# Definition:Division/Field

*This page is about Division over Field. For other uses, see division.*

## Definition

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 $b$.

## Standard Number Fields

The concept of **division over a field** is usually seen in the context of the **standard number fields**:

### Rational Numbers

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 $\Q$.

### Real Numbers

Let $\struct {\R, +, \times}$ be the field of real numbers.

The operation of **division** is defined on $\R$ as:

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

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

### Complex Numbers

Let $\struct {\C, +, \times}$ be the field of complex numbers.

The operation of **division** is defined on $\C$ as:

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

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

## 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

- Definition:Division over Euclidean Domain for how the concept is extended to the general Euclidean domain

- Results about
**division over a field**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

- 1973: C.R.J. Clapham:
*Introduction to Mathematical Analysis*... (previous) ... (next): Chapter $1$: Axioms for the Real Numbers: $2$. Fields - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**field**:**1.** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**field**:**1.**