## Number bases

 Quick links 3.3.1 Number bases 3.3.2 Converting number bases 3.3.3 Units of information 3.3.4 Binary arithmetic 3.3.5 Character encoding 3.3.6 Representing images 3.3.7 Representing sound 3.3.8 Data compression

### Syllabus content

Understand the following number bases:

• • decimal (base 10)
• • binary (base 2)

Understand that computers use binary to represent all data and instructions.   Students should be familiar with the idea that a bit pattern could represent different types of data including text, image, sound and integer.

Explain why hexadecimal is often used in computer science.

## Number bases

### Number bases

We usually use Arabic numbers, which are base 10, or decimal. Computers use binary, which is base 2. This page deals with numbers in other bases. Enter a number to convert it to a different base, or count in a base.

### How numbers in other bases work

In Arabic numbers (decimal, or base 10), there are 10 digits: 0,1,2,3,4,5,6,7,8,9. You need one digit each to count up to 9, but two digits for ten, and three digits for a hundred, which is ten times ten. In Binary, base 2, you need two digits for two, as you only have two digits, 0 and 1. Base 5 has five digits, and the number five becomes 10. For base 16, you will need sixteen digits, and there are only ten numerals. So we use the letters A,B,C,D,E,F. These represent the decimal numbers 10, 11, 12, 13, 14 and 15. Look at the table below and find the pattern for these bases.

Base 10 Base 2 Base 3 Base 4 Base 5 Base 8 Base 16
1 1 1 1 1 1 1
2 10 2 2 2 2 2
3 11 10 3 3 3 3
4 100 11 10 4 4 4
5 101 12 11 10 5 5
6 110 20 12 11 6 6
7 111 21 13 12 7 7
8 1000 22 20 13 10 8
9 1001 100 21 14 11 9
10 1010 101 22 20 12 A
11 1011 102 23 21 13 B
12 1100 110 30 22 14 C
13 1101 111 31 23 15 D
14 1110 112 32 24 16 E
15 1111 120 33 30 17 F
16 10000 121 100 31 20 10
17 10001 122 101 32 21 11
18 10010 200 102 33 22 12
19 10011 201 103 34 23 13
20 10100 202 110 40 24 14

Base systems like binary and hexadecimal seem a bit strange at first. The key is understanding how different systems “tick over” like an odometer when they are full. Base 10, our decimal system, “ticks over” when it gets 10 items, creating a new digit. We wait 60 seconds before “ticking over” to a new minute. Hex and binary are similar, but tick over every 16 and 2 items, respectively.