# Definition:Division/Field/Real Numbers

< Definition:Division | Field(Redirected from Definition:Real Division)

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## Definition

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 $\R$.

## 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
**real division**can be found**here**.

## Sources

- 1971: Wilfred Kaplan and Donald J. Lewis:
*Calculus and Linear Algebra*... (previous) ... (next): Introduction: Review of Algebra, Geometry, and Trigonometry: $\text{0-1}$: The Real Numbers