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 (this is a stronger result!) and
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e r is irrational for rational r ≠ 0.
“π is irrational”
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References
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Aigner, M., Ziegler, G.M. (2001). Some irrational numbers. In: Proofs from THE BOOK. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-04315-8_6
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DOI: https://doi.org/10.1007/978-3-662-04315-8_6
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