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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?

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    $\begingroup$ Is it the presence of the word "universe" in the title of the question which justifies the mathematical physics tag?! $\endgroup$
    – José Figueroa-O'Farrill
    Aug 12, 2010 at 12:01
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    $\begingroup$ 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). $\endgroup$
    – Andrea Ferretti
    Aug 12, 2010 at 12:03
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    $\begingroup$ Somewhat related: en.wikipedia.org/wiki/Indiana_Pi_Bill $\endgroup$
    – José Figueroa-O'Farrill
    Aug 12, 2010 at 12:04
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    $\begingroup$ I've started a meta thread to discuss closing (or not) this question: tea.mathoverflow.net/discussion/601/… $\endgroup$
    – José Figueroa-O'Farrill
    Aug 12, 2010 at 12:13
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    $\begingroup$ I have a funny feeling about this question, that there's a fantastic answer lurking out there somewhere. $\endgroup$
    – Rob Grey
    Aug 12, 2010 at 13:54

3 Answers 3

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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.

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    $\begingroup$ 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. $\endgroup$
    – Stopple
    Aug 12, 2010 at 20:07
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The fact that π is irrational has few direct applications. However the techniques used to prove this, or rather used to prove the stronger statement that it is transcendental, have many applications. For example, Baker proved that 1 and the logs of algebraic numbers are linearly independent over algebraic numbers except in trivial cases. (This includes the fact that π is irrational as a special case because π = log(-1)/i.) Baker used his theorem to give effective bounds on the solutions of Diophantine equations and to solve Gauss's class number problem for imaginary quadratic fields, among other things. See Baker's book on transcendental number theory for more details.

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The fact that one cannot square the circle was proven as a corollary of the fact that pi is transcendental. http://en.wikipedia.org/wiki/Squaring_the_circle

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  • $\begingroup$ thanks, but I was expecting less obvious examples $\endgroup$
    – JME
    Aug 12, 2010 at 14:05
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    $\begingroup$ I was about to write the same answer. @JME: Next time, you will be probably better off if you include the facts/examples which you are aware of in your question. It is not very nice to reply “that’s obvious” to a person who answered your question. $\endgroup$
    – Tsuyoshi Ito
    Aug 12, 2010 at 15:21
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    $\begingroup$ @Tsuyoshi yes, you are absolutely right. Johan, I am deeply sorry for my heartless comment, it was not in bad spirit. $\endgroup$
    – JME
    Aug 12, 2010 at 16:46
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    $\begingroup$ No offense taken. $\endgroup$
    – Johan
    Aug 18, 2010 at 20:13

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