In the general case, you need infinite bits to represent an irrational number. 100% of all irrational numbers are not computable by a Turing machine, because th

In the general case, you need infinite bits to represent an irrational number. 100% of all irrational numbers are not computable by a Turing machine, because the set of all algorithms on a Turing machine is countable and irrationals are not. So it seems a bit silly to talk about "the general case" of irrational numbers; it's extremely unlikely you care about an irrational number that is not computable. If an irrational number doesn't have a "neat representation", then it doesn't come up in your problem.