= Solution
The <Liouville approximation theorem> states that if $\alpha$ is a real irrational <algebraic number> of degree $d$, there is a constant $C(\alpha)>0$ such that, for all integers $p,q$ with $q\ne0$,
$$
\boxed{\left|\alpha-\frac pq\right|\geq\frac{C(\alpha)}{|q|^d}.}
$$
Reduction of the fraction is not necessary: an unreduced denominator only weakens this bound. More generally the same type of bound holds for an <algebraic number> of degree $d$ whenever $p/q\ne\alpha$. The exception is essential for a rational $\alpha$, since an exact rational representation gives zero error. A <rational number> of degree one obeys the bound directly: if $\alpha=a/b$ in lowest terms and $p/q\ne\alpha$, then $|\alpha-p/q|=|aq-bp|/(|bq|)\geq1/(|bq|)$.
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