Pi is IRRATIONAL: simplest proof on toughest test

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In the last video of 2017 I showed you Lambert’s long but easy-to-motivate 1761 proof that pi is irrational. For today’s video Marty and I have tried to streamline an ingenious proof due to the famous French mathematician Charles Hermite into the hopefully simplest and shortest completely self-contained proof of the irrationality of pi. There are a few other versions of this proof floating around and we’ve incorporated the best ideas from these versions into what I’ll show you today; I’ll list some of these other versions below. I also talk about the problem of pi + e and pi x e being irrational at the end of the video, really nice stuff.

Articles to check out:
Ivan Niven’s simple proof (we essentially use his sequence of integers) https://projecteuclid.org/euclid.bams/1183510788

A really nice article by Li Zhou http://arxiv.org/abs/0911.1929. Among many other things this article, has an introductory section on the origins of the strange integral at the core of the proof that I am presenting in this video.

Another article by Li Zhou http://arxiv.org/abs/0911.1933. (the 6 line proof of Theorem 2 is essentially what I am presenting in this video).

All these articles are great, but I think the one article that deserves most credit for having brought Hermite’s beautiful proof to the attention of the wider mathematical community is this article by Jan Stefvens: Zur Irrationalität von pi, Mitt. Math. Ges. Hamburg 18 (1999), 151-158. This one also has a very nice account of Lambert’s and Niven’s proofs.

In the video I mention that another version of this proof made an appearance in the toughest Australian maths exam in 2003; here is the link to this exam
http://educationstandards.nsw.edu.au/wps/wcm/connect/dede688e-11d3-4752-b40d-b83e42941906/maths-ext2-hsc-exam-2003.pdf?MOD=AJPERES&CACHEID=ROOTWORKSPACE-dede688e-11d3-4752-b40d-b83e42941906-lGd8Xdw

As usual thanks to Marty and Danil for all their help with this video.

Enjoy!

Next episode

S04E03 - Visualising Pythagoras: ultimate proofs and crazy contortions