What can be added to a rational number to explain the sum of two rational numbers?

Nothing. All that doing so achieves is to show that in certain circumstances, if you add a given amount to something then you get something else. That sort of result cannot be generalised.

For example:

11 = 1 = 1*1

22 = 4 = 2*2

But from that, you cannot generalise to

33 = 3*3