Definition:Rational Number/Canonical Form
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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
- 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