Category:Definitions/Division
This category contains definitions related to Division.
Related results can be found in Category:Division.
The concept of division can be defined in the following ways, according to context:
Division over a Field
Let $\struct {F, +, \times}$ be a field.
Let the zero of $F$ be $0_F$.
The operation of division is defined as:
- $\forall a, b \in F \setminus \set {0_F}: a / b := a \times b^{-1}$
where $b^{-1}$ is the multiplicative inverse of $b$.
Division over a Euclidean Domain
Let $\struct {D, +, \circ}$ be a Euclidean domain:
- whose zero is $0_D$
- whose Euclidean valuation is denoted $\nu$.
Let $a, b \in D$ such that $b \ne 0_D$.
By the definition of Euclidean valuation:
- $\exists q, r \in D: a = q \circ b + r$
such that either:
- $\map \nu r < \map \nu b$
or:
- $r = 0_D$
The process of finding $q$ and $r$ is known as division of $a$ by $b$, and we write:
- $a \div b = q \rem r$
Division Modulo $m$
Let $m \in \Z$ be an integer.
Let $\Z_m$ be the set of integers modulo $m$:
- $\Z_m = \set {0, 1, \ldots, m - 1}$
The operation of division modulo $m$ is defined on $\Z_m$ as:
- $a \div_m b$ equals the integer $q \in \Z_m$ such that $b \times_m q \equiv a \pmod m$
and is possible only if $q$ is unique modulo $m$.
This happens if and only if $a$ and $m$ are coprime.
Subcategories
This category has the following 11 subcategories, out of 11 total.
D
- Definitions/Dividends (3 P)
E
- Definitions/Exact Division (2 P)
I
M
P
R
- Definitions/Real Division (2 P)
Pages in category "Definitions/Division"
The following 26 pages are in this category, out of 26 total.
D
- Definition:Dividend
- Definition:Division
- Definition:Division over Euclidean Domain
- Definition:Division over Field
- Definition:Division over Field/Quotient
- Definition:Division Product
- Definition:Division/Dividend
- Definition:Division/Divisor
- Definition:Division/Field
- Definition:Division/Field/Complex Numbers
- Definition:Division/Field/Rational Numbers
- Definition:Division/Field/Real Numbers
- Definition:Division/Notation
- Definition:Division/Standard Number Field
- Definition:Divisor of Division Operation
- Definition:Divisor of Division Operation/Also known as