The trouble you have is that you really wanted reals and the rationals are not a much better substitute than the floats (which are mostly just one particularly specialised flavour of rational)
It doesn't feel immediately like there ought to be a problem, but witness, mathematically almost all of the reals are Normal and unfortunately Normal doesn't mean "normal" it's a technical mathematical word, that for our purposes means roughly "completely batshit".
Normal numbers are non-computable, so we definitely don't want to try to use them in software, which is a problem because again, almost all of the reals are Normal.
Huh? This isn't making a lot of sense to me. Why do you say the rationals aren't much better then floats? In situations like this surely we absolutely want to be working in the rationals, and they're a very easy way to do these calculations correctly without having to worry about tolerance bounds for arbitrary-precision float calculations
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.
It doesn't feel immediately like there ought to be a problem, but witness, mathematically almost all of the reals are Normal and unfortunately Normal doesn't mean "normal" it's a technical mathematical word, that for our purposes means roughly "completely batshit".
Normal numbers are non-computable, so we definitely don't want to try to use them in software, which is a problem because again, almost all of the reals are Normal.