Number Bases (AQA GCSE Computer Science)

What is decimal (base 10)?

• Decimal is a number system that is made up of 10 digits (0-9)
• Decimal 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
• Using combinations of the 2 digits we can represent any number

• In this example, (1 x 8) + (1 x 4) = 12
• To represent bigger numbers we add more binary digits (bits)

128	64	32	16	8	4	2	1
27	26	25	24	23	22	21	20

Why do computers use binary?

• The CPU is made up of billions of tiny transistors, transistors can only be in a state of on or off
• Computers use binary numbers to represent data (1 = on, 0 = off)

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

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

Why is hexadecimal used?

• In Computer Science hexadecimal is often preferred when working with large values
• It takes fewer digits to represent a given value in hexadecimal than in binary
• It is beneficial to use hexadecimal over binary because:
  • The more bits there are in a binary number, the harder it is to read
  • Numbers with more bits are more prone to errors when being copied
• Examples of where hexadecimal can be seen:
  • MAC addresses

• • Colour values