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A popular (and maybe the only) approach to showing that $\pi$ is transcendental is to first prove that for every non-zero algebraic number $a$, the number $e^a$ is transcendental.

That requires tools from complex analysis.

But is there a known elementary proof that $\pi$ is transcendental? By an elementary proof I mean proof that does not use complex analysis.

For example, there are known proofs that $e$ is transcendental which do not use complex analysis.

Also, can it be proved that complex analysis must be used to prove some given theorem?

Redbox
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  • See https://math.stackexchange.com/a/31800/589 – lhf May 22 '20 at 18:10
  • The proof in your link involves the use of complex numbers (or $\mathbb{C}\setminus\mathbb{R}$ numbers, to be more precise). I don't think that proof could be called a "real-analytic" one. – Redbox May 22 '20 at 20:56
  • Luckily the proof linked above does not use complex analysis in a significant way. – Paramanand Singh May 23 '20 at 09:34
  • You can have a look at the book Transcendental Numbers by M. Ram Murty, Purusottam Rath – ShBh May 23 '20 at 10:23

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Not an answer, just a few comments that got too long.

The original (cleaned to modern standards) proof that $e$ is transcendental is just stuff with polynomials and the exponential on the reals, so not sure if it is accurate to claim that it uses "complex analysis"; even for $\pi$, the complex analysis used is minimal (maximum modulus essentially and maybe the stuff about rational conjugates being potentially complex numbers (like for $2^{1/3}$ say) but the question is fair;

This being said this question on the need to use complex analysis or if you want the possibility of not using complex analysis, was popular in the 1920's regarding PNT and put to rest by Selberg/Erdos (though their proof of the PNT was much harder than the complex analysis one and that is still true today after major simplifications), but imho it misses the point as complex analysis usually simplifies things and purely real modern techniques are quite hard as one finds out easily by looking at harmonic functions in $\ge 3$ dimensions where the harmonic conjugate is not available anymore and proofs are sometimes much harder, so not sure what is the point

Conrad
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  • I just wanted to know if it is possible at all. – Redbox May 22 '20 at 18:20
  • that's why I am saying the question is fair and I am curious too (haven't heard of any such but one never knows...); my point is that in many situations, complex analysis is a simplifying tool and is more natural – Conrad May 22 '20 at 18:21
  • I agree with that. – Redbox May 22 '20 at 18:24
  • What's "PNT"??? – Lee Mosher May 22 '20 at 19:01
  • @Leo prime number theorem – Conrad May 22 '20 at 19:06
  • @LeeMosher Prenuptial tension. – Calum Gilhooley May 23 '20 at 13:56
  • All known proofs of the transcendence of $\pi$ ultimately involve the relation $e^{i\pi} = -1$, so I think it is not realistic to expect a proof of transcendence to avoid complex numbers entirely. The usual proof of transcendence of $\pi$ uses contour integration in the complex plane: integrals from $0$ to some nonzero complex numbers. It's enough to integrate along the line segments connecting $0$ to those numbers and use $e^z$ along such segments. So some minimal amount of analysis in the complex plane is used. – KCd Sep 25 '22 at 21:14