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Does anyone has a link to a site that confirms that $\pi$ is a transcendental number?

Or, can anyone show how to prove that $\pi$ is a transcendental number?

Thank you in anticipation!

J. M. ain't a mathematician
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matekkerdes
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  • http://www.math.sc.edu/~filaseta/gradcourses/Math785/Math785Notes6.pdf – Yuval Filmus Apr 08 '11 at 21:04
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    http://math.stackexchange.com/questions/12872/how-hard-is-the-proof-of-pi-or-e-being-transcendental this question might be of some interest – InterestedGuest Apr 08 '11 at 21:17
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    Also http://mathoverflow.net/questions/21367/proof-that-pi-is-transcendental-that-doesnt-use-the-infinitude-of-primes – lhf Apr 09 '11 at 01:01

2 Answers2

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As suggested by Yuval's comment, the most straightforward way of showing that $\pi$ is transcendental proceeds through the Lindemann–Weierstrass theorem that $e^x$ is transcendental if $x$ is (nonzero and) algebraic; since $e^{i\pi}=-1$ is algebraic, then $i\pi$ must be transcendental, and therefore $\pi$ must be (since $i$ isn't rational, but it is algebraic!). You can find a rough proof of the theorem at its Wikipedia page.

Steven Stadnicki
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Try the short paper The transcendence of $\pi$ by Niven and his book Irrational Numbers.

lhf
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    This paper has been reproduced in Appendix A of Jonathan M. Borwein and Scott T. Chapman, "I Prefer Pi: A Brief History and Anthology of Articles in the American Mathematical Monthly", American Mathematical Monthly, Volume 122, Issue 03, pp. 189 - 296, March 2015 (downloadable from https://carma.newcastle.edu.au/jon/31415.pdf from the first author's web page at https://carma.newcastle.edu.au/jon/index-papers.shtml). – MarnixKlooster ReinstateMonica Oct 01 '15 at 05:57