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?
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).
> 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...
If one embraces rational intervals throughout, they can be the computational foundation and the ux could have the option of displaying the interval for the complete truth or, to gain an intuitive sense, pick a number in the interval to display, such as the median or mediant. Presumably this would be a a user choice in any given context.
