x^4 + y^4 = z^4 is itself a great example of this. In C, we can just specify any pair from x,y,z and the third is determined by rearranging. In R, it's not tr

x^4 + y^4 = z^4 is itself a great example of this. In C, we can just specify any pair from x,y,z and the third is determined by rearranging. In R, it's not true! (y=2 and z=1 runs into trouble.) However, there are still infinitely many solutions and they're still easy to find. In Z, we have the solution 0,0,0. In N, we meet disaster! To squeeze some meaning out of this, C is in some ways nicer because it is designed for the very purpose of containing solutions to things. We only got Q because ratios aren't integers so we added them in; we got R because sqrt(2) is famously irrational so we take the completion of Q... As we go on we just sort of smooth things out and get a better behaved, richer space. The "gaps" in N can make proving things feel impossible (sometimes rightly so).