It makes more sense to define it the other way around: Q = Z u Q’, where Q’ is the set of all rational numbers that aren’t integers. Redefining the smaller se

It makes more sense to define it the other way around: Q = Z u Q’, where Q’ is the set of all rational numbers that aren’t integers. Redefining the smaller set can’t work because there may be more than one larger set, e.g. split complex numbers vs regular complex numbers. But you can define a larger set to strictly extend a smaller set.