/
, _divide
Divide expressions
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x / y _divide(x
,y
)
x/y
computes the quotient of x
and y
.
x/y
is equivalent to the function call _divide(x,
y)
.
For numbers of type Type::Numeric
, the quotient is returned
as a number.
If neither x
nor y
are
elements of library
domains with "_divide"
methods, x/y
is
internally represented as x * y^(1)
= _mult(x,
_power(y, 1))
.
If x
or y
is an element
of a domain with a slot"_divide"
,
then this method is used to compute x/y
. Many library domains overload
the /
operator by an appropriate "_divide"
slot.
Quotients are processed as follows:
x/y
is searched for elements of library domains
from left to right. Let z
(either x
or y
)
be the first term that is not of one of the basic types provided by
the kernel (numbers, expressions, etc.). If the domain d
= z::dom
= domtype(z)
has
a slot"_divide"
, it
is called in the form d::_divide(x, y)
. The result
returned by d::_divide
is the result of x/y
.
Cf. examples Example 4 and Example 5.
Polynomials of type DOM_POLY
can be divided by /
,
if they have the same indeterminates and the same coefficient ring,
and if exact division is possible. The function divide
can be used to compute the quotient
of polynomials with a remainder term.
For finite sets X
, Y
,
the quotient X/Y
is the set .
The quotient of numbers is simplified to a number:
1234/234, 7.5/7, 6*I/2
Internally, a symbolic quotient x/y
is represented
as the product x * y^(1)
:
type(x/y), op(x/y, 0), op(x/y, 1), op(x/y, 2)
op(op(x/y, 2), 0), op(op(x/y, 2), 1), op(op(x/y, 2), 2)
For finite sets X
, Y
,
the quotient X/Y
is the set :
{a, b, c} / {2, 3}
Polynomials of type DOM_POLY
can be divided by /
if
they have the same indeterminates, the same coefficient ring, and
if exact division is possible:
poly(x^2  1, [x]) / poly(x  1, [x])
poly(x^2  1, [x]) / poly(x  2, [x])
The function divide
provides
division with a remainder:
divide(poly(x^2  1, [x]), poly(x  2, [x]))
The polynomials must have the same indeterminates and the same coefficient ring:
poly(x^2  1, [x, y]) / poly(x  1, [x])
Error: Invalid argument. [divide]
Various library domains such as matrix domains
overload _divide
. The matrix domain defines x/y
as x
* (1/y)
, where 1/y
is the inverse of y
:
x := Dom::Matrix(Dom::Integer)([[1, 2], [3, 4]]): y := Dom::Matrix(Dom::Rational)([[10, 11], [12, 13]]): x/y
The inverse of x
has rational entries. Therefore, 1/x
returns FAIL
, because
the component ring of x
is Dom::Integer
. Consequently,
also y/x
returns FAIL
:
y/x
delete x, y:
This example demonstrates the behavior of _divide
on
userdefined domains. In the first case below, the userdefined domain
does not have a "_divide"
slot.
Thus x/y
is transformed to x * (1/y)
:
Do := newDomain("Do"): x := new(Do, 1): y := new(Do, 2): x/y; op(x/y, 0..2)
After the slot"_divide"
is
defined in the domain Do
, this method is used to
divide elements:
Do::_divide := proc() begin "The Result" end: x/y
delete Do, x, y:

arithmetical
expressions, polynomials of
type 
Arithmetical expression, a polynomial, or a set.
x
, y