Negative Numbers: Two’s Complement

The twenty-fifth intro explains the convention all subtractions in a computer rest on: two's complement. The experiment uses eight lamps and a signed total.

Experiment: a minus on eight lamps

The lamps only count 0–255, but you need to show "minus five". By a convention: 5 − 10 is computed as 5 + (256 − 10) = 251. Build 251 on the lamps (everything except the "4" lamp) and watch the total: it reads −5. A lit "128" lamp — the highest one — means the number is negative and its value counts down from 256.

The rule: the top bit is the sign

This convention is called two's complement. If the highest lamp is lit, the number reads as negative: 11111011 is not 251 but −5. The extreme case — all eight lamps lit: 11111111 is −1. Signed reading: subtract the unsigned value from 256. For example, 11111100 = 256 − 252 = −4.

Transfer: read the signed number

The lamps show 11111100. Without the sign it is 252, but the question turns the sign on, so read downward from 256: 256 − 252 = −4. Only the highest lamp sets the minus; the magnitude is the other seven.

Where it is used

On this level the processor learns to recognise a "minus": a lit top bit of the result raises the N (negative) flag, and the JN instruction jumps if the minus is there. That is how "if the result is below zero" checks work — with no new parts at all, just the convention about the top bit.