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I haven't found a way, except using advanced methods, to prove the irrationality of $\pi$.

It is possible to prove that $\pi$ is irrational, using only elementary algebra?

Michael Hardy
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Andrey França
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  • There isn't a truly easy proof (that i am aware of), but Hermite's argument isn't too bad. You can find it here – lulu Aug 29 '16 at 23:00
  • An algebraic definition of $\pi$ (as opposed to a geometric definition like "the ratio of any circle's circumference to its diameter) is a bit involved and so any proof of its irrationality would likely also be involved. – D_S Aug 29 '16 at 23:07
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    I would typically expect a proof to use the fact that $\pi$ is a positive root (in fact, the least positive root) of $\sin,$ but then defining $\sin$ in a satisfactory way is already involved (and then there's the additional point of proving that there actually exists a least positive root of $\sin$). Typically, satisfactory definitions already involved a fair amount of analysis, so that elementary algebra is already insufficient. The fact is that, since $\pi$ is (in fact) transcendental, it isn't an altogether algebraic object. Some analysis would seem to be needed. – Will R Aug 29 '16 at 23:18
  • Elementary algebra only? No. There isn't. – fleablood Aug 30 '16 at 00:33
  • I think neither https://math.stackexchange.com/questions/1364970/help-with-proving-that-pi-is-irrational nor https://math.stackexchange.com/questions/1076966/simple-proof-that-pi-is-irrational-using-prime-factors-of-denominator have an answer that answers this question. Maybe the task was to give a short simple proof that doesn't rely on a previously proven result that's difficult to prove without proving it. – Timothy Jul 28 '18 at 17:51

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