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Types of number

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Cambridge O Level Mathematics (Syllabus D) 4024

Types of number

The big idea: Every number belongs to a family: natural numbers, integers, primes, squares, cubes, rationals and irrationals. Knowing the family tells you what you can do with it, and prime factors are the key that unlocks HCF and LCM.

Summary — What This Chapter Covers

Natural numbers are 1, 2, 3, …; integers are the whole numbers, positive, zero and negative.
Convert between numbers written in figures and in words (six billion is 6 000 000 000).
Recognise prime, square and cube numbers, and write a number as a product of its prime factors.
Find common factors and common multiples, including the HCF and the LCM of two numbers.
Tell rational from irrational numbers, and find the reciprocal of a number.

1. Natural Numbers, Integers and Numbers in Words

Syllabus 1.1 · Paper 1 & 2

Counting numbers and the whole numbers either side of zero

Natural numbers are the counting numbers 1, 2, 3, 4, … Integers are all the whole numbers: positive, zero and negative (…, −2, −1, 0, 1, 2, …). Every natural number is an integer, but −5 and 0 are integers that are not natural numbers.

To write a large number in figures, group the digits in threes from the right: units, thousands, millions, billions. One billion is 1 000 000 000 (nine zeros).

Words	Figures
six billion	6 000 000 000
ten thousand and seven	10 007
four million, two hundred thousand	4 200 000

Worked Example

Write 3 040 506 in words.

Practice Question 1

(a) Write two million, three hundred and five thousand and eighty in figures.
(b) From the list −4, 0, 7, 2.5, 13, write down the natural numbers.

2. Prime, Square and Cube Numbers

Syllabus 1.1 · Paper 1 & 2

Three families you must recognise on sight

A prime number has exactly two factors, 1 and itself. A square number is an integer multiplied by itself; a cube number is an integer multiplied by itself twice.

Family	First members
Primes	2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47
Squares	1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225
Cubes	1, 8, 27, 64, 125, … , 1000 (= 10³)

Facts that get tested

1 is not prime (it has only one factor). 2 is the only even prime. 1 and 64 are both square and cube numbers.

Practice Question 2

From the list 1, 9, 27, 29, 36, 51, 64, write down
(a) a prime number (b) all the square numbers (c) all the cube numbers.

3. Products of Prime Factors

Syllabus 1.1 · Paper 1 & 2

Every integer above 1 breaks into primes in one way

Divide by primes repeatedly (or draw a factor tree) until only primes remain. Write the answer as a product, using index notation for repeated primes, and check by multiplying back.

Worked Example

Express 72 as a product of its prime factors.

Practice Question 3

Express 540 as a product of its prime factors. Give your answer in index form.

4. Common Factors, Common Multiples, HCF and LCM

Syllabus 1.1 · Paper 1 & 2

Use the prime factors of both numbers

common factor highest common factor (HCF) common multiple lowest common multiple (LCM)

From prime factors both lower every

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Also in the full note

Can it be written as a fraction of two integers?
One divided by the number

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Types of number

Types of number

Counting numbers and the whole numbers either side of zero

Three families you must recognise on sight

Every integer above 1 breaks into primes in one way

Use the prime factors of both numbers

Past papers that test Types of number

More Mathematics D topics