period which contains multiples of all three cycles is the least common multiple of 15 years, 19 years and 28 years, which (since they’re pairwise relatively prime) is 15*19*28 = 7980 years. This is the length of a Julian cycle. Working backwards, the previous year in which all three cycles began was 4713 BC, and so Scaliger chose that year as the beginning of a Julian cycle. Since at the time there were no historical records from before 4713 BC, using this year as a starting point had the advantage of avoiding negative year numbers. In 1849, the astronomer John Herschel (son of William Herschel) suggested using the number of days since the beginning of the Julian cycle as an astronomical dating system; this idea was taken up by other astronomers. (At the time, noon was the start of the astronomical day. Herschel originally suggested counting the days since Jan 1, 4713 BC at noon Alexandria time; this was later amended to noon GMT.) Julian day numbering is largely used in astronomy.

The Unix operating system measures time as an integer number of
seconds since midnight, Jan 1, 1970. To convert a Calc date
value into a Unix time stamp, first subtract 719163 (the code
for ‘`<Jan 1, 1970>`’), then multiply by 86400 (the number of
seconds in a day) and press `R` to round to the nearest
integer. If you have a date form, you can simply subtract the
day ‘`<Jan 1, 1970>`’ instead of unpacking and subtracting
719163. Likewise, divide by 86400 and add ‘`<Jan 1, 1970>`’
to convert from Unix time to a Calc date form. (Note that
Unix normally maintains the time in the GMT time zone; you may
need to subtract five hours to get New York time, or eight hours
for California time. The same is usually true of Julian day
counts.) The built-in `t U` command performs these
conversions.

### 4.10 Modulo Forms

A *modulo form* is a real number which is taken modulo (i.e., within
an integer multiple of) some value `M`. Arithmetic modulo `M`
often arises in number theory. Modulo forms are written
‘`a` `mod`

`M`’,
where `a` and `M` are real numbers or HMS forms, and
‘`0 <= a < M`’.
In many applications ‘

`a`’ and ‘

`M`’ will be integers but this is not required.

To create a modulo form during numeric entry, press the shift-`M`
key to enter the word ‘`mod`’. As a special convenience, pressing
shift-`M` a second time automatically enters the value of ‘`M`’
that was most recently used before. During algebraic entry, either
type ‘`mod`’ by hand or press `M-m` (that’s ` META-m`).
Once again, pressing this a second time enters the current modulo.

Modulo forms are not to be confused with the modulo operator ‘`%`’.
The expression ‘`27 % 10`’ means to compute 27 modulo 10 to produce
the result 7. Further computations treat this 7 as just a regular integer.
The expression ‘`27 mod 10`’ produces the result ‘`7 mod 10`’;
further computations with this value are again reduced modulo 10 so that
the result always lies in the desired range.

When two modulo forms with identical ‘`M`’’s are added or multiplied,
the Calculator simply adds or multiplies the values, then reduces modulo
‘`M`’. If one argument is a modulo form and the other a plain number,
the plain number is treated like a compatible modulo form. It is also
possible to raise modulo forms to powers; the result is the value raised
to the power, then reduced modulo ‘`M`’. (When all values involved
are integers, this calculation is done much more efficiently than
actually computing the power and then reducing.)

Two modulo forms ‘`a` `mod`

`M`’ and ‘`b` `mod`

`M`’
can be divided if ‘`a`’, ‘`b`’, and ‘`M`’ are all
integers. The result is the modulo form which, when multiplied by
‘`b` `mod`

`M`’, produces ‘`a` `mod`

`M`’. If
there is no solution to this equation (which can happen only when
‘`M`’ is non-prime), or if any of the arguments are non-integers, the
division is left in symbolic form. Other operations, such as square
roots, are not yet supported for modulo forms. (Note that, although
‘`(`

`a` `mod`

`M``)^.5`

’ will compute a “modulo square root”
in the sense of reducing
‘`sqrt(a)`’
modulo ‘`M`’, this is not a useful definition from the
number-theoretical point of view.)

It is possible to mix HMS forms and modulo forms. For example, an
HMS form modulo 24 could be used to manipulate clock times; an HMS
form modulo 360 would be suitable for angles. Making the modulo ‘`M`’
also be an HMS form eliminates troubles that would arise if the angular
mode were inadvertently set to Radians, in which case
‘`2@ 0' 0" mod 24`’ would be interpreted as two degrees modulo
24 radians!

Modulo forms cannot have variables or formulas for components. If you
enter the formula ‘`(x + 2) mod 5`’, Calc propagates the modulus
to each of the coefficients: ‘`(1 mod 5) x + (2 mod 5)`’.

You can use `v p` and `%` to modify modulo forms.
See Packing and Unpacking. See Basic Arithmetic.

The algebraic function ‘`makemod(a, m)`’ builds the modulo form
‘`a mod m`’.

### 4.11 Error Forms

An *error form* is a number with an associated standard
deviation, as in ‘`2.3 +/- 0.12`’. The notation
‘`x` `+/-`

sigma’
stands for an uncertain value which follows
a normal or Gaussian distribution of mean ‘`x`’ and standard
deviation or “error”
‘`sigma`’.
Both the mean and the error can be either numbers or
formulas. Generally these are real numbers but the mean may also be
complex. If the error is negative or complex, it is changed to its
absolute value. An error form with zero error is converted to a
regular number by the Calculator.

All arithmetic and transcendental functions accept error forms as input.
Operations on the mean-value part work just like operations on regular
numbers. The error part for any function ‘`f(x)`’ (such as
‘`sin(x)`’)
is defined by the error of ‘`x`’ times the derivative of ‘`f`’
evaluated at the mean value of ‘`x`’. For a two-argument function
‘`f(x,y)`’ (such as addition) the error is the square root of the sum
of the squares of the errors due to ‘`x`’ and ‘`y`’.
Note that this
definition assumes the errors in ‘`x`’ and ‘`y`’ are uncorrelated.
A side effect of this definition is that ‘`(2 +/- 1) * (2 +/- 1)`’
is not the same as ‘`(2 +/- 1)^2`’; the former represents the product
of two independent values which happen to have the same probability
distributions, and the latter is the product of one random value with itself.
The former will produce an answer with less error, since on the average
the two independent errors can be expected to cancel out.

Consult a good text on error analysis for a discussion of the proper use
of standard deviations. Actual errors often are neither Gaussian-distributed
nor uncorrelated, and the above formulas are valid only when errors
are small. As an example, the error arising from
‘`sin(`

`x` `+/-`

`sigma``)`

’
is
‘`sigma` `abs(cos(`

`x``))`

’.
When ‘`x`’ is close to zero,
‘`cos(x)`’
is close to one so the error in the sine is close to
‘`sigma`’;
this makes sense, since
‘`sin(x)`’
is approximately ‘`x`’ near zero, so a given error in ‘`x`’ will
produce about the same error in the sine. Likewise, near 90 degrees
‘`cos(x)`’
is nearly zero and so the computed error is
small: The sine curve is nearly flat in that region, so an error in ‘`x`’
has relatively little effect on the value of
‘`sin(x)`’.
However, consider ‘`sin(90 +/- 1000)`’. The cosine of 90 is zero, so
Calc will report zero error! We get an obviously wrong result because
we have violated the small-error approximation underlying the error
analysis. If the error in ‘`x`’ had been small, the error in
‘`sin(x)`’
would indeed have been negligible.

To enter an error form during regular numeric entry, use the `p`
(“plus-or-minus”) key to type the ‘`+/-`’ symbol. (If you try actually
typing ‘`+/-`’ the `+` key will be interpreted as the Calculator’s
`+` command!) Within an algebraic formula, you can press `M-+` to
type the ‘`+/-`’ symbol, or type it out by hand.

Error forms and complex numbers can be mixed; the formulas shown above are used for complex numbers, too; note that if the error part evaluates to a complex number its absolute value (or the square root of the sum of the squares of the absolute values of the two error contributions) is used. Mathematically, this corresponds to a radially symmetric Gaussian distribution of numbers on the complex plane. However, note that Calc considers an error form with real components to represent a real number, not a complex distribution around a real mean.

Error forms may also be composed of HMS forms. For best results, both the mean and the error should be HMS forms if either one is.

The algebraic function ‘`sdev(a, b)`’ builds the error form ‘`a +/- b`’.

### 4.12 Interval Forms

An *interval* is a subset of consecutive real numbers. For example,
the interval ‘`[2 .. 4]`’ represents all the numbers from 2 to 4,
inclusive. If you multiply it by the interval ‘`[0.5 .. 2]`’ you
obtain ‘`[1 .. 8]`’. This calculation represents the fact that if
you multiply some number in the range ‘`[2 .. 4]`’ by some other
number in the range ‘`[0.5 .. 2]`’, your result will lie in the range
from 1 to 8. Interval arithmetic is used to get a worst-case estimate
of the possible range of values a computation will produce, given the
set of possible values of the input.

Calc supports several varieties of intervals, including *closed*
intervals of the type shown above, *open* intervals such as
‘`(2 .. 4)`’, which represents the range of numbers from 2 to 4
*exclusive*, and *semi-open* intervals in which one end
uses a round parenthesis and the other a square bracket. In mathematical
terms,
‘`[2 .. 4]`’ means ‘`2 <= x <= 4`’, whereas
‘`[2 .. 4)`’ represents ‘`2 <= x < 4`’,
‘`(2 .. 4]`’ represents ‘`2 < x <= 4`’, and
‘`(2 .. 4)`’ represents ‘`2 < x < 4`’.

The lower and upper limits of an interval must be either real numbers
(or HMS or date forms), or symbolic expressions which are assumed to be
real-valued, or ‘`-inf`’ and ‘`inf`’. In general the lower limit
must be less than the upper limit. A closed interval containing only
one value, ‘`[3 .. 3]`’, is converted to a plain number (3)
automatically. An interval containing no values at all (such as
‘`[3 .. 2]`’ or ‘`[2 .. 2)`’) can be represented but is not
guaranteed to behave well when used in arithmetic. Note that the
interval ‘`[3 .. inf)`’ represents all real numbers greater than
or equal to 3, and ‘`(-inf .. inf)`’ represents all real numbers.
In fact, ‘`[-inf .. inf]`’ represents all real numbers including
the real infinities.

Intervals are entered in the notation shown here, either as algebraic
formulas, or using incomplete forms. (See Incomplete Objects.)
In algebraic formulas, multiple periods in a row are collected from
left to right, so that ‘`1...1e2`’ is interpreted as ‘`1.0 .. 1e2`’
rather than ‘`1 .. 0.1e2`’. Add spaces or zeros if you want to
get the other interpretation. If you omit the lower or upper limit,
a default of ‘`-inf`’ or ‘`inf`’ (respectively) is furnished.

Infinite mode also affects operations on intervals
(see Infinities). Calc will always introduce an open infinity,
as in ‘`1 / (0 .. 2] = [0.5 .. inf)`’. But closed infinities,
‘`1 / [0 .. 2] = [0.5 .. inf]`’, arise only in Infinite mode;
otherwise they are left unevaluated. Note that the “direction” of
a zero is not an issue in this case since the zero is always assumed
to be continuous with the rest of the interval. For intervals that
contain zero inside them Calc is forced to give the result,
‘`1 / (-2 .. 2) = [-inf .. inf]`’.

While it may seem that intervals and error forms are similar, they are
based on entirely different concepts of inexact quantities. An error
form
‘`x` `+/-`

`sigma`’
means a variable is random, and its value could
be anything but is “probably” within one
`sigma`
of the mean value ‘`x`’. An interval
‘`[`

`a` `..`

`b``]`

’ means a
variable’s value is unknown, but guaranteed to lie in the specified
range. Error forms are statistical or “average case” approximations;
interval arithmetic tends to produce “worst case” bounds on an
answer.

Intervals may not contain complex numbers, but they may contain HMS forms or date forms.

See Set Operations using Vectors, for commands that interpret interval forms as subsets of the set of real numbers.

The algebraic function ‘`intv(n, a, b)`’ builds an interval form
from ‘`a`’ to ‘`b`’; ‘`n`’ is an integer code which must
be 0 for ‘`(..)`’, 1 for ‘`(..]`’, 2 for ‘`[..)`’, or
3 for ‘`[..]`’.

Please note that in fully rigorous interval arithmetic, care would be
taken to make sure that the computation of the lower bound rounds toward
minus infinity, while upper bound computations round toward plus
infinity. Calc’s arithmetic always uses a round-to-nearest mode,
which means that roundoff errors could creep into an interval
calculation to produce intervals slightly smaller than they ought to
be. For example, entering ‘`[1..2]`’ and pressing `Q 2 ^`
should yield the interval ‘`[1..2]`’ again, but in fact it yields the
(slightly too small) interval ‘`[1..1.9999999]`’ due to roundoff
error.

### 4.13 Incomplete Objects

When `(` or `[` is typed to begin entering a complex number or
vector, respectively, the effect is to push an *incomplete* complex
number or vector onto the stack. The `,` key adds the value(s) at
the top of the stack onto the current incomplete object. The `)`
and `]` keys “close” the incomplete object after adding any values
on the top of the stack in front of the incomplete object.

As a result, the sequence of keystrokes `[ 2 , 3 RET 2 * , 9 ]`
pushes the vector ‘

`[2, 6, 9]`’ onto the stack. Likewise,

`( 1 , 2 Q )`pushes the complex number ‘

`(1, 1.414)`’ (approximately).

If several values lie on the stack in front of the incomplete object,
all are collected and appended to the object. Thus the `,` key
is redundant: `[ 2 RET 3 RET 2 * 9 ]`. Some people
prefer the equivalent

`SPC`key to

`RET`.

As a special case, typing `,` immediately after `(`, `[`, or
`,` adds a zero or duplicates the preceding value in the list being
formed. Typing `DEL` during incomplete entry removes the last item
from the list.

The `;` key is used in the same way as `,` to create polar complex
numbers: `( 1 ; 2 )`. When entering a vector, `;` is useful for
creating a matrix. In particular, `[ [ 1 , 2 ; 3 , 4 ; 5 , 6 ] ]` is
equivalent to `[ [ 1 , 2 ] , [ 3 , 4 ] , [ 5 , 6 ] ]`.

Incomplete entry is also used to enter intervals. For example,
`[ 2 .. 4 )` enters a semi-open interval. Note that when you type
the first period, it will be interpreted as a decimal point, but when
you type a second period immediately afterward, it is re-interpreted as
part of the interval symbol. Typing `..` corresponds to executing
the `calc-dots`

command.

If you find incomplete entry distracting, you may wish to enter vectors and complex numbers as algebraic formulas by pressing the apostrophe key.

### 4.14 Variables

A *variable* is somewhere between a storage register on a conventional
calculator, and a variable in a programming language. (In fact, a Calc
variable is really just an Emacs Lisp variable that contains a Calc number
or formula.) A variable’s name is normally composed of letters and digits.
Calc also allows apostrophes and `#`

signs in variable names.
(The Calc variable `foo`

corresponds to the Emacs Lisp variable
`var-foo`

, but unless you access the variable from within Emacs
Lisp, you don’t need to worry about it. Variable names in algebraic
formulas implicitly have ‘`var-`’ prefixed to their names. The
‘`#`’ character in variable names used in algebraic formulas
corresponds to a dash ‘`-`’ in the Lisp variable name. If the name
contains any dashes, the prefix ‘`var-`’ is *not* automatically
added. Thus the two formulas ‘`foo + 1`’ and ‘`var#foo + 1`’ both
refer to the same variable.)

In a command that takes a variable name, you can either type the full
name of a variable, or type a single digit to use one of the special
convenience variables `q0`

through `q9`

. For example,
`3 s s 2` stores the number 3 in variable `q2`

, and
`3 s s foo RET` stores that number in variable

`foo`

.
To push a variable itself (as opposed to the variable’s value) on the
stack, enter its name as an algebraic expression using the apostrophe
(`'`) key.

The `=` (`calc-evaluate`

) key “evaluates” a formula by
replacing all variables in the formula which have been given values by a
`calc-store`

or `calc-let`

command by their stored values.
Other variables are left alone. Thus a variable that has not been
stored acts like an abstract variable in algebra; a variable that has
been stored acts more like a register in a traditional calculator.
With a positive numeric prefix argument, `=` evaluates the top
`n` stack entries; with a negative argument, `=` evaluates
the `n`th stack entry.

A few variables are called *special constants*. Their names are
‘`e`’, ‘`pi`’, ‘`i`’, ‘`phi`’, and ‘`gamma`’.
(See Scientific Functions.) When they are evaluated with `=`,
their values are calculated if necessary according to the current precision
or complex polar mode. If you wish to use these symbols for other purposes,
simply undefine or redefine them using `calc-store`

.

The variables ‘`inf`’, ‘`uinf`’, and ‘`nan`’ stand for
infinite or indeterminate values. It’s best not to use them as
regular variables, since Calc uses special algebraic rules when
it manipulates them. Calc displays a warning message if you store
a value into any of these special variables.

See Storing and Recalling, for a discussion of commands dealing with variables.

### 4.15 Formulas

When you press the apostrophe key you may enter any expression or formula in algebraic form. (Calc uses the terms “expression” and “formula” interchangeably.) An expression is built up of numbers, variable names, and function calls, combined with various arithmetic operators. Parentheses may be used to indicate grouping. Spaces are ignored within formulas, except that spaces are not permitted within variable names or numbers. Arithmetic operators, in order from highest to lowest precedence, and with their equivalent function names, are:

‘`_`’ [`subscr`

] (subscripts);

postfix ‘`%`’ [`percent`

] (as in ‘`25% = 0.25`’);

prefix ‘`!`’ [`lnot`

] (logical “not,” as in ‘`!x`’);

‘`+/-`’ [`sdev`

] (the standard deviation symbol) and
‘`mod`’ [`makemod`

] (the symbol for modulo forms);

postfix ‘`!`’ [`fact`

] (factorial, as in ‘`n!`’)
and postfix ‘`!!`’ [`dfact`

] (double factorial);

‘`^`’ [`pow`

] (raised-to-the-power-of);

prefix ‘`+`’ and ‘`-`’ [`neg`

] (as in ‘`-x`’);

‘`*`’ [`mul`

];

‘`/`’ [`div`

], ‘`%`’ [`mod`

] (modulo), and
‘`\`’ [`idiv`

] (integer division);

infix ‘`+`’ [`add`

] and ‘`-`’ [`sub`

] (as in ‘`x-y`’);

‘`|`’ [`vconcat`

] (vector concatenation);

relations ‘`=`’ [`eq`

], ‘`!=`’ [`neq`

], ‘`<`’ [`lt`

],
‘`>`’ [`gt`

], ‘`<=`’ [`leq`

], and ‘`>=`’ [`geq`

];

‘`&&`’ [`land`

] (logical “and”);

‘`||`’ [`lor`

] (logical “or”);

the C-style “if” operator ‘`a?b:c`’ [`if`

];

‘`!!!`’ [`pnot`

] (rewrite pattern “not”);

‘`&&&`’ [`pand`

] (rewrite pattern “and”);

‘`|||`’ [`por`

] (rewrite pattern “or”);

‘`:=`’ [`assign`

] (for assignments and rewrite rules);

‘`::`’ [`condition`

] (rewrite pattern condition);

‘`=>`’ [`evalto`

].

Note that, unlike in usual computer notation, multiplication binds more
strongly than division: ‘`a*b/c*d`’ is equivalent to
‘`(a*b)/(c*d)`’.

The multiplication sign ‘`*`’ may be omitted in many cases. In particular,
if the righthand side is a number, variable name, or parenthesized
expression, the ‘`*`’ may be omitted. Implicit multiplication has the
same precedence as the explicit ‘`*`’ operator. The one exception to
the rule is that a variable name followed by a parenthesized expression,
as in ‘`f(x)`’,
is interpreted as a function call, not an implicit ‘`*`’. In many
cases you must use a space if you omit the ‘`*`’: ‘`2a`’ is the
same as ‘`2*a`’, and ‘`a b`’ is the same as ‘`a*b`’, but ‘`ab`’
is a variable called `ab`

, *not* the product of ‘`a`’ and
‘`b`’! Also note that ‘`f (x)`’ is still a function call.

The rules are slightly different for vectors written with square brackets.
In vectors, the space character is interpreted (like the comma) as a
separator of elements of the vector. Thus ‘`[ 2a b+c d ]`’ is
equivalent to ‘`[2*a, b+c, d]`’, whereas ‘`2a b+c d`’ is equivalent
to ‘`2*a*b + c*d`’.
Note that spaces around the brackets, and around explicit commas, are
ignored. To force spaces to be interpreted as multiplication you can
enclose a formula in parentheses as in ‘`[(a b) 2(c d)]`’, which is
interpreted as ‘`[a*b, 2*c*d]`’. An implicit comma is also inserted
between ‘`][`’, as in the matrix ‘`[[1 2][3 4]]`’.

Vectors that contain commas (not embedded within nested parentheses or
brackets) do not treat spaces specially: ‘`[a b, 2 c d]`’ is a vector
of two elements. Also, if it would be an error to treat spaces as
separators, but not otherwise, then Calc will ignore spaces:
‘`[a - b]`’ is a vector of one element, but ‘`[a -b]`’ is
a vector of two elements. Finally, vectors entered with curly braces
instead of square brackets do not give spaces any special treatment.
When Calc displays a vector that does not contain any commas, it will
insert parentheses if necessary to make the meaning clear:
‘`[(a b)]`’.

The expression ‘`5%-2`’ is ambiguous; is this five-percent minus two,
or five modulo minus-two? Calc always interprets the leftmost symbol as
an infix operator preferentially (modulo, in this case), so you would
need to write ‘`(5%)-2`’ to get the former interpretation.

A function call is, e.g., ‘`sin(1+x)`’. (The Calc algebraic function
`foo`

corresponds to the Emacs Lisp function `calcFunc-foo`

,
but unless you access the function from within Emacs Lisp, you don’t
need to worry about it.) Most mathematical Calculator commands like
`calc-sin`

have function equivalents like `sin`

.
If no Lisp function is defined for a function called by a formula, the
call is left as it is during algebraic manipulation: ‘`f(x+y)`’ is
left alone. Beware that many innocent-looking short names like `in`

and `re`

have predefined meanings which could surprise you; however,
single letters or single letters followed by digits are always safe to
use for your own function names. See Index of Algebraic Functions.

In the documentation for particular commands, the notation `H S`
(`calc-sinh`

) [`sinh`

] means that the key sequence `H S`, the
command `M-x calc-sinh`, and the algebraic function `sinh(x)`

all
represent the same operation.

Commands that interpret (“parse”) text as algebraic formulas include
algebraic entry (`'`), editing commands like ``` which parse
the contents of the editing buffer when you finish, the `C-x * g`
and `C-x * r` commands, the `C-y` command, the X window system
“paste” mouse operation, and Embedded mode. All of these operations
use the same rules for parsing formulas; in particular, language modes
(see Language Modes) affect them all in the same way.

When you read a large amount of text into the Calculator (say a vector
which represents a big set of rewrite rules; see Rewrite Rules),
you may wish to include comments in the text. Calc’s formula parser
ignores the symbol ‘`%%`’ and anything following it on a line:

[ a + b, %% the sum of "a" and "b" c + d, %% last line is coming up: e + f ]

This is parsed exactly the same as ‘`[ a + b, c + d, e + f ]`’.

See Syntax Tables, for a way to create your own operators and other input notations. See Compositions, for a way to create new display formats.

See Algebra, for commands for manipulating formulas symbolically.

## 5 Stack and Trail Commands

This chapter describes the Calc commands for manipulating objects on the stack and in the trail buffer. (These commands operate on objects of any type, such as numbers, vectors, formulas, and incomplete objects.)

### 5.1 Stack Manipulation Commands

To duplicate the top object on the stack, press `RET` or `SPC`
(two equivalent keys for the `calc-enter`

command).
Given a positive numeric prefix argument, these commands duplicate
several elements at the top of the stack.
Given a negative argument,
these commands duplicate the specified element of the stack.
Given an argument of zero, they duplicate the entire stack.
For example, with ‘`10 20 30`’ on the stack,
`RET` creates ‘`10 20 30 30`’,
`C-u 2 RET` creates ‘

`10 20 30 20 30`’,

`C-u - 2`creates ‘

`RET``10 20 30 20`’, and

`C-u 0`creates ‘

`RET``10 20 30 10 20 30`’.

The `LFD` (`calc-over`

) command (on a key marked Line-Feed if you
have it, else on `C-j`) is like `calc-enter`

except that the sign of the numeric prefix argument is interpreted
oppositely. Also, with no prefix argument the default argument is 2.
Thus with ‘`10 20 30`’ on the stack, `LFD` and `C-u 2 LFD`
are both equivalent to

`C-u - 2`, producing ‘

`RET``10 20 30 20`’.

To remove the top element from the stack, press `DEL` (`calc-pop`

).
The `C-d` key is a synonym for `DEL`.
(If the top element is an incomplete object with at least one element, the
last element is removed from it.) Given a positive numeric prefix argument,
several elements are removed. Given a negative argument, the specified
element of the stack is deleted. Given an argument of zero, the entire
stack is emptied.
For example, with ‘`10 20 30`’ on the stack,
`DEL` leaves ‘`10 20`’,
`C-u 2 DEL` leaves ‘

`10`’,

`C-u - 2`leaves ‘

`DEL``10 30`’, and

`C-u 0`leaves an empty stack.

`DEL`The `M- DEL` (

`calc-pop-above`

) command is to `DEL`what

`LFD`is to

`RET`: It interprets the sign of the numeric prefix argument in the opposite way, and the default argument is 2. Thus

`M-`by itself removes the second-from-top stack element, leaving the first, third, fourth, and so on;

`DEL``M-3 M-`deletes the third stack element.

`DEL`The above commands do not depend on the location of the cursor.
If the customizable variable `calc-context-sensitive-enter`

is
non-`nil`

(see Customizing Calc), these commands will become
context sensitive. For example, instead of duplicating the top of the stack,
`RET` will copy the element at the cursor to the top of the
stack. With a positive numeric prefix, a copy of the element at the
cursor and the appropriate number of preceding elements will be placed
at the top of the stack. A negative prefix will still duplicate the
specified element of the stack regardless of the cursor position.
Similarly, `DEL` will remove the corresponding elements from the
stack.

To exchange the top two elements of the stack, press `TAB`
(`calc-roll-down`

). Given a positive numeric prefix argument, the
specified number of elements at the top of the stack are rotated downward.
Given a negative argument, the entire stack is rotated downward the specified
number of times. Given an argument of zero, the entire stack is reversed
top-for-bottom.
For example, with ‘`10 20 30 40 50`’ on the stack,
`TAB` creates ‘`10 20 30 50 40`’,
`C-u 3 TAB` creates ‘

`10 20 50 30 40`’,

`C-u - 2`creates ‘

`TAB``40 50 10 20 30`’, and

`C-u 0`creates ‘

`TAB``50 40 30 20 10`’.

The command `M- TAB` (

`calc-roll-up`

) is analogous to `TAB`except that it rotates upward instead of downward. Also, the default with no prefix argument is to rotate the top 3 elements. For example, with ‘

`10 20 30 40 50`’ on the stack,

`M-`creates ‘

`TAB``10 20 40 50 30`’,

`C-u 4 M-`creates ‘

`TAB``10 30 40 50 20`’,

`C-u - 2 M-`creates ‘

`TAB``30 40 50 10 20`’, and

`C-u 0 M-`creates ‘

`TAB``50 40 30 20 10`’.

A good way to view the operation of `TAB` and `M- TAB` is in
terms of moving a particular element to a new position in the stack.
With a positive argument

`n`,

`TAB`moves the top stack element down to level

`n`, making room for it by pulling all the intervening stack elements toward the top.

`M-`moves the element at level

`TAB``n`up to the top. (Compare with

`LFD`, which copies instead of moving the element in level

`n`.)

With a negative argument *- n*,

`TAB`rotates the stack to move the object in level

`n`to the deepest place in the stack, and the object in level

*to the top.*

`n`+1`M-`rotates the deepest stack element to be in level

`TAB``n`, also putting the top stack element in level

*.*

`n`+1See Selecting Sub-Formulas, for a way to apply these commands to any portion of a vector or formula on the stack.

The command `C-x C-t` (`calc-transpose-lines`

) will transpose
the stack object determined by the point with the stack object at the
next higher level. For example, with ‘`10 20 30 40 50`’ on the
stack and the point on the line containing ‘`30`’, `C-x C-t`
creates ‘`10 20 40 30 50`’. More generally, `C-x C-t` acts on
the stack objects determined by the current point (and mark) similar
to how the text-mode command `transpose-lines`

acts on
lines. With argument `n`, `C-x C-t` will move the stack object
at the level above the current point and move it past N other objects;
for example, with ‘`10 20 30 40 50`’ on the stack and the point on
the line containing ‘`30`’, `C-u 2 C-x C-t` creates
‘`10 40 20 30 50`’. With an argument of 0, `C-x C-t` will switch
the stack objects at the levels determined by the point and the mark.

### 5.2 Editing Stack Entries

The ``` (`calc-edit`

) command creates a temporary buffer
(`*Calc Edit*`) for editing the top-of-stack value using regular
Emacs commands. Note that ``` is a grave accent, not an apostrophe.
With a numeric prefix argument, it edits the specified number of stack
entries at once. (An argument of zero edits the entire stack; a
negative argument edits one specific stack entry.)

When you are done editing, press `C-c C-c` to finish and return
to Calc. The `RET` and `LFD` keys also work to finish most
sorts of editing, though in some cases Calc leaves `RET` with its
usual meaning (“insert a newline”) if it’s a situation where you
might want to insert new lines into the editing buffer.

When you finish editing, the Calculator parses the lines of text in
the `*Calc Edit*` buffer as numbers or formulas, replaces the
original stack elements in the original buffer with these new values,
then kills the `*Calc Edit*` buffer. The original Calculator buffer
continues to exist during editing, but for best results you should be
careful not to change it until you have finished the edit. You can
also cancel the edit by killing the buffer with `C-x k`.

The formula is normally reevaluated as it is put onto the stack.
For example, editing ‘`a + 2`’ to ‘`3 + 2`’ and pressing
`C-c C-c` will push 5 on the stack. If you use `LFD` to
finish, Calc will put the result on the stack without evaluating it.

If you give a prefix argument to `C-c C-c`,
Calc will not kill the `*Calc Edit*` buffer. You can switch
back to that buffer and continue editing if you wish. However, you
should understand that if you initiated the edit with ```, the
`C-c C-c` operation will be programmed to replace the top of the
stack with the new edited value, and it will do this even if you have
rearranged the stack in the meanwhile. This is not so much of a problem
with other editing commands, though, such as `s e`
(`calc-edit-variable`

; see Other Operations on Variables).

If the `calc-edit`

command involves more than one stack entry,
each line of the `*Calc Edit*` buffer is interpreted as a
separate formula. Otherwise, the entire buffer is interpreted as
one formula, with line breaks ignored. (You can use `C-o` or
`C-q C-j` to insert a newline in the buffer without pressing `RET`.)

The ``` key also works during numeric or algebraic entry. The
text entered so far is moved to the `*Calc Edit*` buffer for
more extensive editing than is convenient in the minibuffer.

### 5.3 Trail Commands

The commands for manipulating the Calc Trail buffer are two-key sequences
beginning with the `t` prefix.

The `t d` (`calc-trail-display`

) command turns display of the
trail on and off. Normally the trail display is toggled on if it was off,
off if it was on. With a numeric prefix of zero, this command always
turns the trail off; with a prefix of one, it always turns the trail on.
The other trail-manipulation commands described here automatically turn
the trail on. Note that when the trail is off values are still recorded
there; they are simply not displayed. To set Emacs to turn the trail
off by default, type `t d` and then save the mode settings with
`m m` (`calc-save-modes`

).

The `t i` (`calc-trail-in`

) and `t o`
(`calc-trail-out`

) commands switch the cursor into and out of the
Calc Trail window. In practice they are rarely used, since the commands
shown below are a more convenient way to move around in the
trail, and they work “by remote control” when the cursor is still
in the Calculator window.

There is a *trail pointer* which selects some entry of the trail at
any given time. The trail pointer looks like a ‘`>`’ symbol right
before the selected number. The following commands operate on the
trail pointer in various ways.

The `t y` (`calc-trail-yank`

) command reads the selected value in
the trail and pushes it onto the Calculator stack. It allows you to
re-use any previously computed value without retyping. With a numeric
prefix argument `n`, it yanks the value `n` lines above the current
trail pointer.

The `t <` (`calc-trail-scroll-left`

) and `t >`
(`calc-trail-scroll-right`

) commands horizontally scroll the trail
window left or right by one half of its width.

The `t n` (`calc-trail-next`

) and `t p`
(`calc-trail-previous)`

commands move the trail pointer down or up
one line. The `t f` (`calc-trail-forward`

) and `t b`
(`calc-trail-backward`

) commands move the trail pointer down or up
one screenful at a time. All of these commands accept numeric prefix
arguments to move several lines or screenfuls at a time.

The `t [` (`calc-trail-first`

) and `t ]`
(`calc-trail-last`

) commands move the trail pointer to the first or
last line of the trail. The `t h` (`calc-trail-here`

) command
moves the trail pointer to the cursor position; unlike the other trail
commands, `t h` works only when Calc Trail is the selected window.

The `t s` (`calc-trail-isearch-forward`

) and `t r`
(`calc-trail-isearch-backward`

) commands perform an incremental
search forward or backward through the trail. You can press `RET`
to terminate the search; the trail pointer moves to the current line.
If you cancel the search with `C-g`, the trail pointer stays where
it was when the search began.

The `t m` (`calc-trail-marker`

) command allows you to enter a
line of text of your own choosing into the trail. The text is inserted
after the line containing the trail pointer; this usually means it is
added to the end of the trail. Trail markers are useful mainly as the
targets for later incremental searches in the trail.

The `t k` (`calc-trail-kill`

) command removes the selected line
from the trail. The line is saved in the Emacs kill ring suitable for
yanking into another buffer, but it is not easy to yank the text back
into the trail buffer. With a numeric prefix argument, this command
kills the `n` lines below or above the selected one.

The `t .` (`calc-full-trail-vectors`

) command is described
elsewhere; see Vector and Matrix Display Formats.

### 5.4 Keep Arguments

The `K` (`calc-keep-args`

) command acts like a prefix for
the following command. It prevents that command from removing its
arguments from the stack. For example, after `2 RET 3 +`,
the stack contains the sole number 5, but after

`2`, the stack contains the arguments and the result: ‘

`RET`3 K +`2 3 5`’.

With the exception of keyboard macros, this works for all commands that
take arguments off the stack. (To avoid potentially unpleasant behavior,
a `K` prefix before a keyboard macro will be ignored. A `K`
prefix called *within* the keyboard macro will still take effect.)
As another example, `K a s` simplifies a formula, pushing the
simplified version of the formula onto the stack after the original
formula (rather than replacing the original formula). Note that you
could get the same effect by typing ` RET a s`, copying the
formula and then simplifying the copy. One difference is that for a very
large formula the time taken to format the intermediate copy in

`could be noticeable;`

`RET`a s`K a s`would avoid this extra work.

Even stack manipulation commands are affected. `TAB` works by
popping two values and pushing them back in the opposite order,
so `2 RET 3 K TAB` produces ‘

`2 3 3 2`’.

A few Calc commands provide other ways of doing the same thing.
For example, `' sin($)` replaces the number on the stack with
its sine using algebraic entry; to push the sine and keep the
original argument you could use either `' sin($1)` or
`K ' sin($)`. See Algebraic Entry. Also, the `s s`
command is effectively the same as `K s t`. See Storing Variables.

If you execute a command and then decide you really wanted to keep
the argument, you can press `M- RET` (

`calc-last-args`

).
This command pushes the last arguments that were popped by any command
onto the stack. Note that the order of things on the stack will be
different than with `K`:

`2`leaves ‘

`RET`3 + M-`RET``5 2 3`’ on the stack instead of ‘

`2 3 5`’. See Undoing Mistakes.

## 6 Mode Settings

This chapter describes commands that set modes in the Calculator.
They do not affect the contents of the stack, although they may change
the *appearance* or *interpretation* of the stack’s contents.

- General Mode Commands
- Precision
- Inverse and Hyperbolic Flags
- Calculation Modes
- Simplification Modes
- Declarations
- Display Modes
- Language Modes
- The
`Modes`

Variable - The Calc Mode Line

### 6.1 General Mode Commands

You can save all of the current mode settings in your Calc init file
(the file given by the variable `calc-settings-file`

, typically
`~/.emacs.d/calc.el`) with the `m m` (`calc-save-modes`

)
command. This will cause Emacs to reestablish these modes each time
it starts up. The modes saved in the file include everything
controlled by the `m` and `d` prefix keys, the current
precision and binary word size, whether or not the trail is displayed,
the current height of the Calc window, and more. The current
interface (used when you type `C-x * *`) is also saved. If there
were already saved mode settings in the file, they are replaced.
Otherwise, the new mode information is appended to the end of the
file.

The `m R` (`calc-mode-record-mode`

) command tells Calc to
record all the mode settings (as if by pressing `m m`) every
time a mode setting changes. If the modes are saved this way, then this
“automatic mode recording” mode is also saved.
Type `m R` again to disable this method of recording the mode
settings. To turn it off permanently, the `m m` command will also be
necessary. (If Embedded mode is enabled, other options for recording
the modes are available; see Mode Settings in Embedded Mode.)

The `m F` (`calc-settings-file-name`

) command allows you to
choose a different file than the current value of `calc-settings-file`

for `m m`, `Z P`, and similar commands to save permanent information.
You are prompted for a file name. All Calc modes are then reset to
their default values, then settings from the file you named are loaded
if this file exists, and this file becomes the one that Calc will
use in the future for commands like `m m`. The default settings
file name is `~/.emacs.d/calc.el`. You can see the current file name by
giving a blank response to the `m F` prompt. See also the
discussion of the `calc-settings-file`

variable; see Customizing Calc.

If the file name you give is your user init file (typically
`~/.emacs`), `m F` will not automatically load the new file. This
is because your user init file may contain other things you don’t want
to reread. You can give
a numeric prefix argument of 1 to `m F` to force it to read the
file no matter what. Conversely, an argument of *-1* tells
`m F` *not* to read the new file. An argument of 2 or *-2*
tells `m F` not to reset the modes to their defaults beforehand,
which is useful if you intend your new file to have a variant of the
modes present in the file you were using before.

The `m x` (`calc-always-load-extensions`

) command enables a mode
in which the first use of Calc loads the entire program, including all
extensions modules. Otherwise, the extensions modules will not be loaded
until the various advanced Calc features are used. Since this mode only
has effect when Calc is first loaded, `m x` is usually followed by
`m m` to make the mode-setting permanent. To load all of Calc just
once, rather than always in the future, you can press `C-x * L`.

The `m S` (`calc-shift-prefix`

) command enables a mode in which
all of Calc’s letter prefix keys may be typed shifted as well as unshifted.
If you are typing, say, `a S` (`calc-solve-for`

) quite often
you might find it easier to turn this mode on so that you can type
`A S` instead. When this mode is enabled, the commands that used to
be on those single shifted letters (e.g., `A` (`calc-abs`

)) can
now be invoked by pressing the shifted letter twice: `A A`. Note
that the `v` prefix key always works both shifted and unshifted, and
the `z` and `Z` prefix keys are always distinct. Also, the `h`
prefix is not affected by this mode. Press `m S` again to disable
shifted-prefix mode.

### 6.2 Precision

The `p` (`calc-precision`

) command controls the precision to
which floating-point calculations are carried. The precision must be
at least 3 digits and may be arbitrarily high, within the limits of
memory and time. This affects only floats: Integer and rational
calculations are always carried out with as many digits as necessary.

The `p` key prompts for the current precision. If you wish you
can instead give the precision as a numeric prefix argument.

Many internal calculations are carried to one or two digits higher
precision than normal. Results are rounded down afterward to the
current precision. Unless a special display mode has been selected,
floats are always displayed with their full stored precision, i.e.,
what you see is what you get. Reducing the current precision does not
round values already on the stack, but those values will be rounded
down before being used in any calculation. The `c 0` through
`c 9` commands (see Conversions) can be used to round an
existing value to a new precision.

It is important to distinguish the concepts of *precision* and
*accuracy*. In the normal usage of these words, the number
123.4567 has a precision of 7 digits but an accuracy of 4 digits.
The precision is the total number of digits not counting leading
or trailing zeros (regardless of the position of the decimal point).
The accuracy is simply the number of digits after the decimal point
(again not counting trailing zeros). In Calc you control the precision,
not the accuracy of computations. If you were to set the accuracy
instead, then calculations like ‘`exp(100)`’ would generate many
more digits than you would typically need, while ‘`exp(-100)`’ would
probably round to zero! In Calc, both these computations give you
exactly 12 (or the requested number of) significant digits.

The only Calc features that deal with accuracy instead of precision
are fixed-point display mode for floats (`d f`; see Float Formats),
and the rounding functions like `floor`

and `round`

(see Integer Truncation). Also, `c 0` through `c 9`
deal with both precision and accuracy depending on the magnitudes
of the numbers involved.

If you need to work with a particular fixed accuracy (say, dollars and
cents with two digits after the decimal point), one solution is to work
with integers and an “implied” decimal point. For example, $8.99
divided by 6 would be entered `899 RET 6 /`, yielding 149.833
(actually $1.49833 with our implied decimal point); pressing

`R`would round this to 150 cents, i.e., $1.50.

See Floats, for still more on floating-point precision and related issues.

### 6.3 Inverse and Hyperbolic Flags

There is no single-key equivalent to the `calc-arcsin`

function.
Instead, you must first press `I` (`calc-inverse`

) to set
the *Inverse Flag*, then press `S` (`calc-sin`

).
The `I` key actually toggles the Inverse Flag. When this flag
is set, the word ‘`Inv`’ appears in the mode line.

Likewise, the `H` key (`calc-hyperbolic`

) sets or clears the
Hyperbolic Flag, which transforms `calc-sin`

into `calc-sinh`

.
If both of these flags are set at once, the effect will be
`calc-arcsinh`

. (The Hyperbolic flag is also used by some
non-trigonometric commands; for example `H L` computes a base-10,
instead of base-*e*, logarithm.)

Command names like `calc-arcsin`

are provided for completeness, and
may be executed with `x` or `M-x`. Their effect is simply to
toggle the Inverse and/or Hyperbolic flags and then execute the
corresponding base command (`calc-sin`

in this case).

The `O` key (`calc-option`

) sets another flag, the
*Option Flag*, which also can alter the subsequent Calc command in
various ways.

The Inverse, Hyperbolic and Option flags apply only to the next
Calculator command, after which they are automatically cleared. (They
are also cleared if the next keystroke is not a Calc command.) Digits
you type after `I`, `H` or `O` (or `K`) are treated as
prefix arguments for the next command, not as numeric entries. The
same is true of `C-u`, but not of the minus sign (`K -` means
to subtract and keep arguments).

Another Calc prefix flag, `K` (keep-arguments), is discussed
elsewhere. See Keep Arguments.

### 6.4 Calculation Modes

The commands in this section are two-key sequences beginning with
the `m` prefix. (That’s the letter `m`, not the `META` key.)
The ‘`m a`’ (`calc-algebraic-mode`

) command is described elsewhere
(see Algebraic Entry).

- Angular Modes
- Polar Mode
- Fraction Mode
- Infinite Mode
- Symbolic Mode
- Matrix and Scalar Modes
- Automatic Recomputation
- Working Messages

#### 6.4.1 Angular Modes

The Calculator supports three notations for angles: radians, degrees,
and degrees-minutes-seconds. When a number is presented to a function
like `sin`

that requires an angle, the current angular mode is
used to interpret the number as either radians or degrees. If an HMS
form is presented to `sin`

, it is always interpreted as
degrees-minutes-seconds.

Functions that compute angles produce a number in radians, a number in
degrees, or an HMS form depending on the current angular mode. If the
result is a complex number and the current mode is HMS, the number is
instead expressed in degrees. (Complex-number calculations would
normally be done in Radians mode, though. Complex numbers are converted
to degrees by calculating the complex result in radians and then
multiplying by 180 over ‘`pi`’.)

The `m r` (`calc-radians-mode`

), `m d` (`calc-degrees-mode`

),
and `m h` (`calc-hms-mode`

) commands control the angular mode.
The current angular mode is displayed on the Emacs mode line.
The default angular mode is Degrees.

#### 6.4.2 Polar Mode

The Calculator normally “prefers” rectangular complex numbers in the
sense that rectangular form is used when the proper form can not be
decided from the input. This might happen by multiplying a rectangular
number by a polar one, by taking the square root of a negative real
number, or by entering `( 2 SPC 3 )`.

The `m p` (`calc-polar-mode`

) command toggles complex-number
preference between rectangular and polar forms. In Polar mode, all
of the above example situations would produce polar complex numbers.

#### 6.4.3 Fraction Mode

Division of two integers normally yields a floating-point number if the
result cannot be expressed as an integer. In some cases you would
rather get an exact fractional answer. One way to accomplish this is
to use the `:` (`calc-fdiv`

) [`fdiv`

] command, which
divides the two integers on the top of the stack to produce a fraction:
`6 RET 4 :` produces ‘

`3:2`’ even though

`6`produces ‘

`RET`4 /`1.5`’.

To set the Calculator to produce fractional results for normal integer
divisions, use the `m f` (`calc-frac-mode`

) command.
For example, ‘`8/4`’ produces ‘`2`’ in either mode,
but ‘`6/4`’ produces ‘`3:2`’ in Fraction mode, ‘`1.5`’ in
Float mode.

At any time you can use `c f` (`calc-float`

) to convert a
fraction to a float, or `c F` (`calc-fraction`

) to convert a
float to a fraction. See Conversions.

#### 6.4.4 Infinite Mode

The Calculator normally treats results like ‘`1 / 0`’ as errors;
formulas like this are left in unsimplified form. But Calc can be
put into a mode where such calculations instead produce “infinite”
results.

The `m i` (`calc-infinite-mode`

) command turns this mode
on and off. When the mode is off, infinities do not arise except
in calculations that already had infinities as inputs. (One exception
is that infinite open intervals like ‘`[0 .. inf)`’ can be
generated; however, intervals closed at infinity (‘`[0 .. inf]`’)
will not be generated when Infinite mode is off.)

With Infinite mode turned on, ‘`1 / 0`’ will generate `uinf`

,
an undirected infinity. See Infinities, for a discussion of the
difference between `inf`

and `uinf`

. Also, ‘`0 / 0`’
evaluates to `nan`

, the “indeterminate” symbol. Various other
functions can also return infinities in this mode; for example,
‘`ln(0) = -inf`’, and ‘`gamma(-7) = uinf`’. Once again,
note that ‘`exp(inf) = inf`’ regardless of Infinite mode because
this calculation has infinity as an input.

The `m i` command with a numeric prefix argument of zero,
i.e., `C-u 0 m i`, turns on a Positive Infinite mode in
which zero is treated as positive instead of being directionless.
Thus, ‘`1 / 0 = inf`’ and ‘`-1 / 0 = -inf`’ in this mode.
Note that zero never actually has a sign in Calc; there are no
separate representations for *+0* and *-0*. Positive
Infinite mode merely changes the interpretation given to the
single symbol, ‘`0`’. One consequence of this is that, while
you might expect ‘`1 / -0 = -inf`’, actually ‘`1 / -0`’
is equivalent to ‘`1 / 0`’, which is equal to positive `inf`

.

#### 6.4.5 Symbolic Mode

Calculations are normally performed numerically wherever possible.
For example, the `calc-sqrt`

command, or `sqrt`

function in an
algebraic expression, produces a numeric answer if the argument is a
number or a symbolic expression if the argument is an expression:
`2 Q` pushes 1.4142 but `' x+1 RET Q` pushes ‘

`sqrt(x+1)`’.

In *Symbolic mode*, controlled by the `m s` (`calc-symbolic-mode`

)
command, functions which would produce inexact, irrational results are
left in symbolic form. Thus `16 Q` pushes 4, but `2 Q` pushes
‘`sqrt(2)`’.

The shift-`N` (`calc-eval-num`

) command evaluates numerically
the expression at the top of the stack, by temporarily disabling
`calc-symbolic-mode`

and executing `=` (`calc-evaluate`

).
Given a numeric prefix argument, it also
sets the floating-point precision to the specified value for the duration
of the command.

To evaluate a formula numerically without expanding the variables it
contains, you can use the key sequence `m s a v m s` (this uses
`calc-alg-evaluate`

, which resimplifies but doesn’t evaluate
variables.)

#### 6.4.6 Matrix and Scalar Modes

Calc sometimes makes assumptions during algebraic manipulation that
are awkward or incorrect when vectors and matrices are involved.
Calc has two modes, *Matrix mode* and *Scalar mode*, which
modify its behavior around vectors in useful ways.

Press `m v` (`calc-matrix-mode`

) once to enter Matrix mode.
In this mode, all objects are assumed to be matrices unless provably
otherwise. One major effect is that Calc will no longer consider
multiplication to be commutative. (Recall that in matrix arithmetic,
‘`A*B`’ is not the same as ‘`B*A`’.) This assumption affects
rewrite rules and algebraic simplification. Another effect of this
mode is that calculations that would normally produce constants like
0 and 1 (e.g., ‘`a - a`’ and ‘`a / a`’, respectively) will now
produce function calls that represent “generic” zero or identity
matrices: ‘`idn(0)`’, ‘`idn(1)`’. The `idn`

function
‘`idn( a,n)`’ returns

`a`times an

`n`x

`n`identity matrix; if

`n`is omitted, it doesn’t know what dimension to use and so the

`idn`

call remains in symbolic
form. However, if this generic identity matrix is later combined
with a matrix whose size is known, it will be converted into
a true identity matrix of the appropriate size. On the other hand,
if it is combined with a scalar (as in ‘`idn(1) + 2`’), Calc will assume it really was a scalar after all and produce, e.g., 3.

Press `m v` a second time to get Scalar mode. Here, objects are
assumed *not* to be vectors or matrices unless provably so.
For example, normally adding a variable to a vector, as in
‘`[x, y, z] + a`’, will leave the sum in symbolic form because
as far as Calc knows, ‘`a`’ could represent either a number or
another 3-vector. In Scalar mode, ‘`a`’ is assumed to be a
non-vector, and the addition is evaluated to ‘`[x+a, y+a, z+a]`’.

Press `m v` a third time to return to the normal mode of operation.

If you press `m v` with a numeric prefix argument `n`, you
get a special “dimensioned” Matrix mode in which matrices of
unknown size are assumed to be `n`x`n` square matrices.
Then, the function call ‘`idn(1)`’ will expand into an actual
matrix rather than representing a “generic” matrix. Simply typing
`C-u m v` will get you a square Matrix mode, in which matrices of
unknown size are assumed to be square matrices of unspecified size.

Of course these modes are approximations to the true state of affairs, which is probably that some quantities will be matrices and others will be scalars. One solution is to “declare” certain variables or functions to be scalar-valued. See Declarations, to see how to make declarations in Calc.

There is nothing stopping you from declaring a variable to be
scalar and then storing a matrix in it; however, if you do, the
results you get from Calc may not be valid. Suppose you let Calc
get the result ‘`[x+a, y+a, z+a]`’ shown above, and then stored
‘`[1, 2, 3]`’ in ‘`a`’. The result would not be the same as
for ‘`[x, y, z] + [1, 2, 3]`’, but that’s because you have broken
your earlier promise to Calc that ‘`a`’ would be scalar.

Another way to mix scalars and matrices is to use selections
(see Selecting Sub-Formulas). Use Matrix mode when operating on
your formula normally; then, to apply Scalar mode to a certain part
of the formula without affecting the rest just select that part,
change into Scalar mode and press `=` to resimplify the part
under this mode, then change back to Matrix mode before deselecting.

#### 6.4.7 Automatic Recomputation

The *evaluates-to* operator, ‘`=>`’, has the special
property that any ‘`=>`’ formulas on the stack are recomputed
whenever variable values or mode settings that might affect them
are changed. See The Evaluates-To Operator.

The `m C` (`calc-auto-recompute`

) command turns this
automatic recomputation on and off. If you turn it off, Calc will
not update ‘`=>`’ operators on the stack (nor those in the
attached Embedded mode buffer, if there is one). They will not
be updated unless you explicitly do so by pressing `=` or until
you press `m C` to turn recomputation back on. (While automatic
recomputation is off, you can think of `m C m C` as a command
to update all ‘`=>`’ operators while leaving recomputation off.)

To update ‘`=>`’ operators in an Embedded buffer while
automatic recomputation is off, use `C-x * u`.
See Embedded Mode.

#### 6.4.8 Working Messages

Since the Calculator is written entirely in Emacs Lisp, which is not
designed for heavy numerical work, many operations are quite slow.
The Calculator normally displays the message ‘`Working...`’ in the
echo area during any command that may be slow. In addition, iterative
operations such as square roots and trigonometric functions display the
intermediate result at each step. Both of these types of messages can
be disabled if you find them distracting.

Type `m w` (`calc-working`

) with a numeric prefix of 0 to
disable all “working” messages. Use a numeric prefix of 1 to enable
only the plain ‘`Working...`’ message. Use a numeric prefix of 2 to
see intermediate results as well. With no numeric prefix this displays
the current mode.

While it may seem that the “working” messages will slow Calc down considerably, experiments have shown that their impact is actually quite small. But if your terminal is slow you may find that it helps to turn the messages off.

### 6.5 Simplification Modes

The current *simplification mode* controls how numbers and formulas
are “normalized” when being taken from or pushed onto the stack.
Some normalizations are unavoidable, such as rounding floating-point
results to the current precision, and reducing fractions to simplest
form. Others, such as simplifying a formula like ‘`a+a`’ (or ‘`2+3`’),
are done automatically but can be turned off when necessary.

When you press a key like `+` when ‘`2`’ and ‘`3`’ are on the
stack, Calc pops these numbers, normalizes them, creates the formula
‘`2+3`’, normalizes it, and pushes the result. Of course the standard
rules for normalizing ‘`2+3`’ will produce the result ‘`5`’.

Simplification mode commands consist of the lower-case `m` prefix key
followed by a shifted letter.

The `m O` (`calc-no-simplify-mode`

) command turns off all optional
simplifications. These would leave a formula like ‘`2+3`’ alone. In
fact, nothing except simple numbers are ever affected by normalization
in this mode. Explicit simplification commands, such as `=` or
`a s`, can still be given to simplify any formulas.
See Programming with Formulas, for a sample use of
No-Simplification mode.

The `m N` (`calc-num-simplify-mode`

) command turns off simplification
of any formulas except those for which all arguments are constants. For
example, ‘`1+2`’ is simplified to ‘`3`’, and ‘`a+(2-2)`’ is
simplified to ‘`a+0`’ but no further, since one argument of the sum
is not a constant. Unfortunately, ‘`(a+2)-2`’ is *not* simplified
because the top-level ‘`-`’ operator’s arguments are not both
constant numbers (one of them is the formula ‘`a+2`’).
A constant is a number or other numeric object (such as a constant
error form or modulo form), or a vector all of whose
elements are constant.

The `m I` (`calc-basic-simplify-mode`

) command does some basic
simplifications for all formulas. This includes many easy and
fast algebraic simplifications such as ‘`a+0`’ to ‘`a`’, and
‘`a + 2 a`’ to ‘`3 a`’, as well as evaluating functions like
‘` deriv(x^2, x)`’ to ‘

`2 x`’.

The `m B` (`calc-bin-simplify-mode`

) mode applies the basic
simplifications to a result and then, if the result is an integer,
uses the `b c` (`calc-clip`

) command to clip the integer according
to the current binary word size. See Binary Number Functions. Real numbers
are rounded to the nearest integer and then clipped; other kinds of
results (after the basic simplifications) are left alone.

The `m A` (`calc-alg-simplify-mode`

) mode does standard
algebraic simplifications. See Algebraic Simplifications.

The `m E` (`calc-ext-simplify-mode`

) mode does “extended”, or
“unsafe”, algebraic simplification. See “Unsafe” Simplifications.

The `m U` (`calc-units-simplify-mode`

) mode does units
simplification. See Simplification of Units. These include the
algebraic simplifications, plus variable names which
are identifiable as unit names (like ‘`mm`’ for “millimeters”)
are simplified with their unit definitions in mind.

A common technique is to set the simplification mode down to the lowest
amount of simplification you will allow to be applied automatically, then
use manual commands like `a s` and `c c` (`calc-clean`

) to
perform higher types of simplifications on demand.

### 6.6 Declarations

A *declaration* is a statement you make that promises you will
use a certain variable or function in a restricted way. This may
give Calc the freedom to do things that it couldn’t do if it had to
take the fully general situation into account.

#### 6.6.1 Declaration Basics

The `s d` (`calc-declare-variable`

) command is the easiest
way to make a declaration for a variable. This command prompts for
the variable name, then prompts for the declaration. The default
at the declaration prompt is the previous declaration, if any.
You can edit this declaration, or press `C-k` to erase it and
type a new declaration. (Or, erase it and press `RET` to clear
the declaration, effectively “undeclaring” the variable.)

A declaration is in general a vector of *type symbols* and
*range* values. If there is only one type symbol or range value,
you can write it directly rather than enclosing it in a vector.
For example, `s d foo RET real RET` declares

`foo`

to
be a real number, and `s d bar`declares

`RET`[int, const, [1..6]]`RET``bar`

to be a constant integer between 1 and 6.
(Actually, you can omit the outermost brackets and Calc will
provide them for you: `s d bar`.)

`RET`int, const, [1..6]`RET`Declarations in Calc are kept in a special variable called `Decls`

.
This variable encodes the set of all outstanding declarations in
the form of a matrix. Each row has two elements: A variable or
vector of variables declared by that row, and the declaration
specifier as described above. You can use the `s D` command to
edit this variable if you wish to see all the declarations at once.
See Other Operations on Variables, for a description of this command
and the `s p` command that allows you to save your declarations
permanently if you wish.

Items being declared can also be function calls. The arguments in
the call are ignored; the effect is to say that this function returns
values of the declared type for any valid arguments. The `s d`
command declares only variables, so if you wish to make a function
declaration you will have to edit the `Decls`

matrix yourself.

For example, the declaration matrix

[ [ foo, real ] [ [j, k, n], int ] [ f(1,2,3), [0 .. inf) ] ]

declares that `foo`

represents a real number, `j`

, `k`

and `n`

represent integers, and the function `f`

always
returns a real number in the interval shown.

If there is a declaration for the variable `All`

, then that
declaration applies to all variables that are not otherwise declared.
It does not apply to function names. For example, using the row
‘`[All, real]`’ says that all your variables are real unless they
are explicitly declared without `real`

in some other row.
The `s d` command declares `All`

if you give a blank
response to the variable-name prompt.

#### 6.6.2 Kinds of Declarations

The type-specifier part of a declaration (that is, the second prompt
in the `s d` command) can be a type symbol, an interval, or a
vector consisting of zero or more type symbols followed by zero or
more intervals or numbers that represent the set of possible values
for the variable.

[ [ a, [1, 2, 3, 4, 5] ] [ b, [1 .. 5] ] [ c, [int, 1 .. 5] ] ]

Here `a`

is declared to contain one of the five integers shown;
`b`

is any number in the interval from 1 to 5 (any real number
since we haven’t specified), and `c`

is any integer in that
interval. Thus the declarations for `a`

and `c`

are
nearly equivalent (see below).

The type-specifier can be the empty vector ‘`[]`’ to say that
nothing is known about a given variable’s value. This is the same
as not declaring the variable at all except that it overrides any
`All`

declaration which would otherwise apply.

The initial value of `Decls`

is the empty vector ‘`[]`’.
If `Decls`

has no stored value or if the value stored in it
is not valid, it is ignored and there are no declarations as far
as Calc is concerned. (The `s d` command will replace such a
malformed value with a fresh empty matrix, ‘`[]`’, before recording
the new declaration.) Unrecognized type symbols are ignored.

The following type symbols describe what sorts of numbers will be stored in a variable:

`int`

Integers.

`numint`

Numerical integers. (Integers or integer-valued floats.)

`frac`

Fractions. (Rational numbers which are not integers.)

`rat`

Rational numbers. (Either integers or fractions.)

`float`

Floating-point numbers.

`real`

Real numbers. (Integers, fractions, or floats. Actually, intervals and error forms with real components also count as reals here.)

`pos`

Positive real numbers. (Strictly greater than zero.)

`nonneg`

Nonnegative real numbers. (Greater than or equal to zero.)

`number`

Numbers. (Real or complex.)

Calc uses this information to determine when certain simplifications
of formulas are safe. For example, ‘`(x^y)^z`’ cannot be
simplified to ‘`x^(y z)`’ in general; for example,
‘`((-3)^2)^1:2`’ is 3, but ‘`(-3)^(2*1:2) = (-3)^1`’ is *-3*.
However, this simplification *is* safe if `z`

is known
to be an integer, or if `x`

is known to be a nonnegative
real number. If you have given declarations that allow Calc to
deduce either of these facts, Calc will perform this simplification
of the formula.

Calc can apply a certain amount of logic when using declarations.
For example, ‘`(x^y)^(2n+1)`’ will be simplified if `n`

has been declared `int`

; Calc knows that an integer times an
integer, plus an integer, must always be an integer. (In fact,
Calc would simplify ‘`(-x)^(2n+1)`’ to ‘`-(x^(2n+1))`’ since
it is able to determine that ‘`2n+1`’ must be an odd integer.)

Similarly, ‘`(abs(x)^y)^z`’ will be simplified to ‘`abs(x)^(y z)`’
because Calc knows that the `abs`

function always returns a
nonnegative real. If you had a `myabs`

function that also had
this property, you could get Calc to recognize it by adding the row
‘`[myabs(), nonneg]`’ to the `Decls`

matrix.

One instance of this simplification is ‘`sqrt(x^2)`’ (since the
`sqrt`

function is effectively a one-half power). Normally
Calc leaves this formula alone. After the command
`s d x RET real RET`, however, it can simplify the formula to
‘

`abs(x)`’. And after

`s d x`, Calc can simplify this formula all the way to ‘

`RET`nonneg`RET``x`’.

If there are any intervals or real numbers in the type specifier,
they comprise the set of possible values that the variable or
function being declared can have. In particular, the type symbol
`real`

is effectively the same as the range ‘`[-inf .. inf]`’
(note that infinity is included in the range of possible values);
`pos`

is the same as ‘`(0 .. inf]`’, and `nonneg`

is
the same as ‘`[0 .. inf]`’. Saying ‘`[real, [-5 .. 5]]`’ is
redundant because the fact that the variable is real can be
deduced just from the interval, but ‘`[int, [-5 .. 5]]`’ and
‘`[rat, [-5 .. 5]]`’ are useful combinations.

Note that the vector of intervals or numbers is in the same format used by Calc’s set-manipulation commands. See Set Operations using Vectors.

The type specifier ‘`[1, 2, 3]`’ is equivalent to
‘`[numint, 1, 2, 3]`’, *not* to ‘`[int, 1, 2, 3]`’.
In other words, the range of possible values means only that
the variable’s value must be numerically equal to a number in
that range, but not that it must be equal in type as well.
Calc’s set operations act the same way; ‘`in(2, [1., 2., 3.])`’
and ‘`in(1.5, [1:2, 3:2, 5:2])`’ both report “true.”

If you use a conflicting combination of type specifiers, the
results are unpredictable. An example is ‘`[pos, [0 .. 5]]`’,
where the interval does not lie in the range described by the
type symbol.

“Real” declarations mostly affect simplifications involving powers
like the one described above. Another case where they are used
is in the `a P` command which returns a list of all roots of a
polynomial; if the variable has been declared real, only the real
roots (if any) will be included in the list.

“Integer” declarations are used for simplifications which are valid
only when certain values are integers (such as ‘`(x^y)^z`’
shown above).

Calc’s algebraic simplifications also make use of declarations when
simplifying equations and inequalities. They will cancel `x`

from both sides of ‘`a x = b x`’ only if it is sure `x`

is non-zero, say, because it has a `pos`

declaration.
To declare specifically that `x`

is real and non-zero,
use ‘`[[-inf .. 0), (0 .. inf]]`’. (There is no way in the
current notation to say that `x`

is nonzero but not necessarily
real.) The `a e` command does “unsafe” simplifications,
including canceling ‘`x`’ from the equation when ‘`x`’ is
not known to be nonzero.

Another set of type symbols distinguish between scalars and vectors.

`scalar`

The value is not a vector.

`vector`

The value is a vector.

`matrix`

The value is a matrix (a rectangular vector of vectors).

`sqmatrix`

The value is a square matrix.

These type symbols can be combined with the other type symbols
described above; ‘`[int, matrix]`’ describes an object which
is a matrix of integers.

Scalar/vector declarations are used to determine whether certain
algebraic operations are safe. For example, ‘`[a, b, c] + x`’
is normally not simplified to ‘`[a + x, b + x, c + x]`’, but
it will be if `x`

has been declared `scalar`

. On the
other hand, multiplication is usually assumed to be commutative,
but the terms in ‘`x y`’ will never be exchanged if both `x`

and `y`

are known to be vectors or matrices. (Calc currently
never distinguishes between `vector`

and `matrix`

declarations.)

See Matrix and Scalar Modes, for a discussion of Matrix mode and
Scalar mode, which are similar to declaring ‘`[All, matrix]`’
or ‘`[All, scalar]`’ but much more convenient.

One more type symbol that is recognized is used with the `H a d`
command for taking total derivatives of a formula. See Calculus.

`const`

The value is a constant with respect to other variables.

Calc does not check the declarations for a variable when you store
a value in it. However, storing *-3.5* in a variable that has
been declared `pos`

, `int`

, or `matrix`

may have
unexpected effects; Calc may evaluate ‘`sqrt(x^2)`’ to ‘`3.5`’
if it substitutes the value first, or to ‘`-3.5`’ if `x`

was declared `pos`

and the formula ‘`sqrt(x^2)`’ is
simplified to ‘`x`’ before the value is substituted. Before
using a variable for a new purpose, it is best to use `s d`
or `s D` to check to make sure you don’t still have an old
declaration for the variable that will conflict with its new meaning.

#### 6.6.3 Functions for Declarations

Calc has a set of functions for accessing the current declarations
in a convenient manner. These functions return 1 if the argument
can be shown to have the specified property, or 0 if the argument
can be shown *not* to have that property; otherwise they are
left unevaluated. These functions are suitable for use with rewrite
rules (see Conditional Rewrite Rules) or programming constructs
(see Conditionals in Keyboard Macros). They can be entered only using
algebraic notation. See Logical Operations, for functions
that perform other tests not related to declarations.

For example, ‘`dint(17)`’ returns 1 because 17 is an integer, as
do ‘`dint(n)`’ and ‘`dint(2 n - 3)`’ if `n`

has been declared
`int`

, but ‘`dint(2.5)`’ and ‘`dint(n + 0.5)`’ return 0.
Calc consults knowledge of its own built-in functions as well as your
own declarations: ‘`dint(floor(x))`’ returns 1.

The `dint`

function checks if its argument is an integer.
The `dnatnum`

function checks if its argument is a natural
number, i.e., a nonnegative integer. The `dnumint`

function
checks if its argument is numerically an integer, i.e., either an
integer or an integer-valued float. Note that these and the other
data type functions also accept vectors or matrices composed of
suitable elements, and that real infinities ‘`inf`’ and ‘`-inf`’
are considered to be integers for the purposes of these functions.

The `drat`

function checks if its argument is rational, i.e.,
an integer or fraction. Infinities count as rational, but intervals
and error forms do not.

The `dreal`

function checks if its argument is real. This
includes integers, fractions, floats, real error forms, and intervals.

The `dimag`

function checks if its argument is imaginary,
i.e., is mathematically equal to a real number times ‘`i`’.

The `dpos`

function checks for positive (but nonzero) reals.
The `dneg`

function checks for negative reals. The `dnonneg`

function checks for nonnegative reals, i.e., reals greater than or
equal to zero. Note that Calc’s algebraic simplifications, which are
effectively applied to all conditions in rewrite rules, can simplify
an expression like ‘`x > 0`’ to 1 or 0 using `dpos`

.
So the actual functions `dpos`

, `dneg`

, and `dnonneg`

are rarely necessary.

The `dnonzero`

function checks that its argument is nonzero.
This includes all nonzero real or complex numbers, all intervals that
do not include zero, all nonzero modulo forms, vectors all of whose
elements are nonzero, and variables or formulas whose values can be
deduced to be nonzero. It does not include error forms, since they
represent values which could be anything including zero. (This is
also the set of objects considered “true” in conditional contexts.)

The `deven`

function returns 1 if its argument is known to be
an even integer (or integer-valued float); it returns 0 if its argument
is known not to be even (because it is known to be odd or a non-integer).
Calc’s algebraic simplifications use this to simplify a test of the form
‘`x % 2 = 0`’. There is also an analogous `dodd`

function.

The `drange`

function returns a set (an interval or a vector
of intervals and/or numbers; see Set Operations using Vectors) that describes
the set of possible values of its argument. If the argument is
a variable or a function with a declaration, the range is copied
from the declaration. Otherwise, the possible signs of the
expression are determined using a method similar to `dpos`

,
etc., and a suitable set like ‘`[0 .. inf]`’ is returned. If
the expression is not provably real, the `drange`

function
remains unevaluated.

The `dscalar`

function returns 1 if its argument is provably
scalar, or 0 if its argument is provably non-scalar. It is left
unevaluated if this cannot be determined. (If Matrix mode or Scalar
mode is in effect, this function returns 1 or 0, respectively,
if it has no other information.) When Calc interprets a condition
(say, in a rewrite rule) it considers an unevaluated formula to be
“false.” Thus, ‘`dscalar(a)`’ is “true” only if `a`

is
provably scalar, and ‘`!dscalar(a)`’ is “true” only if `a`

is provably non-scalar; both are “false” if there is insufficient
information to tell.

### 6.7 Display Modes

The commands in this section are two-key sequences beginning with the
`d` prefix. The `d l` (`calc-line-numbering`

) and `d b`
(`calc-line-breaking`

) commands are described elsewhere;
see Stack Basics and see Normal Language Modes, respectively.
Display formats for vectors and matrices are also covered elsewhere;
see Vector and Matrix Display Formats.

One thing all display modes have in common is their treatment of the
`H` prefix. This prefix causes any mode command that would normally
refresh the stack to leave the stack display alone. The word “Dirty”
will appear in the mode line when Calc thinks the stack display may not
reflect the latest mode settings.

The `d RET` (

`calc-refresh-top`

) command reformats the
top stack entry according to all the current modes. Positive prefix
arguments reformat the top `n`entries; negative prefix arguments reformat the specified entry, and a prefix of zero is equivalent to

`d`(

`SPC``calc-refresh`

), which reformats the entire stack.
For example, `H d s M-2 d`changes to scientific notation but reformats only the top two stack entries in the new mode.

`RET`The `I` prefix has another effect on the display modes. The mode
is set only temporarily; the top stack entry is reformatted according
to that mode, then the original mode setting is restored. In other
words, `I d s` is equivalent to `H d s d RET H d (old mode)`.

- Radix Modes
- Grouping Digits
- Float Formats
- Complex Formats
- Fraction Formats
- HMS Formats
- Date Formats
- Truncating the Stack
- Justification
- Labels

#### 6.7.1 Radix Modes

Calc normally displays numbers in decimal (*base-10* or *radix-10*)
notation. Calc can actually display in any radix from two (binary) to 36.
When the radix is above 10, the letters `A`

to `Z`

are used as
digits. When entering such a number, letter keys are interpreted as
potential digits rather than terminating numeric entry mode.

The key sequences `d 2`, `d 8`, `d 6`, and `d 0` select
binary, octal, hexadecimal, and decimal as the current display radix,
respectively. Numbers can always be entered in any radix, though the
current radix is used as a default if you press `#` without any initial
digits. A number entered without a `#` is *always* interpreted
as decimal.

To set the radix generally, use `d r` (`calc-radix`

) and enter
an integer from 2 to 36. You can specify the radix as a numeric prefix
argument; otherwise you will be prompted for it.

Integers normally are displayed with however many digits are necessary to
represent the integer and no more. The `d z` (`calc-leading-zeros`

)
command causes integers to be padded out with leading zeros according to the
current binary word size. (See Binary Number Functions, for a discussion of
word size.) If the absolute value of the word size is ‘`w`’, all integers
are displayed with at least enough digits to represent
‘`(2^w)-1`’
in the current radix. (Larger integers will still be displayed in their
entirety.)

Calc can display ‘`w`’-bit integers using two’s complement
notation, although this is most useful with the binary, octal and
hexadecimal display modes. This option is selected by using the
`O` option prefix before setting the display radix, and a negative word
size might be appropriate (see Binary Number Functions). In two’s
complement notation, the integers in the (nearly) symmetric interval
from
‘`-2^(w-1)`’
to
‘`2^(w-1)-1`’
are represented by the integers from ‘`0`’ to ‘`2^w-1`’:
the integers from ‘`0`’ to
‘`2^(w-1)-1`’
are represented by themselves and the integers from
‘`-2^(w-1)`’
to ‘`-1`’ are represented by the integers from
‘`2^(w-1)`’
to ‘`2^w-1`’ (the integer ‘`k`’ is represented by ‘`k+2^w`’).
Calc will display a two’s complement integer by the radix (either
‘`2`’, ‘`8`’ or ‘`16`’), two `#` symbols, and then its
representation (including any leading zeros necessary to include all
‘`w`’ bits). In a two’s complement display mode, numbers that
are not displayed in two’s complement notation (i.e., that aren’t
integers from
‘`-2^(w-1)`’
to
‘`2^(w-1)-1`’)
will be represented using Calc’s usual notation (in the appropriate
radix).

#### 6.7.2 Grouping Digits

Long numbers can be hard to read if they have too many digits. For
example, the factorial of 30 is 33 digits long! Press `d g`
(`calc-group-digits`

) to enable *Grouping* mode, in which digits
are displayed in clumps of 3 or 4 (depending on the current radix)
separated by commas.

The `d g` command toggles grouping on and off.
With a numeric prefix of 0, this command displays the current state of
the grouping flag; with an argument of minus one it disables grouping;
with a positive argument ‘`N`’ it enables grouping on every ‘`N`’
digits. For floating-point numbers, grouping normally occurs only
before the decimal point. A negative prefix argument ‘`-N`’ enables
grouping every ‘`N`’ digits both before and after the decimal point.

The `d ,` (`calc-group-char`

) command allows you to choose any
character as the grouping separator. The default is the comma character.
If you find it difficult to read vectors of large integers grouped with
commas, you may wish to use spaces or some other character instead.
This command takes the next character you type, whatever it is, and
uses it as the digit separator. As a special case, `d , \` selects
‘`\,`’ (TeX’s thin-space symbol) as the digit separator.

Please note that grouped numbers will not generally be parsed correctly
if re-read in textual form, say by the use of `C-x * y` and `C-x * g`.
(See Kill and Yank Functions, for details on these commands.) One exception is
the ‘`\,`’ separator, which doesn’t interfere with parsing because it
is ignored by TeX language mode.

#### 6.7.3 Float Formats

Floating-point quantities are normally displayed in standard decimal form, with scientific notation used if the exponent is especially high or low. All significant digits are normally displayed. The commands in this section allow you to choose among several alternative display formats for floats.

The `d n` (`calc-normal-notation`

) command selects the normal
display format. All significant figures in a number are displayed.
With a positive numeric prefix, numbers are rounded if necessary to
that number of significant digits. With a negative numerix prefix,
the specified number of significant digits less than the current
precision is used. (Thus `C-u -2 d n` displays 10 digits if the
current precision is 12.)

The `d f` (`calc-fix-notation`

) command selects fixed-point
notation. The numeric argument is the number of digits after the
decimal point, zero or more. This format will relax into scientific
notation if a nonzero number would otherwise have been rounded all the
way to zero. Specifying a negative number of digits is the same as
for a positive number, except that small nonzero numbers will be rounded
to zero rather than switching to scientific notation.

The `d s` (`calc-sci-notation`

) command selects scientific
notation. A positive argument sets the number of significant figures
displayed, of which one will be before and the rest after the decimal
point. A negative argument works the same as for `d n` format.
The default is to display all significant digits.

The `d e` (`calc-eng-notation`

) command selects engineering
notation. This is similar to scientific notation except that the
exponent is rounded down to a multiple of three, with from one to three
digits before the decimal point. An optional numeric prefix sets the
number of significant digits to display, as for `d s`.

It is important to distinguish between the current *precision* and
the current *display format*. After the commands `C-u 10 p`
and `C-u 6 d n` the Calculator computes all results to ten
significant figures but displays only six. (In fact, intermediate
calculations are often carried to one or two more significant figures,
but values placed on the stack will be rounded down to ten figures.)
Numbers are never actually rounded to the display precision for storage,
except by commands like `C-k` and `C-x * y` which operate on the
actual displayed text in the Calculator buffer.

The `d .` (`calc-point-char`

) command selects the character used
as a decimal point. Normally this is a period; users in some countries
may wish to change this to a comma. Note that this is only a display
style; on entry, periods must always be used to denote floating-point
numbers, and commas to separate elements in a list.

#### 6.7.4 Complex Formats

There are three supported notations for complex numbers in rectangular
form. The default is as a pair of real numbers enclosed in parentheses
and separated by a comma: ‘`(a,b)`’. The `d c`
(`calc-complex-notation`

) command selects this style.

The other notations are `d i` (`calc-i-notation`

), in which
numbers are displayed in ‘`a+bi`’ form, and `d j`
(`calc-j-notation`

) which displays the form ‘`a+bj`’ preferred
in some disciplines.

Complex numbers are normally entered in ‘`(a,b)`’ format.
If you enter ‘`2+3i`’ as an algebraic formula, it will be stored as
the formula ‘`2 + 3 * i`’. However, if you use `=` to evaluate
this formula and you have not changed the variable ‘`i`’, the ‘`i`’
will be interpreted as ‘`(0,1)`’ and the formula will be simplified
to ‘`(2,3)`’. Other commands (like `calc-sin`

) will *not*
interpret the formula ‘`2 + 3 * i`’ as a complex number.
See Variables, under “special constants.”

#### 6.7.5 Fraction Formats

Display of fractional numbers is controlled by the `d o`
(`calc-over-notation`

) command. By default, a number like
eight thirds is displayed in the form ‘`8:3`’. The `d o` command
prompts for a one- or two-character format. If you give one character,
that character is used as the fraction separator. Common separators are
‘`:`’ and ‘`/`’. (During input of numbers, the `:` key must be
used regardless of the display format; in particular, the `/` is used
for RPN-style division, *not* for entering fractions.)

If you give two characters, fractions use “integer-plus-fractional-part”
notation. For example, the format ‘`+/`’ would display eight thirds
as ‘`2+2/3`’. If two colons are present in a number being entered,
the number is interpreted in this form (so that the entries `2:2:3`
and `8:3` are equivalent).

It is also possible to follow the one- or two-character format with
a number. For example: ‘`:10`’ or ‘`+/3`’. In this case,
Calc adjusts all fractions that are displayed to have the specified
denominator, if possible. Otherwise it adjusts the denominator to
be a multiple of the specified value. For example, in ‘`:6`’ mode
the fraction ‘`1:6`’ will be unaffected, but ‘`2:3`’ will be
displayed as ‘`4:6`’, ‘`1:2`’ will be displayed as ‘`3:6`’,
and ‘`1:8`’ will be displayed as ‘`3:24`’. Integers are also
affected by this mode: 3 is displayed as ‘`18:6`’. Note that the
format ‘`:1`’ writes fractions the same as ‘`:`’, but it writes
integers as ‘`n:1`’.

The fraction format does not affect the way fractions or integers are stored, only the way they appear on the screen. The fraction format never affects floats.

#### 6.7.6 HMS Formats

The `d h` (`calc-hms-notation`

) command controls the display of
HMS (hours-minutes-seconds) forms. It prompts for a string which
consists basically of an “hours” marker, optional punctuation, a
“minutes” marker, more optional punctuation, and a “seconds” marker.
Punctuation is zero or more spaces, commas, or semicolons. The hours
marker is one or more non-punctuation characters. The minutes and
seconds markers must be single non-punctuation characters.

The default HMS format is ‘`@ ' "`’, producing HMS values of the form
‘`23@ 30' 15.75"`’. The format ‘`deg, ms`’ would display this same
value as ‘`23deg, 30m15.75s`’. During numeric entry, the `h` or `o`
keys are recognized as synonyms for `@` regardless of display format.
The `m` and `s` keys are recognized as synonyms for `'` and
`"`, respectively, but only if an `@` (or `h` or `o`) has
already been typed; otherwise, they have their usual meanings
(`m-` prefix and `s-` prefix). Thus, `5 "`, `0 @ 5 "`, and
`0 h 5 s` are some of the ways to enter the quantity “five seconds.”
The `'` key is recognized as “minutes” only if `@` (or `h` or
`o`) has already been pressed; otherwise it means to switch to algebraic
entry.

#### 6.7.7 Date Formats

The `d d` (`calc-date-notation`

) command controls the display
of date forms (see Date Forms). It prompts for a string which
contains letters that represent the various parts of a date and time.
To show which parts should be omitted when the form represents a pure
date with no time, parts of the string can be enclosed in ‘`< >`’
marks. If you don’t include ‘`< >`’ markers in the format, Calc
guesses at which parts, if any, should be omitted when formatting
pure dates.

The default format is: ‘`<H:mm:SSpp >Www Mmm D, YYYY`’.
An example string in this format is ‘`3:32pm Wed Jan 9, 1991`’.
If you enter a blank format string, this default format is
reestablished.

Calc uses ‘`< >`’ notation for nameless functions as well as for
dates. See Specifying Operators. To avoid confusion with nameless
functions, your date formats should avoid using the ‘`#`’ character.

#### 6.7.7.1 ISO 8601

The same date can be written down in different formats and Calc tries to allow you to choose your preferred format. Some common formats are ambiguous, however; for example, 10/11/2012 means October 11, 2012 in the United States but it means November 10, 2012 in Europe. To help avoid such ambiguities, the International Organization for Standardization (ISO) provides the ISO 8601 standard, which provides three different but easily distinguishable and unambiguous ways to represent a date.

The ISO 8601 calendar date representation is

YYYY-MM-DD

where `YYYY` is the four digit year, `MM` is the two-digit month
number (01 for January to 12 for December), and `DD` is the
two-digit day of the month (01 to 31). (Note that `YYYY` does not
correspond to Calc’s date formatting code, which will be introduced
later.) The year, which should be padded with zeros to ensure it has at
least four digits, is the Gregorian year, except that the year before
0001 (1 AD) is the year 0000 (1 BC). The date October 11, 2012 is
written 2012-10-11 in this representation and November 10, 2012 is
written 2012-11-10.

The ISO 8601 ordinal date representation is

YYYY-DDD

where `YYYY` is the year, as above, and `DDD` is the day of the year.
The date December 31, 2011 is written 2011-365 in this representation
and January 1, 2012 is written 2012-001.

The ISO 8601 week date representation is

YYYY-Www-D

where `YYYY` is the ISO week-numbering year, `ww` is the two
digit week number (preceded by a literal “W”), and `D` is the day
of the week (1 for Monday through 7 for Sunday). The ISO week-numbering
year is based on the Gregorian year but can differ slightly. The first
week of an ISO week-numbering year is the week with the Gregorian year’s
first Thursday in it (equivalently, the week containing January 4);
any day of that week (Monday through Sunday) is part of the same ISO
week-numbering year, any day from the previous week is part of the
previous year. For example, January 4, 2013 is on a Friday, and so
the first week for the ISO week-numbering year 2013 starts on
Monday, December 31, 2012. The day December 31, 2012 is then part of the
Gregorian year 2012 but ISO week-numbering year 2013. In the week
date representation, this week goes from 2013-W01-1 (December 31,
2012) to 2013-W01-7 (January 6, 2013).

All three ISO 8601 representations arrange the numbers from most significant to least significant; as well as being unambiguous representations, they are easy to sort since chronological order in this formats corresponds to lexicographical order. The hyphens are sometimes omitted.

The ISO 8601 standard uses a 24 hour clock; a particular time is
represented by `hh`:`mm`:`ss` where `hh` is the
two-digit hour (from 00 to 24), `mm` is the two-digit minute (from
00 to 59) and `ss` is the two-digit second. The seconds or minutes
and seconds can be omitted, and decimals can be added. If a date with a
time is represented, they should be separated by a literal “T”, so noon
on December 13, 2012 can be represented as 2012-12-13T12:00.

#### 6.7.7.2 Date Formatting Codes

When displaying a date, the current date format is used. All
characters except for letters and ‘`<`’ and ‘`>`’ are
copied literally when dates are formatted. The portion between
‘`< >`’ markers is omitted for pure dates, or included for
date/time forms. Letters are interpreted according to the table
below.

When dates are read in during algebraic entry, Calc first tries to match the input string to the current format either with or without the time part. The punctuation characters (including spaces) must match exactly; letter fields must correspond to suitable text in the input. If this doesn’t work, Calc checks if the input is a simple number; if so, the number is interpreted as a number of days since Dec 31, 1 BC. Otherwise, Calc tries a much more relaxed and flexible algorithm which is described in the next section.

Weekday names are ignored during reading.

Two-digit year numbers are interpreted as lying in the range
from 1941 to 2039. Years outside that range are always
entered and displayed in full. Year numbers with a leading
‘`+`’ sign are always interpreted exactly, allowing the
entry and display of the years 1 through 99 AD.

Here is a complete list of the formatting codes for dates:

- Y
Year: “91” for 1991, “7” for 2007, “+23” for 23 AD.

- YY
Year: “91” for 1991, “07” for 2007, “+23” for 23 AD.

- BY
Year: “91” for 1991, “ 7” for 2007, “+23” for 23 AD.

- YYY
Year: “1991” for 1991, “23” for 23 AD.

- YYYY
Year: “1991” for 1991, “+23” for 23 AD.

- ZYYY
Year: “1991” for 1991, “0023” for 23 AD, “0000” for 1 BC.

- IYYY
Year: ISO 8601 week-numbering year.

- aa
Year: “ad” or blank.

- AA
Year: “AD” or blank.

- aaa
Year: “ad ” or blank. (Note trailing space.)

- AAA
Year: “AD ” or blank.

- aaaa
Year: “a.d.” or blank.

- AAAA
Year: “A.D.” or blank.

- bb
Year: “bc” or blank.

- BB
Year: “BC” or blank.

- bbb
Year: “ bc” or blank. (Note leading space.)

- BBB
Year: “ BC” or blank.

- bbbb
Year: “b.c.” or blank.

- BBBB
Year: “B.C.” or blank.

- M
Month: “8” for August.

- MM
Month: “08” for August.

- BM
Month: “ 8” for August.

- MMM
Month: “AUG” for August.

- Mmm
Month: “Aug” for August.

- mmm
Month: “aug” for August.

- MMMM
Month: “AUGUST” for August.

- Mmmm
Month: “August” for August.

- D
Day: “7” for 7th day of month.

- DD
Day: “07” for 7th day of month.

- BD
Day: “ 7” for 7th day of month.

- W
Weekday: “0” for Sunday, “6” for Saturday.

- w
Weekday: “1” for Monday, “7” for Sunday.

- WWW
Weekday: “SUN” for Sunday.

- Www
Weekday: “Sun” for Sunday.

- www
Weekday: “sun” for Sunday.

- WWWW
Weekday: “SUNDAY” for Sunday.

- Wwww
Weekday: “Sunday” for Sunday.

- Iww
Week number: ISO 8601 week number, “W01” for week 1.

- d
Day of year: “34” for Feb. 3.

- ddd
Day of year: “034” for Feb. 3.

- bdd
Day of year: “ 34” for Feb. 3.

- T
Letter: Literal “T”.

- h
Hour: “5” for 5 AM; “17” for 5 PM.

- hh
Hour: “05” for 5 AM; “17” for 5 PM.

- bh
Hour: “ 5” for 5 AM; “17” for 5 PM.

- H
Hour: “5” for 5 AM and 5 PM.

- HH
Hour: “05” for 5 AM and 5 PM.

- BH
Hour: “ 5” for 5 AM and 5 PM.

- p
AM/PM: “a” or “p”.

- P
AM/PM: “A” or “P”.

- pp
AM/PM: “am” or “pm”.

- PP
AM/PM: “AM” or “PM”.

- pppp
AM/PM: “a.m.” or “p.m.”.

- PPPP
AM/PM: “A.M.” or “P.M.”.

- m
Minutes: “7” for 7.

- mm
Minutes: “07” for 7.

- bm
Minutes: “ 7” for 7.

- s
Seconds: “7” for 7; “7.23” for 7.23.

- ss
Seconds: “07” for 7; “07.23” for 7.23.

- bs
Seconds: “ 7” for 7; “ 7.23” for 7.23.

- SS
Optional seconds: “07” for 7; blank for 0.

- BS
Optional seconds: “ 7” for 7; blank for 0.

- N
Numeric date/time: “726842.25” for 6:00am Wed Jan 9, 1991.

- n
Numeric date: “726842” for any time on Wed Jan 9, 1991.

- J
Julian date/time: “2448265.75” for 6:00am Wed Jan 9, 1991.

- j
Julian date: “2448266” for any time on Wed Jan 9, 1991.

- U
Unix time: “663400800” for 6:00am Wed Jan 9, 1991.

- X
Brackets suppression. An “X” at the front of the format causes the surrounding ‘

`< >`’ delimiters to be omitted when formatting dates. Note that the brackets are still required for algebraic entry.

If “SS” or “BS” (optional seconds) is preceded by a colon, the colon is also omitted if the seconds part is zero.

If “bb,” “bbb” or “bbbb” or their upper-case equivalents
appear in the format, then negative year numbers are displayed
without a minus sign. Note that “aa” and “bb” are mutually
exclusive. Some typical usages would be ‘`YYYY AABB`’;
‘`AAAYYYYBBB`’; ‘`YYYYBBB`’.

The formats “YY,” “YYYY,” “MM,” “DD,” “ddd,” “hh,” “HH,” “mm,” “ss,” and “SS” actually match any number of digits during reading unless several of these codes are strung together with no punctuation in between, in which case the input must have exactly as many digits as there are letters in the format.

The “j,” “J,” and “U” formats do not make any time zone
adjustment. They effectively use ‘`julian(x,0)`’ and
‘`unixtime(x,0)`’ to make the conversion; see Date Arithmetic.

#### 6.7.7.3 Free-Form Dates

When reading a date form during algebraic entry, Calc falls back on the algorithm described here if the input does not exactly match the current date format. This algorithm generally “does the right thing” and you don’t have to worry about it, but it is described here in full detail for the curious.

Calc does not distinguish between upper- and lower-case letters while interpreting dates.

First, the time portion, if present, is located somewhere in the text and then removed. The remaining text is then interpreted as the date.

A time is of the form ‘`hh:mm:ss`’, possibly with the seconds
part omitted and possibly with an AM/PM indicator added to indicate
12-hour time. If the AM/PM is present, the minutes may also be
omitted. The AM/PM part may be any of the words ‘`am`’,
‘`pm`’, ‘`noon`’, or ‘`midnight`’; each of these may be
abbreviated to one letter, and the alternate forms ‘`a.m.`’,
‘`p.m.`’, and ‘`mid`’ are also understood. Obviously
‘`noon`’ and ‘`midnight`’ are allowed only on 12:00:00.
The words ‘`noon`’, ‘`mid`’, and ‘`midnight`’ are also
recognized with no number attached. Midnight will represent the
beginning of a day.

If there is no AM/PM indicator, the time is interpreted in 24-hour format.

When reading the date portion, Calc first checks to see if it is an ISO 8601 week-numbering date; if the string contains an integer representing the year, a “W” followed by two digits for the week number, and an integer from 1 to 7 representing the weekday (in that order), then all other characters are ignored and this information determines the date. Otherwise, all words and numbers are isolated from the string; other characters are ignored. All words must be either month names or day-of-week names (the latter of which are ignored). Names can be written in full or as three-letter abbreviations.

Large numbers, or numbers with ‘`+`’ or ‘`-`’ signs,
are interpreted as years. If one of the other numbers is
greater than 12, then that must be the day and the remaining
number in the input is therefore the month. Otherwise, Calc
assumes the month, day and year are in the same order that they
appear in the current date format. If the year is omitted, the
current year is taken from the system clock.

If there are too many or too few numbers, or any unrecognizable words, then the input is rejected.

If there are any large numbers (of five digits or more) other than the year, they are ignored on the assumption that they are something like Julian dates that were included along with the traditional date components when the date was formatted.

One of the words ‘`ad`’, ‘`a.d.`’, ‘`bc`’, or ‘`b.c.`’
may optionally be used; the latter two are equivalent to a
minus sign on the year value.

If you always enter a four-digit year, and use a name instead of a number for the month, there is no danger of ambiguity.

#### 6.7.7.4 Standard Date Formats

There are actually ten standard date formats, numbered 0 through 9.
Entering a blank line at the `d d` command’s prompt gives
you format number 1, Calc’s usual format. You can enter any digit
to select the other formats.

To create your own standard date formats, give a numeric prefix
argument from 0 to 9 to the `d d` command. The format you
enter will be recorded as the new standard format of that
number, as well as becoming the new current date format.
You can save your formats permanently with the `m m`
command (see Mode Settings).

- 0
‘

`N`’ (Numerical format)- 1
‘

`<H:mm:SSpp >Www Mmm D, YYYY`’ (American format)- 2
‘

`D Mmm YYYY<, h:mm:SS>`’ (European format)- 3
‘

`Www Mmm BD< hh:mm:ss> YYYY`’ (Unix written date format)- 4
‘

`M/D/Y< H:mm:SSpp>`’ (American slashed format)- 5
‘

`D.M.Y< h:mm:SS>`’ (European dotted format)- 6
‘

`M-D-Y< H:mm:SSpp>`’ (American dashed format)- 7
‘

`D-M-Y< h:mm:SS>`’ (European dashed format)- 8
‘

`j<, h:mm:ss>`’ (Julian day plus time)- 9
‘

`YYddd< hh:mm:ss>`’ (Year-day format)- 10
‘

`ZYYY-MM-DD Www< hh:mm>`’ (Org mode format)- 11
‘

`IYYY-Iww-w<Thh:mm:ss>`’ (ISO 8601 week numbering format)

#### 6.7.8 Truncating the Stack

The `d t` (`calc-truncate-stack`

) command moves the ‘`.`’
line that marks the top-of-stack up or down in the Calculator buffer.
The number right above that line is considered to the be at the top of
the stack. Any numbers below that line are “hidden” from all stack
operations (although still visible to the user). This is similar to the
Emacs “narrowing” feature, except that the values below the ‘`.`’
are *visible*, just temporarily frozen. This feature allows you to
keep several independent calculations running at once in different parts
of the stack, or to apply a certain command to an element buried deep in
the stack.

Pressing `d t` by itself moves the ‘`.`’ to the line the cursor
is on. Thus, this line and all those below it become hidden. To un-hide
these lines, move down to the end of the buffer and press `d t`.
With a positive numeric prefix argument ‘`n`’, `d t` hides the
bottom ‘`n`’ values in the buffer. With a negative argument, it hides
all but the top ‘`n`’ values. With an argument of zero, it hides zero
values, i.e., moves the ‘`.`’ all the way down to the bottom.

The `d [` (`calc-truncate-up`

) and `d ]`
(`calc-truncate-down`

) commands move the ‘`.`’ up or down one
line at a time (or several lines with a prefix argument).

#### 6.7.9 Justification

Values on the stack are normally left-justified in the window. You can
control this arrangement by typing `d <` (`calc-left-justify`

),
`d >` (`calc-right-justify`

), or `d =`
(`calc-center-justify`

). For example, in Right-Justification mode,
stack entries are displayed flush-right against the right edge of the
window.

If you change the width of the Calculator window you may have to type
`d SPC` (

`calc-refresh`

) to re-align right-justified or centered
text.
Right-justification is especially useful together with fixed-point
notation (see `d f`

; `calc-fix-notation`

). With these modes
together, the decimal points on numbers will always line up.

With a numeric prefix argument, the justification commands give you
a little extra control over the display. The argument specifies the
horizontal “origin” of a display line. It is also possible to
specify a maximum line width using the `d b` command (see Normal Language Modes). For reference, the precise rules for formatting and
breaking lines are given below. Notice that the interaction between
origin and line width is slightly different in each justification
mode.

In Left-Justified mode, the line is indented by a number of spaces given by the origin (default zero). If the result is longer than the maximum line width, if given, or too wide to fit in the Calc window otherwise, then it is broken into lines which will fit; each broken line is indented to the origin.

In Right-Justified mode, lines are shifted right so that the rightmost character is just before the origin, or just before the current window width if no origin was specified. If the line is too long for this, then it is broken; the current line width is used, if specified, or else the origin is used as a width if that is specified, or else the line is broken to fit in the window.

In Centering mode, the origin is the column number of the center of each stack entry. If a line width is specified, lines will not be allowed to go past that width; Calc will either indent less or break the lines if necessary. If no origin is specified, half the line width or Calc window width is used.

Note that, in each case, if line numbering is enabled the display is indented an additional four spaces to make room for the line number. The width of the line number is taken into account when positioning according to the current Calc window width, but not when positioning by explicit origins and widths. In the latter case, the display is formatted as specified, and then uniformly shifted over four spaces to fit the line numbers.

#### 6.7.10 Labels

The `d {` (`calc-left-label`

) command prompts for a string,
then displays that string to the left of every stack entry. If the
entries are left-justified (see Justification), then they will
appear immediately after the label (unless you specified an origin
greater than the length of the label). If the entries are centered
or right-justified, the label appears on the far left and does not
affect the horizontal position of the stack entry.

Give a blank string (with `d { RET`) to turn the label off.

The `d }` (`calc-right-label`

) command similarly adds a
label on the righthand side. It does not affect positioning of
the stack entries unless they are right-justified. Also, if both
a line width and an origin are given in Right-Justified mode, the
stack entry is justified to the origin and the righthand label is
justified to the line width.

One application of labels would be to add equation numbers to formulas you are manipulating in Calc and then copying into a document (possibly using Embedded mode). The equations would typically be centered, and the equation numbers would be on the left or right as you prefer.

### 6.8 Language Modes

The commands in this section change Calc to use a different notation for entry and display of formulas, corresponding to the conventions of some other common language such as Pascal or LaTeX. Objects displayed on the stack or yanked from the Calculator to an editing buffer will be formatted in the current language; objects entered in algebraic entry or yanked from another buffer will be interpreted according to the current language.

The current language has no effect on things written to or read from the
trail buffer, nor does it affect numeric entry. Only algebraic entry is
affected. You can make even algebraic entry ignore the current language
and use the standard notation by giving a numeric prefix, e.g., `C-u '`.

For example, suppose the formula ‘`2*a[1] + atan(a[2])`’ occurs in a C
program; elsewhere in the program you need the derivatives of this formula
with respect to ‘`a[1]`’ and ‘`a[2]`’. First, type `d C`
to switch to C notation. Now use `C-u C-x * g`

to grab the formula
into the Calculator, `a d a[1] RET` to differentiate with respect
to the first variable, and

`C-x * y`to yank the formula for the derivative back into your C program. Press

`U`to undo the differentiation and repeat with

`a d a[2]`for the other derivative.

`RET`Without being switched into C mode first, Calc would have misinterpreted
the brackets in ‘`a[1]`’ and ‘`a[2]`’, would not have known that
`atan`

was equivalent to Calc’s built-in `arctan`

function,
and would have written the formula back with notations (like implicit
multiplication) which would not have been valid for a C program.

As another example, suppose you are maintaining a C program and a LaTeX document, each of which needs a copy of the same formula. You can grab the formula from the program in C mode, switch to LaTeX mode, and yank the formula into the document in LaTeX math-mode format.

Language modes are selected by typing the letter `d` followed by a
shifted letter key.

- Normal Language Modes
- C, FORTRAN, and Pascal Modes
- TeX and LaTeX Language Modes
- Eqn Language Mode
- Yacas Language Mode
- Maxima Language Mode
- Giac Language Mode
- Mathematica Language Mode
- Maple Language Mode
- Compositions
- Syntax Tables

#### 6.8.1 Normal Language Modes

The `d N` (`calc-normal-language`

) command selects the usual
notation for Calc formulas, as described in the rest of this manual.
Matrices are displayed in a multi-line tabular format, but all other
objects are written in linear form, as they would be typed from the
keyboard.

The `d O` (`calc-flat-language`

) command selects a language
identical with the normal one, except that matrices are written in
one-line form along with everything else. In some applications this
form may be more suitable for yanking data into other buffers.

Even in one-line mode, long formulas or vectors will still be split
across multiple lines if they exceed the width of the Calculator window.
The `d b` (`calc-line-breaking`

) command turns this line-breaking
feature on and off. (It works independently of the current language.)
If you give a numeric prefix argument of five or greater to the `d b`
command, that argument will specify the line width used when breaking
long lines.

The `d B` (`calc-big-language`

) command selects a language
which uses textual approximations to various mathematical notations,
such as powers, quotients, and square roots:

____________ | a + 1 2 | ----- + c \| b

in place of ‘`sqrt((a+1)/b + c^2)`’.

Subscripts like ‘`a_i`’ are displayed as actual subscripts in Big
mode. Double subscripts, ‘`a_i_j`’ (‘`subscr(subscr(a, i), j)`’)
are displayed as ‘`a`’ with subscripts separated by commas:
‘`i, j`’. They must still be entered in the usual underscore
notation.

One slight ambiguity of Big notation is that

3 - - 4

can represent either the negative rational number ‘`-3:4`’, or the
actual expression ‘`-(3/4)`’; but the latter formula would normally
never be displayed because it would immediately be evaluated to
‘`-3:4`’ or ‘`-0.75`’, so this ambiguity is not a problem in
typical use.

Non-decimal numbers are displayed with subscripts. Thus there is no
way to tell the difference between ‘`16#C2`’ and ‘`C2_16`’,
though generally you will know which interpretation is correct.
Logarithms ‘`log(x,b)`’ and ‘`log10(x)`’ also use subscripts
in Big mode.

In Big mode, stack entries often take up several lines. To aid
readability, stack entries are separated by a blank line in this mode.
You may find it useful to expand the Calc window’s height using
`C-x ^` (`enlarge-window`

) or to make the Calc window the only
one on the screen with `C-x 1` (`delete-other-windows`

).

Long lines are currently not rearranged to fit the window width in
Big mode, so you may need to use the `<` and `>` keys
to scroll across a wide formula. For really big formulas, you may
even need to use `{` and `}` to scroll up and down.

The `d U` (`calc-unformatted-language`

) command altogether disables
the use of operator notation in formulas. In this mode, the formula
shown above would be displayed:

sqrt(add(div(add(a, 1), b), pow(c, 2)))

These four modes differ only in display format, not in the format expected for algebraic entry. The standard Calc operators work in all four modes, and unformatted notation works in any language mode (except that Mathematica mode expects square brackets instead of parentheses).

#### 6.8.2 C, FORTRAN, and Pascal Modes

The `d C` (`calc-c-language`

) command selects the conventions
of the C language for display and entry of formulas. This differs from
the normal language mode in a variety of (mostly minor) ways. In
particular, C language operators and operator precedences are used in
place of Calc’s usual ones. For example, ‘`a^b`’ means ‘`xor(a,b)`’
in C mode; a value raised to a power is written as a function call,
‘`pow(a,b)`’.

In C mode, vectors and matrices use curly braces instead of brackets.
Octal and hexadecimal values are written with leading ‘`0`’ or ‘`0x`’
rather than using the ‘`#`’ symbol. Array subscripting is
translated into `subscr`

calls, so that ‘`a[i]`’ in C
mode is the same as ‘`a_i`’ in Normal mode. Assignments
turn into the `assign`

function, which Calc normally displays
using the ‘`:=`’ symbol.

The variables `pi`

and `e`

would be displayed ‘`pi`’
and ‘`e`’ in Normal mode, but in C mode they are displayed as
‘`M_PI`’ and ‘`M_E`’, corresponding to the names of constants
typically provided in the `<math.h>` header. Functions whose
names are different in C are translated automatically for entry and
display purposes. For example, entering ‘`asin(x)`’ will push the
formula ‘`arcsin(x)`’ onto the stack; this formula will be displayed
as ‘`asin(x)`’ as long as C mode is in effect.

The `d P` (`calc-pascal-language`

) command selects Pascal
conventions. Like C mode, Pascal mode interprets array brackets and uses
a different table of operators. Hexadecimal numbers are entered and
displayed with a preceding dollar sign. (Thus the regular meaning of
`$2` during algebraic entry does not work in Pascal mode, though
`$` (and `$$`, etc.) not followed by digits works the same as
always.) No special provisions are made for other non-decimal numbers,
vectors, and so on, since there is no universally accepted standard way
of handling these in Pascal.

The `d F` (`calc-fortran-language`

) command selects FORTRAN
conventions. Various function names are transformed into FORTRAN
equivalents. Vectors are written as ‘`/1, 2, 3/`’, and may be
entered this way or using square brackets. Since FORTRAN uses round
parentheses for both function calls and array subscripts, Calc displays
both in the same way; ‘`a(i)`’ is interpreted as a function call
upon reading, and subscripts must be entered as ‘`subscr(a, i)`’.
If the variable `a`

has been declared to have type
`vector`

or `matrix`

, however, then ‘`a(i)`’ will be
parsed as a subscript. (See Declarations.) Usually it doesn’t
matter, though; if you enter the subscript expression ‘`a(i)`’ and
Calc interprets it as a function call, you’ll never know the difference
unless you switch to another language mode or replace `a`

with an
actual vector (or unless `a`

happens to be the name of a built-in
function!).

Underscores are allowed in variable and function names in all of these
language modes. The underscore here is equivalent to the ‘`#`’ in
Normal mode, or to hyphens in the underlying Emacs Lisp variable names.

FORTRAN and Pascal modes normally do not adjust the case of letters in
formulas. Most built-in Calc names use lower-case letters. If you use a
positive numeric prefix argument with `d P` or `d F`, these
modes will use upper-case letters exclusively for display, and will
convert to lower-case on input. With a negative prefix, these modes
convert to lower-case for display and input.

#### 6.8.3 TeX and LaTeX Language Modes

The `d T` (`calc-tex-language`

) command selects the conventions
of “math mode” in Donald Knuth’s TeX typesetting language,
and the `d L` (`calc-latex-language`

) command selects the
conventions of “math mode” in LaTeX, a typesetting language that
uses TeX as its formatting engine. Calc’s LaTeX language mode can
read any formula that the TeX language mode can, although LaTeX
mode may display it differently.

Formulas are entered and displayed in the appropriate notation;
‘`sin(a/b)`’
will appear as ‘`\sin\left( {a \over b} \right)`’ in TeX mode and
‘`\sin\left(\frac{a}{b}\right)`’ in LaTeX mode.
Math formulas are often enclosed by ‘`$ $`’ signs in TeX and
LaTeX; these should be omitted when interfacing with Calc. To Calc,
the ‘`$`’ sign has the same meaning it always does in algebraic
formulas (a reference to an existing entry on the stack).

Complex numbers are displayed as in ‘`3 + 4i`’. Fractions and
quotients are written using `\over`

in TeX mode (as in
`{a \over b}`

) and `\frac`

in LaTeX mode (as in
`\frac{a}{b}`

); binomial coefficients are written with
`\choose`

in TeX mode (as in `{a \choose b}`

) and
`\binom`

in LaTeX mode (as in `\binom{a}{b}`

).
Interval forms are written with `\ldots`

, and error forms are
written with `\pm`

. Absolute values are written as in
‘`|x + 1|`’, and the floor and ceiling functions are written with
`\lfloor`

, `\rfloor`

, etc. The words `\left`

and
`\right`

are ignored when reading formulas in TeX and LaTeX
modes. Both `inf`

and `uinf`

are written as `\infty`

;
when read, `\infty`

always translates to `inf`

.

Function calls are written the usual way, with the function name followed
by the arguments in parentheses. However, functions for which TeX
and LaTeX have special names (like `\sin`

) will use curly braces
instead of parentheses for very simple arguments. During input, curly
braces and parentheses work equally well for grouping, but when the
document is formatted the curly braces will be invisible. Thus the
printed result is
‘`sin 2x`’
but
‘`sin(2 + x)`’.

The TeX specific unit names (see Predefined Units) will not use
the ‘`tex`’ prefix; the unit name for a TeX point will be
‘`pt`’ instead of ‘`texpt`’, for example.

Function and variable names not treated specially by TeX and LaTeX
are simply written out as-is, which will cause them to come out in
italic letters in the printed document. If you invoke `d T` or
`d L` with a positive numeric prefix argument, names of more than
one character will instead be enclosed in a protective commands that
will prevent them from being typeset in the math italics; they will be
written ‘`\hbox{ name}`’ in TeX mode and
‘

`\text{`’ in LaTeX mode. The ‘

`name`}`\hbox{ }`’ and ‘

`\text{ }`’ notations are ignored during reading. If you use a negative prefix argument, such function names are written ‘

`\`’, and function names that begin with

`name``\`

during
reading have the `\`

removed. (Note that in this mode, long
variable names are still written with `\hbox`

or `\text`

.
However, you can always make an actual variable name like `\bar`

in
any TeX mode.)
During reading, text of the form ‘`\matrix{ ... }`’ is replaced
by ‘`[ ... ]`’. The same also applies to `\pmatrix`

and
`\bmatrix`

. In LaTeX mode this also applies to
‘`\begin{matrix} ... \end{matrix}`’,
‘`\begin{bmatrix} ... \end{bmatrix}`’,
‘`\begin{pmatrix} ... \end{pmatrix}`’, as well as
‘`\begin{smallmatrix} ... \end{smallmatrix}`’.
The symbol ‘`&`’ is interpreted as a comma,
and the symbols ‘`\cr`’ and ‘`\\`’ are interpreted as semicolons.
During output, matrices are displayed in ‘`\matrix{ a & b \\ c & d}`’
format in TeX mode and in
‘`\begin{pmatrix} a & b \\ c & d \end{pmatrix}`’ format in
LaTeX mode; you may need to edit this afterwards to change to your
preferred matrix form. If you invoke `d T` or `d L` with an
argument of 2 or −2, then matrices will be displayed in two-dimensional
form, such as

\begin{pmatrix} a & b \\ c & d \end{pmatrix}

This may be convenient for isolated matrices, but could lead to expressions being displayed like

\begin{pmatrix} \times x a & b \\ c & d \end{pmatrix}

While this wouldn’t bother Calc, it is incorrect LaTeX. (Similarly for TeX.)

Accents like `\tilde`

and `\bar`

translate into function
calls internally (‘`tilde(x)`’, ‘`bar(x)`’). The `\underline`

sequence is treated as an accent. The `\vec`

accent corresponds
to the function name `Vec`

, because `vec`

is the name of
a built-in Calc function. The following table shows the accents
in Calc, TeX, LaTeX and *eqn* (described in the next section):

Calc TeX LaTeX eqn ---- --- ----- --- acute \acute \acute Acute \Acute bar \bar \bar bar Bar \Bar breve \breve \breve Breve \Breve check \check \check Check \Check dddot \dddot ddddot \ddddot dot \dot \dot dot Dot \Dot dotdot \ddot \ddot dotdot DotDot \Ddot dyad dyad grave \grave \grave Grave \Grave hat \hat \hat hat Hat \Hat Prime prime tilde \tilde \tilde tilde Tilde \Tilde under \underline \underline under Vec \vec \vec vec VEC \Vec

The ‘`=>`’ (evaluates-to) operator appears as a `\to`

symbol:
‘`{ a \to b}`’. TeX defines

`\to`

as an
alias for `\rightarrow`

. However, if the ‘`=>`’ is the top-level expression being formatted, a slightly different notation is used: ‘

`\evalto`’. The

`a`\to`b``\evalto`

word is ignored by Calc’s input routines, and is undefined in TeX.
You will typically want to include one of the following definitions
at the top of a TeX file that uses `\evalto`

:
\def\evalto{} \def\evalto#1\to{}

The first definition formats evaluates-to operators in the usual
way. The second causes only the `b` part to appear in the
printed document; the `a` part and the arrow are hidden.
Another definition you may wish to use is ‘`\let\to=\Rightarrow`’
which causes `\to`

to appear more like Calc’s ‘`=>`’ symbol.
See The Evaluates-To Operator, for a discussion of `evalto`

.

The complete set of TeX control sequences that are ignored during reading is:

\hbox \mbox \text \left \right \, \> \: \; \! \quad \qquad \hfil \hfill \displaystyle \textstyle \dsize \tsize \scriptstyle \scriptscriptstyle \ssize \ssize \rm \bf \it \sl \roman \bold \italic \slanted \cal \mit \Cal \Bbb \frak \goth \evalto

Note that, because these symbols are ignored, reading a TeX or LaTeX formula into Calc and writing it back out may lose spacing and font information.

Also, the “discretionary multiplication sign” ‘`\*`’ is read
the same as ‘`*`’.

The TeX version of this manual includes some printed examples at the end of this section.

#### 6.8.4 Eqn Language Mode

*Eqn* is another popular formatter for math formulas. It is
designed for use with the TROFF text formatter, and comes standard
with many versions of Unix. The `d E` (`calc-eqn-language`

)
command selects *eqn* notation.

The *eqn* language’s main idiosyncrasy is that whitespace plays
a significant part in the parsing of the language. For example,
‘`sqrt x+1 + y`’ treats ‘`x+1`’ as the argument of the
`sqrt`

operator. *Eqn* also understands more conventional
grouping using curly braces: ‘`sqrt{x+1} + y`’. Braces are
required only when the argument contains spaces.

In Calc’s *eqn* mode, however, curly braces are required to
delimit arguments of operators like `sqrt`

. The first of the
above examples would treat only the ‘`x`’ as the argument of
`sqrt`

, and in fact ‘`sin x+1`’ would be interpreted as
‘`sin * x + 1`’, because `sin`

is not a special operator
in the *eqn* language. If you always surround the argument
with curly braces, Calc will never misunderstand.

Calc also understands parentheses as grouping characters. Another
peculiarity of *eqn*’s syntax makes it advisable to separate
words with spaces from any surrounding characters that aren’t curly
braces, so Calc writes ‘`sin ( x + y )`’ in *eqn* mode.
(The spaces around `sin`

are important to make *eqn*
recognize that `sin`

should be typeset in a roman font, and
the spaces around `x`

and `y`

are a good idea just in
case the *eqn* document has defined special meanings for these
names, too.)

Powers and subscripts are written with the `sub`

and `sup`

operators, respectively. Note that the caret symbol ‘`^`’ is
treated the same as a space in *eqn* mode, as is the ‘`~`’
symbol (these are used to introduce spaces of various widths into
the typeset output of *eqn*).

As in LaTeX mode, Calc’s formatter omits parentheses around the
arguments of functions like `ln`

and `sin`

if they are
“simple-looking”; in this case Calc surrounds the argument with
braces, separated by a ‘`~`’ from the function name: ‘`sin~{x}`’.

Font change codes (like ‘`roman x`’) and positioning codes
(like ‘

`~`’ and ‘

`down`’) are ignored by the

`n``x`*eqn*reader. Also ignored are the words

`left`

, `right`

,
`mark`

, and `lineup`

. Quotation marks in *eqn*mode input are treated the same as curly braces: ‘

`sqrt "1+x"`’ is equivalent to ‘

`sqrt {1+x}`’; this is only an approximation to the true meaning of quotes in

*eqn*, but it is good enough for most uses.

Accent codes (‘` x dot`’) are handled by treating them as
function calls (‘

`dot(`’) internally. See TeX and LaTeX Language Modes, for a table of these accent functions. The

`x`)`prime`

accent is treated specially if it occurs on
a variable or function name: ‘`f prime prime ( x prime )`’ is stored internally as ‘

`f''(x')`’. For example, taking the derivative of ‘

`f(2 x)`’ with

`a d x`will produce ‘

`2 f'(2 x)`’, which

*eqn*mode will display as ‘

`2 f prime ( 2 x )`’.

Assignments are written with the ‘`<-`’ (left-arrow) symbol,
and `evalto`

operators are written with ‘`->`’ or
‘`evalto ... ->`’ (see TeX and LaTeX Language Modes, for a discussion
of this). The regular Calc symbols ‘`:=`’ and ‘`=>`’ are also
recognized for these operators during reading.

Vectors in *eqn* mode use regular Calc square brackets, but
matrices are formatted as ‘`matrix { ccol { a above b } ... }`’.
The words `lcol`

and `rcol`

are recognized as synonyms
for `ccol`

during input, and are generated instead of `ccol`

if the matrix justification mode so specifies.

#### 6.8.5 Yacas Language Mode

The `d Y` (`calc-yacas-language`

) command selects the
conventions of Yacas, a free computer algebra system. While the
operators and functions in Yacas are similar to those of Calc, the names
of built-in functions in Yacas are capitalized. The Calc formula
‘`sin(2 x)`’, for example, is entered and displayed ‘`Sin(2 x)`’
in Yacas mode, and ‘`arcsin(x^2)`’ is ‘`ArcSin(x^2)`’ in Yacas
mode. Complex numbers are written are written ‘`3 + 4 I`’.
The standard special constants are written `Pi`

, `E`

,
`I`

, `GoldenRatio`

and `Gamma`

. `Infinity`

represents both `inf`

and `uinf`

, and `Undefined`

represents `nan`

.

Certain operators on functions, such as `D`

for differentiation
and `Integrate`

for integration, take a prefix form in Yacas. For
example, the derivative of ‘`e^x sin(x)`’ can be computed with
‘`D(x) Exp(x)*Sin(x)`’.

Other notable differences between Yacas and standard Calc expressions
are that vectors and matrices use curly braces in Yacas, and subscripts
use square brackets. If, for example, ‘`A`’ represents the list
‘`{a,2,c,4}`’, then ‘`A[3]`’ would equal ‘`c`’.

#### 6.8.6 Maxima Language Mode

The `d X` (`calc-maxima-language`

) command selects the
conventions of Maxima, another free computer algebra system. The
function names in Maxima are similar, but not always identical, to Calc.
For example, instead of ‘`arcsin(x)`’, Maxima will use
‘`asin(x)`’. Complex numbers are written ‘`3 + 4 %i`’. The
standard special constants are written `%pi`

, `%e`

,
`%i`

, `%phi`

and `%gamma`

. In Maxima, `inf`

means
the same as in Calc, but `infinity`

represents Calc’s `uinf`

.

Underscores as well as percent signs are allowed in function and
variable names in Maxima mode. The underscore again is equivalent to
the ‘`#`’ in Normal mode, and the percent sign is equivalent to
‘`o'o`’.

Maxima uses square brackets for lists and vectors, and matrices are
written as calls to the function `matrix`

, given the row vectors of
the matrix as arguments. Square brackets are also used as subscripts.

#### 6.8.7 Giac Language Mode

The `d A` (`calc-giac-language`

) command selects the
conventions of Giac, another free computer algebra system. The function
names in Giac are similar to Maxima. Complex numbers are written
‘`3 + 4 i`’. The standard special constants in Giac are the same as
in Calc, except that `infinity`

represents both Calc’s `inf`

and `uinf`

.

Underscores are allowed in function and variable names in Giac mode.
Brackets are used for subscripts. In Giac, indexing of lists begins at
0, instead of 1 as in Calc. So if ‘`A`’ represents the list
‘`[a,2,c,4]`’, then ‘`A[2]`’ would equal ‘`c`’. In general,
‘`A[n]`’ in Giac mode corresponds to ‘`A_(n+1)`’ in Normal mode.

The Giac interval notation ‘`2 .. 3`’ has no surrounding brackets;
Calc reads ‘`2 .. 3`’ as the closed interval ‘`[2 .. 3]`’ and
writes any kind of interval as ‘`2 .. 3`’. This means you cannot see
the difference between an open and a closed interval while in Giac mode.

#### 6.8.8 Mathematica Language Mode

The `d M` (`calc-mathematica-language`

) command selects the
conventions of Mathematica. Notable differences in Mathematica mode
are that the names of built-in functions are capitalized, and function
calls use square brackets instead of parentheses. Thus the Calc
formula ‘`sin(2 x)`’ is entered and displayed ‘`Sin[2 x]`’ in
Mathematica mode.

Vectors and matrices use curly braces in Mathematica. Complex numbers
are written ‘`3 + 4 I`’. The standard special constants in Calc are
written `Pi`

, `E`

, `I`

, `GoldenRatio`

, `EulerGamma`

,
`Infinity`

, `ComplexInfinity`

, and `Indeterminate`

in
Mathematica mode.
Non-decimal numbers are written, e.g., ‘`16^^7fff`’. Floating-point
numbers in scientific notation are written ‘`1.23*10.^3`’.
Subscripts use double square brackets: ‘`a[[i]]`’.

#### 6.8.9 Maple Language Mode

The `d W` (`calc-maple-language`

) command selects the
conventions of Maple.

Maple’s language is much like C. Underscores are allowed in symbol
names; square brackets are used for subscripts; explicit ‘`*`’s for
multiplications are required. Use either ‘`^`’ or ‘`**`’ to
denote powers.

Maple uses square brackets for lists and curly braces for sets. Calc
interprets both notations as vectors, and displays vectors with square
brackets. This means Maple sets will be converted to lists when they
pass through Calc. As a special case, matrices are written as calls
to the function `matrix`

, given a list of lists as the argument,
and can be read in this form or with all-capitals `MATRIX`

.

The Maple interval notation ‘`2 .. 3`’ is like Giac’s interval
notation, and is handled the same by Calc.

Maple writes complex numbers as ‘`3 + 4*I`’. Its special constants
are `Pi`

, `E`

, `I`

, and `infinity`

(all three of
`inf`

, `uinf`

, and `nan`

display as `infinity`

).
Floating-point numbers are written ‘`1.23*10.^3`’.

Among things not currently handled by Calc’s Maple mode are the
various quote symbols, procedures and functional operators, and
inert (‘`&`’) operators.

#### 6.8.10 Compositions

There are several *composition functions* which allow you to get
displays in a variety of formats similar to those in Big language
mode. Most of these functions do not evaluate to anything; they are
placeholders which are left in symbolic form by Calc’s evaluator but
are recognized by Calc’s display formatting routines.

Two of these, `string`

and `bstring`

, are described elsewhere.
See Strings. For example, ‘`string("ABC")`’ is displayed as
‘`ABC`’. When viewed on the stack it will be indistinguishable from
the variable `ABC`

, but internally it will be stored as
‘`string([65, 66, 67])`’ and can still be manipulated this way; for
example, the selection and vector commands `j 1 v v j u` would
select the vector portion of this object and reverse the elements, then
deselect to reveal a string whose characters had been reversed.

The composition functions do the same thing in all language modes
(although their components will of course be formatted in the current
language mode). The one exception is Unformatted mode (`d U`),
which does not give the composition functions any special treatment.
The functions are discussed here because of their relationship to
the language modes.

- Composition Basics
- Horizontal Compositions
- Vertical Compositions
- Other Compositions
- Information about Compositions
- User-Defined Compositions

#### 6.8.10.1 Composition Basics

Compositions are generally formed by stacking formulas together
horizontally or vertically in various ways. Those formulas are
themselves compositions. TeX users will find this analogous
to TeX’s “boxes.” Each multi-line composition has a
*baseline*; horizontal compositions use the baselines to
decide how formulas should be positioned relative to one another.
For example, in the Big mode formula

2 a + b 17 + ------ c

the second term of the sum is four lines tall and has line three as its baseline. Thus when the term is combined with 17, line three is placed on the same level as the baseline of 17.

Another important composition concept is *precedence*. This is
an integer that represents the binding strength of various operators.
For example, ‘`*`’ has higher precedence (195) than ‘`+`’ (180),
which means that ‘`(a * b) + c`’ will be formatted without the
parentheses, but ‘`a * (b + c)`’ will keep the parentheses.

The operator table used by normal and Big language modes has the following precedences:

_ 1200 (subscripts) % 1100 (as in n%) ! 1000 (as in !n) mod 400 +/- 300 !! 210 (as in n!!) ! 210 (as in n!) ^ 200 - 197 (as in -n) * 195 (or implicit multiplication) / % \ 190 + - 180 (as in a+b) | 170 < = 160 (and other relations) && 110 || 100 ? : 90 !!! 85 &&& 80 ||| 75 := 50 :: 45 => 40

The general rule is that if an operator with precedence ‘`n`’
occurs as an argument to an operator with precedence ‘`m`’, then
the argument is enclosed in parentheses if ‘`n < m`’. Top-level
expressions and expressions which are function arguments, vector
components, etc., are formatted with precedence zero (so that they
normally never get additional parentheses).

For binary left-associative operators like ‘`+`’, the righthand
argument is actually formatted with one-higher precedence than shown
in the table. This makes sure ‘`(a + b) + c`’ omits the parentheses,
but the unnatural form ‘`a + (b + c)`’ keeps its parentheses.
Right-associative operators like ‘`^`’ format the lefthand argument
with one-higher precedence.

The `cprec`

function formats an expression with an arbitrary
precedence. For example, ‘`cprec(abc, 185)`’ will combine into
sums and products as follows: ‘`7 + abc`’, ‘`7 (abc)`’ (because
this `cprec`

form has higher precedence than addition, but lower
precedence than multiplication).

A final composition issue is *line breaking*. Calc uses two
different strategies for “flat” and “non-flat” compositions.
A non-flat composition is anything that appears on multiple lines
(not counting line breaking). Examples would be matrices and Big
mode powers and quotients. Non-flat compositions are displayed
exactly as specified. If they come out wider than the current
window, you must use horizontal scrolling (`<` and `>`) to
view them.

Flat compositions, on the other hand, will be broken across several
lines if they are too wide to fit the window. Certain points in a
composition are noted internally as *break points*. Calc’s
general strategy is to fill each line as much as possible, then to
move down to the next line starting at the first break point that
didn’t fit. However, the line breaker understands the hierarchical
structure of formulas. It will not break an “inner” formula if
it can use an earlier break point from an “outer” formula instead.
For example, a vector of sums might be formatted as:

[ a + b + c, d + e + f, g + h + i, j + k + l, m ]

If the ‘`m`’ can fit, then so, it seems, could the ‘`g`’.
But Calc prefers to break at the comma since the comma is part
of a “more outer” formula. Calc would break at a plus sign
only if it had to, say, if the very first sum in the vector had
itself been too large to fit.

Of the composition functions described below, only `choriz`

generates break points. The `bstring`

function (see Strings)
also generates breakable items: A break point is added after every
space (or group of spaces) except for spaces at the very beginning or
end of the string.

Composition functions themselves count as levels in the formula
hierarchy, so a `choriz`

that is a component of a larger
`choriz`

will be less likely to be broken. As a special case,
if a `bstring`

occurs as a component of a `choriz`

or
`choriz`

-like object (such as a vector or a list of arguments
in a function call), then the break points in that `bstring`

will be on the same level as the break points of the surrounding
object.

#### 6.8.10.2 Horizontal Compositions

The `choriz`

function takes a vector of objects and composes
them horizontally. For example, ‘`choriz([17, a b/c, d])`’ formats
as ‘`17a b / cd`’ in Normal language mode, or as

a b 17---d c

in Big language mode. This is actually one case of the general
function ‘`choriz( vec, sep, prec)`’, where
either or both of

`sep`and

`prec`may be omitted.

`Prec`gives the

*precedence*to use when formatting each of the components of

`vec`. The default precedence is the precedence from the surrounding environment.

`Sep` is a string (i.e., a vector of character codes as might
be entered with `" "`

notation) which should separate components
of the composition. Also, if `sep` is given, the line breaker
will allow lines to be broken after each occurrence of `sep`.
If `sep` is omitted, the composition will not be breakable
(unless any of its component compositions are breakable).

For example, ‘`2 choriz([a, b c, d = e], " + ", 180)`’ is
formatted as ‘`2 a + b c + (d = e)`’. To get the `choriz`

to have precedence 180 “outwards” as well as “inwards,”
enclose it in a `cprec`

form: ‘`2 cprec(choriz(...), 180)`’
formats as ‘`2 (a + b c + (d = e))`’.

The baseline of a horizontal composition is the same as the baselines of the component compositions, which are all aligned.

#### 6.8.10.3 Vertical Compositions

The `cvert`

function makes a vertical composition. Each
component of the vector is centered in a column. The baseline of
the result is by default the top line of the resulting composition.
For example, ‘`f(cvert([a, bb, ccc]), cvert([a^2 + 1, b^2]))`’
formats in Big mode as

f( a , 2 ) bb a + 1 ccc 2 b

There are several special composition functions that work only as
components of a vertical composition. The `cbase`

function
controls the baseline of the vertical composition; the baseline
will be the same as the baseline of whatever component is enclosed
in `cbase`

. Thus ‘`f(cvert([a, cbase(bb), ccc]),
cvert([a^2 + 1, cbase(b^2)]))`’ displays as

2 a + 1 a 2 f(bb , b ) ccc

There are also `ctbase`

and `cbbase`

functions which
make the baseline of the vertical composition equal to the top
or bottom line (rather than the baseline) of that component.
Thus ‘`cvert([cbase(a / b)]) + cvert([ctbase(a / b)]) +
cvert([cbbase(a / b)])`’ gives

a a - - + a + b b - b

There should be only one `cbase`

, `ctbase`

, or `cbbase`

function in a given vertical composition. These functions can also
be written with no arguments: ‘`ctbase()`’ is a zero-height object
which means the baseline is the top line of the following item, and
‘`cbbase()`’ means the baseline is the bottom line of the preceding
item.

The `crule`

function builds a “rule,” or horizontal line,
across a vertical composition. By itself ‘`crule()`’ uses ‘`-`’
characters to build the rule. You can specify any other character,
e.g., ‘`crule("=")`’. The argument must be a character code or
vector of exactly one character code. It is repeated to match the
width of the widest item in the stack. For example, a quotient
with a thick line is ‘`cvert([a + 1, cbase(crule("=")), b^2])`’:

a + 1 ===== 2 b

Finally, the functions `clvert`

and `crvert`

act exactly
like `cvert`

except that the items are left- or right-justified
in the stack. Thus ‘`clvert([a, bb, ccc]) + crvert([a, bb, ccc])`’
gives:

a + a bb bb ccc ccc

Like `choriz`

, the vertical compositions accept a second argument
which gives the precedence to use when formatting the components.
Vertical compositions do not support separator strings.

#### 6.8.10.4 Other Compositions

The `csup`

function builds a superscripted expression. For
example, ‘`csup(a, b)`’ looks the same as ‘`a^b`’ does in Big
language mode. This is essentially a horizontal composition of
‘`a`’ and ‘`b`’, where ‘`b`’ is shifted up so that its
bottom line is one above the baseline.

Likewise, the `csub`

function builds a subscripted expression.
This shifts ‘`b`’ down so that its top line is one below the
bottom line of ‘`a`’ (note that this is not quite analogous to
`csup`

). Other arrangements can be obtained by using
`choriz`

and `cvert`

directly.

The `cflat`

function formats its argument in “flat” mode,
as obtained by ‘`d O`’, if the current language mode is normal
or Big. It has no effect in other language modes. For example,
‘`a^(b/c)`’ is formatted by Big mode like ‘`csup(a, cflat(b/c))`’
to improve its readability.

The `cspace`

function creates horizontal space. For example,
‘`cspace(4)`’ is effectively the same as ‘`string(" ")`’.
A second string (i.e., vector of characters) argument is repeated
instead of the space character. For example, ‘`cspace(4, "ab")`’
looks like ‘`abababab`’. If the second argument is not a string,
it is formatted in the normal way and then several copies of that
are composed together: ‘`cspace(4, a^2)`’ yields

2 2 2 2 a a a a

If the number argument is zero, this is a zero-width object.

The `cvspace`

function creates vertical space, or a vertical
stack of copies of a certain string or formatted object. The
baseline is the center line of the resulting stack. A numerical
argument of zero will produce an object which contributes zero
height if used in a vertical composition.

There are also `ctspace`

and `cbspace`

functions which
create vertical space with the baseline the same as the baseline
of the top or bottom copy, respectively, of the second argument.
Thus ‘`cvspace(2, a/b) + ctspace(2, a/b) + cbspace(2, a/b)`’
displays as:

a - a b - a a b + - + - a b b - a b - b

#### 6.8.10.5 Information about Compositions

The functions in this section are actual functions; they compose their arguments according to the current language and other display modes, then return a certain measurement of the composition as an integer.

The `cwidth`

function measures the width, in characters, of a
composition. For example, ‘`cwidth(a + b)`’ is 5, and
‘`cwidth(a / b)`’ is 5 in Normal mode, 1 in Big mode, and 11 in
TeX mode (for ‘`{a \over b}`’). The argument may involve
the composition functions described in this section.

The `cheight`

function measures the height of a composition.
This is the total number of lines in the argument’s printed form.

The functions `cascent`

and `cdescent`

measure the amount
of the height that is above (and including) the baseline, or below
the baseline, respectively. Thus ‘`cascent( x) + cdescent(x)`’
always equals ‘

`cheight(`’. For a one-line formula like ‘

`x`)`a + b`’,

`cascent`

returns 1 and `cdescent`

returns 0.
For ‘`a / b`’ in Big mode,

`cascent`

returns 2 and `cdescent`

returns 1. The only formula for which `cascent`

will return zero
is ‘`cvspace(0)`’ or equivalents.

#### 6.8.10.6 User-Defined Compositions

The `Z C` (`calc-user-define-composition`

) command lets you
define the display format for any algebraic function. You provide a
formula containing a certain number of argument variables on the stack.
Any time Calc formats a call to the specified function in the current
language mode and with that number of arguments, Calc effectively
replaces the function call with that formula with the arguments
replaced.

Calc builds the default argument list by sorting all the variable names
that appear in the formula into alphabetical order. You can edit this
argument list before pressing `RET` if you wish. Any variables in
the formula that do not appear in the argument list will be displayed
literally; any arguments that do not appear in the formula will not
affect the display at all.

You can define formats for built-in functions, for functions you have
defined with `Z F` (see Programming with Formulas), or for functions
which have no definitions but are being used as purely syntactic objects.
You can define different formats for each language mode, and for each
number of arguments, using a succession of `Z C` commands. When
Calc formats a function call, it first searches for a format defined
for the current language mode (and number of arguments); if there is
none, it uses the format defined for the Normal language mode. If
neither format exists, Calc uses its built-in standard format for that
function (usually just ‘` func(args)`’).

If you execute `Z C` with the number 0 on the stack instead of a
formula, any defined formats for the function in the current language
mode will be removed. The function will revert to its standard format.

For example, the default format for the binomial coefficient function
‘`choose(n, m)`’ in the Big language mode is

n ( ) m

You might prefer the notation,

C n m

To define this notation, first make sure you are in Big mode, then put the formula

choriz([cvert([cvspace(1), n]), C, cvert([cvspace(1), m])])

on the stack and type `Z C`. Answer the first prompt with
`choose`

. The second prompt will be the default argument list
of ‘`(C m n)`’. Edit this list to be ‘`(n m)`’ and press
`RET`. Now, try it out: For example, turn simplification
off with `m O` and enter ‘`choose(a,b) + choose(7,3)`’
as an algebraic entry.

C + C a b 7 3

As another example, let’s define the usual notation for Stirling
numbers of the first kind, ‘`stir1(n, m)`’. This is just like
the regular format for binomial coefficients but with square brackets
instead of parentheses.

choriz([string("["), cvert([n, cbase(cvspace(1)), m]), string("]")])

Now type `Z C stir1 RET`, edit the argument list to
‘

`(n m)`’, and type

`RET`.

The formula provided to `Z C` usually will involve composition
functions, but it doesn’t have to. Putting the formula ‘`a + b + c`’
onto the stack and typing `Z C foo RET RET` would define
the function ‘

`foo(x,y,z)`’ to display like ‘

`x + y + z`’. This “sum” will act exactly like a real sum for all formatting purposes (it will be parenthesized the same, and so on). However it will be computationally unrelated to a sum. For example, the formula ‘

`2 * foo(1, 2, 3)`’ will display as ‘

`2 (1 + 2 + 3)`’. Operator precedences have caused the “sum” to be written in parentheses, but the arguments have not actually been summed. (Generally a display format like this would be undesirable, since it can easily be confused with a real sum.)

The special function `eval`

can be used inside a `Z C`
composition formula to cause all or part of the formula to be
evaluated at display time. For example, if the formula is
‘`a + eval(b + c)`’, then ‘`foo(1, 2, 3)`’ will be displayed
as ‘`1 + 5`’. Evaluation will use the default simplifications,
regardless of the current simplification mode. There are also
`evalsimp`

and `evalextsimp`

which simplify as if by
`a s` and `a e` (respectively). Note that these “functions”
operate only in the context of composition formulas (and also in
rewrite rules, where they serve a similar purpose; see Rewrite Rules). On the stack, a call to `eval`

will be left in
symbolic form.

It is not a good idea to use `eval`

except as a last resort.
It can cause the display of formulas to be extremely slow. For
example, while ‘`eval(a + b)`’ might seem quite fast and simple,
there are several situations where it could be slow. For example,
‘`a`’ and/or ‘`b`’ could be polar complex numbers, in which
case doing the sum requires trigonometry. Or, ‘`a`’ could be
the factorial ‘`fact(100)`’ which is unevaluated because you
have typed `m O`; `eval`

will evaluate it anyway to
produce a large, unwieldy integer.

You can save your display formats permanently using the `Z P`
command (see Creating User Keys).

#### 6.8.11 Syntax Tables

Syntax tables do for input what compositions do for output: They allow you to teach custom notations to Calc’s formula parser. Calc keeps a separate syntax table for each language mode.

(Note that the Calc “syntax tables” discussed here are completely unrelated to the syntax tables described in the Emacs manual.)

The `Z S` (`calc-edit-user-syntax`

) command edits the
syntax table for the current language mode. If you want your
syntax to work in any language, define it in the Normal language
mode. Type `C-c C-c` to finish editing the syntax table, or
`C-x k` to cancel the edit. The `m m` command saves all
the syntax tables along with the other mode settings;
see General Mode Commands.

#### 6.8.11.1 Syntax Table Basics

*Parsing* is the process of converting a raw string of characters,
such as you would type in during algebraic entry, into a Calc formula.
Calc’s parser works in two stages. First, the input is broken down
into *tokens*, such as words, numbers, and punctuation symbols
like ‘`+`’, ‘`:=`’, and ‘`+/-`’. Space between tokens is
ignored (except when it serves to separate adjacent words). Next,
the parser matches this string of tokens against various built-in
syntactic patterns, such as “an expression followed by ‘`+`’
followed by another expression” or “a name followed by ‘`(`’,
zero or more expressions separated by commas, and ‘`)`’.”

A *syntax table* is a list of user-defined *syntax rules*,
which allow you to specify new patterns to define your own
favorite input notations. Calc’s parser always checks the syntax
table for the current language mode, then the table for the Normal
language mode, before it uses its built-in rules to parse an
algebraic formula you have entered. Each syntax rule should go on
its own line; it consists of a *pattern*, a ‘`:=`’ symbol,
and a Calc formula with an optional *condition*. (Syntax rules
resemble algebraic rewrite rules, but the notation for patterns is
completely different.)

A syntax pattern is a list of tokens, separated by spaces. Except for a few special symbols, tokens in syntax patterns are matched literally, from left to right. For example, the rule,

foo ( ) := 2+3

would cause Calc to parse the formula ‘`4+foo()*5`’ as if it
were ‘`4+(2+3)*5`’. Notice that the parentheses were written
as two separate tokens in the rule. As a result, the rule works
for both ‘`foo()`’ and ‘`foo ( )`’. If we had written
the rule as ‘`foo () := 2+3`’, then Calc would treat ‘`()`’
as a single, indivisible token, so that ‘`foo( )`’ would
not be recognized by the rule. (It would be parsed as a regular
zero-argument function call instead.) In fact, this rule would
also make trouble for the rest of Calc’s parser: An unrelated
formula like ‘`bar()`’ would now be tokenized into ‘`bar ()`’
instead of ‘`bar ( )`’, so that the standard parser for function
calls would no longer recognize it!

While it is possible to make a token with a mixture of letters
and punctuation symbols, this is not recommended. It is better to
break it into several tokens, as we did with ‘`foo()`’ above.

The symbol ‘`#`’ in a syntax pattern matches any Calc expression.
On the righthand side, the things that matched the ‘`#`’s can
be referred to as ‘`#1`’, ‘`#2`’, and so on (where ‘`#1`’
matches the leftmost ‘`#`’ in the pattern). For example, these
rules match a user-defined function, prefix operator, infix operator,
and postfix operator, respectively:

foo ( # ) := myfunc(#1) foo # := myprefix(#1) # foo # := myinfix(#1,#2) # foo := mypostfix(#1)

Thus ‘`foo(3)`’ will parse as ‘`myfunc(3)`’, and ‘`2+3 foo`’
will parse as ‘`mypostfix(2+3)`’.

It is important to write the first two rules in the order shown,
because Calc tries rules in order from first to last. If the
pattern ‘`foo #`’ came first, it would match anything that could
match the ‘`foo ( # )`’ rule, since an expression in parentheses
is itself a valid expression. Thus the ‘`foo ( # )`’ rule would
never get to match anything. Likewise, the last two rules must be
written in the order shown or else ‘`3 foo 4`’ will be parsed as
‘`mypostfix(3) * 4`’. (Of course, the best way to avoid these
ambiguities is not to use the same symbol in more than one way at
the same time! In case you’re not convinced, try the following
exercise: How will the above rules parse the input ‘`foo(3,4)`’,
if at all? Work it out for yourself, then try it in Calc and see.)

Calc is quite flexible about what sorts of patterns are allowed.
The only rule is that every pattern must begin with a literal
token (like ‘`foo`’ in the first two patterns above), or with
a ‘`#`’ followed by a literal token (as in the last two
patterns). After that, any mixture is allowed, although putting
two ‘`#`’s in a row will not be very useful since two
expressions with nothing between them will be parsed as one
expression that uses implicit multiplication.

As a more practical example, Maple uses the notation
‘`sum(a(i), i=1..10)`’ for sums, which Calc’s Maple mode doesn’t
recognize at present. To handle this syntax, we simply add the
rule,

sum ( # , # = # .. # ) := sum(#1,#2,#3,#4)

to the Maple mode syntax table. As another example, C mode can’t
read assignment operators like ‘`++`’ and ‘`*=`’. We can
define these operators quite easily:

# *= # := muleq(#1,#2) # ++ := postinc(#1) ++ # := preinc(#1)

To complete the job, we would use corresponding composition functions
and `Z C` to cause these functions to display in their respective
Maple and C notations. (Note that the C example ignores issues of
operator precedence, which are discussed in the next section.)

You can enclose any token in quotes to prevent its usual interpretation in syntax patterns:

# ":=" # := becomes(#1,#2)

Quotes also allow you to include spaces in a token, although once again it is generally better to use two tokens than one token with an embedded space. To include an actual quotation mark in a quoted token, precede it with a backslash. (This also works to include backslashes in tokens.)

# "bad token" # "/\"\\" # := silly(#1,#2,#3)

This will parse ‘`3 bad token 4 /"\ 5`’ to ‘`silly(3,4,5)`’.

The token `#` has a predefined meaning in Calc’s formula parser;
it is not valid to use ‘`"#"`’ in a syntax rule. However, longer
tokens that include the ‘`#`’ character are allowed. Also, while
‘`"$"`’ and ‘`"\""`’ are allowed as tokens, their presence in
the syntax table will prevent those characters from working in their
usual ways (referring to stack entries and quoting strings,
respectively).

Finally, the notation ‘`%%`’ anywhere in a syntax table causes
the rest of the line to be ignored as a comment.

#### 6.8.11.2 Precedence

Different operators are generally assigned different *precedences*.
By default, an operator defined by a rule like

# foo # := foo(#1,#2)

will have an extremely low precedence, so that ‘`2*3+4 foo 5 == 6`’
will be parsed as ‘`(2*3+4) foo (5 == 6)`’. To change the
precedence of an operator, use the notation ‘`#/ p`’ in
place of ‘

`#`’, where

`p`is an integer precedence level. For example, 185 lies between the precedences for ‘

`+`’ and ‘

`*`’, so if we change this rule to

#/185 foo #/186 := foo(#1,#2)

then ‘`2+3 foo 4*5`’ will be parsed as ‘`2+(3 foo (4*5))`’.
Also, because we’ve given the righthand expression slightly higher
precedence, our new operator will be left-associative:
‘`1 foo 2 foo 3`’ will be parsed as ‘`(1 foo 2) foo 3`’.
By raising the precedence of the lefthand expression instead, we
can create a right-associative operator.

See Composition Basics, for a table of precedences of the
standard Calc operators. For the precedences of operators in other
language modes, look in the Calc source file `calc-lang.el`.

#### 6.8.11.3 Advanced Syntax Patterns

To match a function with a variable number of arguments, you could write

foo ( # ) := myfunc(#1) foo ( # , # ) := myfunc(#1,#2) foo ( # , # , # ) := myfunc(#1,#2,#3)

but this isn’t very elegant. To match variable numbers of items, Calc uses some notations inspired regular expressions and the “extended BNF” style used by some language designers.

foo ( { # }*, ) := apply(myfunc,#1)

The token ‘`{`’ introduces a repeated or optional portion.
One of the three tokens ‘`}*`’, ‘`}+`’, or ‘`}?`’
ends the portion. These will match zero or more, one or more,
or zero or one copies of the enclosed pattern, respectively.
In addition, ‘`}*`’ and ‘`}+`’ can be followed by a
separator token (with no space in between, as shown above).
Thus ‘`{ # }*,`’ matches nothing, or one expression, or
several expressions separated by commas.

A complete ‘`{ ... }`’ item matches as a vector of the
items that matched inside it. For example, the above rule will
match ‘`foo(1,2,3)`’ to get ‘`apply(myfunc,[1,2,3])`’.
The Calc `apply`

function takes a function name and a vector
of arguments and builds a call to the function with those
arguments, so the net result is the formula ‘`myfunc(1,2,3)`’.

If the body of a ‘`{ ... }`’ contains several ‘`#`’s
(or nested ‘`{ ... }`’ constructs), then the items will be
strung together into the resulting vector. If the body
does not contain anything but literal tokens, the result will
always be an empty vector.

foo ( { # , # }+, ) := bar(#1) foo ( { { # }*, }*; ) := matrix(#1)

will parse ‘`foo(1, 2, 3, 4)`’ as ‘`bar([1, 2, 3, 4])`’, and
‘`foo(1, 2; 3, 4)`’ as ‘`matrix([[1, 2], [3, 4]])`’. Also, after
some thought it’s easy to see how this pair of rules will parse
‘`foo(1, 2, 3)`’ as ‘`matrix([[1, 2, 3]])`’, since the first
rule will only match an even number of arguments. The rule

foo ( # { , # , # }? ) := bar(#1,#2)

will parse ‘`foo(2,3,4)`’ as ‘`bar(2,[3,4])`’, and
‘`foo(2)`’ as ‘`bar(2,[])`’.

The notation ‘`{ ... }?.`’ (note the trailing period) works
just the same as regular ‘`{ ... }?`’, except that it does not
count as an argument; the following two rules are equivalent:

foo ( # , { also }? # ) := bar(#1,#3) foo ( # , { also }?. # ) := bar(#1,#2)

Note that in the first case the optional text counts as ‘`#2`’,
which will always be an empty vector, but in the second case no
empty vector is produced.

Another variant is ‘`{ ... }?$`’, which means the body is
optional only at the end of the input formula. All built-in syntax
rules in Calc use this for closing delimiters, so that during
algebraic entry you can type `[sqrt(2), sqrt(3 RET`, omitting
the closing parenthesis and bracket. Calc does this automatically
for trailing ‘

`)`’, ‘

`]`’, and ‘

`>`’ tokens in syntax rules, but you can use ‘

`{ ... }?$`’ explicitly to get this effect with any token (such as ‘

`"}"`’ or ‘

`end`’). Like ‘

`{ ... }?.`’, this notation does not count as an argument. Conversely, you can use quotes, as in ‘

`")"`’, to prevent a closing-delimiter token from being automatically treated as optional.

Calc’s parser does not have full backtracking, which means some patterns will not work as you might expect:

foo ( { # , }? # , # ) := bar(#1,#2,#3)

Here we are trying to make the first argument optional, so that
‘`foo(2,3)`’ parses as ‘`bar([],2,3)`’. Unfortunately, Calc
first tries to match ‘`2,`’ against the optional part of the
pattern, finds a match, and so goes ahead to match the rest of the
pattern. Later on it will fail to match the second comma, but it
doesn’t know how to go back and try the other alternative at that
point. One way to get around this would be to use two rules:

foo ( # , # , # ) := bar([#1],#2,#3) foo ( # , # ) := bar([],#1,#2)

More precisely, when Calc wants to match an optional or repeated part of a pattern, it scans forward attempting to match that part. If it reaches the end of the optional part without failing, it “finalizes” its choice and proceeds. If it fails, though, it backs up and tries the other alternative. Thus Calc has “partial” backtracking. A fully backtracking parser would go on to make sure the rest of the pattern matched before finalizing the choice.

#### 6.8.11.4 Conditional Syntax Rules

It is possible to attach a *condition* to a syntax rule. For
example, the rules

foo ( # ) := ifoo(#1) :: integer(#1) foo ( # ) := gfoo(#1)

will parse ‘`foo(3)`’ as ‘`ifoo(3)`’, but will parse
‘`foo(3.5)`’ and ‘`foo(x)`’ as calls to `gfoo`

. Any
number of conditions may be attached; all must be true for the
rule to succeed. A condition is “true” if it evaluates to a
nonzero number. See Logical Operations, for a list of Calc
functions like `integer`

that perform logical tests.

The exact sequence of events is as follows: When Calc tries a
rule, it first matches the pattern as usual. It then substitutes
‘`#1`’, ‘`#2`’, etc., in the conditions, if any. Next, the
conditions are simplified and evaluated in order from left to right,
using the algebraic simplifications (see Simplifying Formulas).
Each result is true if it is a nonzero number, or an expression
that can be proven to be nonzero (see Declarations). If the
results of all conditions are true, the expression (such as
‘`ifoo(#1)`’) has its ‘`#`’s substituted, and that is the
result of the parse. If the result of any condition is false, Calc
goes on to try the next rule in the syntax table.

Syntax rules also support `let`

conditions, which operate in
exactly the same way as they do in algebraic rewrite rules.
See Other Features of Rewrite Rules, for details. A `let`

condition is always true, but as a side effect it defines a
variable which can be used in later conditions, and also in the
expression after the ‘`:=`’ sign:

foo ( # ) := hifoo(x) :: let(x := #1 + 0.5) :: dnumint(x)

The `dnumint`

function tests if a value is numerically an
integer, i.e., either a true integer or an integer-valued float.
This rule will parse `foo`

with a half-integer argument,
like ‘`foo(3.5)`’, to a call like ‘`hifoo(4.)`’.

The lefthand side of a syntax rule `let`

must be a simple
variable, not the arbitrary pattern that is allowed in rewrite
rules.

The `matches`

function is also treated specially in syntax
rule conditions (again, in the same way as in rewrite rules).
See Matching Commands. If the matching pattern contains
meta-variables, then those meta-variables may be used in later
conditions and in the result expression. The arguments to
`matches`

are not evaluated in this situation.

sum ( # , # ) := sum(#1,a,b,c) :: matches(#2, a=[b..c])

This is another way to implement the Maple mode `sum`

notation.
In this approach, we allow ‘`#2`’ to equal the whole expression
‘`i=1..10`’. Then, we use `matches`

to break it apart into
its components. If the expression turns out not to match the pattern,
the syntax rule will fail. Note that `Z S` always uses Calc’s
Normal language mode for editing expressions in syntax rules, so we
must use regular Calc notation for the interval ‘`[b..c]`’ that
will correspond to the Maple mode interval ‘`1..10`’.

### 6.9 The `Modes`

Variable

The `m g` (`calc-get-modes`

) command pushes onto the stack
a vector of numbers that describes the various mode settings that
are in effect. With a numeric prefix argument, it pushes only the
`n`th mode, i.e., the `n`th element of this vector. Keyboard
macros can use the `m g` command to modify their behavior based
on the current mode settings.

The modes vector is also available in the special variable
`Modes`

. In other words, `m g` is like `s r Modes RET`.
It will not work to store into this variable; in fact, if you do,

`Modes`

will cease to track the current modes. (The `m g`command will continue to work, however.)

In general, each number in this vector is suitable as a numeric
prefix argument to the associated mode-setting command. (Recall
that the `~` key takes a number from the stack and gives it as
a numeric prefix to the next command.)

The elements of the modes vector are as follows:

- Current precision. Default is 12; associated command is
`p`. - Binary word size. Default is 32; associated command is
`b w`. - Stack size (not counting the value about to be pushed by
`m g`). This is zero if`m g`is executed with an empty stack. - Number radix. Default is 10; command is
`d r`. - Floating-point format. This is the number of digits, plus the
constant 0 for normal notation, 10000 for scientific notation,
20000 for engineering notation, or 30000 for fixed-point notation.
These codes are acceptable as prefix arguments to the
`d n`command, but note that this may lose information: For example,`d s`and`C-u 12 d s`have similar (but not quite identical) effects if the current precision is 12, but they both produce a code of 10012, which will be treated by`d n`as`C-u 12 d s`. If the precision then changes, the float format will still be frozen at 12 significant figures. - Angular mode. Default is 1 (degrees). Other values are 2 (radians)
and 3 (HMS). The
`m d`command accepts these prefixes. - Symbolic mode. Value is 0 or 1; default is 0. Command is
`m s`. - Fraction mode. Value is 0 or 1; default is 0. Command is
`m f`. - Polar mode. Value is 0 (rectangular) or 1 (polar); default is 0.
Command is
`m p`. - Matrix/Scalar mode. Default value is
*-1*. Value is 0 for Scalar mode,*-2*for Matrix mode,*-3*for square Matrix mode, or`N`for`N`x`N`Matrix mode. Command is`m v`. - Simplification mode. Default is 1. Value is
*-1*for off (`m O`), 0 for`m N`, 2 for`m B`, 3 for`m A`, 4 for`m E`, or 5 for`m U`. The`m D`command accepts these prefixes. - Infinite mode. Default is
*-1*(off). Value is 1 if the mode is on, or 0 if the mode is on with positive zeros. Command is`m i`.

For example, the sequence `M-1 m g RET 2 + ~ p` increases the
precision by two, leaving a copy of the old precision on the stack.
Later,

`~ p`will restore the original precision using that stack value. (This sequence might be especially useful inside a keyboard macro.)

As another example, `M-3 m g 1 - ~ DEL` deletes all but the
oldest (bottommost) stack entry.

Yet another example: The HP-48 “round” command rounds a number
to the current displayed precision. You could roughly emulate this
in Calc with the sequence `M-5 m g 10000 % ~ c c`. (This
would not work for fixed-point mode, but it wouldn’t be hard to
do a full emulation with the help of the `Z [` and `Z ]`
programming commands. See Conditionals in Keyboard Macros.)

### 6.10 The Calc Mode Line

This section is a summary of all symbols that can appear on the Calc mode line, the highlighted bar that appears under the Calc stack window (or under an editing window in Embedded mode).

The basic mode line format is:

--%*-Calc: 12 Degother modes(Calculator)

The ‘`%*`’ indicates that the buffer is “read-only”; it shows that
regular Emacs commands are not allowed to edit the stack buffer
as if it were text.

The word ‘`Calc:`’ changes to ‘`CalcEmbed:`’ if Embedded mode
is enabled. The words after this describe the various Calc modes
that are in effect.

The first mode is always the current precision, an integer.
The second mode is always the angular mode, either `Deg`

,
`Rad`

, or `Hms`

.

Here is a complete list of the remaining symbols that can appear on the mode line:

`Alg`

Algebraic mode (

`m a`; see Algebraic Entry).`Alg[(`

Incomplete algebraic mode (

`C-u m a`).`Alg*`

Total algebraic mode (

`m t`).`Symb`

Symbolic mode (

`m s`; see Symbolic Mode).`Matrix`

Matrix mode (

`m v`; see Matrix and Scalar Modes).`Matrix`

`n`Dimensioned Matrix mode (

`C-u`; see Matrix and Scalar Modes).`n`m v`SqMatrix`

Square Matrix mode (

`C-u m v`; see Matrix and Scalar Modes).`Scalar`

Scalar mode (

`m v`; see Matrix and Scalar Modes).`Polar`

Polar complex mode (

`m p`; see Polar Mode).`Frac`

Fraction mode (

`m f`; see Fraction Mode).`Inf`

Infinite mode (

`m i`; see Infinite Mode).`+Inf`

Positive Infinite mode (

`C-u 0 m i`).`NoSimp`

Default simplifications off (

`m O`; see Simplification Modes).`NumSimp`

Default simplifications for numeric arguments only (

`m N`).`BinSimp`

`w`Binary-integer simplification mode; word size

`w`(`m B`,`b w`).`BasicSimp`

Basic simplification mode (

`m I`).`ExtSimp`

Extended algebraic simplification mode (

`m E`).`UnitSimp`

Units simplification mode (

`m U`).`Bin`

Current radix is 2 (

`d 2`; see Radix Modes).`Oct`

Current radix is 8 (

`d 8`).`Hex`

Current radix is 16 (

`d 6`).`Radix`

`n`Current radix is

`n`(`d r`).`Zero`

Leading zeros (

`d z`; see Radix Modes).`Big`

Big language mode (

`d B`; see Normal Language Modes).`Flat`

One-line normal language mode (

`d O`).`Unform`

Unformatted language mode (

`d U`).`C`

C language mode (

`d C`; see C, FORTRAN, and Pascal Modes).`Pascal`

Pascal language mode (

`d P`).`Fortran`

FORTRAN language mode (

`d F`).`TeX`

TeX language mode (

`d T`; see TeX and LaTeX Language Modes).`LaTeX`

LaTeX language mode (

`d L`; see TeX and LaTeX Language Modes).`Eqn`

*Eqn*language mode (`d E`; see Eqn Language Mode).`Math`

Mathematica language mode (

`d M`; see Mathematica Language Mode).`Maple`

Maple language mode (

`d W`; see Maple Language Mode).`Norm`

`n`Normal float mode with

`n`digits (`d n`; see Float Formats).`Fix`

`n`Fixed point mode with

`n`digits after the point (`d f`).`Sci`

Scientific notation mode (

`d s`).`Sci`

`n`Scientific notation with

`n`digits (`d s`).`Eng`

Engineering notation mode (

`d e`).`Eng`

`n`Engineering notation with

`n`digits (`d e`).`Left`

`n`Left-justified display indented by

`n`(`d <`; see Justification).`Right`

Right-justified display (

`d >`).`Right`

`n`Right-justified display with width

`n`(`d >`).`Center`

Centered display (

`d =`).`Center`

`n`Centered display with center column

`n`(`d =`).`Wid`

`n`Line breaking with width

`n`(`d b`; see Normal Language Modes).`Wide`

No line breaking (

`d b`).`Break`

Selections show deep structure (

`j b`; see Making Selections).`Save`

Record modes in

`~/.emacs.d/calc.el`(`m R`; see General Mode Commands).`Local`

Record modes in Embedded buffer (

`m R`).`LocEdit`

Record modes as editing-only in Embedded buffer (

`m R`).`LocPerm`

Record modes as permanent-only in Embedded buffer (

`m R`).`Global`

Record modes as global in Embedded buffer (

`m R`).`Manual`

Automatic recomputation turned off (

`m C`; see Automatic Recomputation).`Graph`

GNUPLOT process is alive in background (see Graphics).

`Sel`

Top-of-stack has a selection (Embedded only; see Making Selections).

`Dirty`

The stack display may not be up-to-date (see Display Modes).

`Inv`

“Inverse” prefix was pressed (

`I`; see Inverse and Hyperbolic Flags).`Hyp`

“Hyperbolic” prefix was pressed (

`H`).`Keep`

“Keep-arguments” prefix was pressed (

`K`).`Narrow`

Stack is truncated (

`d t`; see Truncating the Stack).

In addition, the symbols `Active`

and `~Active`

can appear
as minor modes on an Embedded buffer’s mode line. See Embedded Mode.

## 7 Arithmetic Functions

This chapter describes the Calc commands for doing simple calculations
on numbers, such as addition, absolute value, and square roots. These
commands work by removing the top one or two values from the stack,
performing the desired operation, and pushing the result back onto the
stack. If the operation cannot be performed, the result pushed is a
formula instead of a number, such as ‘`2/0`’ (because division by zero
is invalid) or ‘`sqrt(x)`’ (because the argument ‘`x`’ is a formula).

Most of the commands described here can be invoked by a single keystroke.
Some of the more obscure ones are two-letter sequences beginning with
the `f` (“functions”) prefix key.

See Numeric Prefix Arguments, for a discussion of the effect of numeric prefix arguments on commands in this chapter which do not otherwise interpret a prefix argument.

- Basic Arithmetic
- Integer Truncation
- Complex Number Functions
- Conversions
- Date Arithmetic
- Financial Functions
- Binary Number Functions

### 7.1 Basic Arithmetic

The `+` (`calc-plus`

) command adds two numbers. The numbers may
be any of the standard Calc data types. The resulting sum is pushed back
onto the stack.

If both arguments of `+` are vectors or matrices (of matching dimensions),
the result is a vector or matrix sum. If one argument is a vector and the
other a scalar (i.e., a non-vector), the scalar is added to each of the
elements of the vector to form a new vector. If the scalar is not a
number, the operation is left in symbolic form: Suppose you added ‘`x`’
to the vector ‘`[1,2]`’. You may want the result ‘`[1+x,2+x]`’, or
you may plan to substitute a 2-vector for ‘`x`’ in the future. Since
the Calculator can’t tell which interpretation you want, it makes the
safest assumption. See Reducing and Mapping Vectors, for a way to add ‘`x`’
to every element of a vector.

If either argument of `+` is a complex number, the result will in general
be complex. If one argument is in rectangular form and the other polar,
the current Polar mode determines the form of the result. If Symbolic
mode is enabled, the sum may be left as a formula if the necessary
conversions for polar addition are non-trivial.

If both arguments of `+` are HMS forms, the forms are added according to
the usual conventions of hours-minutes-seconds notation. If one argument
is an HMS form and the other is a number, that number is converted from
degrees or radians (depending on the current Angular mode) to HMS format
and then the two HMS forms are added.

If one argument of `+` is a date form, the other can be either a
real number, which advances the date by a certain number of days, or
an HMS form, which advances the date by a certain amount of time.
Subtracting two date forms yields the number of days between them.
Adding two date forms is meaningless, but Calc interprets it as the
subtraction of one date form and the negative of the other. (The
negative of a date form can be understood by remembering that dates
are stored as the number of days before or after Jan 1, 1 AD.)

If both arguments of `+` are error forms, the result is an error form
with an appropriately computed standard deviation. If one argument is an
error form and the other is a number, the number is taken to have zero error.
Error forms may have symbolic formulas as their mean and/or error parts;
adding these will produce a symbolic error form result. However, adding an
error form to a plain symbolic formula (as in ‘`(a +/- b) + c`’) will not
work, for the same reasons just mentioned for vectors. Instead you must
write ‘`(a +/- b) + (c +/- 0)`’.

If both arguments of `+` are modulo forms with equal values of ‘`M`’,
or if one argument is a modulo form and the other a plain number, the
result is a modulo form which represents the sum, modulo ‘`M`’, of
the two values.

If both arguments of `+` are intervals, the result is an interval
which describes all possible sums of the possible input values. If
one argument is a plain number, it is treated as the interval
‘`[x .. x]`’.

If one argument of `+` is an infinity and the other is not, the
result is that same infinity. If both arguments are infinite and in
the same direction, the result is the same infinity, but if they are
infinite in different directions the result is `nan`

.

The `-` (`calc-minus`

) command subtracts two values. The top
number on the stack is subtracted from the one behind it, so that the
computation `5 RET 2 -` produces 3, not

*-3*. All options available for

`+`are available for

`-`as well.

The `*` (`calc-times`

) command multiplies two numbers. If one
argument is a vector and the other a scalar, the scalar is multiplied by
the elements of the vector to produce a new vector. If both arguments
are vectors, the interpretation depends on the dimensions of the
vectors: If both arguments are matrices, a matrix multiplication is
done. If one argument is a matrix and the other a plain vector, the
vector is interpreted as a row vector or column vector, whichever is
dimensionally correct. If both arguments are plain vectors, the result
is a single scalar number which is the dot product of the two vectors.

If one argument of `*` is an HMS form and the other a number, the
HMS form is multiplied by that amount. It is an error to multiply two
HMS forms together, or to attempt any multiplication involving date
forms. Error forms, modulo forms, and intervals can be multiplied;
see the comments for addition of those forms. When two error forms
or intervals are multiplied they are considered to be statistically
independent; thus, ‘`[-2 .. 3] * [-2 .. 3]`’ is ‘`[-6 .. 9]`’,
whereas ‘`[-2 .. 3] ^ 2`’ is ‘`[0 .. 9]`’.

The `/` (`calc-divide`

) command divides two numbers.

When combining multiplication and division in an algebraic formula, it
is good style to use parentheses to distinguish between possible
interpretations; the expression ‘`a/b*c`’ should be written
‘`(a/b)*c`’ or ‘`a/(b*c)`’, as appropriate. Without the
parentheses, Calc will interpret ‘`a/b*c`’ as ‘`a/(b*c)`’, since
in algebraic entry Calc gives division a lower precedence than
multiplication. (This is not standard across all computer languages, and
Calc may change the precedence depending on the language mode being used.
See Language Modes.) This default ordering can be changed by setting
the customizable variable `calc-multiplication-has-precedence`

to
`nil`

(see Customizing Calc); this will give multiplication and
division equal precedences. Note that Calc’s default choice of
precedence allows ‘`a b / c d`’ to be used as a shortcut for

a b ---. c d

When dividing a scalar ‘`B`’ by a square matrix ‘`A`’, the
computation performed is ‘`B`’ times the inverse of ‘`A`’. This
also occurs if ‘`B`’ is itself a vector or matrix, in which case the
effect is to solve the set of linear equations represented by ‘`B`’.
If ‘`B`’ is a matrix with the same number of rows as ‘`A`’, or a
plain vector (which is interpreted here as a column vector), then the
equation ‘`A X = B`’ is solved for the vector or matrix ‘`X`’.
Otherwise, if ‘`B`’ is a non-square matrix with the same number of
*columns* as ‘`A`’, the equation ‘`X A = B`’ is solved. If
you wish a vector ‘`B`’ to be interpreted as a row vector to be
solved as ‘`X A = B`’, make it into a one-row matrix with `C-u 1
v p` first. To force a left-handed solution with a square matrix
‘`B`’, transpose ‘`A`’ and ‘`B`’ before dividing, then
transpose the result.

HMS forms can be divided by real numbers or by other HMS forms. Error forms can be divided in any combination of ways. Modulo forms where both values and the modulo are integers can be divided to get an integer modulo form result. Intervals can be divided; dividing by an interval that encompasses zero or has zero as a limit will result in an infinite interval.

The `^` (`calc-power`

) command raises a number to a power. If
the power is an integer, an exact result is computed using repeated
multiplications. For non-integer powers, Calc uses Newton’s method or
logarithms and exponentials. Square matrices can be raised to integer
powers. If either argument is an error (or interval or modulo) form,
the result is also an error (or interval or modulo) form.

If you press the `I` (inverse) key first, the `I ^` command
computes an Nth root: `125 RET 3 I ^` computes the number 5.
(This is entirely equivalent to

`125`.)

`RET`1:3 ^The `\` (`calc-idiv`

) command divides two numbers on the stack
to produce an integer result. It is equivalent to dividing with
`/`, then rounding down with `F` (`calc-floor`

), only a bit
more convenient and efficient. Also, since it is an all-integer
operation when the arguments are integers, it avoids problems that
`/ F` would have with floating-point roundoff.

The `%` (`calc-mod`

) command performs a “modulo” (or “remainder”)
operation. Mathematically, ‘`a%b = a - (a\b)*b`’, and is defined
for all real numbers ‘`a`’ and ‘`b`’ (except ‘`b=0`’). For
positive ‘`b`’, the result will always be between 0 (inclusive) and
‘`b`’ (exclusive). Modulo does not work for HMS forms and error forms.
If ‘`a`’ is a modulo form, its modulo is changed to ‘`b`’, which
must be positive real number.

The `:` (`calc-fdiv`

) [`fdiv`

] command
divides the two integers on the top of the stack to produce a fractional
result. This is a convenient shorthand for enabling Fraction mode (with
`m f`) temporarily and using ‘`/`’. Note that during numeric entry
the `:` key is interpreted as a fraction separator, so to divide 8 by 6
you would have to type `8 RET 6 RET :`. (Of course, in
this case, it would be much easier simply to enter the fraction directly
as

`8:6`!)

`RET`The `n` (`calc-change-sign`

) command negates the number on the top
of the stack. It works on numbers, vectors and matrices, HMS forms, date
forms, error forms, intervals, and modulo forms.

The `A` (`calc-abs`

) [`abs`

] command computes the absolute
value of a number. The result of `abs`

is always a nonnegative
real number: With a complex argument, it computes the complex magnitude.
With a vector or matrix argument, it computes the Frobenius norm, i.e.,
the square root of the sum of the squares of the absolute values of the
elements. The absolute value of an error form is defined by replacing
the mean part with its absolute value and leaving the error part the same.
The absolute value of a modulo form is undefined. The absolute value of
an interval is defined in the obvious way.

The `f A` (`calc-abssqr`

) [`abssqr`

] command computes the
absolute value squared of a number, vector or matrix, or error form.

The `f s` (`calc-sign`

) [`sign`

] command returns 1 if its
argument is positive, *-1* if its argument is negative, or 0 if its
argument is zero. In algebraic form, you can also write ‘`sign(a,x)`’
which evaluates to ‘`x * sign(a)`’, i.e., either ‘`x`’, ‘`-x`’, or
zero depending on the sign of ‘`a`’.

The `&` (`calc-inv`

) [`inv`

] command computes the
reciprocal of a number, i.e., ‘`1 / x`’. Operating on a square
matrix, it computes the inverse of that matrix.

The `Q` (`calc-sqrt`

) [`sqrt`

] command computes the square
root of a number. For a negative real argument, the result will be a
complex number whose form is determined by the current Polar mode.

The `f h` (`calc-hypot`

) [`hypot`

] command computes the square
root of the sum of the squares of two numbers. That is, ‘`hypot(a,b)`’
is the length of the hypotenuse of a right triangle with sides ‘`a`’
and ‘`b`’. If the arguments are complex numbers, their squared
magnitudes are used.

The `f Q` (`calc-isqrt`

) [`isqrt`

] command computes the
integer square root of an integer. This is the true square root of the
number, rounded down to an integer. For example, ‘`isqrt(10)`’
produces 3. Note that, like `\` [`idiv`

], this uses exact
integer arithmetic throughout to avoid roundoff problems. If the input
is a floating-point number or other non-integer value, this is exactly
the same as ‘`floor(sqrt(x))`’.

The `f n` (`calc-min`

) [`min`

] and `f x` (`calc-max`

)
[`max`

] commands take the minimum or maximum of two real numbers,
respectively. These commands also work on HMS forms, date forms,
intervals, and infinities. (In algebraic expressions, these functions
take any number of arguments and return the maximum or minimum among
all the arguments.)

The `f M` (`calc-mant-part`

) [`mant`

] function extracts
the “mantissa” part ‘`m`’ of its floating-point argument; `f X`
(`calc-xpon-part`

) [`xpon`

] extracts the “exponent” part
‘`e`’. The original number is equal to
‘`m * 10^e`’,
where ‘`m`’ is in the interval ‘`[1.0 .. 10.0)`’ except that
‘`m=e=0`’ if the original number is zero. For integers
and fractions, `mant`

returns the number unchanged and `xpon`

returns zero. The `v u` (`calc-unpack`

) command can also be
used to “unpack” a floating-point number; this produces an integer
mantissa and exponent, with the constraint that the mantissa is not
a multiple of ten (again except for the ‘`m=e=0`’ case).

The `f S` (`calc-scale-float`

) [`scf`

] function scales a number
by a given power of ten. Thus, ‘`scf(mant(x), xpon(x)) = x`’ for any
real ‘`x`’. The second argument must be an integer, but the first
may actually be any numeric value. For example, ‘`scf(5,-2) = 0.05`’
or ‘`1:20`’ depending on the current Fraction mode.

The `f [` (`calc-decrement`

) [`decr`

] and `f ]`
(`calc-increment`

) [`incr`

] functions decrease or increase
a number by one unit. For integers, the effect is obvious. For
floating-point numbers, the change is by one unit in the last place.
For example, incrementing ‘`12.3456`’ when the current precision
is 6 digits yields ‘`12.3457`’. If the current precision had been
8 digits, the result would have been ‘`12.345601`’. Incrementing
‘`0.0`’ produces
‘`10^-p`’,
where ‘`p`’ is the current
precision. These operations are defined only on integers and floats.
With numeric prefix arguments, they change the number by ‘`n`’ units.

Note that incrementing followed by decrementing, or vice-versa, will
almost but not quite always cancel out. Suppose the precision is
6 digits and the number ‘`9.99999`’ is on the stack. Incrementing
will produce ‘`10.0000`’; decrementing will produce ‘`9.9999`’.
One digit has been dropped. This is an unavoidable consequence of the
way floating-point numbers work.

Incrementing a date/time form adjusts it by a certain number of seconds. Incrementing a pure date form adjusts it by a certain number of days.

### 7.2 Integer Truncation

There are four commands for truncating a real number to an integer, differing mainly in their treatment of negative numbers. All of these commands have the property that if the argument is an integer, the result is the same integer. An integer-valued floating-point argument is converted to integer form.

If you press `H` (`calc-hyperbolic`

) first, the result will be
expressed as an integer-valued floating-point number.

The `F` (`calc-floor`

) [`floor`

or `ffloor`

] command
truncates a real number to the next lower integer, i.e., toward minus
infinity. Thus `3.6 F` produces 3, but `_3.6 F` produces
*-4*.

The `I F` (`calc-ceiling`

) [`ceil`

or `fceil`

]
command truncates toward positive infinity. Thus `3.6 I F` produces
4, and `_3.6 I F` produces *-3*.

The `R` (`calc-round`

) [`round`

or `fround`

] command
rounds to the nearest integer. When the fractional part is .5 exactly,
this command rounds away from zero. (All other rounding in the
Calculator uses this convention as well.) Thus `3.5 R` produces 4
but `3.4 R` produces 3; `_3.5 R` produces *-4*.

The `I R` (`calc-trunc`

) [`trunc`

or `ftrunc`

]
command truncates toward zero. In other words, it “chops off”
everything after the decimal point. Thus `3.6 I R` produces 3 and
`_3.6 I R` produces *-3*.

These functions may not be applied meaningfully to error forms, but they
do work for intervals. As a convenience, applying `floor`

to a
modulo form floors the value part of the form. Applied to a vector,
these functions operate on all elements of the vector one by one.
Applied to a date form, they operate on the internal numerical
representation of dates, converting a date/time form into a pure date.

There are two more rounding functions which can only be entered in
algebraic notation. The `roundu`

function is like `round`

except that it rounds up, toward plus infinity, when the fractional
part is .5. This distinction matters only for negative arguments.
Also, `rounde`

rounds to an even number in the case of a tie,
rounding up or down as necessary. For example, ‘`rounde(3.5)`’ and
‘`rounde(4.5)`’ both return 4, but ‘`rounde(5.5)`’ returns 6.
The advantage of round-to-even is that the net error due to rounding
after a long calculation tends to cancel out to zero. An important
subtle point here is that the number being fed to `rounde`

will
already have been rounded to the current precision before `rounde`

begins. For example, ‘`rounde(2.500001)`’ with a current precision
of 6 will incorrectly, or at least surprisingly, yield 2 because the
argument will first have been rounded down to ‘`2.5`’ (which
`rounde`

sees as an exact tie between 2 and 3).

Each of these functions, when written in algebraic formulas, allows
a second argument which specifies the number of digits after the
decimal point to keep. For example, ‘`round(123.4567, 2)`’ will
produce the answer 123.46, and ‘`round(123.4567, -1)`’ will
produce 120 (i.e., the cutoff is one digit to the *left* of
the decimal point). A second argument of zero is equivalent to
no second argument at all.

To compute the fractional part of a number (i.e., the amount which, when
added to ‘`floor(`

`n``)`

’, will produce `n`) just take `n`
modulo 1 using the `%`

command.

Note also the `\` (integer quotient), `f I` (integer logarithm),
and `f Q` (integer square root) commands, which are analogous to
`/`, `B`, and `Q`, respectively, except that they take integer
arguments and return the result rounded down to an integer.

### 7.3 Complex Number Functions

The `J` (`calc-conj`

) [`conj`

] command computes the
complex conjugate of a number. For complex number ‘`a+bi`’, the
complex conjugate is ‘`a-bi`’. If the argument is a real number,
this command leaves it the same. If the argument is a vector or matrix,
this command replaces each element by its complex conjugate.

The `G` (`calc-argument`

) [`arg`

] command computes the
“argument” or polar angle of a complex number. For a number in polar
notation, this is simply the second component of the pair
‘`(`

`r``;`

`theta``)`

’.
The result is expressed according to the current angular mode and will
be in the range *-180* degrees (exclusive) to *+180* degrees
(inclusive), or the equivalent range in radians.

The `calc-imaginary`

command multiplies the number on the
top of the stack by the imaginary number ‘`i = (0,1)`’. This
command is not normally bound to a key in Calc, but it is available
on the `IMAG` button in Keypad mode.

The `f r` (`calc-re`

) [`re`

] command replaces a complex number
by its real part. This command has no effect on real numbers. (As an
added convenience, `re`

applied to a modulo form extracts
the value part.)

The `f i` (`calc-im`

) [`im`

] command replaces a complex number
by its imaginary part; real numbers are converted to zero. With a vector
or matrix argument, these functions operate element-wise.

The `v p` (`calc-pack`

) command can pack the top two numbers on
the stack into a composite object such as a complex number. With
a prefix argument of *-1*, it produces a rectangular complex number;
with an argument of *-2*, it produces a polar complex number.
(Also, see Building Vectors.)

The `v u` (`calc-unpack`

) command takes the complex number
(or other composite object) on the top of the stack and unpacks it
into its separate components.

### 7.4 Conversions

The commands described in this section convert numbers from one form
to another; they are two-key sequences beginning with the letter `c`.

The `c f` (`calc-float`

) [`pfloat`

] command converts the
number on the top of the stack to floating-point form. For example,
‘`23`’ is converted to ‘`23.0`’, ‘`3:2`’ is converted to
‘`1.5`’, and ‘`2.3`’ is left the same. If the value is a composite
object such as a complex number or vector, each of the components is
converted to floating-point. If the value is a formula, all numbers
in the formula are converted to floating-point. Note that depending
on the current floating-point precision, conversion to floating-point
format may lose information.

As a special exception, integers which appear as powers or subscripts
are not floated by `c f`. If you really want to float a power,
you can use a `j s` command to select the power followed by `c f`.
Because `c f` cannot examine the formula outside of the selection,
it does not notice that the thing being floated is a power.
See Selecting Sub-Formulas.

The normal `c f` command is “pervasive” in the sense that it
applies to all numbers throughout the formula. The `pfloat`

algebraic function never stays around in a formula; ‘`pfloat(a + 1)`’
changes to ‘`a + 1.0`’ as soon as it is evaluated.

With the Hyperbolic flag, `H c f` [`float`

] operates
only on the number or vector of numbers at the top level of its
argument. Thus, ‘`float(1)`’ is 1.0, but ‘`float(a + 1)`’
is left unevaluated because its argument is not a number.

You should use `H c f` if you wish to guarantee that the final
value, once all the variables have been assigned, is a float; you
would use `c f` if you wish to do the conversion on the numbers
that appear right now.

The `c F` (`calc-fraction`

) [`pfrac`

] command converts a
floating-point number into a fractional approximation. By default, it
produces a fraction whose decimal representation is the same as the
input number, to within the current precision. You can also give a
numeric prefix argument to specify a tolerance, either directly, or,
if the prefix argument is zero, by using the number on top of the stack
as the tolerance. If the tolerance is a positive integer, the fraction
is correct to within that many significant figures. If the tolerance is
a non-positive integer, it specifies how many digits fewer than the current
precision to use. If the tolerance is a floating-point number, the
fraction is correct to within that absolute amount.

The `pfrac`

function is pervasive, like `pfloat`

.
There is also a non-pervasive version, `H c F` [`frac`

],
which is analogous to `H c f` discussed above.

The `c d` (`calc-to-degrees`

) [`deg`

] command converts a
number into degrees form. The value on the top of the stack may be an
HMS form (interpreted as degrees-minutes-seconds), or a real number which
will be interpreted in radians regardless of the current angular mode.

The `c r` (`calc-to-radians`

) [`rad`

] command converts an
HMS form or angle in degrees into an angle in radians.

The `c h` (`calc-to-hms`

) [`hms`

] command converts a real
number, interpreted according to the current angular mode, to an HMS
form describing the same angle. In algebraic notation, the `hms`

function also accepts three arguments: ‘`hms( h, m, s)`’.
(The three-argument version is independent of the current angular mode.)

The `calc-from-hms`

command converts the HMS form on the top of the
stack into a real number according to the current angular mode.

The `c p` (`calc-polar`

) command converts the complex number on
the top of the stack from polar to rectangular form, or from rectangular
to polar form, whichever is appropriate. Real numbers are left the same.
This command is equivalent to the `rect`

or `polar`

functions in algebraic formulas, depending on the direction of
conversion. (It uses `polar`

, except that if the argument is
already a polar complex number, it uses `rect`

instead. The
`I c p` command always uses `rect`

.)

The `c c` (`calc-clean`

) [`pclean`

] command “cleans” the
number on the top of the stack. Floating point numbers are re-rounded
according to the current precision. Polar numbers whose angular
components have strayed from the *-180* to *+180* degree range
are normalized. (Note that results will be undesirable if the current
angular mode is different from the one under which the number was
produced!) Integers and fractions are generally unaffected by this
operation. Vectors and formulas are cleaned by cleaning each component
number (i.e., pervasively).

If the simplification mode is set below basic simplification, it is raised
for the purposes of this command. Thus, `c c` applies the basic
simplifications even if their automatic application is disabled.
See Simplification Modes.

A numeric prefix argument to `c c` sets the floating-point precision
to that value for the duration of the command. A positive prefix (of at
least 3) sets the precision to the specified value; a negative or zero
prefix decreases the precision by the specified amount.

The keystroke sequences `c 0` through `c 9` are equivalent
to `c c` with the corresponding negative prefix argument. If roundoff
errors have changed 2.0 into 1.999999, typing `c 1` to clip off one
decimal place often conveniently does the trick.

The `c c` command with a numeric prefix argument, and the `c 0`
through `c 9` commands, also “clip” very small floating-point
numbers to zero. If the exponent is less than or equal to the negative
of the specified precision, the number is changed to 0.0. For example,
if the current precision is 12, then `c 2` changes the vector
‘`[1e-8, 1e-9, 1e-10, 1e-11]`’ to ‘`[1e-8, 1e-9, 0, 0]`’.
Numbers this small generally arise from roundoff noise.

If the numbers you are using really are legitimately this small,
you should avoid using the `c 0` through `c 9` commands.
(The plain `c c` command rounds to the current precision but
does not clip small numbers.)

One more property of `c 0` through `c 9`, and of `c c` with
a prefix argument, is that integer-valued floats are converted to
plain integers, so that `c 1` on ‘`[1., 1.5, 2., 2.5, 3.]`’
produces ‘`[1, 1.5, 2, 2.5, 3]`’. This is not done for huge
numbers (‘`1e100`’ is technically an integer-valued float, but
you wouldn’t want it automatically converted to a 100-digit integer).

With the Hyperbolic flag, `H c c` and `H c 0` through `H c 9`
operate non-pervasively [`clean`

].

### 7.5 Date Arithmetic

The commands described in this section perform various conversions
and calculations involving date forms (see Date Forms). They
use the `t` (for time/date) prefix key followed by shifted
letters.

The simplest date arithmetic is done using the regular `+` and `-`
commands. In particular, adding a number to a date form advances the
date form by a certain number of days; adding an HMS form to a date
form advances the date by a certain amount of time; and subtracting two
date forms produces a difference measured in days. The commands
described here provide additional, more specialized operations on dates.

Many of these commands accept a numeric prefix argument; if you give
plain `C-u` as the prefix, these commands will instead take the
additional argument from the top of the stack.

#### 7.5.1 Date Conversions

The `t D` (`calc-date`

) [`date`

] command converts a
date form into a number, measured in days since Jan 1, 1 AD. The
result will be an integer if `date` is a pure date form, or a
fraction or float if `date` is a date/time form. Or, if its
argument is a number, it converts this number into a date form.

With a numeric prefix argument, `t D` takes that many objects
(up to six) from the top of the stack and interprets them in one
of the following ways:

The ‘`date( year, month, day)`’ function
builds a pure date form out of the specified year, month, and
day, which must all be integers.

`Year`is a year number, such as 1991 (

*not*the same as 91!).

`Month`must be an integer in the range 1 to 12;

`day`must be in the range 1 to 31. If the specified month has fewer than 31 days and

`day`is too large, the equivalent day in the following month will be used.

The ‘`date( month, day)`’ function builds a
pure date form using the current year, as determined by the
real-time clock.

The ‘`date( year, month, day, hms)`’
function builds a date/time form using an

`hms`form.

The ‘`date( year, month, day, hour,
minute, second)`’ function builds a date/time form.

`hour`should be an integer in the range 0 to 23;

`minute`should be an integer in the range 0 to 59;

`second`should be any real number in the range ‘

`[0 .. 60)`’. The last two arguments default to zero if omitted.

The `t J` (`calc-julian`

) [`julian`

] command converts
a date form into a Julian day count, which is the number of days
since noon (GMT) on Jan 1, 4713 BC. A pure date is converted to an
integer Julian count representing noon of that day. A date/time form
is converted to an exact floating-point Julian count, adjusted to
interpret the date form in the current time zone but the Julian
day count in Greenwich Mean Time. A numeric prefix argument allows
you to specify the time zone; see Time Zones. Use a prefix of
zero to suppress the time zone adjustment. Note that pure date forms
are never time-zone adjusted.

This command can also do the opposite conversion, from a Julian day count (either an integer day, or a floating-point day and time in the GMT zone), into a pure date form or a date/time form in the current or specified time zone.

The `t U` (`calc-unix-time`

) [`unixtime`

] command
converts a date form into a Unix time value, which is the number of
seconds since midnight on Jan 1, 1970, or vice-versa. The numeric result
will be an integer if the current precision is 12 or less; for higher
precision, the result may be a float with (`precision`−12)
digits after the decimal. Just as for `t J`, the numeric time
is interpreted in the GMT time zone and the date form is interpreted
in the current or specified zone. Some systems use Unix-like
numbering but with the local time zone; give a prefix of zero to
suppress the adjustment if so.

The `t C` (`calc-convert-time-zones`

) [`tzconv`

]
command converts a date form from one time zone to another. You
are prompted for each time zone name in turn; you can answer with
any suitable Calc time zone expression (see Time Zones).
If you answer either prompt with a blank line, the local time
zone is used for that prompt. You can also answer the first
prompt with `$` to take the two time zone names from the
stack (and the date to be converted from the third stack level).

#### 7.5.2 Date Functions

The `t N` (`calc-now`

) [`now`

] command pushes the
current date and time on the stack as a date form. The time is
reported in terms of the specified time zone; with no numeric prefix
argument, `t N` reports for the current time zone.

The `t P` (`calc-date-part`

) command extracts one part
of a date form. The prefix argument specifies the part; with no
argument, this command prompts for a part code from 1 to 9.
The various part codes are described in the following paragraphs.

The `M-1 t P` [`year`

] function extracts the year number
from a date form as an integer, e.g., 1991. This and the
following functions will also accept a real number for an
argument, which is interpreted as a standard Calc day number.
Note that this function will never return zero, since the year
1 BC immediately precedes the year 1 AD.

The `M-2 t P` [`month`

] function extracts the month number
from a date form as an integer in the range 1 to 12.

The `M-3 t P` [`day`

] function extracts the day number
from a date form as an integer in the range 1 to 31.

The `M-4 t P` [`hour`

] function extracts the hour from
a date form as an integer in the range 0 (midnight) to 23. Note
that 24-hour time is always used. This returns zero for a pure
date form. This function (and the following two) also accept
HMS forms as input.

The `M-5 t P` [`minute`

] function extracts the minute
from a date form as an integer in the range 0 to 59.

The `M-6 t P` [`second`

] function extracts the second
from a date form. If the current precision is 12 or less,
the result is an integer in the range 0 to 59. For higher
precision, the result may instead be a floating-point number.

The `M-7 t P` [`weekday`

] function extracts the weekday
number from a date form as an integer in the range 0 (Sunday)
to 6 (Saturday).

The `M-8 t P` [`yearday`

] function extracts the day-of-year
number from a date form as an integer in the range 1 (January 1)
to 366 (December 31 of a leap year).

The `M-9 t P` [`time`

] function extracts the time portion
of a date form as an HMS form. This returns ‘`0@ 0' 0"`’
for a pure date form.

The `t M` (`calc-new-month`

) [`newmonth`

] command
computes a new date form that represents the first day of the month
specified by the input date. The result is always a pure date
form; only the year and month numbers of the input are retained.
With a numeric prefix argument `n` in the range from 1 to 31,
`t M` computes the `n`th day of the month. (If `n`
is greater than the actual number of days in the month, or if
`n` is zero, the last day of the month is used.)

The `t Y` (`calc-new-year`

) [`newyear`

] command
computes a new pure date form that represents the first day of
the year specified by the input. The month, day, and time
of the input date form are lost. With a numeric prefix argument
`n` in the range from 1 to 366, `t Y` computes the
`n`th day of the year (366 is treated as 365 in non-leap
years). A prefix argument of 0 computes the last day of the
year (December 31). A negative prefix argument from *-1* to
*-12* computes the first day of the `n`th month of the year.

The `t W` (`calc-new-week`

) [`newweek`

] command
computes a new pure date form that represents the Sunday on or before
the input date. With a numeric prefix argument, it can be made to
use any day of the week as the starting day; the argument must be in
the range from 0 (Sunday) to 6 (Saturday). This function always
subtracts between 0 and 6 days from the input date.

Here’s an example use of `newweek`

: Find the date of the next
Wednesday after a given date. Using `M-3 t W` or ‘`newweek(d, 3)`’
will give you the *preceding* Wednesday, so ‘`newweek(d+7, 3)`’
will give you the following Wednesday. A further look at the definition
of `newweek`

shows that if the input date is itself a Wednesday,
this formula will return the Wednesday one week in the future. An
exercise for the reader is to modify this formula to yield the same day
if the input is already a Wednesday. Another interesting exercise is
to preserve the time-of-day portion of the input (`newweek`

resets
the time to midnight; hint: how can `newweek`

be defined in terms
of the `weekday`

function?).

The ‘`pwday( date)`’ function (not on any key) computes the
day-of-month number of the Sunday on or before

`date`. With two arguments, ‘

`pwday(`’ computes the day number of the Sunday on or before day number

`date`,`day`)`day`of the month specified by

`date`. The

`day`must be in the range from 7 to 31; if the day number is greater than the actual number of days in the month, the true number of days is used instead. Thus ‘

`pwday(`’ finds the first Sunday of the month, and ‘

`date`, 7)`pwday(`’ finds the last Sunday of the month. With a third

`date`, 31)`weekday`argument,

`pwday`

can be made to look
for any day of the week instead of Sunday.
The `t I` (`calc-inc-month`

) [`incmonth`

] command
increases a date form by one month, or by an arbitrary number of
months specified by a numeric prefix argument. The time portion,
if any, of the date form stays the same. The day also stays the
same, except that if the new month has fewer days the day
number may be reduced to lie in the valid range. For example,
‘`incmonth(<Jan 31, 1991>)`’ produces ‘`<Feb 28, 1991>`’.
Because of this, `t I t I` and `M-2 t I` do not always give
the same results (‘`<Mar 28, 1991>`’ versus ‘`<Mar 31, 1991>`’
in this case).

The ‘`incyear( date, step)`’ function increases
a date form by the specified number of years, which may be
any positive or negative integer. Note that ‘

`incyear(d, n)`’ is equivalent to ‘

`incmonth(d, 12*n)`’, but these do not have simple equivalents in terms of day arithmetic because months and years have varying lengths. If the

`step`argument is omitted, 1 year is assumed. There is no keyboard command for this function; use

`C-u 12 t I`instead.

There is no `newday`

function at all because `F` [`floor`

]
serves this purpose. Similarly, instead of `incday`

and
`incweek`

simply use ‘`d + n`’ or ‘`d + 7 n`’.

See Basic Arithmetic, for the `f ]` [`incr`

] command
which can adjust a date/time form by a certain number of seconds.

#### 7.5.3 Business Days

Often time is measured in “business days” or “working days,” where weekends and holidays are skipped. Calc’s normal date arithmetic functions use calendar days, so that subtracting two consecutive Mondays will yield a difference of 7 days. By contrast, subtracting two consecutive Mondays would yield 5 business days (assuming two-day weekends and the absence of holidays).

The `t +` (`calc-business-days-plus`

) [`badd`

]
and `t -` (`calc-business-days-minus`

) [`bsub`

]
commands perform arithmetic using business days. For `t +`,
one argument must be a date form and the other must be a real
number (positive or negative). If the number is not an integer,
then a certain amount of time is added as well as a number of
days; for example, adding 0.5 business days to a time in Friday
evening will produce a time in Monday morning. It is also
possible to add an HMS form; adding ‘`12@ 0' 0"`’ also adds
half a business day. For `t -`, the arguments are either a
date form and a number or HMS form, or two date forms, in which
case the result is the number of business days between the two
dates.

By default, Calc considers any day that is not a Saturday or
Sunday to be a business day. You can define any number of
additional holidays by editing the variable `Holidays`

.
(There is an `s H` convenience command for editing this
variable.) Initially, `Holidays`

contains the vector
‘`[sat, sun]`’. Entries in the `Holidays`

vector may
be any of the following kinds of objects:

- Date forms (pure dates, not date/time forms). These specify particular days which are to be treated as holidays.
- Intervals of date forms. These specify a range of days, all of which are holidays (e.g., Christmas week). See Interval Forms.
- Nested vectors of date forms. Each date form in the vector is considered to be a holiday.
- Any Calc formula which evaluates to one of the above three things.
If the formula involves the variable ‘
`y`’, it stands for a yearly repeating holiday; ‘`y`’ will take on various year numbers like 1992. For example, ‘`date(y, 12, 25)`’ specifies Christmas day, and ‘`newweek(date(y, 11, 7), 4) + 21`’ specifies Thanksgiving (which is held on the fourth Thursday of November). If the formula involves the variable ‘`m`’, that variable takes on month numbers from 1 to 12: ‘`date(y, m, 15)`’ is a holiday that takes place on the 15th of every month. - A weekday name, such as
`sat`

or`sun`

. This is really a variable whose name is a three-letter, lower-case day name. - An interval of year numbers (integers). This specifies the span of
years over which this holiday list is to be considered valid. Any
business-day arithmetic that goes outside this range will result
in an error message. Use this if you are including an explicit
list of holidays, rather than a formula to generate them, and you
want to make sure you don’t accidentally go beyond the last point
where the holidays you entered are complete. If there is no
limiting interval in the
`Holidays`

vector, the default ‘`[1 .. 2737]`’ is used. (This is the absolute range of years for which Calc’s business-day algorithms will operate.) - An interval of HMS forms. This specifies the span of hours that
are to be considered one business day. For example, if this
range is ‘
`[9@ 0' 0" .. 17@ 0' 0"]`’ (i.e., 9am to 5pm), then the business day is only eight hours long, so that`1.5 t +`on ‘`<4:00pm Fri Dec 13, 1991>`’ will add one business day and four business hours to produce ‘`<12:00pm Tue Dec 17, 1991>`’. Likewise,`t -`will now express differences in time as fractions of an eight-hour day. Times before 9am will be treated as 9am by business date arithmetic, and times at or after 5pm will be treated as 4:59:59pm. If there is no HMS interval in`Holidays`

, the full 24-hour day ‘`[0 0' 0" .. 24 0' 0"]`’ is assumed. (Regardless of the type of bounds you specify, the interval is treated as inclusive on the low end and exclusive on the high end, so that the work day goes from 9am up to, but not including, 5pm.)

If the `Holidays`

vector is empty, then `t +` and
`t -` will act just like `+` and `-` because there will
then be no difference between business days and calendar days.

Calc expands the intervals and formulas you give into a complete
list of holidays for internal use. This is done mainly to make
sure it can detect multiple holidays. (For example,
‘`<Jan 1, 1989>`’ is both New Year’s Day and a Sunday, but
Calc’s algorithms take care to count it only once when figuring
the number of holidays between two dates.)

Since the complete list of holidays for all the years from 1 to 2737 would be huge, Calc actually computes only the part of the list between the smallest and largest years that have been involved in business-day calculations so far. Normally, you won’t have to worry about this. Keep in mind, however, that if you do one calculation for 1992, and another for 1792, even if both involve only a small range of years, Calc will still work out all the holidays that fall in that 200-year span.

If you add a (positive) number of days to a date form that falls on a
weekend or holiday, the date form is treated as if it were the most
recent business day. (Thus adding one business day to a Friday,
Saturday, or Sunday will all yield the following Monday.) If you
subtract a number of days from a weekend or holiday, the date is
effectively on the following business day. (So subtracting one business
day from Saturday, Sunday, or Monday yields the preceding Friday.) The
difference between two dates one or both of which fall on holidays
equals the number of actual business days between them. These
conventions are consistent in the sense that, if you add `n`
business days to any date, the difference between the result and the
original date will come out to `n` business days. (It can’t be
completely consistent though; a subtraction followed by an addition
might come out a bit differently, since `t +` is incapable of
producing a date that falls on a weekend or holiday.)

There is a `holiday`

function, not on any keys, that takes
any date form and returns 1 if that date falls on a weekend or
holiday, as defined in `Holidays`

, or 0 if the date is a
business day.

#### 7.5.4 Time Zones

Time zones and daylight saving time are a complicated business. The conversions to and from Julian and Unix-style dates automatically compute the correct time zone and daylight saving adjustment to use, provided they can figure out this information. This section describes Calc’s time zone adjustment algorithm in detail, in case you want to do conversions in different time zones or in case Calc’s algorithms can’t determine the right correction to use.

Adjustments for time zones and daylight saving time are done by
`t U`, `t J`, `t N`, and `t C`, but not by any other
commands. In particular, ‘`<may 1 1991> - <apr 1 1991>`’ evaluates
to exactly 30 days even though there is a daylight-saving
transition in between. This is also true for Julian pure dates:
‘`julian(<may 1 1991>) - julian(<apr 1 1991>)`’. But Julian
and Unix date/times will adjust for daylight saving time: using Calc’s
default daylight saving time rule (see the explanation below),
‘`julian(<12am may 1 1991>) - julian(<12am apr 1 1991>)`’
evaluates to ‘`29.95833`’ (that’s 29 days and 23 hours)
because one hour was lost when daylight saving commenced on
April 7, 1991.

In brief, the idiom ‘`julian( date1) - julian(date2)`’
computes the actual number of 24-hour periods between two dates, whereas
‘

`’ computes the number of calendar days between two dates without taking daylight saving into account.`

`date1`-`date2`The `calc-time-zone`

[`tzone`

] command converts the time
zone specified by its numeric prefix argument into a number of
seconds difference from Greenwich mean time (GMT). If the argument
is a number, the result is simply that value multiplied by 3600.
Typical arguments for North America are 5 (Eastern) or 8 (Pacific). If
Daylight Saving time is in effect, one hour should be subtracted from
the normal difference.

If you give a prefix of plain `C-u`, `calc-time-zone`

(like other
date arithmetic commands that include a time zone argument) takes the
zone argument from the top of the stack. (In the case of `t J`
and `t U`, the normal argument is then taken from the second-to-top
stack position.) This allows you to give a non-integer time zone
adjustment. The time-zone argument can also be an HMS form, or
it can be a variable which is a time zone name in upper- or lower-case.
For example ‘`tzone(PST) = tzone(8)`’ and ‘`tzone(pdt) = tzone(7)`’
(for Pacific standard and daylight saving times, respectively).

North American and European time zone names are defined as follows; note that for each time zone there is one name for standard time, another for daylight saving time, and a third for “generalized” time in which the daylight saving adjustment is computed from context.

YST PST MST CST EST AST NST GMT WET MET MEZ 9 8 7 6 5 4 3.5 0 -1 -2 -2 YDT PDT MDT CDT EDT ADT NDT BST WETDST METDST MESZ 8 7 6 5 4 3 2.5 -1 -2 -3 -3 YGT PGT MGT CGT EGT AGT NGT BGT WEGT MEGT MEGZ 9/8 8/7 7/6 6/5 5/4 4/3 3.5/2.5 0/-1 -1/-2 -2/-3 -2/-3

To define time zone names that do not appear in the above table,
you must modify the Lisp variable `math-tzone-names`

. This
is a list of lists describing the different time zone names; its
structure is best explained by an example. The three entries for
Pacific Time look like this:

( ( "PST" 8 0 ) ; Name as an upper-case string, then standard ( "PDT" 8 -1 ) ; adjustment, then daylight saving adjustment. ( "PGT" 8 "PST" "PDT" ) ) ; Generalized time zone.

With no arguments, `calc-time-zone`

or ‘`tzone()`’ will by
default get the time zone and daylight saving information from the
calendar (see The Calendar and the Diary in The GNU Emacs Manual). To use a different time zone, or if the
calendar does not give the desired result, you can set the Calc variable
`TimeZone`

(which is by default `nil`

) to an appropriate
time zone name. (The easiest way to do this is to edit the
`TimeZone`

variable using Calc’s `s T` command, then use the
`s p` (`calc-permanent-variable`

) command to save the value of
`TimeZone`

permanently.)
If the time zone given by `TimeZone`

is a generalized time zone,
e.g., `EGT`

, Calc examines the date being converted to tell whether
to use standard or daylight saving time. But if the current time zone
is explicit, e.g., `EST`

or `EDT`

, then that adjustment is
used exactly and Calc’s daylight saving algorithm is not consulted.
The special time zone name `local`

is equivalent to no argument; i.e., it uses the information obtained
from the calendar.

The `t J` and `t U`

commands with no numeric prefix
arguments do the same thing as ‘`tzone()`’; namely, use the
information from the calendar if `TimeZone`

is `nil`

,
otherwise use the time zone given by `TimeZone`

.

When Calc computes the daylight saving information itself (i.e., when
the `TimeZone`

variable is set), it will by default consider
daylight saving time to begin at 2 a.m. on the second Sunday of March
(for years from 2007 on) or on the last Sunday in April (for years
before 2007), and to end at 2 a.m. on the first Sunday of
November. (for years from 2007 on) or the last Sunday in October (for
years before 2007). These are the rules that have been in effect in
much of North America since 1966 and take into account the rule change
that began in 2007. If you are in a country that uses different rules
for computing daylight saving time, you have two choices: Write your own
daylight saving hook, or control time zones explicitly by setting the
`TimeZone`

variable and/or always giving a time-zone argument for
the conversion functions.

The Lisp variable `math-daylight-savings-hook`

holds the
name of a function that is used to compute the daylight saving
adjustment for a given date. The default is
`math-std-daylight-savings`

, which computes an adjustment
(either 0 or *-1*) using the North American rules given above.

The daylight saving hook function is called with four arguments:
The date, as a floating-point number in standard Calc format;
a six-element list of the date decomposed into year, month, day,
hour, minute, and second, respectively; a string which contains
the generalized time zone name in upper-case, e.g., `"WEGT"`

;
and a special adjustment to be applied to the hour value when
converting into a generalized time zone (see below).

The Lisp function `math-prev-weekday-in-month`

is useful for
daylight saving computations. This is an internal version of
the user-level `pwday`

function described in the previous
section. It takes four arguments: The floating-point date value,
the corresponding six-element date list, the day-of-month number,
and the weekday number (0–6).

The default daylight saving hook ignores the time zone name, but a more sophisticated hook could use different algorithms for different time zones. It would also be possible to use different algorithms depending on the year number, but the default hook always uses the algorithm for 1987 and later. Here is a listing of the default daylight saving hook:

(defun math-std-daylight-savings (date dt zone bump) (cond ((< (nth 1 dt) 4) 0) ((= (nth 1 dt) 4) (let ((sunday (math-prev-weekday-in-month date dt 7 0))) (cond ((< (nth 2 dt) sunday) 0) ((= (nth 2 dt) sunday) (if (>= (nth 3 dt) (+ 3 bump)) -1 0)) (t -1)))) ((< (nth 1 dt) 10) -1) ((= (nth 1 dt) 10) (let ((sunday (math-prev-weekday-in-month date dt 31 0))) (cond ((< (nth 2 dt) sunday) -1) ((= (nth 2 dt) sunday) (if (>= (nth 3 dt) (+ 2 bump)) 0 -1)) (t 0)))) (t 0)) )

The `bump`

parameter is equal to zero when Calc is converting
from a date form in a generalized time zone into a GMT date value.
It is *-1* when Calc is converting in the other direction. The
adjustments shown above ensure that the conversion behaves correctly
and reasonably around the 2 a.m. transition in each direction.

There is a “missing” hour between 2 a.m. and 3 a.m. at the
beginning of daylight saving time; converting a date/time form that
falls in this hour results in a time value for the following hour,
from 3 a.m. to 4 a.m. At the end of daylight saving time, the
hour from 1 a.m. to 2 a.m. repeats itself; converting a date/time
form that falls in this hour results in a time value for the first
manifestation of that time (*not* the one that occurs one hour
later).

If `math-daylight-savings-hook`

is `nil`

, then the
daylight saving adjustment is always taken to be zero.

In algebraic formulas, ‘`tzone( zone, date)`’
computes the time zone adjustment for a given zone name at a
given date. The

`date`is ignored unless

`zone`is a generalized time zone. If

`date`is a date form, the daylight saving computation is applied to it as it appears. If

`date`is a numeric date value, it is adjusted for the daylight-saving version of

`zone`before being given to the daylight saving hook. This odd-sounding rule ensures that the daylight-saving computation is always done in local time, not in the GMT time that a numeric

`date`is typically represented in.

The ‘`dsadj( date, zone)`’ function computes the
daylight saving adjustment that is appropriate for

`date`in time zone

`zone`. If

`zone`is explicitly in or not in daylight saving time (e.g.,

`PDT`

or `PST`

) the
`date`is ignored. If

`zone`is a generalized time zone, the algorithms described above are used. If

`zone`is omitted, the computation is done for the current time zone.

### 7.6 Financial Functions

Calc’s financial or business functions use the `b` prefix
key followed by a shifted letter. (The `b` prefix followed by
a lower-case letter is used for operations on binary numbers.)

Note that the rate and the number of intervals given to these functions must be on the same time scale, e.g., both months or both years. Mixing an annual interest rate with a time expressed in months will give you very wrong answers!

It is wise to compute these functions to a higher precision than you really need, just to make sure your answer is correct to the last penny; also, you may wish to check the definitions at the end of this section to make sure the functions have the meaning you expect.

- Percentages
- Future Value
- Present Value
- Related Financial Functions
- Depreciation Functions
- Definitions

#### 7.6.1 Percentages

The `M-%` (`calc-percent`

) command takes a percentage value,
say 5.4, and converts it to an equivalent actual number. For example,
`5.4 M-%` enters 0.054 on the stack. (That’s the `META` or
`ESC` key combined with `%`.)

Actually, `M-%` creates a formula of the form ‘`5.4%`’.
You can enter ‘`5.4%`’ yourself during algebraic entry. The
‘`%`’ operator simply means, “the preceding value divided by
100.” The ‘`%`’ operator has very high precedence, so that
‘`1+8%`’ is interpreted as ‘`1+(8%)`’, not as ‘`(1+8)%`’.
(The ‘`%`’ operator is just a postfix notation for the
`percent`

function, just like ‘`20!`’ is the notation for
‘`fact(20)`’, or twenty-factorial.)

The formula ‘`5.4%`’ would normally evaluate immediately to
0.054, but the `M-%` command suppresses evaluation as it puts
the formula onto the stack. However, the next Calc command that
uses the formula ‘`5.4%`’ will evaluate it as its first step.
The net effect is that you get to look at ‘`5.4%`’ on the stack,
but Calc commands see it as ‘`0.054`’, which is what they expect.

In particular, ‘`5.4%`’ and ‘`0.054`’ are suitable values
for the `rate` arguments of the various financial functions,
but the number ‘`5.4`’ is probably *not* suitable—it
represents a rate of 540 percent!

The key sequence `M-% *` effectively means “percent-of.”
For example, `68 RET 25 M-% *` computes 17, which is 25% of
68 (and also 68% of 25, which comes out to the same thing).

The `c %` (`calc-convert-percent`

) command converts the
value on the top of the stack from numeric to percentage form.
For example, if 0.08 is on the stack, `c %` converts it to
‘`8%`’. The quantity is the same, it’s just represented
differently. (Contrast this with `M-%`, which would convert
this number to ‘`0.08%`’.) The `=` key is a convenient way
to convert a formula like ‘`8%`’ back to numeric form, 0.08.

To compute what percentage one quantity is of another quantity,
use `/ c %`. For example, `17 RET 68 / c %` displays
‘

`25%`’.

The `b %` (`calc-percent-change`

) [`relch`

] command
calculates the percentage change from one number to another.
For example, `40 RET 50 b %` produces the answer ‘

`25%`’, since 50 is 25% larger than 40. A negative result represents a decrease:

`50`produces ‘

`RET`40 b %`-20%`’, since 40 is 20% smaller than 50. (The answers are different in magnitude because, in the first case, we’re increasing by 25% of 40, but in the second case, we’re decreasing by 20% of 50.) The effect of

`40`is to compute ‘

`RET`50 b %`(50-40)/40`’, converting the answer to percentage form as if by

`c %`.

#### 7.6.2 Future Value

The `b F` (`calc-fin-fv`

) [`fv`

] command computes
the future value of an investment. It takes three arguments
from the stack: ‘`fv( rate, n, payment)`’.
If you give payments of

`payment`every year for

`n`years, and the money you have paid earns interest at

`rate`per year, then this function tells you what your investment would be worth at the end of the period. (The actual interval doesn’t have to be years, as long as

`n`and

`rate`are expressed in terms of the same intervals.) This function assumes payments occur at the

*end*of each interval.

The `I b F` [`fvb`

] command does the same computation,
but assuming your payments are at the beginning of each interval.
Suppose you plan to deposit $1000 per year in a savings account
earning 5.4% interest, starting right now. How much will be
in the account after five years? `fvb(5.4%, 5, 1000) = 5870.73`

.
Thus you will have earned $870 worth of interest over the years.
Using the stack, this calculation would have been
`5.4 M-% 5 RET 1000 I b F`. Note that the rate is expressed
as a number between 0 and 1,

*not*as a percentage.

The `H b F` [`fvl`

] command computes the future value
of an initial lump sum investment. Suppose you could deposit
those five thousand dollars in the bank right now; how much would
they be worth in five years? `fvl(5.4%, 5, 5000) = 6503.89`

.

The algebraic functions `fv`

and `fvb`

accept an optional
fourth argument, which is used as an initial lump sum in the sense
of `fvl`

. In other words, `fv(`

.
`rate`, `n`,
`payment`, `initial`) = fv(`rate`, `n`, `payment`)
+ fvl(`rate`, `n`, `initial`)

To illustrate the relationships between these functions, we could
do the `fvb`

calculation “by hand” using `fvl`

. The
final balance will be the sum of the contributions of our five
deposits at various times. The first deposit earns interest for
five years: `fvl(5.4%, 5, 1000) = 1300.78`

. The second
deposit only earns interest for four years: ```
fvl(5.4%, 4, 1000) =
1234.13
```

. And so on down to the last deposit, which earns one
year’s interest: `fvl(5.4%, 1, 1000) = 1054.00`

. The sum of
these five values is, sure enough, $5870.73, just as was computed
by `fvb`

directly.

What does `fv(5.4%, 5, 1000) = 5569.96`

mean? The payments
are now at the ends of the periods. The end of one year is the same
as the beginning of the next, so what this really means is that we’ve
lost the payment at year zero (which contributed $1300.78), but we’re
now counting the payment at year five (which, since it didn’t have
a chance to earn interest, counts as $1000). Indeed, ‘`5569.96 =
5870.73 - 1300.78 + 1000`’ (give or take a bit of roundoff error).

#### 7.6.3 Present Value

The `b P` (`calc-fin-pv`

) [`pv`

] command computes
the present value of an investment. Like `fv`

, it takes
three arguments: `pv(`

.
It computes the present value of a series of regular payments.
Suppose you have the chance to make an investment that will
pay $2000 per year over the next four years; as you receive
these payments you can put them in the bank at 9% interest.
You want to know whether it is better to make the investment, or
to keep the money in the bank where it earns 9% interest right
from the start. The calculation `rate`, `n`, `payment`)`pv(9%, 4, 2000)`

gives the
result 6479.44. If your initial investment must be less than this,
say, $6000, then the investment is worthwhile. But if you had to
put up $7000, then it would be better just to leave it in the bank.

Here is the interpretation of the result of `pv`

: You are
trying to compare the return from the investment you are
considering, which is `fv(9%, 4, 2000) = 9146.26`

, with
the return from leaving the money in the bank, which is
`fvl(9%, 4, `

where `x`)`x` is the amount of money
you would have to put up in advance. The `pv`

function
finds the break-even point, ‘`x = 6479.44`’, at which
`fvl(9%, 4, 6479.44)`

is also equal to 9146.26. This is
the largest amount you should be willing to invest.

The `I b P` [`pvb`

] command solves the same problem,
but with payments occurring at the beginning of each interval.
It has the same relationship to `fvb`

as `pv`

has
to `fv`

. For example `pvb(9%, 4, 2000) = 7062.59`

,
a larger number than `pv`

produced because we get to start
earning interest on the return from our investment sooner.

The `H b P` [`pvl`

] command computes the present value of
an investment that will pay off in one lump sum at the end of the
period. For example, if we get our $8000 all at the end of the
four years, `pvl(9%, 4, 8000) = 5667.40`

. This is much
less than `pv`

reported, because we don’t earn any interest
on the return from this investment. Note that `pvl`

and
`fvl`

are simple inverses: `fvl(9%, 4, 5667.40) = 8000`

.

You can give an optional fourth lump-sum argument to `pv`

and `pvb`

; this is handled in exactly the same way as the
fourth argument for `fv`

and `fvb`

.

The `b N` (`calc-fin-npv`

) [`npv`

] command computes
the net present value of a series of irregular investments.
The first argument is the interest rate. The second argument is
a vector which represents the expected return from the investment
at the end of each interval. For example, if the rate represents
a yearly interest rate, then the vector elements are the return
from the first year, second year, and so on.

Thus, `npv(9%, [2000,2000,2000,2000]) = pv(9%, 4, 2000) = 6479.44`

.
Obviously this function is more interesting when the payments are
not all the same!

The `npv`

function can actually have two or more arguments.
Multiple arguments are interpreted in the same way as for the
vector statistical functions like `vsum`

.
See Single-Variable Statistics. Basically, if there are several
payment arguments, each either a vector or a plain number, all these
values are collected left-to-right into the complete list of payments.
A numeric prefix argument on the `b N` command says how many
payment values or vectors to take from the stack.

The `I b N` [`npvb`

] command computes the net present
value where payments occur at the beginning of each interval
rather than at the end.

#### 7.6.4 Related Financial Functions

The functions in this section are basically inverses of the present value functions with respect to the various arguments.

The `b M` (`calc-fin-pmt`

) [`pmt`

] command computes
the amount of periodic payment necessary to amortize a loan.
Thus `pmt(`

equals the
value of `rate`, `n`, `amount`)`payment` such that `pv(`

.
`rate`, `n`,
`payment`) = `amount`

The `I b M` [`pmtb`

] command does the same computation
but using `pvb`

instead of `pv`

. Like `pv`

and
`pvb`

, these functions can also take a fourth argument which
represents an initial lump-sum investment.

The `H b M` key just invokes the `fvl`

function, which is
the inverse of `pvl`

. There is no explicit `pmtl`

function.

The `b #` (`calc-fin-nper`

) [`nper`

] command computes
the number of regular payments necessary to amortize a loan.
Thus `nper(`

equals
the value of `rate`, `payment`, `amount`)`n` such that `pv(`

. If `rate`, `n`,
`payment`) = `amount``payment` is too small
ever to amortize a loan for `amount` at interest rate `rate`,
the `nper`

function is left in symbolic form.

The `I b #` [`nperb`

] command does the same computation
but using `pvb`

instead of `pv`

. You can give a fourth
lump-sum argument to these functions, but the computation will be
rather slow in the four-argument case.

The `H b #` [`nperl`

] command does the same computation
using `pvl`

. By exchanging `payment` and `amount` you
can also get the solution for `fvl`

. For example,
`nperl(8%, 2000, 1000) = 9.006`

, so if you place $1000 in a
bank account earning 8%, it will take nine years to grow to $2000.

The `b T` (`calc-fin-rate`

) [`rate`

] command computes
the rate of return on an investment. This is also an inverse of `pv`

:
`rate(`

computes the value of
`n`, `payment`, `amount`)`rate` such that `pv(`

. The result is expressed as a formula like ‘`rate`, `n`, `payment`) =
`amount``6.3%`’.

The `I b T` [`rateb`

] and `H b T` [`ratel`

]
commands solve the analogous equations with `pvb`

or `pvl`

in place of `pv`

. Also, `rate`

and `rateb`

can
accept an optional fourth argument just like `pv`

and `pvb`

.
To redo the above example from a different perspective,
`ratel(9, 2000, 1000) = 8.00597%`

, which says you will need an
interest rate of 8% in order to double your account in nine years.

The `b I` (`calc-fin-irr`

) [`irr`

] command is the
analogous function to `rate`

but for net present value.
Its argument is a vector of payments. Thus `irr(`

computes the `payments`)`rate` such that `npv(`

;
this rate is known as the `rate`, `payments`) = 0*internal rate of return*.

The `I b I` [`irrb`

] command computes the internal rate of
return assuming payments occur at the beginning of each period.

#### 7.6.5 Depreciation Functions

The functions in this section calculate *depreciation*, which is
the amount of value that a possession loses over time. These functions
are characterized by three parameters: `cost`, the original cost
of the asset; `salvage`, the value the asset will have at the end
of its expected “useful life”; and `life`, the number of years
(or other periods) of the expected useful life.

There are several methods for calculating depreciation that differ in the way they spread the depreciation over the lifetime of the asset.

The `b S` (`calc-fin-sln`

) [`sln`

] command computes the
“straight-line” depreciation. In this method, the asset depreciates
by the same amount every year (or period). For example,
‘`sln(12000, 2000, 5)`’ returns 2000. The asset costs $12000
initially and will be worth $2000 after five years; it loses $2000
per year.

The `b Y` (`calc-fin-syd`

) [`syd`

] command computes the
accelerated “sum-of-years’-digits” depreciation. Here the depreciation
is higher during the early years of the asset’s life. Since the
depreciation is different each year, `b Y` takes a fourth `period`
parameter which specifies which year is requested, from 1 to `life`.
If `period` is outside this range, the `syd`

function will
return zero.

The `b D` (`calc-fin-ddb`

) [`ddb`

] command computes an
accelerated depreciation using the double-declining balance method.
It also takes a fourth `period` parameter.

For symmetry, the `sln`

function will accept a `period`
parameter as well, although it will ignore its value except that the
return value will as usual be zero if `period` is out of range.

For example, pushing the vector ‘`[1,2,3,4,5]`’ (perhaps with `v x 5`)
and then mapping `V M ' [sln(12000,2000,5,$), syd(12000,2000,5,$),
ddb(12000,2000,5,$)] RET` produces a matrix that allows us to compare
the three depreciation methods:

[ [ 2000, 3333, 4800 ] [ 2000, 2667, 2880 ] [ 2000, 2000, 1728 ] [ 2000, 1333, 592 ] [ 2000, 667, 0 ] ]

(Values have been rounded to nearest integers in this figure.)
We see that `sln`

depreciates by the same amount each year,
`syd` depreciates more at the beginning and less at the end,
and `ddb` weights the depreciation even more toward the beginning.

Summing columns with `V R : +` yields ‘`[10000, 10000, 10000]`’;
the total depreciation in any method is (by definition) the
difference between the cost and the salvage value.

#### 7.6.6 Definitions

For your reference, here are the actual formulas used to compute Calc’s financial functions.

Calc will not evaluate a financial function unless the `rate` or
`n` argument is known. However, `payment` or `amount` can
be a variable. Calc expands these functions according to the
formulas below for symbolic arguments only when you use the `a "`
(`calc-expand-formula`

) command, or when taking derivatives or
integrals or solving equations involving the functions.

These formulas are shown using the conventions of Big display
mode (`d B`); for example, the formula for `fv`

written
linearly is ‘`pmt * ((1 + rate)^n) - 1) / rate`’.

n (1 + rate) - 1 fv(rate, n, pmt) = pmt * --------------- rate n ((1 + rate) - 1) (1 + rate) fvb(rate, n, pmt) = pmt * ---------------------------- rate n fvl(rate, n, pmt) = pmt * (1 + rate) -n 1 - (1 + rate) pv(rate, n, pmt) = pmt * ---------------- rate -n (1 - (1 + rate) ) (1 + rate) pvb(rate, n, pmt) = pmt * ----------------------------- rate -n pvl(rate, n, pmt) = pmt * (1 + rate) -1 -2 -3 npv(rate, [a, b, c]) = a*(1 + rate) + b*(1 + rate) + c*(1 + rate) -1 -2 npvb(rate, [a, b, c]) = a + b*(1 + rate) + c*(1 + rate) -n (amt - x * (1 + rate) ) * rate pmt(rate, n, amt, x) = ------------------------------- -n 1 - (1 + rate) -n (amt - x * (1 + rate) ) * rate pmtb(rate, n, amt, x) = ------------------------------- -n (1 - (1 + rate) ) (1 + rate) amt * rate nper(rate, pmt, amt) = - log(1 - ------------, 1 + rate) pmt amt * rate nperb(rate, pmt, amt) = - log(1 - ---------------, 1 + rate) pmt * (1 + rate) amt nperl(rate, pmt, amt) = - log(---, 1 + rate) pmt 1/n pmt ratel(n, pmt, amt) = ------ - 1 1/n amt cost - salv sln(cost, salv, life) = ----------- life (cost - salv) * (life - per + 1) syd(cost, salv, life, per) = -------------------------------- life * (life + 1) / 2 book * 2 ddb(cost, salv, life, per) = --------, book = cost - depreciation so far life

In `pmt`

and `pmtb`

, ‘`x=0`’ if omitted.

These functions accept any numeric objects, including error forms, intervals, and even (though not very usefully) complex numbers. The above formulas specify exactly the behavior of these functions with all sorts of inputs.

Note that if the first argument to the `log`

in `nper`

is
negative, `nper`

leaves itself in symbolic form rather than
returning a (financially meaningless) complex number.

‘`rate(num, pmt, amt)`’ solves the equation
‘`pv(rate, num, pmt) = amt`’ for ‘`rate`’ using `H a R`
(`calc-find-root`

), with the interval ‘`[.01% .. 100%]`’
for an initial guess. The `rateb`

function is the same except
that it uses `pvb`

. Note that `ratel`

can be solved
directly; its formula is shown in the above list.

Similarly, ‘`irr(pmts)`’ solves the equation ‘`npv(rate, pmts) = 0`’
for ‘`rate`’.

If you give a fourth argument to `nper`

or `nperb`

, Calc
will also use `H a R` to solve the equation using an initial
guess interval of ‘`[0 .. 100]`’.

A fourth argument to `fv`

simply sums the two components
calculated from the above formulas for `fv`

and `fvl`

.
The same is true of `fvb`

, `pv`

, and `pvb`

.

The `ddb` function is computed iteratively; the “book” value
starts out equal to `cost`, and decreases according to the above
formula for the specified number of periods. If the book value
would decrease below `salvage`, it only decreases to `salvage`
and the depreciation is zero for all subsequent periods. The `ddb`

function returns the amount the book value decreased in the specified
period.