The square root of two is still a computable Real. We choose not to cope with that, but it's not actually impossible it was merely inconvenient. I've mentioned elsewhere that my Rust care realistic is quite happy to work with these numbers e.g. take the square root of ten, and the square root of forty, multiply them together and get the quite ordinary integer twenty.

The non computable reals are a huge problem because, as their name suggests, we can't compute them - and in the strict sense that's Almost All reals, but none of the ones you're thinking of are non-computable so you'll likely be fine.

For the merely rational numbers like a third, or sixteen hundred and five sevenths, it's even more so a matter of choosing not to address it rather than it being out of reach.

The problem with computables is that equivalence between them is only semi-decidable. (If the two numbers are different, it is decidable, but if they are not, it isn't. The problem is that you don't know if they are different a priori, so you might get lucky and find difference, but you might as well not.)

We know for sure that algebraic numbers behave nicely in terms of equivalence, and there are other, bigger number systems that are conjectured to behave nicely ( https://en.wikipedia.org/wiki/Period_(algebraic_geometry) ), but the problem with these and computers is that they are hard to represent.