Well, I do know the answer, as it happens --- so does APG, no doubt. Anybody determined to cheat (not that anybody would, of course) can always consult https://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_theorem WFL On 8/12/15, Keith F. Lynch <kfl@keithlynch.net> wrote:
Fred Lunnon <fred.lunnon@gmail.com> wrote:
However, I did slip up again: gamma has not actually been proven transcendental! I should have employed exp(1), ...
Indeed, exp(x) is transcendental for all non-zero algebraic x. It follows from that that if exp(x) is algebraic, x must be transcendental. Hence, for instance, the natural log of 2 must be transcendental.
But since the transcendental numbers are of a higher cardinality than the algebraic numbers, it must be that in "most" cases x and exp(x) are both transcendental.
Puzzle: Give an example where x and exp(x) are both known to be transcendental. I'll give a solution in a week if nobody posts one sooner.
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