Like... yeah, sure, you can derive ℚ from ℤ, just like how you can derive ℤ from ℕ. And it's pretty natural to use the process of division that you learned as a kid while working in ℚ, to the point where, yeah, sure, if you wanted to solve x = 6 * 3⁻¹ in ℚ (which, if you're talking about "whole numbers", you probably are), you'd do it by following that particular procedure.
But if you're working in, say, ℂ? Or something more exotic? Then you'll find that your intuition of what "division" is to be less helpful.