Definition:Division/Rational Numbers

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Definition

Let $\left({\Q, +, \times}\right)$ be the field of rational numbers.


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

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

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