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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!

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    Also http://mathoverflow.net/questions/21367/proof-that-pi-is-transcendental-that-doesnt-use-the-infinitude-of-primes2011-04-09

2 Answers 2

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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.

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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](https://carma.newcastle.edu.au/jon/31415.pdf): A Brief History and Antholo$g$y of Articles in the American Mathematical Monthly", American Mathematical Monthly, Volume 122, $I$ssue 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).2015-10-01