If is any number and and are integers, then there is a rational number for which
(1)
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If is irrational and is any whole number, there is a fraction with and for which
(2)
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Furthermore, there are an infinite number of fractions for which
(3)
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(Hilbert and Cohn-Vossen 1999, pp. 40-44).
Hurwitz has shown that for an irrational number
(4)
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there are infinitely rational numbers if , but if , there are some for which this approximation holds for only finitely many .