Definition:Rational Number/Canonical Form

Definition

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

• Definition:Cancellation of Fraction

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

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

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.

Sources

• 1971: Wilfred Kaplan and Donald J. Lewis: Calculus and Linear Algebra ... (previous) ... (next): Introduction: Review of Algebra, Geometry, and Trigonometry: $\text{0-1}$: The Real Numbers
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): reduced fraction
• 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.4$: Integer Functions and Elementary Number Theory