Definition:Division

Definition

The concept of division can be defined in the following ways, according to context:

Division over a Field

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 $b$.

Division over a Euclidean Domain

Let $\struct {D, +, \circ}$ be a Euclidean domain:

whose zero is $0_D$

whose Euclidean valuation is denoted $\nu$.

Let $a, b \in D$ such that $b \ne 0_D$.

By the definition of Euclidean valuation:

$\exists q, r \in D: a = q \circ b + r$

such that either:

$\map \nu r < \map \nu b$

or:

$r = 0_D$

The process of finding $q$ and $r$ is known as division of $a$ by $b$, and we write:

$a \div b = q \rem r$

Division Modulo $m$

Let $m \in \Z$ be an integer.

Let $\Z_m$ be the set of integers modulo $m$:

$\Z_m = \set {0, 1, \ldots, m - 1}$

The operation of division modulo $m$ is defined on $\Z_m$ as:

$a \div_m b$ equals the integer $q \in \Z_m$ such that $b \times_m q \equiv a \pmod m$

and is possible only if $q$ is unique modulo $m$.

This happens if and only if $a$ and $m$ are coprime.

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.

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.

Linguistic Note

The verb form of the word division is divide.

Thus to divide is to perform an act of division.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): division
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): division