In a purely numerical setting, you can only distinguish these two cases when you evaluate the expression with enough accuracy. This may feel like a weakness, but if you think about this it is a much more "honest" way of handling inaccuracy than just rounding like you would do with floating point arithmetic.

A good way to think about the framework, is that for any expression you can compute a rational lower and upper bound for the "true" real solution. With enough computation you can get them arbitrarily close, but when an intermediate result is not rational, you will never be able to compute the true solution (even if it happens to be rational; a good example is that for sqrt(2) * sqrt(2) you will only be able to get a solution of the form 2 ± ϵ for some arbitrarily small ϵ).