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Comment by mjw | Hacker News Reader

This is a really nice proof, and I love this kind of "you could have discovered X if you'd just had this core idea!" kind of exposition.

Perhaps a hard ask, but do you know of any similarly explicable proofs for transcendental-ness of "well known" numbers?

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As Colin says, transcendence proofs tend to be hard except for numbers carefully constructed to make transcendence proofs easy :-).

His mention of the (beautiful but difficult) Gelfond-Schneider theorem reminds me of the following much easier, much less deep but rather pretty observation.

Question: Is it possible to take an irrational number, raise it to an irrational power, and get a rational number?

Answer: Yes. Write s=sqrt(2) -- which is of course irrational -- and consider s^s. If it's rational then we're done. Otherwise, (s^s)^s is an irrational number to an irrational power; but it equals s^(s^2) = s^2 = 2, which is rational!

Thanks to Gelfond and Schneider, we know that actually s^s is not only irrational but transcendental.

(In 1919, the great mathematician David Hilbert mentioned three big unsolved problems in number theory: the Riemann Hypothesis, Fermat's "last theorem", and the transcendence of 2^sqrt(2). He said RH would probably be proved within a few years, FLT maybe in his lifetime, and the transcendence of 2^sqrt(2) probably not in the lifetime of anyone in his audience. In fact he got the order exactly wrong: the transcendence of 2^sqrt(2) was proved in 1930, FLT in 1995, and RH still isn't done despite its great importance.)

Another answer I've seen to the irrational^irrational question is sqrt(10)^log(4)=2, where log is the common logarithm. This avoids upsetting the constructivists.

But upsetting the constructivists is the whole point! (That is: what's so cute about that proof is that it tells you that there's a solution to irrational^irrational=rational, and that it's one of two things, but it doesn't tell you which. Also, I think it's a little shorter and more elegant than using sqrt(10)^log10(4). Don't you?)

Results for specific known numbers such as e and pi are known to be hard to come by.

http://en.wikipedia.org/wiki/Transcendental_number#Numbers_p...

The main tools are the Gelfond-Schneider theorem and the Lindeman-Weierstrass theorem:

http://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_theor...

http://en.wikipedia.org/wiki/Lindemann%E2%80%93Weierstrass_t...

gjm11 will know much more about these than I, and is better placed to offer insight into them.

e can be done reasonably with material from a good first year college calculus course. Such a proof is given around page 437 of the 3rd edition of Spivak's "Calculus".

Felix Klein's lecture made Lindemann's pi and Hermite's e transcendental proof accessible and interesting to German `HS' students, and was my own first persuasion. It was a Dover soft cover: Felix Klein's Famous Problems of Elementary Geometry, Dover, 1956.

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