Definition:Rational Number/Canonical Form

Theorem
Let $r \in \Q$ be a rational number.

The canonical form of $r$ is the expression $\dfrac p q$, where:
 * $r = \dfrac p q: p \in \Z, q \in \Z_{>0}, p \perp q$

where $p \perp q$ denotes that $p$ and $q$ have no common divisor except $1$.

That is, in its canonical form, $r$ is expressed as $\dfrac p q$ where:


 * $p$ is an integer
 * $q$ is a strictly positive integer
 * $p$ and $q$ are coprime.

Also known as
The canonical form of a rational number is also known as a reduced rational number or reduced fraction.

Some sources refer to a fraction in its lowest terms.

Also see

 * Existence of Canonical Form of Rational Number
 * Canonical Form of Rational Number is Unique

Motivation
To put this into a more everyday context, we note that rendering rational numbers (or fractions) into their canonical form is, of course, an exercise much beloved of grade-school teachers.