\begin{definition}[Definition:Best Rational Approximation]
Let $x \in \R$ be an (irrational) real number.
The rational number $a = \dfrac p q$ is a '''best rational approximation''' to $x$ {{iff}}:
:$(1): \quad a$ is in canonical form, that is $p$ is coprime to $q$: $p \perp q$
:$(2): \quad \left\vert{x - \dfrac p q}\right\vert = \min \left\{ {\left\vert{x - \dfrac {p'} {q'} }\right\vert: q' \le q}\right\}$
That is:
:$\left\vert{x - \dfrac p q}\right\vert$ is smaller than for any $\dfrac {p'} {q'}$ where $q' \le q$
where $\left\vert{x}\right\vert$ denotes the absolute value of $x$.
\end{definition}