Canonical Form of Rational Number is Unique
Jump to navigation
Jump to search
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 Absolute Value of Integer is 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 $r$.
$\blacksquare$
Also see
Sources
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 2.4$: The rational numbers and some finite fields: Exercise $1$