I fully agree with Vincents suggestions.
 Also, the choice of real numbers (as example of additive group) is a particularly good example; I vividly recall trying to convince my students of the fact that 0.99... = 1.00... which is true in real numbers but not true as say strings of digits. This case is very much related to the case of signed zero; in both of them the source of the "problem" is the representation and the "solution" is an equivalence relation.