# Definition:Division

## Definition

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.

## Notation

The operation of division can be denoted as:

$a / b$, which is probably the most common in the general informal context
$\dfrac a b$, which is the preferred style on $\mathsf{Pr} \infty \mathsf{fWiki}$
$a : b$, which is usually used when discussing ratios
$a \div b$, which is rarely seen outside grade school, but can be useful in contexts where it is important to be specific.

## Also see

• Results about division can be found here.

## Historical Note

The symbol for division that was introduced by Gottfried Wilhelm von Leibniz was in fact a colon $:$

This can still be seen in the context of ratios and proportions.

## Linguistic Note

The verb form of the word division is divide.

Thus to divide is to perform an act of division.