On the Irrationality of Pi.


One of the couple of mathematics related details that really bugged me during high school (other than the fact that we constantly had to put those annoying arrows in front of ALL of our “vectors”), was the irrationality of \pi.

There was no easy proof that I could use, with my current knowledge at the time, that could prove it was irrational, unlike simpler numbers such as \sqrt{2}. I could not even find any teacher at the time who could explain to me using basic calculus why the ratio between the diameter of a circle and its radius was never rational.

Skip many years ahead, and I came upon this interesting, albeit “short” proof:


Although the proof can be summarized through eight remarks and a few properties about integrals, trying to prove each remark can be a bit challenging. I would estimate that the average high school grade 11 or 12 student would not be able to get past the second remark without great difficulty.

That is why I compiled this, a fairly easy to follow guide detailing each of the remarks and their derivations. In the .pdf, I also left a couple exercises for the reader to do just for fun (but if really need to know the answer, don’t hesitate to private message me). However, if you would like to do the proofs yourself, send me a message or an email and I would be happy to post your proof up on the site.

If you want to know how I made the article, please check out the link in the “Resources” page under “Typesetting”.