Explain why x == ~(~x + 1) + 1 (two's complement and back!)

Question

As we all know usually negative numbers in memory represents as two\'s complement numbers like that

from x to ~x + 1

and to get back we don

Answer 1

This is because if you increment ~x (assuming no overflow). Then converting it to back to x, you've incremented relative to ~x, but decremented relative to x. Same thing applies vice versa. Assuming your variable x has a specific value, every time you increment it, relative to ~x you'll notice it decrements.

From a programmer's point of view, this is what you'd essentially witness.

Let short int x = 1         (0x0001)
then ~x = 65534             (0xFFFE)
~x + 1 =  65534 + 1         (0xFFFF)
~(~x+1) = 0                 (0x0000)
~(~x+1) + 1 = 0 + 1         (0x0001)