Does there exist a function that does not include any trigonometric function in its definition that has similar properties (periodicity, for instance) as trigonometric functions? I can't think of any, and this strikes me as a bit surprising.

Edit: I thought of a simple answer: piecewise functions can achieve this!

Do we have any mathematicians in the house? I've been wondering for a while why math is usually focused around real numbers instead of computable numbers - that is the set of numbers that you can actually be computed to arbitrary, finite precision in finite time. Note that they necessarily include pi, e, sqrt(2) and every number you could ever compute. If you've seen it, it's computable.

What do we lose, beyond cantor's argument, by restricting math to computable numbers? By corollary of binary files and therefore algorithms being countable, the computable numbers are countable too, different from reals.

Bonus points if you can name a real, non-computable number. (My partner replied with "a number gained by randomly sampling decimal places ad infinitum" - a reply as cheeky as the question.) Also bonus points for naming further niceness properties we would get by restricting to computables.

I've read the wikipedia article on computable numbers and a bit beyond.