To be honest, I never really understood the importance of algebraic numbers. If we lived in an universe where π was algebraic, would there be a palpable difference between that universe and ours?
My choice of π for this question isn’t really that important, any other ”famous” transcendental number (i.e. e) could work.
I’m aware there are a lot of open problems about deciding whether some number is transcendental or algebraic (for an example Apery’s constant, Euler-Mascheroni constant and even π+e).
Are those problems important only because they are hard to tackle? Are they important at all? If tomorrow was published a proof of algebraicity of those numbers, what would we gain from it?
EDIT: OK, maybe I took too much ”artistic freedom” with the title of my question. I wasn’t really curious about alternate universes. Bottom line was: why are those proofs important? Why is ”being a transcendental number” important property of a number?
No such universe is possible, it would be a universe in which 1 is equal to 2.
That said, a rational approximation to π with error <10−200 is undoubtedly good enough for all practical purposes.
Lindemann’s proof that π is transcendental was a great achievement, but knowing the result has no consequences outside mathematics.