reduce-historical/r38/doc/help/arith.tex

\section{Arithmetic Operations}

\begin{Introduction}{ARITHMETIC\_OPERATIONS}
This section considers operations defined in REDUCE that concern numbers,
or operators that can operate on numbers in addition, in most cases, to
more general expressions.
\end{Introduction}

\begin{Operator}{ABS}
\index{absolute value}
The \name{abs} operator returns the absolute value of its argument.

\begin{Syntax}
\name{abs}\(\meta{expression}\)
\end{Syntax}

\meta{expression} can be any REDUCE scalar expression.

\begin{Examples}
abs(-a);                    &        ABS(A) \\
abs(-5);                    &        5 \\
a := -10;                   &        A := -10 \\
abs(a);                     &        10 \\
abs(-a);                    &        10
\end{Examples}
\begin{Comments}
If the argument has had no numeric value assigned to it, such as an
identifier or polynomial, \name{abs} returns an expression involving
\name{abs} of its argument, doing as much simplification of the argument
as it can, such as dropping any preceding minus sign.
\end{Comments}
\end{Operator}


\begin{Switch}{ADJPREC}
\index{input}\index{precision}
When a real number is input, it is normally truncated to the 
\nameref{precision} in
effect at the time the number is read.  If it is desired to keep the full
precision of all numbers input, the switch \name{adjprec}
(for \meta{adjust precision}) can be turned on.  While on, \name{adjprec}
will automatically increase the precision, when necessary, to match that
of any integer or real input, and a message printed to inform the user of
the precision increase.

\begin{Examples}
on rounded; \\
1.23456789012345; & 1.23456789012 \\
on adjprec; \\
1.23456789012345; \\
*** precision increased to 15 \\
1.23456789012345
\end{Examples}
\end{Switch}


\begin{Operator}{ARG}
\index{complex}\index{polar angle}
If \nameref{complex} and \nameref{rounded} are on, and {\em arg}
evaluates to a complex number, \name{arg} returns the polar angle of {\em
arg}, measured in radians.  Otherwise an expression in {\em arg} is
returned.

\begin{Examples}
arg(3+4i)  & ARG(3 + 4*I) \\
on rounded, complex; \\
ws; & 0.927295218002 \\
arg a; & ARG(A)
\end{Examples}

\end{Operator}


\begin{Operator}{CEILING}
\index{integer}

\begin{Syntax}
\name{ceiling}\(\meta{expression}\)
\end{Syntax}

This operator returns the ceiling (i.e., the least integer greater than or
equal to its argument) if its argument has a numerical value.  For
negative numbers, this is equivalent to \nameref{fix}.  For non-numeric
arguments, the value is an expression in the original operator.

\begin{Examples}
ceiling 3.4; & 4 \\
fix 3.4; & 3 \\
ceiling(-5.2); & -5 \\
fix(-5.2); & -5 \\
ceiling a; & CEILING(A)
\end{Examples}

\end{Operator}


\begin{Operator}{CHOOSE}
\name{choose}(\meta{m},\meta{m}) returns the number of ways of choosing
\meta{m} objects from a collection of \meta{n} distinct objects --- in other
words the binomial coefficient.  If \meta{m} and \meta{n} are not positive
integers, or $m > n$, the expression is returned unchanged.
than or equal to
\begin{Examples}
choose(2,3); & 3 \\
choose(3,2); & CHOOSE(3,2) \\
choose(a,b); & CHOOSE(A,B)
\end{Examples}
\end{Operator}


\begin{Operator}{DEG2DMS}
\index{degrees}\index{radians}
\begin{Syntax}
\name{deg2dms}\(\meta{expression}\)
\end{Syntax}

In \nameref{rounded} mode, if \meta{expression} is a real number, the
operator \name{deg2dms} will interpret it as degrees, and convert it to a
list containing the equivalent degrees, minutes and seconds.  In all other
cases, an expression in terms of the original operator is returned.

\begin{Examples}
deg2dms 60; & DEG2DMS(60) \\
on rounded; \\
ws; & \{60,0,0\} \\
deg2dms 42.4; & \{42,23,60.0\} \\
deg2dms a; & DEG2DMS(A)
\end{Examples}

\end{Operator}

\begin{Operator}{DEG2RAD}
\index{degrees}\index{radians}

\begin{Syntax}
\name{deg2rad}\(\meta{expression}\)
\end{Syntax}

In \nameref{rounded} mode, if \meta{expression} is a real number, the
operator \name{deg2rad} will interpret it as degrees, and convert it to
the equivalent radians.  In all other cases, an expression in terms of the
original operator is returned.

\begin{Examples}
deg2rad 60; & DEG2RAD(60) \\
on rounded; \\
ws; & 1.0471975512 \\
deg2rad a; & DEG2RAD(A)
\end{Examples}

\end{Operator}


\begin{Operator}{DIFFERENCE}
The \name{difference} operator may be used as either an infix or prefix binary
subtraction operator.  It is identical to \name{-} as a binary operator.

\begin{Syntax}
\name{difference}\(\meta{expression},\meta{expression}\) or

\meta{expression} \name{difference} \meta{expression}
      \{\name{difference} \meta{expression}\}\optional
\end{Syntax}

\meta{expression} can be a number or any other valid REDUCE expression.  Matrix
expressions are allowed if they are of the same dimensions.

\begin{Examples}

difference(10,4);                                     &         6 \\

15 difference 5 difference 2;                         &         8 \\

a difference b;                                       &         A - B

\end{Examples}

\begin{Comments}
The \name{difference} operator is left associative, as shown in the second
example above.

\end{Comments}
\end{Operator}


\begin{Operator}{DILOG}
\index{dilogarithm function}
The \name{dilog} operator is known to the differentiation and integration
operators, but has numeric value attached only at \name{dilog(0)}.  Dilog is
defined by
\begin{TEX}
\begin{displaymath}
  dilog(x) = -\int{{log(x)\,dx}\over{x-1}}
\end{displaymath}
\end{TEX}
\begin{INFO}
  dilog(x) = -int(log(x),x)/(x-1)
\end{INFO}
\begin{Examples}
df(dilog(x**2),x);          &      - \rfrac{2*LOG(X^{2})*X}{X^{2}  - 1}\\

int(dilog(x),x);            &     DILOG(X)*X - DILOG(X) + LOG(X)*X - X \\

dilog(0);                   &     \rfrac{PI^{2}}{6}
\end{Examples}

\end{Operator}


\begin{Operator}{DMS2DEG}
\index{degrees}\index{radians}

\begin{Syntax}
\name{dms2deg}\(\meta{list}\)
\end{Syntax}

In \nameref{rounded} mode, if \meta{list} is a list of three real numbers,
the operator \name{dms2deg} will interpret the list as degrees, minutes
and seconds and convert it to the equivalent degrees.  In all other cases,
an expression in terms of the original operator is returned.

\begin{Examples}
dms2deg \{42,3,7\}; & DMS2DEG(\{42,3,7\}) \\
on rounded; \\
ws; & 42.0519444444 \\
dms2deg a; & DMS2DEG(A)
\end{Examples}

\end{Operator}


\begin{Operator}{DMS2RAD}
\index{degrees}\index{radians}

\begin{Syntax}
\name{dms2rad}\(\meta{list}\)
\end{Syntax}

In \nameref{rounded} mode, if \meta{list} is a list of three real numbers,
the operator \name{dms2rad} will interpret the list as degrees, minutes
and seconds and convert it to the equivalent radians.  In all other cases,
an expression in terms of the original operator is returned.

\begin{Examples}
dms2rad \{42,3,7\}; & DMS2RAD(\{42,3,7\}) \\
on rounded; \\
ws; & 0.733944887421 \\
dms2rad a; & DMS2RAD(A)
\end{Examples}

\end{Operator}


\begin{Operator}{FACTORIAL}
\index{gamma}
\begin{Syntax}
\name{factorial}\(\meta{expression}\)
\end{Syntax}

If the argument of \name{factorial} is a positive integer or zero, its
factorial is returned.  Otherwise the result is expressed in terms of the
original operator.  For more general operations, the \nameref{gamma} operator
is available in the \nameref{Special Function Package}.

\begin{Examples}
factorial 4; & 24 \\
factorial 30 ; & 265252859812191058636308480000000 \\
factorial(a) ; FACTORIAL(A)
\end{Examples}

\end{Operator}


\begin{Operator}{FIX}
\index{integer}
\begin{Syntax}
\name{fix}\(\meta{expression}\)
\end{Syntax}

The operator \name{fix} returns the integer part of its argument, if that
argument has a numerical value.  For positive numbers, this is equivalent
to \nameref{floor}, and, for negative numbers, \nameref{ceiling}.  For
non-numeric arguments, the value is an expression in the original operator.

\begin{Examples}
fix 3.4; & 3 \\
floor 3.4; & 3 \\
ceiling 3.4; & 4 \\
fix(-5.2); & -5 \\
floor(-5.2); & -6 \\
ceiling(-5.2); & -5 \\
fix(a); & FIX(A)
\end{Examples}

\end{Operator}


\begin{Operator}{FIXP}
\index{integer}
The \name{fixp} logical operator returns true if its argument is an integer.
\begin{Syntax}
\name{fixp}\(\meta{expression}\) or \name{fixp} \meta{simple\_expression}
\end{Syntax}

\meta{expression} can be any valid REDUCE expression, \meta{simple\_expression}
must be a single identifier or begin with a prefix operator.

\begin{Examples}
if fixp 1.5 then write "ok" else write "not";
			     &             not \\
if fixp(a) then write "ok" else write "not";
			     &             not \\
a := 15;                     &             A := 15 \\
if fixp(a) then write "ok" else write "not";
			     &             ok
\end{Examples}

\begin{Comments}
Logical operators can only be used inside conditional expressions such as
\name{if}\ldots\name{then} or \name{while}\ldots\name{do}.
\end{Comments}
\end{Operator}


\begin{Operator}{FLOOR}
\index{integer}

\begin{Syntax}
\name{floor}\(\meta{expression}\)
\end{Syntax}

This operator returns the floor (i.e., the greatest integer less than or
equal to its argument) if its argument has a numerical value.  For
positive numbers, this is equivalent to \nameref{fix}.  For non-numeric
arguments, the value is an expression in the original operator.

\begin{Examples}
floor 3.4; & 3 \\
fix 3.4; & 3 \\
floor(-5.2); & -6 \\
fix(-5.2); & -5 \\
floor a; & FLOOR(A)
\end{Examples}

\end{Operator}


\begin{Operator}{EXPT}
The \name{expt} operator is both an infix and prefix binary exponentiation
operator.  It is identical to \name{^} or \name{**}.
\begin{Syntax}
\name{expt}\(\meta{expression},\meta{expression}\)
 or \meta{expression} \name{expt} \meta{expression}
\end{Syntax}
\begin{Examples}
a expt b;                    &    A^{B} \\
expt(a,b);                   &    A^{B} \\
(x+y) expt 4;                &    X^{4} + 4*X^{3}*Y + 6*X^{2}*Y^{2} + 4*X*Y^{3} + Y^{4}
\end{Examples}
\begin{Comments}
Scalar expressions may be raised to fractional and floating-point powers.
Square matrix expressions may be raised to positive powers, and also to
negative powers if non-singular.

\name{expt} is left associative.  In other words, \name{a expt b expt c} is
equivalent to \name{a expt (b*c)}, not \name{a expt (b expt c)}, which
would be right associative.
\end{Comments}
\end{Operator}


\begin{Operator}{GCD}
\index{greatest common divisor}\index{polynomial}
The \name{gcd} operator returns the greatest common divisor of two
polynomials.
\begin{Syntax}
\name{gcd}\(\meta{expression},\meta{expression}\)
\end{Syntax}
\meta{expression} must be a polynomial (or integer), otherwise an error
occurs.

\begin{Examples}
gcd(2*x**2 - 2*y**2,4*x + 4*y);                         &       2*(X + Y) \\
gcd(sin(x),x**2 + 1);                                   &       1  \\
gcd(765,68);                                            &       17
\end{Examples}

\begin{Comments}
The operator \name{gcd} described here provides an explicit means to find the
gcd of two expressions.  The switch \name{gcd} described below simplifies
expressions by finding and canceling gcd's at every opportunity.  When
the switch \nameref{ezgcd} is also on, gcd's are figured using the EZ GCD
algorithm, which is usually faster.
%%% The \nameref{heugcd} switch is also available, providing
%%% gcd's by the Heuristic algorithm.
\end{Comments}
\end{Operator}


\begin{Operator}{LN}
\index{logarithm}
\begin{Syntax}
\name{ln}\(\meta{expression}\)
\end{Syntax}
\meta{expression} can be any valid scalar REDUCE expression.

The \name{ln} operator returns the natural logarithm of its argument.
However, unlike \nameref{log}, there are no algebraic rules associated
with it; it will only evaluate when \nameref{rounded} is on, and the
argument is a real number.

\begin{Examples}
ln(x);                     &         LN(X) \\
ln 4;                      &         LN(4) \\
ln(e);                     &         LN(E) \\
df(ln(x),x);               &         DF(LN(X),X) \\
on rounded; \\
ln 4;                      &         1.38629436112 \\
ln e;                      &         1
\end{Examples}

\begin{Comments}
Because of the restricted algebraic properties of \name{ln}, users are
advised to use \nameref{log} whenever possible.
\end{Comments}

\end{Operator}


\begin{Operator}{LOG}
\index{logarithm}
The \name{log} operator returns the natural logarithm of its argument.
\begin{Syntax}
\name{log}\(\meta{expression}\) or \name{log} \meta{expression}
\end{Syntax}

\meta{expression} can be any valid scalar REDUCE expression.

\begin{Examples}
log(x);                     &         LOG(X) \\
log 4;                      &         LOG(4) \\
log(e);                     &         1 \\
on rounded; \\
log 4;                      &         1.38629436112
\end{Examples}

\begin{Comments}
\name{log} returns a numeric value only when \nameref{rounded} is on.  In that
case, use of a negative argument for \name{log} results in an error
message.  No error is given on a negative argument when REDUCE is not in
that mode.
\end{Comments}
\end{Operator}


\begin{Operator}{LOGB}
\index{logarithm}
\begin{Syntax}
\name{logb}\(\meta{expression}\,\meta{integer}\)
\end{Syntax}
\meta{expression} can be any valid scalar REDUCE expression.

The \name{logb} operator returns the logarithm of its first argument using
the second argument as base.  However, unlike \nameref{log}, there are no
algebraic rules associated with it; it will only evaluate when
\nameref{rounded} is on, and the first argument is a real number.

\begin{Examples}
logb(x,2);                 &         LOGB(X,2) \\
logb(4,3);                 &         LOGB(4,3) \\
logb(2,2);                 &         LOGB(2,2) \\
df(logb(x,3),x);           &         DF(LOGB(X,3),X) \\
on rounded; \\
logb(4,3);                 &         1.26185950714 \\
logb(2,2);                 &         1
\end{Examples}

\end{Operator}


\begin{Operator}{MAX}
\index{maximum}
The operator \name{max} is an n-ary prefix operator, which returns the largest
value in its arguments.
\begin{Syntax}
\name{max}\(\meta{expression}\{,\meta{expression}\}\optional\)

\end{Syntax}

\meta{expression} must evaluate to a number.  \name{max} of an empty list
returns 0.

\begin{Examples}
max(4,6,10,-1);              &        10 \\
<<a := 23;b := 2*a;c := 4**2;max(a,b,c)>>;
			     &        46 \\
max(-5,-10,-a);              &        -5
\end{Examples}

\end{Operator}


\begin{Operator}{MIN}
\index{minimum}
The operator \name{min} is an n-ary prefix operator, which returns the
smallest value in its arguments.
\begin{Syntax}
\name{min}\(\meta{expression}\{,\meta{expression}\}\optional\)
\end{Syntax}

\meta{expression} must evaluate to a number. \name{min} of an empty list
returns 0.
\begin{Examples}
min(-3,0,17,2);              &        -3 \\
<<a := 23;b := 2*a;c := 4**2;min(a,b,c)>>;
			     &        16 \\
min(5,10,a);                 &        5
\end{Examples}
\end{Operator}


\begin{Operator}{MINUS}
The \name{minus} operator is a unary minus, returning the negative of its
argument.  It is equivalent to the unary \name{-}.
\begin{Syntax}
\name{minus}\(\meta{expression}\)


\end{Syntax}

\meta{expression} may be any scalar REDUCE expression.

\begin{Examples}
minus(a);                    &          - A \\
minus(-1);                   &          1 \\
minus((x+1)**4);             &          - (X^{4} + 4*X^{3} + 6*X^{2} + 4*X + 1)
\end{Examples}

\end{Operator}


\begin{Operator}{NEXTPRIME}
\index{prime number}
\begin{Syntax}
\name{nextprime}\(\meta{expression}\)
\end{Syntax}

If the argument of \name{nextprime} is an integer, the least prime greater
than that argument is returned.  Otherwise, a type error results.

\begin{Examples}
nextprime 5001; & 5003  \\
nextprime(10^30); & 1000000000000000000000000000057 \\
nextprime a; & ***** A invalid as integer
\end{Examples}

\end{Operator}


\begin{Switch}{NOCONVERT}
Under normal circumstances when \name{rounded} is on, \REDUCE\ converts the
number 1.0 to the integer 1.  If this is not desired, the switch
\name{noconvert} can be turned on.
\begin{Examples}
on rounded; \\
1.0000000000001; & 1 \\
on noconvert; \\
1.0000000000001; & 1.0 \\
\end{Examples}
\end{Switch}

\begin{Operator}{NORM}
\index{complex}
\begin{Syntax}
\name{norm}\(\meta{expression}\)
\end{Syntax}

If \name{rounded} is on, and the argument is a real number, \meta{norm}
returns its absolute value.  If \name{complex} is also on, \meta{norm}
returns the square root of the sum of squares of the real and imaginary
parts of the argument.  In all other cases, a result is returned in
terms of the original operator.

\begin{Examples}
norm (-2); & NORM(-2) \\
on rounded;\\
ws; &  2.0 \\
norm(3+4i); & NORM(4*I+3) \\
on complex;\\
ws; &  5.0\\
\end{Examples}

\end{Operator}


\begin{Operator}{PERM}
\index{permutation}
\begin{Syntax}
perm(\meta{expression1},\meta{expression2})
\end{Syntax}

If \meta{expression1} and \meta{expression2} evaluate to positive integers,
\name{perm} returns the number of permutations possible in selecting
\meta{expression1} objects from \meta{expression2} objects.
In other cases, an expression in the original operator is returned.

\begin{Examples}
perm(1,1); & 1 \\
perm(3,5); & 60 \\
perm(-3,5); & PERM(-3,5) \\
perm(a,b); & PERM(A,B)
\end{Examples}

\end{Operator}

\begin{Operator}{PLUS}
The \name{plus} operator is both an infix and prefix n-ary addition
operator.  It exists because of the way in which {\REDUCE} handles such
operators internally, and is not recommended for use in algebraic mode
programming. \nameref{plussign}, which has the identical effect, should be
used instead.
\begin{Syntax}
\name{plus}\(\meta{expression},\meta{expression}\{,\meta{expression}\}
\optional\) or \\
        \meta{expression} \name{plus} \meta{expression} \{\name{plus} \meta{expression}\}\optional
\end{Syntax}

\meta{expression} can be any valid REDUCE expression, including matrix
expressions of the same dimensions.

\begin{Examples}
a plus b plus c plus d;      &      A + B + C + D \\
4.5 plus 10;                 & \rfrac{29}{2} \\\\
plus(x**2,y**2);             &      X^{2} + Y^{2}
\end{Examples}
\end{Operator}


\begin{Operator}{QUOTIENT}
The \name{quotient} operator is both an infix and prefix binary operator that
returns the quotient of its first argument divided by its second.  It is
also a unary \nameref{recip}rocal operator.  It is identical to \name{/} and 
\nameref{slash}.
\begin{Syntax}
\name{quotient}\(\meta{expression},\meta{expression}\) or
\meta{expression} \name{quotient} \meta{expression}  or
\name{quotient}\(\meta{expression}\) or
\name{quotient} \meta{expression}
\end{Syntax}

\meta{expression} can be any valid REDUCE scalar expression.  Matrix
expressions can also be used if the second expression is invertible and the
matrices are of the correct dimensions.
\begin{Examples}
quotient(a,x+1);              &         \rfrac{A}{X + 1} \\
7 quotient 17;                &         \rfrac{7}{17} \\
on rounded; \\
4.5 quotient 2;               &          2.25 \\
quotient(x**2 + 3*x + 2,x+1); &          X + 2 \\
matrix m,inverse; \\
m := mat((a,b),(c,d));        & \begin{multilineoutput}{6cm}
M(1,1) := A;
M(1,2) := B;
M(2,1) := C
M(2,2) := D
\end{multilineoutput}\\

inverse := quotient m;        & \begin{multilineoutput}{8cm}
INVERSE(1,1) := \rfrac{D}{A*D - B*C}
INVERSE(1,2) := - \rfrac{B}{A*D - B*C}
INVERSE(2,1) := - \rfrac{C}{A*D - B*C}
INVERSE(2,2) := \rfrac{A}{A*D - B*C}
\end{multilineoutput}

\end{Examples}

\begin{Comments}
The \name{quotient} operator is left associative: \name{a quotient b quotient c}
is equivalent to \name{(a quotient b) quotient c}.

If a matrix argument to the unary \name{quotient} is not invertible, or if the
second matrix argument to the binary quotient is not invertible, an error
message is given.
\end{Comments}
\end{Operator}


\begin{Operator}{RAD2DEG}
\index{degrees}\index{radians}
\begin{Syntax}
\name{rad2deg}\(\meta{expression}\)
\end{Syntax}

In \nameref{rounded} mode, if \meta{expression} is a real number, the
operator \name{rad2deg} will interpret it as radians, and convert it to
the equivalent degrees.  In all other cases, an expression in terms of the
original operator is returned.

\begin{Examples}
rad2deg 1; & RAD2DEG(1) \\
on rounded; \\
ws; & 57.2957795131 \\
rad2deg a; & RAD2DEG(A)
\end{Examples}

\end{Operator}


\begin{Operator}{RAD2DMS}
\index{degrees}\index{radians}

\begin{Syntax}
\name{rad2dms}\(\meta{expression}\)
\end{Syntax}

In \nameref{rounded} mode, if \meta{expression} is a real number, the
operator \name{rad2dms} will interpret it as radians, and convert it to a
list containing the equivalent degrees, minutes and seconds.  In all other
cases, an expression in terms of the original operator is returned.

\begin{Examples}
rad2dms 1; & RAD2DMS(1) \\
on rounded; \\
ws; & \{57,17,44.8062470964\} \\
rad2dms a; & RAD2DMS(A)
\end{Examples}

\end{Operator}


\begin{Operator}{RECIP}
\name{recip} is the  alphabetical name for the division operator \name{/}
or \nameref{slash} used as a unary operator.  The use of \name{/} is preferred.

\begin{Examples}
recip a; & \rfrac{1}{A} \\
recip 2; & \rfrac{1}{2}
\end{Examples}

\end{Operator}


\begin{Operator}{REMAINDER}
\index{polynomial}
The \name{remainder} operator returns the remainder after its first
argument is divided by its second argument.

\begin{Syntax}
\name{remainder}\(\meta{expression},\meta{expression}\)
\end{Syntax}

\meta{expression} can be any valid REDUCE polynomial, and is not limited
to numeric values.

\begin{Examples}
remainder(13,6);                                        &         1 \\
remainder(x**2 + 3*x + 2,x+1);                          &         0  \\
remainder(x**3 + 12*x + 4,x**2 + 1);                    &         11*X + 4 \\
remainder(sin(2*x),x*y);                                &         SIN(2*X)
\end{Examples}

\begin{Comments}
In the default case, remainders are calculated over the integers.  If you
need the remainder with respect to another domain, it must be declared
explicitly.

If the first argument to \name{remainder} contains a denominator not equal to
1, an error occurs.
\end{Comments}
\end{Operator}


\begin{Operator}{ROUND}
\index{integer}

\begin{Syntax}
\name{round}\(\meta{expression}\)
\end{Syntax}

If its argument has a numerical value, \name{round} rounds it to the
nearest integer.  For non-numeric arguments, the value is an expression in
the original operator.

\begin{Examples}
round 3.4; & 3 \\
round 3.5; & 4 \\
round a; & ROUND(A)
\end{Examples}

\end{Operator}


\begin{Command}{SETMOD}
\index{modular}
The \name{setmod} command sets the modulus value for subsequent \nameref{modular}
arithmetic.
\begin{Syntax}
\name{setmod} \meta{integer}
\end{Syntax}

\meta{integer} must be positive, and greater than 1.  It need not be a prime
number.

\begin{Examples}
setmod 6;                    &                1 \\
on modular; \\
16;                          &                4 \\
x^2 + 5x + 7;                &                X^{2} + 5*X + 1 \\
x/3;                         &                \rfrac{X}{3} \\
setmod 2;                    &                6 \\
(x+1)^4;                     &                X^{4} + 1 \\
x/3;                         &                X
\end{Examples}
\begin{Comments}
\name{setmod} returns the previous modulus, or 1 if none has been set
before.  \name{setmod} only has effect when \nameref{modular} is on.

Modular operations are done only on numbers such as coefficients of 
polynomials, not on the exponents.   The modulus need not be prime.
Attempts to divide by a power of the modulus produces an error message, since the
operation is equivalent to dividing by 0.  However, dividing by a factor
of a non-prime modulus does not produce an error message.
\end{Comments}
\end{Command}


\begin{Operator}{SIGN}

\begin{Syntax}
\name{sign} \meta{expression}
\end{Syntax}

\name{sign} tries to evaluate the sign of its argument. If this
is possible \name{sign} returns one of 1, 0 or -1.  Otherwise, the result
is the original form or a simplified variant.

\begin{Examples}
        sign(-5)      &  -1\\
        sign(-a^2*b)  &  -SIGN(B)\\
\end{Examples}

\begin{Comments}
Even powers of formal expressions are assumed to be positive only as long
as the switch \nameref{complex} is off.
\end{Comments}
\end{Operator}


\begin{Operator}{SQRT}
\index{square root}
The \name{sqrt} operator returns the square root of its argument.
\begin{Syntax}
\name{sqrt}\(\meta{expression}\)
\end{Syntax}

\meta{expression} can be any REDUCE scalar expression.

\begin{Examples}
sqrt(16*a^3);               &         4*SQRT(A)*A \\
sqrt(17);                    &         SQRT(17) \\
on rounded; \\
sqrt(17);                    &         4.12310562562 \\
off rounded; \\
sqrt(a*b*c^5*d^3*27);      &
                     3*SQRT(D)*SQRT(C)*SQRT(B)*SQRT(A)*SQRT(3)*C^{2}*D
\end{Examples}
\begin{Comments}
\name{sqrt} checks its argument for squared factors and removes them.
% If the \name{reduced} switch is on, square roots of a product become a
% product of square roots.

Numeric values for square roots that are not exact integers are given only
when \nameref{rounded} is on.

Please note that \name{sqrt(a**2)} is given as \name{a}, which may be
incorrect if \name{a} eventually has a negative value.  If you are
programming a calculation in which this is a concern, you can turn on the
\nameref{precise} switch, which causes the absolute value of the square root
to be returned.
\end{Comments}
\end{Operator}


\begin{Operator}{TIMES}
The \name{times} operator is an infix or prefix n-ary multiplication
operator.  It is identical to \name{*}.
\begin{Syntax}
\meta{expression} \name{times} \meta{expression} \{\name{times} \meta{expression}\}\optional

 or \name{times}\(\meta{expression},\meta{expression} \{,\meta{expression}\}\optional\)
\end{Syntax}

\meta{expression} can be any valid REDUCE scalar or matrix expression.
Matrix expressions must be of the correct dimensions.  Compatible scalar
and matrix expressions can be mixed.

\begin{Examples}
var1 times var2;             &         VAR1*VAR2 \\
times(6,5);                  &         30 \\
matrix aa,bb; \\
aa := mat((1),(2),(x))\$ \\
bb := mat((0,3,1))\$ \\
aa times bb times 5;         & 
\begin{multilineoutput}{6cm}
[0   15    5 ]
[            ]
[0   30   10 ]
[            ]
[0  15*X  5*X]
\end{multilineoutput}
\end{Examples}
\end{Operator}