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.

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.