Existence of Canonical Form of Rational Number

Theorem
Let $r \in \Q$.

Then:


 * $\exists p \in \Z, q \in \Z_{>0}: r = \dfrac p q, p \perp q$

That is, every rational number can be expressed in its canonical form.

Proof
As the set of rational numbers is the quotient field of the set of integers, it follows from Divided by Positive Element of Quotient Field that:


 * $\exists s \in \Z, t \in \Z_{>0}: r = \dfrac s t$

Now if $s \perp t$, our task is complete.

Otherwise, let:
 * $\gcd \left\{{s, t}\right\} = d$

where $\gcd \left\{{s, t}\right\}$ denotes the greatest common divisor of $s$ and $t$.

Let $s = p d, t = q d$.

As $t, d \in \Z_{>0}$, so is $q$.

From Divide by GCD for Coprime Integers, $p \perp q$.

Also:


 * $\displaystyle \frac s t = \frac {p d} {q d} = \frac p q \frac d d = \frac p q 1 = \frac p q$

Thus $r = p / q$ where $p \perp q$ and $q \in \Z_{>0}$.

Also see

 * Canonical Form of Rational Number is Unique