Definition:Best Rational Approximation
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Definition
Let $x \in \R$ be an (irrational) real number.
The rational number $a = \dfrac p q$ is a best rational approximation to $x$ if and only if:
- $(1): \quad a$ 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 $x$.
Sequence
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