Some irrational numbers

  • Martin Aigner
  • Günter M. Ziegler


This was already conjectured by Aristotle, when he claimed that diameter and circumference of a circle are not commensurable. The first proof of this fundamental fact was given by Johann Heinrich Lambert in 1766. Our Book Proof is due to Ivan Niven, 1947: an extremely elegant one-page proof that needs only elementary calculus. Its idea is powerful, and quite a bit more can be derived from it, as was shown by Iwamoto and Koksma, respectively:
  • π2 is irrational (this is a stronger result!) and

  • e r is irrational for rational r≠0.


Famous Identity 


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  1. [1]
    T. M. Apostol: A proof that Euler missed: Evaluating ζ(2) the easy way, Math. Intelligencer 5 (1983), 59 - 60.Google Scholar
  2. [2]
    C. Hermite: Sur la fonction exponentielle, Comptes rendus de l’Académie des Sciences (Paris) 77 (1873), 18-24; OEuvres de Charles Hermite, Vol. III, Gauthier-Villars, Paris 1912, pp. 150 - 18I.Google Scholar
  3. [3]
    Y. Iwamoto: A proof that π2 is irrational, J. Osaka Institute of Science and Technology 1 (1949), 147 - 148.Google Scholar
  4. [4]
    J. F. Koksma: On Niven’s proof that 71 is irrational, Nieuw Archiv Wiskunde (2) 23 (1949), 39.Google Scholar
  5. [5]
    I. Niven: A simple proof that it is irrational, Bulletin Amer. Math. Soc. 53 (1947), 509.Google Scholar
  6. [6]
    A. van der Porten: A proof that Euler missed… Apéry’s proof of the irrationality of ζ(3). An informal report,Math. lntelligencer 1 (1979), 195203.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Martin Aigner
    • 1
  • Günter M. Ziegler
    • 2
  1. 1.Institut für Mathematik II (WE2)Freie Universität BerlinBerlinGermany
  2. 2.Fachbereich Mathematik, MA 7-1Technische Universität BerlinBerlinGermany

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