# Prove that $\pi$ is a transcendental number

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!

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Have you tired searching Google? –  JavaMan Apr 8 '11 at 21:01
math.stackexchange.com/questions/12872/… this question might be of some interest –  InterestedGuest Apr 8 '11 at 21:17
–  lhf Apr 9 '11 at 1:01

## 2 Answers

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.

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

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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 carma.newcastle.edu.au/jon/31415.pdf from the first author's web page at carma.newcastle.edu.au/jon/index-papers.shtml). –  Marnix Klooster Oct 1 at 5:57