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,
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 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') command [`fdiv' function in a formula] 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.