# Prove that $\pi$ is irrational [duplicate]

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?

• 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
• 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 0:33
• I think neither math.stackexchange.com/questions/1364970/… nor math.stackexchange.com/questions/1076966/… 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