Canonical Form of Rational Number is Unique

Theorem
The canonical form of a rational number is unique.

Proof
Let $r \in \mathbb{Q}$ be a rational number, and let $\dfrac{p}{q}$ and $\dfrac{p'}{q'}$ be two canonical forms of $r$. Without loss of generality, assume $q \leq q'$. Cross-multiplying yields:


 * $pq' = p'q$

Therefore, $q'$ divides $p'q$. Since $q'$ is coprime to $p'$, we must have that $q'$ divides $q$. Since $q$ and $q'$ are both positive, we have $q' \leq q$. This, combined with our assumption yields $q = q'$. It then follows that $p = p'$.

Also see

 * Existence of Canonical Form of Rational Number