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:
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π2 is irrational and
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er is irrational for rational r ≠ 0.
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References
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.
Y. Iwamoto: A proof that π 2 is irrational, J. Osaka Institute of Science and Technology 1 (1949), 147–148.
J. F. Koksma: On Niven’s proof that π is irrational, Nieuw Archief voor Wiskunde (2) 23 (1949), 39.
J. Liouville: Sur l’irrationalité du nombre e = 2,718…, Journal de Mathématiques Pures et Appl. (1) 5 (1840), 192; Addition, 193–194.
I. Niven: A simple proof that π is irrational, Bulletin Amer. Math. Soc. 53 (1947), 509.
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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
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