Definition:Integer Division
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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
- 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.