1.1b Negative Numbers — Cambridge IGCSE CS 0478

Representing Negative Numbers in Binary

Computers need to represent negative numbers in binary. Two methods are used: sign and magnitude and two's complement. Cambridge 0478 requires you to know both.

Sign and Magnitude

The most significant bit (MSB) — the leftmost bit — is used as a sign bit: 0 means positive, 1 means negative. The remaining bits represent the magnitude (size) of the number.

Binary	Sign bit	Magnitude	Value
00000101	0 (positive)	0000101	+5
10000101	1 (negative)	0000101	–5

Problem with sign and magnitude: There are two representations of zero (+0 = 00000000, –0 = 10000000), and arithmetic is complex to implement in hardware.

Two's Complement

Two's complement is the method actually used in modern computers. It allows straightforward binary addition for both positive and negative numbers.

Range of values (8-bit two's complement)

The MSB has a place value of –128 (negative of 2⁷)
Range: –128 to +127

Bit position	–128	64	32	16	8	4	2	1
Example	1	0	1	1	0	1	0	0

Example: 10110100 in two's complement = –128 + 32 + 16 + 4 = –76

Converting a positive number to a negative two's complement number

Write the positive number in binary
Invert all bits (flip 0s and 1s)
Add 1

Example: Convert +13 to –13 in two's complement (8-bit):

+13 = 00001101
Invert: 11110010
Add 1: 11110011
Check: –128+64+32+16+2+1 = –128+115 = –13 ✓

Adding two's complement numbers

Add as normal binary. Any carry out of the MSB is discarded.

Example: 5 + (–3) = 2

5 = 00000101, –3 = 11111101
Sum: 100000010 → discard carry → 00000010 = 2 ✓

Exam tip: In Cambridge 0478, two's complement is the preferred method for representing negative numbers in hardware. If asked for the range of an n-bit two's complement number, it is –2^(n-1) to 2^(n-1)–1. For 8 bits: –128 to +127.

⚠️ Common Mistakes

Forgetting that the MSB in two's complement has a negative place value (–128 not +128)
Forgetting to add 1 after inverting when converting to two's complement
Confusing sign and magnitude with two's complement
Claiming the range of 8-bit two's complement is –127 to +127 (it's –128 to +127)

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Key points

Sign and magnitude: MSB is the sign bit, 0=positive, 1=negative
Two's complement: invert all bits then add 1 to negate a number
MSB in two's complement has value –2^(n-1)
8-bit two's complement range: –128 to +127
Binary addition works naturally with two's complement

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Worksheet — 1.1b Negative Numbers

5 questions · 11 marks · instantly marked

Q1State what is meant by sign and magnitude representation of a binary number.[2 marks]

✅ Mark scheme

The most significant bit (MSB) is used as the sign bit [1]; 0 indicates positive, 1 indicates negative, and the remaining bits represent the magnitude/size [1]

Q2Convert +19 to its 8-bit two's complement representation. Show your working.[2 marks]

✅ Mark scheme

+19 = 00010011 [1]; (note: two's complement of a positive number is just its binary representation) [1]

Q3Convert –19 to its 8-bit two's complement representation. Show all steps.[3 marks]

✅ Mark scheme

+19 = 00010011 [1]; invert to 11101100 [1]; add 1 = 11101101 [1]

Q4What is the denary value of the two's complement binary number 11001100?[2 marks]

✅ Mark scheme

MSB has value –128 [1]; –128+64+8+4 = –52 [1]

Q5State the range of values that can be represented by an 8-bit two's complement number.[2 marks]

✅ Mark scheme

–128 [1] to +127 [1]

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Representing Negative Numbers in Binary

Sign and Magnitude

Two's Complement

Range of values (8-bit two's complement)

Converting a positive number to a negative two's complement number

Adding two's complement numbers

Key points

Worksheet — 1.1b Negative Numbers

Mini Test — 1.1b Negative Numbers