Canonical Form of Rational Number is Unique

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Theorem

The canonical form of a rational number is unique.


Proof

Let $r \in \Q$ be a rational number.

Let $\dfrac p q$ and $\dfrac {p'} {q'}$ be two canonical forms of $r$.

Without loss of generality, assume $q \le q'$.

By Equality of Rational Numbers:

$p q' = p' q$

Therefore, $q'$ divides $p' q$.

As $\dfrac {p'} {q'}$ is a canonical form of $r$, by definition $q'$ is coprime to $p'$.

By Euclid's Lemma $q'$ divides $q$.

$q$ and $q'$ are both positive.

So from Integer Absolute Value not less than Divisors: Corollary:

$q' \le q$

As $q' \le q$ and $q \le q'$ it follows that $q = q'$.

It then follows that $p = p'$.

Hence $\dfrac p q$ and $\dfrac {p'} {q'}$ are the same unique canonical form of $r$.

$\blacksquare$


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