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.
KeywordsZeta Function Irrational Number Riemann Zeta Function Geometric Series Elementary Calculus
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- T. M. Apostol: A proof that Euler missed: Evaluating ζ(2) the easy way, Math. Intelligencer 5 (1983), 59 - 60.Google Scholar
- 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
- Y. Iwamoto: A proof that π2 is irrational, J. Osaka Institute of Science and Technology 1 (1949), 147 - 148.Google Scholar
- J. F. Koksma: On Niven’s proof that 71 is irrational, Nieuw Archiv Wiskunde (2) 23 (1949), 39.Google Scholar
- I. Niven: A simple proof that it is irrational, Bulletin Amer. Math. Soc. 53 (1947), 509.Google Scholar
- 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
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