Negative numbers: one's and two's complement

5 minAS Level1.1.2

AI-drafted, independently checked · not yet teacher-reviewed

By the end of this lesson

By the end you can write a negative integer in 8-bit one's complement and two's complement, and turn either back into denary.

There's no minus sign in a register

A register holds only bits, so "negative" has to be encoded in the bits themselves. In both schemes below:

Positive numbers are stored exactly as normal, with the most significant bit (MSB) 0.
Negative numbers have the MSB set to 1.

The schemes differ in how the remaining bits are worked out.

Definition

One's complement

A representation in which the negative of a number is formed by inverting every bit of its positive form (each 0 becomes 1 and each 1 becomes 0).

Definition

Two's complement

A representation in which the MSB has a negative place value (−128 in 8 bits) and all other bits keep their usual positive values. The negative of a number is formed by inverting every bit and then adding 1.

Worked example: Denary to one's and two's complement

Represent −42 in 8-bit one's complement and in 8-bit two's complement.

Have a go first?

Quick check

Fill in the blanksQuestion 1 · Apply

Write −20 in 8-bit two's complement. +20 is 0001 0100. Inverting gives . Adding 1 gives .

CalculateQuestion 2 · Apply

Convert the 8-bit two's complement number 1011 0001 to denary.

Answer the 2 questions above to continue.

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Two's complement in one picture: the MSB is worth −128, everything else is ordinary binary.