Number Systems

Data Representation · 4 question types

Practise

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Exam Frequency Analysis

Past paper frequency (2018 to 2024)

This topic accounts for approximately 12% of your exam marks.

stable

High

Stable12%

Binary/hex conversion and binary arithmetic appear in every Paper 1. Consistently 8 to 15 marks.

Computer science uses three number systems. The key idea is the base: the number of distinct digits the system uses.

System	Base	Digits used	Example
Denary (decimal)	10	0 to 9	3268
Binary	2	0 and 1 only	1100
Hexadecimal	16	0 to 9 plus A to F	9C

Denary (base 10)

Denary is the everyday number system used by humans. Each digit's value depends on the column it is in. Reading right to left, the columns are powers of 10: 1, 10, 100, 1000 and so on.

For example, the number 3268 breaks down as:

(3 × 1000) + (2 × 100) + (6 × 10) + (8 × 1) = 3268

A place-value table for the denary number 3268, with column headings 1000, 100, 10 and 1 (powers of ten, 10³ to 10⁰) and the digits 3, 2, 6, 8 written underneath each column — Source: Denary Place Value by Save My Exams

Binary (base 2)

Binary uses only two digits, 0 and 1. Each digit is called a bit (short for binary digit). Each column is a power of 2, doubling as you move left: 1, 2, 4, 8, 16, 32, 64, 128 and so on.

For example, the binary value 1100 breaks down as:

(1 × 8) + (1 × 4) + (0 × 2) + (0 × 1) = 12

A place-value table for the binary number 1100, with column headings 8, 4, 2 and 1 (powers of two, 2³ to 2⁰) and the bits 1, 1, 0, 0 written underneath, giving 8 + 4 = 12 in denary — Source: Binary Place Value by Save My Exams

The 8-bit place values you must memorise:

128	64	32	16	8	4	2	1

For 16-bit binary, the columns continue further: 32 768, 16 384, 8 192, 4 096, 2 048, 1 024, 512, 256, 128, 64, 32, 16, 8, 4, 2, 1. The largest denary number that can be stored in 16 bits is 65 535 (binary 1111 1111 1111 1111).

Hexadecimal (base 16)

Hexadecimal needs 16 different symbols, but only ten digits exist (0 to 9), so the next six values are written as letters:

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

Each column is a power of 16: 1, 16, 256, 4096 and so on. For example, the hexadecimal value 13 in 2-digit form means:

(1 × 16) + (3 × 1) = 19 in denary

A place-value table for a hexadecimal number, with column headings 4096, 256, 16 and 1 (powers of sixteen, 16³ to 16⁰) and the hex digits 0, 0, 9, 2 written underneath each column — Source: Hex Place Value by Save My Exams

The key relationship: one hex digit = one nibble

A group of 4 binary bits is called a (half a byte). The largest 4-bit binary value is 1111, which is 15 in denary, which is F in hexadecimal. So one hex digit covers exactly the same range as one nibble of . This is what makes binary-to-hex conversion fast: split the binary into nibbles from the right, convert each nibble to one hex digit.