= Diophantine approximation
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<Diophantine approximation> asks how closely a real or complex quantity can be approximated by <rational numbers> or <algebraic numbers> of bounded arithmetic complexity. For a real $\alpha$, one compares $|\alpha-p/q|$ with functions of the denominator $q$.
For example, <continued fraction convergents> to $\sqrt2$ give approximations $3/2,7/5,17/12,\ldots$. A nonzero <integer> norm of $p-q\sqrt2$ also supplies a <lower bound>, preventing arbitrarily exceptional accuracy. The <Liouville approximation theorem> extends this idea to algebraic numbers, while <Roth's theorem> sharpens the exponent for algebraic irrational inputs. Approximation quality can therefore distinguish algebraic from transcendental behavior, not merely give convenient decimal values.
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