# 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**.

## Sources

- 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**division** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**division**