Yes, there are definitions, but the definitions come with the system of axioms and a manner in which the numbers have been defined. The definitions are given separately for operations on natural numbers, integers, rational fractions, reals, complex numbers, etc. And all those groups of numbers could be defined in several ways. The definitions of operations would be adapted accordingly.

This isn't as important in middle school, but proofs become crucial in later math. If you're interested in checking out a possible definition, check out an article (probably from this site) on Peano Arithmetic. It only works for whole numbers, and it doesn't do division, but it does do a lot of cool things.