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Stephenmg92 (talk | contribs) Dirichlet's Theorem as it was stated was correct but was not optimal. I have replaced it with its optimal version. Furthermore it was originally only stated when N was an integer. I have replaced this to be for when N is any real number bigger than 1. |
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In [[number theory]], '''Dirichlet's theorem on Diophantine approximation''', also called '''Dirichlet's approximation theorem''', states that for any [[real
:<math> \left | q \alpha -p \right | \leq \frac{1}{[N]+1} < \frac{1}{N}. </math>
Here <math> [N] </math> represents the [[integer part]] of <math> N </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
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