Definition:Rational Number/Canonical Form

Theorem
Let $$r \in \Q$$. Then:

$$\exists p \in \Z, q \in \Z^*_+: r = \frac p q, p \perp q$$

That is, every rational number can be expressed in the form $$\frac p q$$ where:


 * $$p$$ is an integer;
 * $$q$$ is a strictly positive integer;
 * $$p$$ and $$q$$ are coprime, that is, they have no common divisor except $$1$$.

This form of a rational number is known as its canonical form.

Note that for any given rational number, its canonical form is unique.

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


 * $$\exists s \in \Z, t \in \Z^*_+: r = \frac s t$$

Now if $$s \perp t$$, our job is done.

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

Then let $$s = p d, t = q d$$. As $$t, d \in \Z^*_+$$, so is $$q$$.

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

Also:


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

Thus $$r = p / q$$ where $$p \perp q$$ and $$q \in \Z^*_+$$.

Comment
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.