This is going to be a pretty open and loosely defined problem (sincere apologies beforehand).
Are there mathematical functions/equations (over integers) such that:
Given an N (weather its positive or negative is simply unknown) such that when the function is calculated using it, there is a black box way to verify if the original N was positive or negative (using some additional black box functions if necessary).
Essentially, a function that would be computable on N, and the result's magnitude could be used again to infer the sign of the original N (using some other black box operations on it).
The operations permitted are the basic 4 operations (addition, subtraction, multiplication and division).