Definition:Division

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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 $\left({\R, +, \times}\right)$ be the field of real numbers.


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

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

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


Complex Numbers

Let $\left({\C, +, \times}\right)$ 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

  • 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