TITLE: What if $\pi$ was an algebraic number? (significance of algebraic numbers)
QUESTION [51 upvotes]: 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?

REPLY [2 votes]: If tomorrow was published a proof of algebraicity of those numbers, what would we gain from it?

New theory that applies where those numbers appear, which for $\pi$ is most of mathematics (physics, engineering, ...).
When a transcendentally constructed number is algebraic or rational, it can be a sign of additional structure and a theory waiting to be built.  In some cases, a very substantial theory, such as "arithmetic geometry" coming from observations about numerical values of zeta- and L-functions.  If a number as ubiquitous as $\pi$ had previously undetected algebraic structure, it would probably be so for a systematic reason, such as hidden symmetries that once discovered can apply to all kinds of problems where $\pi$ appears.  
If $\pi$ were rational this would potentially be more dramatic. For example, $\pi = 22/7$ might imply some secret $11$-fold symmetry of the circle, or the existence of a previously unsuspected category of spaces in which the circle participates in a $22$- or $7$-fold covering with some interesting new structures.  And the notion of geometry would be updated to reflect these new possibilities.  
This is not that far from what really happened for numbers that generalize $\pi$ (periods of algebraic varieties): there is a compelling point of view that such numbers are just the numerical manifestation of more structured "upgraded" objects, and statements about transcendence and irrationality of the numbers then acquire the meaningful and practical interpretation that all 

relations between the numbers are accounted for by relations between the structures.    

The highlighted sentence (stripped of "the"'s) is the real answer to the question.