Definition:Integer Division

Definition

Let $a, b \in \Z$ be integers such that $b \ne 0$.

From the Division Theorem:

$\exists_1 q, r \in \Z: a = q b + r, 0 \le r < \size b$

where $q$ is the quotient and $r$ is the remainder.

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

$a \div b = q \rem r$

Exact Division

Let:

$a \div b = q \rem 0$

where $\div$ denotes integer division.

This is an instance of exact 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 $\mathsf{Pr} \infty \mathsf{fWiki}$

$a : b$, 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.

Examples

$29$ Divided by $8$

$29 \div 8 = 3 \rem 5$

Division by $-7$

\(\ds 1 \div \paren {-7}\)

\(=\)

\(\ds 0\)

\(\ds \rem 1\)

\(\ds -2 \div \paren {-7}\)

\(=\)

\(\ds 1\)

\(\ds \rem 5\)

\(\ds 61 \div \paren {-7}\)

\(=\)

\(\ds -8\)

\(\ds \rem 5\)

\(\ds -59 \div \paren {-7}\)

\(=\)

\(\ds 9\)

\(\ds \rem 4\)

Also see

• Division Theorem

• Definition:Division over Euclidean Domain of which this is an example, with the Euclidean valuation being the absolute value

• Results about integer division can be found here.

Linguistic Note

The verb form of the word division is divide.

Thus to divide is to perform an act of division.