# Definition:Division/Field/Rational Numbers

## Definition

Let $\struct {\Q, +, \times}$ be the field of rational numbers.

The operation of division is defined on $\Q$ as:

$\forall a, b \in \Q \setminus \set 0: a / b := a \times b^{-1}$

where $b^{-1}$ is the multiplicative inverse of $b$ in $\Q$.

## 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.

## Specific Terminology

### Divisor

Let $c = a / b$ denote the division operation on two elements $a$ and $b$ of a field or a Euclidean domain.

The element $b$ is the divisor of $a$.

### Dividend

Let $c = a / b$ denote the division operation on two elements $a$ and $b$ of a field or a Euclidean domain.

The element $a$ is the dividend of $b$.

### Quotient

Let $c = a / b$ denote the division operation on two elements $a$ and $b$ of a field.

The element $c$ is the quotient of $a$ (divided) by $b$.

## Also see

• Results about rational division can be found here.