Definition:Rational Number/Canonical Form

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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

  • Results about the canonical form of a rational number can be found here.


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.