📁 Paper 2 · 3.3 Data Representation
3.3.1 Number Bases
AQA 8525 · GCSE Computer Science · ~10 min read
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Video
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Worksheet
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Quiz

Why Number Bases?

A number base (or radix) tells you how many different digits are available. Everything stored in a computer is ultimately represented in binary, but humans also use decimal and hexadecimal when working with computers.

BaseNameDigits usedUsed by
2Binary0, 1Computer hardware
10Decimal0–9Everyday human use
16Hexadecimal0–9, A–FProgrammers / colour codes / memory addresses

Binary (Base 2)

Binary uses only two digits: 0 and 1. Each digit is called a bit. Computers use binary because their circuits have exactly two stable states: on (1) and off (0).

The place values in an 8-bit (one byte) binary number are:

2⁷
128
bit 7
2⁶
64
bit 6
2⁵
32
bit 5
2⁴
16
bit 4
8
bit 3
4
bit 2
2
bit 1
2⁰
1
bit 0

Example: 10110101₂ = 128 + 0 + 32 + 16 + 0 + 4 + 0 + 1 = 181₁₀

Hexadecimal (Base 16)

Hexadecimal uses 16 digits. When the digits run out (after 9), letters A–F are used:

0
= 0
1
= 1
2
= 2
3
= 3
4
= 4
5
= 5
6
= 6
7
= 7
8
= 8
9
= 9
A
= 10
B
= 11
C
= 12
D
= 13
E
= 14
F
= 15

Each hex digit represents exactly 4 bits (a nibble). So one byte = two hex digits.

Example: 2A₁₆ — place values are 16¹ and 16⁰ → (2 × 16) + (10 × 1) = 32 + 10 = 42₁₀

Why Hexadecimal?

  • It is much more compact than binary — 1 hex digit = 4 bits
  • It is easier for humans to read and write than long binary strings
  • Used for: memory addresses, colour codes (e.g. #FF5733), MAC addresses, error codes

Example: the binary string 11111111 is written as FF in hex — much shorter!

Counting in Each Base

DecimalBinaryHex
000000
100011
501015
910019
101010A
151111F
160001 000010
2551111 1111FF
Exam tip: When writing hex in an exam, you can prefix with 0x (e.g. 0xFF) or use subscript ₁₆. AQA may show hex values as 3B, FF, 2A etc. — always check the context to know which base is being used. Hex letters can be upper or lower case.
⚠️ Common Mistakes
  • Forgetting that A=10, B=11, …, F=15 — not A=1, B=2
  • Confusing base notation — 10₂ = 2₁₀, not 10!
  • Thinking hex digits go up to 9 only — they go 0–9 then A–F (16 total)
Video coming soon

Key points

  • Binary (base 2): digits 0–1; used by computer hardware due to two electrical states
  • Decimal (base 10): digits 0–9; everyday human number system
  • Hexadecimal (base 16): digits 0–9 plus A–F; A=10 through F=15
  • One hex digit = 4 bits; two hex digits = one byte
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✍️

Worksheet — 3.3.1 Number Bases

8 questions · 18 marks

Q1State what is meant by the 'base' of a number system and give the base used in binary.[2]
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The base (or radix) describes how many different digit symbols are available in the number system [1]; binary uses base 2 (digits 0 and 1 only) [1].
Q2State the decimal values of the hexadecimal digits A, C, and F.[3]
✅ Mark scheme
Mark scheme
A = 10 [1]; C = 12 [1]; F = 15 [1].
Q3Why do computers use binary rather than decimal internally?[2]
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Computer circuits have two stable electrical states — on (1) and off (0) [1]; binary maps perfectly to these two states, making circuits simpler and more reliable [1].
Q4How many bits does one hex digit represent? How many hex digits make up one byte?[2]
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One hex digit represents 4 bits [1]; two hex digits make up one byte (8 bits) [1].
Q5Write the binary number 11111111 in hexadecimal. Show your working.[2]
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Split into nibbles: 1111 | 1111 [1]; each 1111 = 15 = F in hex, so the answer is FF [1].
Q6Give two reasons why hexadecimal is used by programmers instead of binary.[2]
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Any two: hexadecimal is more compact than binary — 2 hex digits replace 8 binary digits [1]; easier for humans to read and write without making errors [1]; it can still be directly converted to/from binary without complex arithmetic [1].
Q7Give two real-world uses of hexadecimal in computing.[2]
✅ Mark scheme
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Any two from: memory addresses [1]; HTML/CSS colour codes (e.g. #FF5733) [1]; MAC addresses [1]; error codes / debugging [1]; assembly language instructions [1].
Q8Complete the table: Decimal 13 in binary is __; Hex digit B in decimal is __; Binary 0100 in hex is __.[3]
✅ Mark scheme
Mark scheme
13 in binary = 1101 [1]; B in decimal = 11 [1]; 0100 in hex = 4 [1].
Check your answers above.
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Mini Test — 3.3.1 Number Bases

Timed exam conditions.

  • 8 questions · 10 minutes
  • 5 MCQ + 3 short answer
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3.3.2 Converting Bases →