# Definition:Rational Number/Canonical Form

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## 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

## 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): Entry:**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