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 ( in 8-bit). Flip bits: . Add 1: . This is in two's complement. To verify: . The overflow bit is discarded, leaving (zero), confirming they are opposites.
In -bit two's complement, the range is to . For 8 bits: to . The leftmost bit still indicates sign (1 for negative), but the encoding makes arithmetic automatic.