[fgdorais]

This is better known as the Table Maker's Dilemma.

Say you have some computable number p, that means you can compute a (rational) approximation p' to p within any given tolerance eps > 0 (i.e. you know |p - p'| < eps). To determine whether p > 0, p = 0, or p < 0, you compute an approximation p' to a certain tolerance eps. If p' > eps then you know p > 0, if p < -eps then you know p < 0, otherwise you need a better approximation... Without further knowledge about p, there is no point where you can assert p = 0.

[ekez]

Thank you for the response. (I assume you mean "p' < -eps" in the second conditional?)

How often are rational approximations computable within any given tolerance?