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In [[number theory]], '''[[Dirichlet]]'s theorem on [[Diophantine approximation]]''', also called '''Dirichlet's approximation theorem''', states that for any [[real number]] α and any [[positive integer]] ''N'', there exists integers ''p'' and ''q'' such that 1 ≤ ''q'' ≤ ''N'' and
:<math> \left| q \alpha -p \right| < \frac{1}{N+1} </math>
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==Method of proof==
This theorem is a consequence of the [[pigeonhole principle]]. [[Dirichlet]] who proved the result used the same principle in other contexts (for example, the [[Pell equation]]) and by naming the principle (in German) popularized its use, though its status in textbook terms comes later.<ref>http://jeff560.tripod.com/p.html for a number of historical references.</ref> The method extends to simultaneous approximation.<ref>{{Springer|id=d/d032940|title=Dirichlet theorem}}</ref>
==See also==
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