[monkeyjoe]

> Rational numbers are numbers that we can have in our hand while irrational numbers are ones which we can never have. It is important to have a setup that respects that difference.

Do you mean physically? Basic shapes like circles, squares and triangles allow us to hold irrational numbers in our hands as distances. Children playing with blocks can sense that root 2 does not conform nicely with other (rational) distances.

[jostylr]

I did not mean physically. I meant, having it explicitly written out in a way that rational numbers can be. If I write 3/4 + 1/2, I can compute out 5/4 and that is infinitely accurate.

If I want to compute pi + e, an infinitely accurate version is pi + e. That's about it. So what we are actually looking for in this computation is an estimation algorithm, one which can be made as accurate as we wish, but finitely so. The natural way to express this is with rational intervals as rationals are precise and intervals give a containment of the numbers.

For arithmetic, we can have a mechanism for figuring out how precise the input approximations need to be in order to get a given precision for the final computation. The perspective presented here naturally leads to that as a matter of defining and establishing the arithmetic of oracles.

As for playing around with physical representations of irrational numbers, keep in mind that there is no way to prove that, say, something that looks like a unit square to us is really a perfect square down to infinite precision. And without that, we can easily have that the unit square is only very close to such a figure, but is actually a rational rectangle with a rational diagonal that very closely approximates the square root of 2.

[ginnungagap]

This is irrelevant to your point, just change the numbers used as example, but we do not know whether pi+e is irrational! (Even though nobody believes it to be rational)