The problem with computables is that equivalence between them is only semi-decidable. (If the two numbers are different, it is decidable, but if they are not, it isn't. The problem is that you don't know if they are different a priori, so you might get lucky and find difference, but you might as well not.)

We know for sure that algebraic numbers behave nicely in terms of equivalence, and there are other, bigger number systems that are conjectured to behave nicely ( https://en.wikipedia.org/wiki/Period_(algebraic_geometry) ), but the problem with these and computers is that they are hard to represent.