So when doing operations on the reals mathematicians like to use the definition of field. For a field we only need multiplication and addition. The inverse is then given by division and subtraction, respectively.

And a question:

n objects ÷ n boxes = 1 object per box

how do you know this is true? I think before you can talk about this you need to define what arithmetic is. What zero is is then a result of the definition of arithmetic https://en.wikipedia.org/wiki/Peano_axioms