This is discussed heavily in topics around constructing the reals and constructivism. Wikipedia has a short section about using the computables instead of the reals: https://en.m.wikipedia.org/wiki/Computable_number
It is pretty fascinating to think about: that almost all real numbers are not computable ("almost all" of course meaning "all but a countable set"). And yet you almost certainly will never run into a noncomputable real except in computer science when you're specifically defining one to illustrate something about computability (like Chaitin's constant).