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?