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Proofs from THE BOOK pp 27–33Cite as

Some irrational numbers

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Abstract

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 and

  • er is irrational for rational r ≠ 0.

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References

  1. C. Hermite: Sur la fonction exponentielle, Comptes rendus de l’Académie des Sciences (Paris) 77 (1873), 18–24; Œuvres de Charles Hermite, Vol. III, Gauthier-Villars, Paris 1912, pp. 150–181.

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  2. Y. Iwamoto: A proof that π 2 is irrational, J. Osaka Institute of Science and Technology 1 (1949), 147–148.

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  3. J. F. Koksma: On Niven’s proof that π is irrational, Nieuw Archief voor Wiskunde (2) 23 (1949), 39.

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  4. J. Liouville: Sur l’irrationalité du nombre e = 2,718…, Journal de Mathématiques Pures et Appl. (1) 5 (1840), 192; Addition, 193–194.

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  5. I. Niven: A simple proof that π is irrational, Bulletin Amer. Math. Soc. 53 (1947), 509.

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Author information

Authors and Affiliations

  1. Institut für Mathematik II (WE2), Freie Universität Berlin, Arnimallee 3, 14195, Berlin, Germany

    Martin Aigner

  2. Institut für Mathematik, MA 6-2, Technische Universität Berlin, Straße des 17. Juni 136, 10623, Berlin, Germany

    Günter M. Ziegler

Authors
  1. Martin Aigner
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  2. Günter M. Ziegler
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© 2004 Springer-Verlag Berlin Heidelberg

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Aigner, M., Ziegler, G.M. (2004). Some irrational numbers. In: Proofs from THE BOOK. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-05412-3_6

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  • DOI: https://doi.org/10.1007/978-3-662-05412-3_6

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-05414-7

  • Online ISBN: 978-3-662-05412-3

  • eBook Packages: Springer Book Archive

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