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Approximation to Irrational Numbers by Rationals

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Part of the Lecture Notes in Mathematics book series (LNM,volume 785)

Abstract

Given a real number α, let [α], the integer part of α, denote the greatest integer ≤ α, and let {α} = α − [α]. Then {α} is the fractional part of α, and satisfies 0 ≤ {α} < 1. Also, let ‖α‖ denote the distance from α to the nearest integer. Then always 0 ≤ ‖α‖ ≤ 1/2.

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References

  • L.G.P. Dirichlet (1842). Verallgemeinerung eines Satzes aus der Lehre von den Kettenbrüchen nebst einigen Anwendungen auf die Theorie der Zahlen. S. B. Preuss. Akad. Wiss. 93–95.

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© 1980 Springer-Verlag Berlin Heidelberg

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(1980). Approximation to Irrational Numbers by Rationals. In: Diophantine Approximation. Lecture Notes in Mathematics, vol 785. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-38645-2_1

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  • DOI: https://doi.org/10.1007/978-3-540-38645-2_1

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-09762-4

  • Online ISBN: 978-3-540-38645-2

  • eBook Packages: Springer Book Archive