Dirichlet's approximation theorem: Difference between revisions

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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 | \le< \frac{1}{N+1} </math>

This is a fundamental result in [[Diophantine approximation]], showing that any real number has a sequence of good rational approximations: in fact an immediate consequence is that for a given irrational α, the inequality