Is two’s complement a good fit?

Can we use C₂(x) as -(x)?

Let us try the rules of negation of two’s complement.

First of all, is it true that C₂(C₂(x)) = x?

C (C (x)) = (2n - 1)- C (x)+ 1 (16) 2 2 n 2n = (2 - 1)- (2 - 1- x + 1)+ 1 (17) = x (18)

However, this alone isn’t enough. Let us see if x + C₂(x) = 0:

x+ C2(x) = x + (2n - 1- x +1) (19) = 2n (20) = 0 (21)

Why does 2ⁿ = 0? This is because we are using modulo-2ⁿ arithmetics. Recall that in modulo arithmetic, 2ⁿ ≡ (2ⁿ mod2ⁿ) ≡ 0.

Now, on to some real tests. How about y + C₂(x) = y - x?

n y+ C2(x) = y + (2 - 1- x+ 1) (22) = y - x+ (2n) (23) = y - x (24)