Why does two's complement have one more negative number than positive?

Because zero occupies one of the non-negative patterns, leaving one fewer pattern for positives than for negatives.

With w bits there are 2^w patterns to share. Zero sits on the MSB = 0 (non-negative) side, so that side holds zero plus the positives, while the whole MSB = 1 side is available for negatives — giving one extra negative value.

For 4-bit two's complement:

• Positive: 0001 to 0111 (1 to 7) → 7 values
• Zero: 0000 → 1 value
• Negative: 1000 to 1111 (−8 to −1) → 8 values

Pattern 1000...0:

• In unsigned: 2^{w-1} (middle value)
• In two's complement: -2^{w-1} (TMin, most negative)

Consequence: -TMin = TMin (negating TMin overflows back to itself!)