Which of the following binary numbers is the same as its 2's complement?

Concept

In an n-bit two's-complement system, the two's complement of a number N is formed by inverting every bit and adding 1, equivalent to computing 2n minus N. A number equals its own two's complement only when 2N is congruent to 0 modulo 2n. Among all n-bit patterns, only two satisfy this: the all-zero pattern (N = 0) and the pattern with just the most significant bit set (N = 100...0, the smallest representable negative value) -- every other pattern's double cannot wrap around exactly to zero modulo 2n.

Application

1. 1010: invert each bit to get 0101, then add 1 to get 0110 -- not equal to 1010.

2. 0101: invert each bit to get 1010, then add 1 to get 1011 -- not equal to 0101.

3. 1000: invert each bit to get 0111, then add 1 to get 1000 -- equal to 1000, matching the predicted MSB-only pattern.

4. 1001: invert each bit to get 0110, then add 1 to get 0111 -- not equal to 1001.

Cross-check

Using the direct formula 2n - N with n = 4: 16 - 8 = 8, which is 1000 in binary -- the same result obtained independently, confirming that 1000 is the only offered value equal to its own two's complement.