Next: Algebraic Simplifications, Up: Simplifying Formulas [Contents][Index]

This section describes basic simplifications which Calc performs in many
situations. For example, both binary simplifications and algebraic
simplifications begin by performing these basic simplifications. You
can type `m I` to restrict the simplifications done on the stack to
these simplifications.

The most basic simplification is the evaluation of functions.
For example, ‘`2 + 3`’ is evaluated to ‘`5`’, and ‘` sqrt(9)`’
is evaluated to ‘

Calc simplifies (evaluates) the arguments to a function before it
simplifies the function itself. Thus ‘` sqrt(5+4)`’ is
simplified to ‘

`sqrt`

function
itself is applied. There are very few exceptions to this rule:
`quote`

, `lambda`

, and `condition`

(the `::`

operator) do not evaluate their arguments, `if`

(the `? :`

operator) does not evaluate all of its arguments, and `evalto`

does not evaluate its lefthand argument.
Most commands apply at least these basic simplifications to all
arguments they take from the stack, perform a particular operation,
then simplify the result before pushing it back on the stack. In the
common special case of regular arithmetic commands like `+` and
`Q` [`sqrt`

], the arguments are simply popped from the stack
and collected into a suitable function call, which is then simplified
(the arguments being simplified first as part of the process, as
described above).

Even the basic set of simplifications are too numerous to describe completely here, but this section will describe the ones that apply to the major arithmetic operators. This list will be rather technical in nature, and will probably be interesting to you only if you are a serious user of Calc’s algebra facilities.

As well as the simplifications described here, if you have stored
any rewrite rules in the variable `EvalRules`

then these rules
will also be applied before any of the basic simplifications.
See Automatic Rewrites, for details.

And now, on with the basic simplifications:

Arithmetic operators like `+` and `*` always take two
arguments in Calc’s internal form. Sums and products of three or
more terms are arranged by the associative law of algebra into
a left-associative form for sums, ‘`((a + b) + c) + d`’, and
(by default) a right-associative form for products,
‘`a * (b * (c * d))`’. Formulas like ‘`(a + b) + (c + d)`’ are
rearranged to left-associative form, though this rarely matters since
Calc’s algebra commands are designed to hide the inner structure of sums
and products as much as possible. Sums and products in their proper
associative form will be written without parentheses in the examples
below.

Sums and products are *not* rearranged according to the
commutative law (‘`a + b`’ to ‘`b + a`’) except in a few
special cases described below. Some algebra programs always
rearrange terms into a canonical order, which enables them to
see that ‘`a b + b a`’ can be simplified to ‘`2 a b`’.
If you are using Basic Simplification mode, Calc assumes you have put
the terms into the order you want and generally leaves that order alone,
with the consequence that formulas like the above will only be
simplified if you explicitly give the `a s` command.
See Algebraic Simplifications.

Differences ‘`a - b`’ are treated like sums ‘`a + (-b)`’
for purposes of simplification; one of the default simplifications
is to rewrite ‘`a + (-b)`’ or ‘`(-b) + a`’, where ‘`-b`’
represents a “negative-looking” term, into ‘`a - b`’ form.
“Negative-looking” means negative numbers, negated formulas like
‘`-x`’, and products or quotients in which either term is
negative-looking.

Other simplifications involving negation are ‘`-(-x)`’ to ‘`x`’;
‘`-(a b)`’ or ‘`-(a/b)`’ where either ‘`a`’ or ‘`b`’ is
negative-looking, simplified by negating that term, or else where
‘`a`’ or ‘`b`’ is any number, by negating that number;
‘`-(a + b)`’ to ‘`-a - b`’, and ‘`-(b - a)`’ to ‘`a - b`’.
(This, and rewriting ‘`(-b) + a`’ to ‘`a - b`’, are the only
cases where the order of terms in a sum is changed by the default
simplifications.)

The distributive law is used to simplify sums in some cases:
‘`a x + b x`’ to ‘`(a + b) x`’, where ‘`a`’ represents
a number or an implicit 1 or *-1* (as in ‘`x`’ or ‘`-x`’)
and similarly for ‘`b`’. Use the `a c`, `a f`, or
`j M` commands to merge sums with non-numeric coefficients
using the distributive law.

The distributive law is only used for sums of two terms, or
for adjacent terms in a larger sum. Thus ‘`a + b + b + c`’
is simplified to ‘`a + 2 b + c`’, but ‘`a + b + c + b`’
is not simplified. The reason is that comparing all terms of a
sum with one another would require time proportional to the
square of the number of terms; Calc omits potentially slow
operations like this in basic simplification mode.

Finally, ‘`a + 0`’ and ‘`0 + a`’ are simplified to ‘`a`’.
A consequence of the above rules is that ‘`0 - a`’ is simplified
to ‘`-a`’.

The products ‘`1 a`’ and ‘`a 1`’ are simplified to ‘`a`’;
‘`(-1) a`’ and ‘`a (-1)`’ are simplified to ‘`-a`’;
‘`0 a`’ and ‘`a 0`’ are simplified to ‘`0`’, except that
in Matrix mode where ‘`a`’ is not provably scalar the result
is the generic zero matrix ‘`idn(0)`’, and that if ‘`a`’ is
infinite the result is ‘`nan`’.

Also, ‘`(-a) b`’ and ‘`a (-b)`’ are simplified to ‘`-(a b)`’,
where this occurs for negated formulas but not for regular negative
numbers.

Products are commuted only to move numbers to the front:
‘`a b 2`’ is commuted to ‘`2 a b`’.

The product ‘`a (b + c)`’ is distributed over the sum only if
‘`a`’ and at least one of ‘`b`’ and ‘`c`’ are numbers:
‘`2 (x + 3)`’ goes to ‘`2 x + 6`’. The formula
‘`(-a) (b - c)`’, where ‘`-a`’ is a negative number, is
rewritten to ‘`a (c - b)`’.

The distributive law of products and powers is used for adjacent
terms of the product: ‘`x^a x^b`’ goes to
‘`x^(a+b)`’
where ‘`a`’ is a number, or an implicit 1 (as in ‘`x`’),
or the implicit one-half of ‘` sqrt(x)`’, and similarly for
‘

The product of a negative power times anything but another negative
power is changed to use division:
‘`x^(-2) y`’
goes to ‘`y / x^2`’ unless Matrix mode is
in effect and neither ‘`x`’ nor ‘`y`’ are scalar (in which
case it is considered unsafe to rearrange the order of the terms).

Finally, ‘`a (b/c)`’ is rewritten to ‘`(a b)/c`’, and also
‘`(a/b) c`’ is changed to ‘`(a c)/b`’ unless in Matrix mode.

Simplifications for quotients are analogous to those for products.
The quotient ‘`0 / x`’ is simplified to ‘`0`’, with the same
exceptions that were noted for ‘`0 x`’. Likewise, ‘`x / 1`’
and ‘`x / (-1)`’ are simplified to ‘`x`’ and ‘`-x`’,
respectively.

The quotient ‘`x / 0`’ is left unsimplified or changed to an
infinite quantity, as directed by the current infinite mode.
See Infinite Mode.

The expression
‘`a / b^(-c)`’
is changed to ‘`a b^c`’, where ‘`-c`’ is any negative-looking
power. Also, ‘`1 / b^c`’ is changed to
‘`b^(-c)`’
for any power ‘`c`’.

Also, ‘`(-a) / b`’ and ‘`a / (-b)`’ go to ‘`-(a/b)`’;
‘`(a/b) / c`’ goes to ‘`a / (b c)`’; and ‘`a / (b/c)`’
goes to ‘`(a c) / b`’ unless Matrix mode prevents this
rearrangement. Similarly, ‘`a / (b:c)`’ is simplified to
‘`(c:b) a`’ for any fraction ‘`b:c`’.

The distributive law is applied to ‘`(a + b) / c`’ only if
‘`c`’ and at least one of ‘`a`’ and ‘`b`’ are numbers.
Quotients of powers and square roots are distributed just as
described for multiplication.

Quotients of products cancel only in the leading terms of the
numerator and denominator. In other words, ‘`a x b / a y b`’
is canceled to ‘`x b / y b`’ but not to ‘`x / y`’. Once
again this is because full cancellation can be slow; use `a s`
to cancel all terms of the quotient.

Quotients of negative-looking values are simplified according
to ‘`(-a) / (-b)`’ to ‘`a / b`’, ‘`(-a) / (b - c)`’
to ‘`a / (c - b)`’, and ‘`(a - b) / (-c)`’ to ‘`(b - a) / c`’.

The formula ‘`x^0`’ is simplified to ‘`1`’, or to ‘`idn(1)`’
in Matrix mode. The formula ‘`0^x`’ is simplified to ‘`0`’
unless ‘`x`’ is a negative number, complex number or zero.
If ‘`x`’ is negative, complex or ‘`0.0`’, ‘`0^x`’ is an
infinity or an unsimplified formula according to the current infinite
mode. The expression ‘`0^0`’ is simplified to ‘`1`’.

Powers of products or quotients ‘`(a b)^c`’, ‘`(a/b)^c`’
are distributed to ‘`a^c b^c`’, ‘`a^c / b^c`’ only if ‘`c`’
is an integer, or if either ‘`a`’ or ‘`b`’ are nonnegative
real numbers. Powers of powers ‘`(a^b)^c`’ are simplified to
‘`a^(b c)`’
only when ‘`c`’ is an integer and ‘`b c`’ also
evaluates to an integer. Without these restrictions these simplifications
would not be safe because of problems with principal values.
(In other words,
‘`((-3)^1:2)^2`’
is safe to simplify, but
‘`((-3)^2)^1:2`’
is not.) See Declarations, for ways to inform Calc that your
variables satisfy these requirements.

As a special case of this rule, ‘` sqrt(x)^n`’ is simplified to
‘

If ‘`a`’ is known to be real, ‘`b`’ is an even integer, and
‘`c`’ is a half- or quarter-integer, then ‘`(a^b)^c`’ is
simplified to ‘` abs(a^(b c))`’.

Also, ‘`(-a)^b`’ is simplified to ‘`a^b`’ if ‘`b`’ is an
even integer, or to ‘`-(a^b)`’ if ‘`b`’ is an odd integer,
for any negative-looking expression ‘`-a`’.

Square roots ‘` sqrt(x)`’ generally act like one-half powers
‘

Also, note that
‘`1 / x^1:2`’
is changed to
‘`x^(-1:2)`’,
but ‘`1 / sqrt(x)`’ is left alone.

Generic identity matrices (see Matrix and Scalar Modes) are simplified by the
following rules: ‘` idn(a) + b`’ to ‘

`idn`

; and ‘The `floor`

function and other integer truncation functions
vanish if the argument is provably integer-valued, so that
‘` floor(round(x))`’ simplifies to ‘

`float`

, `floor`

and its friends,
and `ffloor`

and its friends, are simplified in appropriate
ways. See Integer Truncation.
The expression ‘` abs(-x)`’ changes to ‘

While most functions do not recognize the variable `i`

as an
imaginary number, the `arg`

function does handle the two cases
‘` arg(i)`’ and ‘

The expression ‘` conj(conj(x))`’ simplifies to ‘

`conj`

, `re`

, and
`im`

are simplified, especially if some of the arguments are
provably real or involve the constant `i`

. For example,
‘Functions like `sin`

and `arctan`

generally don’t have
any default simplifications beyond simply evaluating the functions
for suitable numeric arguments and infinity. The algebraic
simplifications described in the next section do provide some
simplifications for these functions, though.

One important simplification that does occur is that
‘` ln(e)`’ is simplified to 1, and ‘

Among the logical functions, `!( a <= b)` changes to

Most other Calc functions have few if any basic simplifications defined, aside of course from evaluation when the arguments are suitable numbers.

Next: Algebraic Simplifications, Up: Simplifying Formulas [Contents][Index]