Two's Complement

Two's complement is the standard way to store signed integers, giving one zero and uniform arithmetic.

The representation

In two's complement the most significant bit has a negative weight of −2ⁿ⁻¹ while the rest keep their normal positive weights. So 8-bit 11111111 equals −128 + 127 = −1, and 10000000 equals −128.

Negating a number

To negate a value, invert every bit and add 1. Negating 8-bit 00000101 (5) gives 11111010 + 1 = 11111011, which reads as −5. Applying the same procedure again returns the original.

Why addition just works

Because negatives are stored modulo 2ⁿ, a subtraction a−b becomes the addition a + (−b) with the same adder used for unsigned numbers. The carry out of the top bit is simply discarded.

Range and asymmetry

With n bits the range is −2ⁿ⁻¹ to 2ⁿ⁻¹−1. The most negative value has no positive counterpart, so negating it overflows back to itself — a subtle edge case worth guarding in code.

Overflow detection

Signed overflow occurs when two operands of the same sign produce a result of the opposite sign. Hardware flags this by comparing the carry into and out of the sign bit.

def neg(x, n):
    return (~x + 1) & ((1 << n) - 1)
print(neg(5, 8))  # 251 = -5 pattern