# Definition:Division/Rational Numbers

## Contents

## 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 $\mathsf{Pr} \infty \mathsf{fWiki}$

- $a \div b$, which is rarely seen outside grade school.

## Specific Terminology

### Divisor

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

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.

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 $b$**.

## Sources

- 1960: Walter Ledermann:
*Complex Numbers*... (previous) ... (next): $\S 1.1$. Number Systems - 1974: Murray R. Spiegel:
*Theory and Problems of Advanced Calculus*(SI ed.) ... (previous) ... (next): Chapter $1$: Numbers: Real Numbers: $3$ - 1981: Murray R. Spiegel:
*Theory and Problems of Complex Variables*(SI ed.) ... (previous) ... (next): Chapter $1$: Complex Numbers: The Real Number System: $3$