I'm not sure how useful that would be. Pi might be represented more precisely, but the result of every operation you do with it is still going to have the issue

I'm not sure how useful that would be. Pi might be represented more precisely, but the result of every operation you do with it is still going to have the issue of varying precision (except for powers of your base/ratio for the representation). It's not exactly what you're talking about but there is some use for "block" floating point, where you manually track the decimal point through your arithmetic via shifts, which allows you to explicitly control the precision you need for a given portion of your arithmetic. The code is disgusting however. I wonder though if there's a numeric representation that could make trigonometric functions into a shift/mask operation, kind of like multiplying by powers of the base for integers/floats. That would be real neat.