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Approximation of Irrationals

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If is irrational and is any positive integer, there is a fraction with and for which .

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Proof. Let be a positive integer. Then by the pigeonhole principle, among the points (where denotes the fractional part of ), there are at least two numbers such that . Then for some integer . The statement is proved if we put .

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Contributed by: Izidor Hafner (March 2011)
Open content licensed under CC BY-NC-SA

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[1] D. Benko, "A New Approach to Hilbert's Third Problem," American Mathematical Monthly, 114(8), 2007 pp. 665–676.

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