# // -- a binary operator, usually used for quotient

• Usage:
x // y

## Description

For ring elements in the integers, polynomial rings, and other rings, there are two types of division: exact division, whose result is often in a larger field, such as the rationals or a function field, and division with remainder, whose result is in the same ring. In Macaulay2, '/' denotes the first kind of division, while '//' denotes the latter kind. The following example shows the difference between // and /.
 ```i1 : 4/2 o1 = 2 o1 : QQ``` ```i2 : 4//2 o2 = 2```
 `i3 : R = QQ[x];` ```i4 : (x^2-3)//(x-1) o4 = x + 1 o4 : R``` ```i5 : (x^2-3)%(x-1) o5 = -2 o5 : R``` ```i6 : (x^2-3)/(x-1) 2 x - 3 o6 = ------ x - 1 o6 : frac R```

## See also

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

## Ways to use // :

• CC // CC
• CC // QQ
• CC // RR
• CC // ZZ
• InfiniteNumber // QQ
• InfiniteNumber // RR
• InfiniteNumber // ZZ
• Matrix // Number
• Number // InfiniteNumber
• Number // Matrix
• Number // RingElement
• QQ // QQ
• QQ // ZZ
• RingElement // Number
• RR // QQ
• RR // RR
• RR // ZZ
• ZZ // MonomialIdeal
• ZZ // ZZ
• Constant // InfiniteNumber (missing documentation)
• InexactNumber // RingElement (missing documentation)
• InfiniteNumber // CC (missing documentation)
• InfiniteNumber // Constant (missing documentation)
• InfiniteNumber // InfiniteNumber (missing documentation)
• List // Number (missing documentation)
• List // RingElement (missing documentation)
• Matrix // GroebnerBasis, see Matrix // Matrix -- factor a map through another
• Matrix // Matrix -- factor a map through another
• Matrix // MonomialIdeal, see Matrix // Matrix -- factor a map through another
• Matrix // RingElement, see Matrix // Matrix -- factor a map through another
• RingElement // GroebnerBasis, see Matrix // Matrix -- factor a map through another
• RingElement // Matrix, see Matrix // Matrix -- factor a map through another
• RingElement // MonomialIdeal, see Matrix // Matrix -- factor a map through another
• RingElement // RingElement, see Matrix // Matrix -- factor a map through another
• RingElement // InexactNumber (missing documentation)
• Thing // Command, see Thing // Function -- apply a function
• Thing // Function -- apply a function
• Thing // SelfInitializingType, see Thing // Function -- apply a function

## For the programmer

The object // is .

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.