Algebraics plus pi avoid almost all the concern you mention, but still have the same problem for software. All those nasty reals are unreachable by your program

Algebraics plus pi avoid almost all the concern you mention, but still have the same problem for software. All those nasty reals are unreachable by your program. The actual problems with irrationals is that it is extremely hard to measure their size without converting to rational approximation, so you are back to the original problem.