Number Systems (Cambridge (CIE) O Level Computer Science): Revision Note
Exam code: 2210
The Denary, Binary & Hexadecimal Number Systems
Examiner Tips and Tricks
All conversions here are capped at 16 bits because that’s Cambridge’s limit. No distractions, just the format you’ll see in Paper 1.
What is denary?
Denary is a number system that is made up of 10 digits (0-9)
Denary is referred to as a base-10 number system
Each digit has a weight factor of 10 raised to a power, the rightmost digit is 1s (100), the next digit to the left 10s (101) and so on
Humans use the denary system for counting, measuring and performing maths calculations
Using combinations of the 10 digits we can represent any number
In this example, (3 x 1000) + (2 x 100) + (6 x 10) + (8 x 1) = 3268
To represent a bigger number we add more digits
What is binary?
Binary is a number system that is made up of two digits (1 and 0)
Binary is referred to as a base-2 number system
Each digit has a weight factor of 2 raised to a power, the rightmost digit is 1s (20), the next digit to the left 2s (21) and so on
Each time a new digit is added, the column value is multiplied by 2
Using combinations of the 2 digits we can represent any number
In this example, Binary 1100 = (1 x 8) + (1 x 4) = 12
To represent bigger numbers we add more binary digits (bits)
32,768 | 16,384 | 8,192 | 4,096 | 2,048 | 1,024 | 512 | 256 | 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
215 | 214 | 213 | 212 | 211 | 210 | 29 | 28 | 27 | 26 | 25 | 24 | 23 | 22 | 21 | 20 |
Examiner Tips and Tricks
The largest denary number that can be represented using 16 bits is:
65,535 (Binary 1111111111111111)
What is hexadecimal?
Hexadecimal is a number system that is made up of 16 digits, 10 numbers (0-9) and 6 letters (A-F)
0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | A | B | C | D | E | F |
Hexadecimal is referred to as a base-16 number system
Each digit has a weight factor of 16 raised to a power, the rightmost digit is 1s (16^0), the next digit to the left 16s (16^1)
In GCSE you are required to work with up to and including 2 digit hexadecimal values
16s | 1s |
|
|---|---|---|
1 | 3 |
|
1 x16 | 3 x 1 | = 19 |
A quick comparison table demonstrates a relationship between hexadecimal and a binary nibble
One hexadecimal digit can represent four bits of binary data
Examiner Tips and Tricks
You must be able to convert binary to hex and back using nibbles. Write out the hex digits (0–F) during the exam to avoid confusion—it’s a top tip that examiners mention every year.
Denary | Binary | Hexadecimal |
|---|---|---|
0 | 0000 | 0 |
1 | 0001 | 1 |
2 | 0010 | 2 |
3 | 0011 | 3 |
4 | 0100 | 4 |
5 | 0101 | 5 |
6 | 0110 | 6 |
7 | 0111 | 7 |
8 | 1000 | 8 |
9 | 1001 | 9 |
10 | 1010 | A |
11 | 1011 | B |
12 | 1100 | C |
13 | 1101 | D |
14 | 1110 | E |
15 | 1111 | F |
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