To be honest, I never really understood the importance of algebraic numbers. If we lived in an universe where $\pi$ was algebraic, would there be a palpable difference between that universe and ours?
My choice of $\pi$ 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 $\pi + e$).
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?
Best Answer
No such universe is possible, it would be a universe in which $1$ is equal to $2$.
That said, a rational approximation to $\pi$ with error $\lt 10^{-200}$ is undoubtedly good enough for all practical purposes.
Lindemann's proof that $\pi$ is transcendental was a great achievement, but knowing the result has no consequences outside mathematics.