[Y_Y]

You didn't say otherwise, but it's interesting to note that the irrationals is quite "big" in the sense that you can do a lot without them. Not only can you make a nice consistent extension of usual structures on N, Z, Q etc by just adding sqrt(2), you can even add all square roots, or all n-th roots, or the solutions to all polynomials without getting plenty of good irrationals. Sets between Q and R are often neglected, but there's a wealth of good maths in there.