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.


Zeta Function Irrational Number Riemann Zeta Function Geometric Series Elementary Calculus 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  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

Personalised recommendations