The usual formal definition for the rational numbers is equivalence classes of pairs of integers. The zero is then the equivalence class of (0, 1) which is not the same as the integer 0.
You could certainly somehow get it to work by starting with the closure of the division operation but would introduce a lot of unnecessary headache along the way.
You could certainly somehow get it to work by starting with the closure of the division operation but would introduce a lot of unnecessary headache along the way.