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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) [ Ссылка ]
A really nice article by Li Zhou [ Ссылка ]. 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 [ Ссылка ]. (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
[ Ссылка ]
As usual thanks to Marty and Danil for all their help with this video.
Enjoy!
