It seems to me that this comes at the price that now in your geometry the diagonal of a unit square does not have a defined length. Only an approximate one of 1.41421, but what does this approximate?
What price? In an algebraic setting you are never going to convert sqrt(2) into a real number, and in any practical setting you're going to have to round because nothing real actually has infinite precision.
It would appear that sqrt(2) does not have a meaning now, as sqrt(-1) in R. So it does not seem to matter if you do not convert it in an algebraic setting. There is no such number that you can reach with the enumerative approach, which will only give you the rational numbers. There are lots of proofs that sqrt(2) is not rational.
But in all probability we are discussing the wrong thing here. Our difference it's likely at a deeper conceptual level than this.
