# Definition: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 $b$.

The concept 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 $\C$.

### Integer Division

Let $a, b \in \Z$ be integers such that $b \ne 0$..

From the Division Theorem:

- $\exists_1 q, r \in \Z: a = q b + r, 0 \le r < \left|{b}\right|$

where $q$ is the quotient and $r$ is the remainder.

The process of finding $q$ and $r$ is known as **(integer) division**.

## 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 $b$**.

## Also see

- Division Theorem for how the concept is extended to the integers

- 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.

## Sources

- 1951: B. Hague:
*An Introduction to Vector Analysis*(5th ed.) ... (previous) ... (next): Chapter $\text I$: Definitions. Elements of Vector Algebra: $1$. Scalar and Vector Quantities - 1973: C.R.J. Clapham:
*Introduction to Mathematical Analysis*... (previous) ... (next): Chapter $1$: Axioms for the Real Numbers: $2$. Fields - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $2$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $2$