← IGCSE Computer Science Paper 1

Why two's complement is used

A way to store negative integers in binary. The leftmost bit shows the sign: 0 means positive, 1 means negative. Positive numbers look the same as ordinary binary.

Part of: IGCSE Computer Science 0478 — 1.1 Number Systems — Module 28

Understanding Why two's complement is used in IGCSE Computer Science Paper 1

What it does: The leftmost bit is the sign bit. Here it is 1, so the number is negative, which lets binary store negatives without a minus symbol. A leftmost bit of 0 means the number is positive (or zero). 10010110 starts with 1, so it is negative. 01101001 starts with 0, so it is positive. Positive numbers look the same as ordinary binary. For example, 00000101 is 5 in both. The sign meanings are swapped: 0 means positive and 1 means negative.

Common mistakes: In two's complement the sign is shown by the leftmost bit, not the rightmost. The line should be: sign ← leftmost bit of bits. A leftmost bit of 1 means the number is negative, so the test must be IF sign = 1 THEN. A sign bit of 0 means positive.

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Learning
Duo
What is two's complement used for?
Storing negative integers in binaryStoring only decimal fractions in binary
Matches
Leftmost bit 1 means the number is positive
↓
In two's complement a leftmost bit of 1 means the number is negative, not positive. Positive numbers have a leftmost bit of 0.
Applying
Meanings
Term
two's complement
Meaning
A way to store negative integers in binary. The leftmost bit shows the sign: 0 means positive, 1 means negative. Positive numbers look the same as ordinary binary.
Practice
// Two's complement stores negatives
Testing
Gaps
Sign bit 0 means the number is
Check Answer
Spots
1bits ← 00000101
2sign ← rightmost bit of bits

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Quick Questions

1. What is two's complement used for?
A. Storing negative integers in binary ✓
B. Storing only decimal fractions in binary
Two's complement is a method that lets a computer store negative whole numbers (integers) using only 0s and 1s.
2. Why can't ordinary binary show a negative number by itself?
A. It can only store numbers bigger than 255
B. It only uses 0 and 1, so there is no minus sign ✓
Binary has only the digits 0 and 1. A computer cannot store a '-' symbol, so it needs a method such as two's complement.
3. In two's complement, the leftmost bit is 1. What does that tell you?
A. The number is positive
B. The number is negative ✓
The leftmost bit shows the sign. 1 means negative.
4. In two's complement, the leftmost bit is 0. What does that tell you?
A. The number is positive ✓
B. The number is negative
A leftmost bit of 0 means the number is positive (or zero).
5. Which 8-bit pattern is a negative number in two's complement?
A. 10010110 ✓
B. 01101001
10010110 starts with 1, so it is negative. 01101001 starts with 0, so it is positive.

Match the Pairs

00001010 → A positive number in 8-bit two's complement: the leftmost bit is 0, so it is just ordinary binary for denary 10
10000001 → A negative number in 8-bit two's complement: the leftmost bit is 1, so the value is below zero
Binary alone has no minus sign, so two's complement uses the leftmost bit to show the sign → 1 0 0 0 0 0 1 1 (leftmost bit = sign)
A positive number in two's complement looks exactly the same as ordinary binary → 00000101 = 5 in binary and in two's complement
Leftmost bit 1 means the number is positive → In two's complement a leftmost bit of 1 means the number is negative, not positive. Positive numbers have a leftmost bit of 0.
-00000101 shows -5 in binary → Binary cannot contain a minus sign because computers only store 0s and 1s. Two's complement uses the leftmost bit to show a negative.

Key Term

two's complement
A way to store negative integers in binary. The leftmost bit shows the sign: 0 means positive, 1 means negative. Positive numbers look the same as ordinary binary.

Fill in the Blanks

1. Sign bit 0 means the number is positive
Hint: Type the word for what a leftmost bit of 0 shows about the number
2. Sign bit 1 means the number is negative
Hint: Type the word for what a leftmost bit of 1 shows about the number
3. +5 in 8 bits: 00000101
Hint: Type the sign bit that marks a positive number
4. Binary only has 0 and 1, so no minus sign
Hint: Type the symbol that ordinary binary cannot show for negative numbers
5. Method for negatives: two's complement
Hint: Type the second word of the method name used to store negative integers in binary

Spot the Error

Question 1
A bits ← 00000101
B sign ← rightmost bit of bits
C OUTPUT sign
Line B — In two's complement the sign is shown by the leftmost bit, not the rightmost. The line should be: sign ← leftmost bit of bits.
Question 2
A bits ← 10000011
B sign ← leftmost bit of bits
C IF sign = 0 THEN
D OUTPUT "negative"
Line C — A leftmost bit of 1 means the number is negative, so the test must be IF sign = 1 THEN. A sign bit of 0 means positive.
Question 3
A denary 6 in binary: 00000110
B in two's complement: 10000110
C # positives look the same
Line B — Positive numbers look the same in two's complement as in ordinary binary. 6 stays 00000110, with a leftmost bit of 0.
Question 4
A Binary uses only 0 and 1
B It has a minus symbol
C So negatives need a sign bit
Line B — Binary has no minus sign, only the digits 0 and 1. That is why two's complement uses the leftmost bit to show negative numbers.
Question 5
A 10000001 has sign bit 1
B so it is negative
C 00000001 has sign bit 1
D so it is positive
Line C — The leftmost bit of 00000001 is 0, not 1. A sign bit of 0 means the number is positive.

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