Number Systems

This lesson covers the three number systems you need for OCR A-Level Computer Science (H446): binary (base 2), denary (base 10), and hexadecimal (base 16). You must be confident converting between all three and understanding where each is used in computing.

Why Different Number Systems?

Computers operate using electrical signals that are either on (1) or off (0). This makes binary the natural language of hardware. However, binary numbers quickly become very long and hard to read, so hexadecimal provides a compact shorthand. Denary is the number system humans use every day.

System	Base	Digits Used	Example
Binary	2	0, 1	11010110
Denary	10	0-9	214
Hexadecimal	16	0-9, A-F	D6

Binary (Base 2)

Each digit in a binary number is called a bit (binary digit). Each position represents a power of 2, starting from the right at 2^0.

Place Values (8-bit)

128	64	32	16	8	4	2	1
2^7	2^6	2^5	2^4	2^3	2^2	2^1	2^0

Key Terms

Term	Definition
Bit	A single binary digit (0 or 1)
Nibble	4 bits
Byte	8 bits
Word	The number of bits a processor can handle in one operation (commonly 32 or 64 bits)

Denary to Binary Conversion

Use the subtraction method: write out powers of 2 from the largest that fits, subtract from the denary number, place a 1 in that column, and repeat.

Worked Example: Convert 173 to Binary

128	64	32	16	8	4	2	1
1	0	1	0	1	1	0	1

173 - 128 = 45 (place 1 under 128)
45 - 32 = 13 (place 1 under 32)
13 - 8 = 5 (place 1 under 8)
5 - 4 = 1 (place 1 under 4)
1 - 1 = 0 (place 1 under 1)

173 in denary = 10101101 in binary

Binary to Denary Conversion

Add up the place values where there is a 1.

Worked Example: Convert 11001010 to Denary

128	64	32	16	8	4	2	1
1	1	0	0	1	0	1	0

128 + 64 + 8 + 2 = 202

Hexadecimal (Base 16)

Hexadecimal uses 16 digits. Since we only have 10 numerical digits (0-9), the letters A-F represent values 10-15.

Hex	Denary	Binary (nibble)
0	0	0000
1	1	0001
2	2	0010
3	3	0011
4	4	0100
5	5	0101
6	6	0110
7	7	0111
8	8	1000
9	9	1001
A	10	1010
B	11	1011
C	12	1100
D	13	1101
E	14	1110
F	15	1111

Binary to Hexadecimal Conversion

Split the binary number into nibbles (groups of 4 bits) starting from the right, then convert each nibble to its hex digit.

Worked Example: Convert 11010110 to Hex

Split: 1101 | 0110
1101 = D, 0110 = 6

11010110 in binary = D6 in hexadecimal

Hexadecimal to Binary Conversion

Convert each hex digit to its 4-bit binary equivalent.

Worked Example: Convert 3F to Binary

3 = 0011, F = 1111

3F in hexadecimal = 00111111 in binary

Hexadecimal to Denary Conversion

Multiply each hex digit by its place value (powers of 16) and add.

Worked Example: Convert A7 to Denary

A = 10, position value 16^1 = 16
7 = 7, position value 16^0 = 1
(10 x 16) + (7 x 1) = 160 + 7 = 167

Denary to Hexadecimal Conversion

Divide by 16 repeatedly, noting remainders.

Worked Example: Convert 254 to Hex

254 / 16 = 15 remainder 14
15 = F, 14 = E

254 in denary = FE in hexadecimal

Uses of Hexadecimal in Computing

Use Case	Example	Why Hex?
Colour codes	#FF5733 (RGB)	Two hex digits per colour channel (red, green, blue), representing 0-255
MAC addresses	00:1A:2B:3C:4D:5E	Six pairs of hex digits identify network hardware
Memory addresses	0x7FFF0000	Compact representation of large binary addresses
Error codes	0x80070005	Easier to read and remember than binary
Assembly/machine code	MOV AX, 0xFF	Compact representation of binary instructions
IPv6 addresses	2001:0db8:85a3::8a2e	Hexadecimal groups separated by colons

Exam Tip: When asked why hexadecimal is used instead of binary, always state that hex is a more compact and human-readable representation of binary data. Each hex digit represents exactly 4 bits (one nibble), making conversion between the two trivial.

Common Conversion Mistakes

Mistake	How to Avoid
Forgetting to pad nibbles with leading zeros	Always write 4 bits per nibble (e.g., 3 = 0011, not 11)
Mixing up hex digits A-F with denary	Remember A=10, B=11, C=12, D=13, E=14, F=15
Reading binary place values left to right as 1, 2, 4...	Place values go right to left: 1, 2, 4, 8, 16, 32, 64, 128
Forgetting that hex is base 16, not base 8	The place values are 16^0=1, 16^1=16, 16^2=256, etc.

Summary

Binary (base 2) uses digits 0 and 1 and is the language of computers.
Denary (base 10) is the everyday human number system.
Hexadecimal (base 16) uses 0-9 and A-F and provides a compact way to represent binary.
You must be able to convert between all three systems fluently.
Hex is widely used for colour codes, MAC addresses, memory addresses, and error codes.

Exam Tip: Practise conversions until they are second nature. In the exam, show your working clearly — you can earn method marks even if the final answer is wrong. Always double-check by converting your answer back to the original base.