How do you work out an answer for x - y when eg x = sqrt(2) and y = sqrt(2) - epsilon for arbitrarily small epsilon? How do you differentiate that from x - x?
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).
The problem with that from a UX perspective is that you won't even get to write out the first digit of the solution because you can never decide whether it should be 1.999...999something (which truncates to 1.99) or 2.000...000something (which truncates to 2.00). This is a well-known peculiarity of "exact" real computation and is basically one especially relevant case of the 'Table-maker's dilemma' https://en.wikipedia.org/wiki/Rounding#Table-maker%27s_dilem...