Diophantine approximation is a branch of number theory that deals with the approximation of real numbers by rational numbers. It specifically studies the extent to which real numbers can be closely approximated by rational numbers, with a focus on the quality of these approximations. The name "Diophantine" comes from the ancient Greek mathematician Diophantus, who is known for his work in algebra, particularly in solving polynomial equations.
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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 , one compares with functions of the denominator .
For example, continued fraction convergents to give approximations . A nonzero integer norm of 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.