% -- a binary operator, usually used for remainder and reduction

Synopsis

Usage:

x % y

Description

The usual meaning for this operator is remainder, or normal form with respect to a Gröbner basis.

For integers, the remainder is non-negative.

i1 : 1232132141242345 % 1000000

o1 = 242345

i2 : (-4)%5

o2 = 1

In polynomial rings, the division algorithm is used.

i3 : A = ZZ[a,b]

o3 = A

o3 : PolynomialRing

i4 : (3*a^3-a*b-4) % (5*a-b)

         3    2
o4 = - 2a  + a b - a*b - 4

o4 : A

i5 : pseudoRemainder(3*a^3-a*b-4, 5*a-b)

       3      2
o5 = 3b  - 25b  - 500

o5 : A

i6 : B = QQ[a,b]

o6 = B

o6 : PolynomialRing

i7 : (3*a^3-a*b-4) % (5*a-b)

      3  3   1 2
o7 = ---b  - -b  - 4
     125     5

o7 : B

In more complicated situations, Gröbner bases are usually needed. See methods for normal forms and remainder.

Ways to use symbol % :

CC % CC
CC % QQ
CC % RR
CC % ZZ
Number % GroebnerBasis
Number % Ideal
Number % RingElement
QQ % QQ
QQ % ZZ
RingElement % Number
RR % QQ
RR % RR
RR % ZZ
ZZ % MonomialIdeal
ZZ % ZZ
Expression % Expression -- see Expression -- the class of all expressions
Expression % Holder -- see Expression -- the class of all expressions
Expression % Thing -- see Expression -- the class of all expressions
Holder % Expression -- see Expression -- the class of all expressions
Holder % Holder -- see Expression -- the class of all expressions
Thing % Expression -- see Expression -- the class of all expressions
InexactNumber % RingElement (missing documentation)
List % Number (missing documentation)
List % RingElement (missing documentation)
Matrix % GroebnerBasis -- calculate the normal form of ring elements and matrices using a (partially computed) Gröbner basis
RingElement % GroebnerBasis -- see Matrix % GroebnerBasis -- calculate the normal form of ring elements and matrices using a (partially computed) Gröbner basis
Matrix % Ideal -- see methods for normal forms and remainder -- normal form of ring elements and matrices
Matrix % Matrix -- see methods for normal forms and remainder -- normal form of ring elements and matrices
Matrix % Module -- see methods for normal forms and remainder -- normal form of ring elements and matrices
Matrix % MonomialIdeal -- see methods for normal forms and remainder -- normal form of ring elements and matrices
Matrix % RingElement -- see methods for normal forms and remainder -- normal form of ring elements and matrices
RingElement % Ideal -- see methods for normal forms and remainder -- normal form of ring elements and matrices
RingElement % Matrix -- see methods for normal forms and remainder -- normal form of ring elements and matrices
RingElement % MonomialIdeal -- see methods for normal forms and remainder -- normal form of ring elements and matrices
RingElement % RingElement -- see methods for normal forms and remainder -- normal form of ring elements and matrices
RingElement % InexactNumber (missing documentation)
Vector % Ideal (missing documentation)

For the programmer

The object % is a keyword.

This operator may be used as a binary operator in an expression like x%y. The user may install binary methods for handling such expressions with code such as

         X % Y := (x,y) -> ...

where X is the class of x and Y is the class of y.

% -- a binary operator, usually used for remainder and reduction

Synopsis

Description

See also

Ways to use symbol % :

For the programmer