Property
The continued fraction representation of a rational is unique.
Proof
Let $r \in \mathbb{Q}$ and $[p_0, p_1, …, p_n], [q_0, q_1, …, q_m]$ be continued fraction representations of $r$. Now, process induction on $\min(n, m)$. The case $\min(n, m) = 0$ is trivial.
By definition, we know that $p_0 = q_0 = \lfloor r \rfloor$. Then,
$$(r - \lfloor r \rfloor)^{-1} = [p_1, …, p_n] = [q_1, …, q_m]$$
the rest can be shown by induction hypothesis.