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?