inf is infinity – a value that is greater than any other value. -inf is therefore smaller than any other value.
nan stands for Not A Number, and this is not equal to 0.
Although positive and negative infinity can be said to be symmetric about 0, the same can be said for any value n, meaning that the result of adding the two yields nan. This idea is discussed in this math.se question.
Because nan is (literally) not a number, you can’t do arithmetic with it, so the result of the second operation is also not a number (nan)