2004 | OriginalPaper | Chapter
Some irrational numbers
Authors : Martin Aigner, Günter M. Ziegler
Published in: Proofs from THE BOOK
Publisher: Springer Berlin Heidelberg
Included in: Professional Book Archive
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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 ander is irrational for rational r ≠ 0.