There’s a very nice proof of the irrationality of Pi at the Vigorous Handwaving blog. Link below. The proof is set to the lyrics of the Major General’s song from the Gilbert and Sullivan comic opera, The Pirates of Penzance. Those lyrics are from Kevin Wald, math at U of Chicago, and are at his website (link in the blog post).

I found the proof very interesting, and very accessible. As the blog post author Matthew Calvin says, it could be assigned as a (guided, sequential) exercise in first year calculus — for advanced students, with a flair for math reasoning and a serious interest in math. I think it might be better as a topic for a math club, accessible at 1st year level so suitable for all, as a sort of 20-min seminar and providing a good discussion topic.

I got interested in the math of the proof. It’s fine. There are a couple of minor discrepancies in the final recursion formula — I get an extra factor of n in one term, and a minus between the two terms, but the overall proof still goes thru successfully. Also, I turned the logic around. Instead of supposing at the start an “on the contrary, suppose that Pi is rational and equals m/n”, I just let n be any positive constant, and derived the recursion formula. It is valid for any positive n, and is a useful thing to know. Then, at the very end of the revised proof, one supposes the contrary, Pi=m/n, and easily shows that is contradictory because with that n all the recursion coefficients would be positive integers, and not converge to 0.

There is more to this method than just this proof. One can replace the nt(pi-t) by a positive function g(t) which is symmetric about pi/2. The ak coefficients remind me of Fourier sine coefficients, though I haven’t sat down to make the exact correspondence yet. The corresponding cosine coefficients are zero because g is symmetric about pi/2 and cos(t) is anti-symmetric about pi/2. The sum function f(t)=exp(g(t)) does not have to be exp() actually — all that one needs for the proof is that the sum function is bounded on (0,pi). One can consider extending into the complex plane. Then think about fiddling with contours and other complex analysis techniques. Alternatively, for proving other numbers are irrational, think about crafting a similar line of reasoning which used integration of a suitable g(t) (or its sum function f) over another interval, let’s say (0,Q) where Q is the number in question. One will get a recursion formula. Under what conditions will the recursion formula allow one to obtain a contradiction, eg the ak all positive integers thus contradicting their convergence to zero, if one supposes that Q is rational?

Further, just as a potentially really enjoyable possibility if it works out, wonder about the feasibility of casting the proof into geometric language. Could Archimedes have come up with a proof that Pi is irrational? He was very comfortable with limiting processes, integrations and the like. And let’s not forget my favourite curve, the Quadratix of Hippias! There is plenty of opportunity to do interesting work with this proof technique. (See Hardy and Wright, Intro to the Theory of Numbers, for geometric proof techniques for sqrt(N), and for a variety of proof techniques used for e and pi.)

I really enjoyed the Vigorous Handwaving blog, and recommend it. Many thanks to Matthew Calvin and to the several authors!

Here is a link to their blog:
http://vigoroushandwaving.wordpress.com/

And a link to the specific post about the irrationality of Pi:

The Pi Proof of Penzance

Best wishes,
Ken Roberts
23-May-2014