Definition:Division/Field

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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

  • 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

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