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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Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Martin Aigner
    • 1
  • Günter M. Ziegler
    • 2
  1. 1.Institut für Mathematik II (WE2)Freie Universität BerlinBerlinGermany
  2. 2.Institut für Mathematik, MA 6-2Technische Universität BerlinBerlinGermany

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