Two's Complement

Two's complement represents negative numbers so that addition works without special cases. To negate a number, flip all bits and add 1.

Example: Negate $00000101$ ($+5$ in 8-bit). Flip bits: $11111010$. Add 1: $11111011$. This is $-5$ in two's complement. To verify: $00000101 + 11111011 = 100000000$. The overflow bit is discarded, leaving $00000000$ (zero), confirming they are opposites.

In $n$-bit two's complement, the range is $-2^{n-1}$ to $2^{n-1} - 1$. For 8 bits: $-128$ to $127$. The leftmost bit still indicates sign (1 for negative), but the encoding makes arithmetic automatic.