# Are there any (interesting) consequences of the irrationality of π? [closed]

I am not sure how appropriate this question is for MO. If it is not, I apologize in advance but I could not resist asking it and if by any chance I get some interesting answers, it will for sure be very useful to keep my students excited about mathematics and physics as September arrives.

We all know very well that $\pi$ (the ratio of the circumference of a circle to its diameter in Euclidean space) is irrational and even transcendental. These are some of the famous results in all mathematics.

So I was wondering what will go wrong if $\pi$ was just an integer number?

Are there important theorems that are based on the fact that it is actually irrational and/or transcendental?

## closed as not a real question by Pete L. Clark, Andrea Ferretti, Felipe Voloch, Loop Space, Robin ChapmanAug 12 '10 at 16:42

It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, visit the help center. If this question can be reworded to fit the rules in the help center, please edit the question.

• Is it the presence of the word "universe" in the title of the question which justifies the mathematical physics tag?! – José Figueroa-O'Farrill Aug 12 '10 at 12:01
• Sorry JME, I don't think this question is suitable for MathOverflow. Please, have a look at the FAQ. Maybe you will find there somewhere else where to post this (although I would not know what to suggest). – Andrea Ferretti Aug 12 '10 at 12:03
• Somewhat related: en.wikipedia.org/wiki/Indiana_Pi_Bill – José Figueroa-O'Farrill Aug 12 '10 at 12:04
• I've started a meta thread to discuss closing (or not) this question: tea.mathoverflow.net/discussion/601/… – José Figueroa-O'Farrill Aug 12 '10 at 12:13
• I have a funny feeling about this question, that there's a fantastic answer lurking out there somewhere. – Rob Grey Aug 12 '10 at 13:54

Since Euler showed that $$\frac{\pi^2}{6}=\prod_{p} \Big(1-\frac{1}{p^2}\Big)^{-1},$$ the fact that $\pi^2$ is irrational implies that there are infinitely many primes.
• Gregory's series en.wikipedia.org/wiki/Gregory's_series for $\pi/4$ also has an Euler product, so the irrationality of $\pi$ itself shows the infinitude of primes. – Stopple Aug 12 '10 at 20:07