By intuition I can see that the 2's complement will be the negative of a number but I want a more rigorous proof to convince myself that no arithmetic will ever fail.
EDIT More clarification:
Consider the domain [0,8)
decimal | 2's comp | integer 0 | 8 | 0 1 | 7 | 1 2 | 6 | 2 3 | 5 | 3 4 | 4 | -4 5 | 3 | -3 6 | 2 | -2 7 | 1 | -1
it seems that integer column picks its -ve numbers from the lower half of 2's complement column and the +ve numbers from the upper half of decimal column.
What magic is going on here?