Sign and magnitude and two's complement to represent negative numbers

Use of sign and magnitude and two’s complement to represent negative numbers in binary

1.4.1 Data Types

Data Types — Note 3 of 10

Candidates should know how to store negative numbers using Sign and Magnitude and two’s complement. Candidates should be able to convert denary numbers to sign and magnitude, and two’s complement – and vice-versa

What you need to know

Negative integers can be represented in binary using sign and magnitude or two's complement. Two's complement is used by computers because binary addition works correctly with it and there is only one representation of zero.

What to know

Key terms

Exam tipCommon mistake

Sign and magnitude: The most significant bit (MSB) is the sign (0 = positive, 1 = negative) and the remaining bits hold the size of the number.
-45 in 8 bits = 10101101. Range in 8 bits: -127 to +127.
Sign and magnitude drawbacks: two representations of zero (00000000 and 10000000), and normal binary addition gives wrong answers.
Two's complement: The MSB has a NEGATIVE place value. In 8 bits the place values are -128 64 32 16 8 4 2 1.
To make a number negative in two's complement: write the positive binary, flip every bit, then add 1. (Or: from the right, copy up to and including the first 1, then flip the remaining bits.)
Two's complement: -45 = 11010011 (-128 + 64 + 16 + 2 + 1). Range in 8 bits: -128 to +127.
Two's complement benefits: one representation of zero; addition and subtraction work with ordinary binary addition.
Exam tip: To convert two's complement to denary, add the place values using -128 for the MSB.

Exam technique

Mistake:Forgetting to add 1 after flipping the bits.

Code examples

Converting -45 to 8-bit two's complement

text

+45           0 0 1 0 1 1 0 1
Flip bits     1 1 0 1 0 0 1 0
Add 1       + 0 0 0 0 0 0 0 1
-45           1 1 0 1 0 0 1 1

Check: -128 + 64 + 16 + 2 + 1 = -45