NEW (Christmas 2019). Two ways to support Mathologer
Mathologer Patreon:   / mathologer  
Mathologer PayPal: paypal.me/mathologer
(see the Patreon page for details)

In the last video of 2017 I showed you Lambert’s long but easytomotivate 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 selfcontained 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...

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), 151158. 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/...

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

Enjoy!