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

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

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

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.