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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

  1. 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.Google Scholar
  2. A. Hurwitz (1891). Über die angenäherte Darstellung der Irrationalzahlen durch rationale Brüche. Math. Ann. 39, 279–284.CrossRefMathSciNetGoogle Scholar
  3. O. Perron (1954). Die Lehre von den Kettenbrüchen. 3. Aufl. B.G. Teubner: Stuttgart.zbMATHGoogle Scholar
  4. J.W.S. Cassels (1957). An introduction to diophantine approximation. Cambridge Tracts 45, Cambridge Univ. Press.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1980

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