Types of Number & Sets · Notes

1.Lesson overview

Syllabus focus

Cambridge IGCSE syllabus reference

1.1 Types of number
1.2 Sets

Edexcel IGCSE syllabus reference

1.1 Integers
1.4 Powers and roots
1.5 Set language and notation

AQA IGCSE syllabus reference

1.1 Structure and calculation

Number types are determined by definitions, and one value may belong to several nested sets. In set problems, place each element by testing intersection, union and complement conditions rather than by appearance.

By the end of this lesson

Identify and use:
• natural numbers
• integers (positive, zero and negative)
• prime numbers
• square numbers
• cube numbers
• common factors
• common multiples
• rational and irrational numbers
• reciprocals.
Understand and use set language, notation and Venn diagrams to describe sets and represent relationships between sets.

2.Learning map

Move from classifying a number to proving its classification, then use disjoint Venn regions to count without double-counting. The map below separates number structure from set structure before reconnecting them in problems.

Number classification map

Definitions and set membership

This is the foundation lesson for Number; later probability work reuses set notation and Venn diagrams. Big idea. A classification tells you what a number is; a set diagram tells you which objects belong to which group.

Natural Numbers & Integers

Natural Numbers $N$ : The counting numbers: 1, 2, 3, 4, 5, … Some definitions include 0.

Prime Numbers

Prime Number: A number greater than 1 that has exactly two factors : 1 and itself. Examples: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, …

Squares & Cubes

Square Number: $n^{2}$ - the result of multiplying a number by itself. For example, $7^{2} = 49$ .

3.Core idea

This is the foundation lesson for Number; later probability work reuses set notation and Venn diagrams.

Big idea. A classification tells you what a number is; a set diagram tells you which objects belong to which group.

Key relationship

n (A \cup B) = n (A) + n (B) - n (A \cap B)

Worked example

In a group,

n (A) = 18

n (B) = 14

and

n (A \cap B) = 6

. Find

n (A \cup B)

1
$n (A \cup B) = n (A) + n (B) - n (A \cap B)$
2
$n (A \cup B) = 18 + 14 - 6 = 26$

4.Natural Numbers & Integers

Key Definitions

Natural Numbers $N$: The counting numbers: 1, 2, 3, 4, 5, … Some definitions include 0.
Integers $Z$: All whole numbers, including negatives: …, -3, -2, -1, 0, 1, 2, 3, …
Positive Integers $Z^{+}$: 1, 2, 3, 4, … (same as natural numbers)
Negative Integers $Z^{-}$: -1, -2, -3, -4, …

Converting Between Numbers and Words

6,000,456,218 = "six billion, four hundred and fifty-six thousand, two hundred and eighteen"

10,007 = "ten thousand and seven".

Classify numbers into natural, integer and other setsOpen full screen

5.Prime Numbers

Definition

Prime Number: A number greater than 1 that has exactly two factors: 1 and itself.
Examples: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, …

Common Mistakes

1 is NOT prime (it has only one factor).

2 is the only even prime.

6.Squares & Cubes

Squares (

n^{2}

) and Cubes (

n^{3}

) to Know

n	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
$n^{2}$	1	4	9	16	25	36	49	64	81	100	121	144	169	196	225
$n^{3}$	1	8	27	64	125	216	343	512	729	1000	1331	1728	2197	2744	3375

You must memorise squares from

1^{2}

1 5^{2}

and cubes of 1, 2, 3, 4, 5, and 10.

Square & Cube Numbers

Square Number: $n^{2}$ — the result of multiplying a number by itself. For example, $7^{2} = 49$ .
Cube Number: $n^{3}$ — the result of multiplying a number by itself twice. E.g. $4^{3} = 64$
Square Root: $n$ — reverse of squaring. E.g. $144 = 12$
Cube Root: $3 n$ — reverse of cubing. E.g. $3125 = 5$

7.Factors & Multiples

Definitions

Factor: A number that divides exactly into another number. E.g. factors of 12 are: 1, 2, 3, 4, 6, 12
Multiple: The result of multiplying a number by an integer. E.g. multiples of 7 are 7, 14, 21, 28, …
Common Factor: A factor shared by two or more numbers
Common Multiple: A multiple shared by two or more numbers

Finding All Factors

Find all factors of 36:

1
$1 \times 36 = 36$
2
$2 \times 18 = 36$
3
$3 \times 12 = 36$
4
$4 \times 9 = 36$
5
$6 \times 6 = 36$
6
Factors of 36: $1, 2, 3, 4, 6, 9, 12, 18, 36$

8.HCF & LCM

HCF & LCM

HCF (Highest Common Factor): The largest number that divides into both numbers. Also called GCD.
LCM (Lowest Common Multiple): The smallest number that both numbers divide into.

Prime factors show what HCF and LCM select

1
12
$2^{2} \times 3$ · factor completely
2
18
$2 \times 3^{2}$ · factor completely
3
HCF
shared minimum powers · $2 \times 3 = 6$
4
LCM
all maximum powers · $2^{2} \times 3^{2} = 36$

HCF uses factors common to both; LCM includes enough prime factors to build both numbers.

Worked Example: Using Prime Factor Trees

Find the HCF and LCM of 72 and 120.

1
Prime factorise: $72 = 2^{3} \times 3^{2}, 120 = 2^{3} \times 3 \times 5$
2
$HCF = 2^{3} \times 3 = 24$
3
$LCM = 2^{3} \times 3^{2} \times 5 = 360$

Quick Check

$HCF \times LCM = product of the two numbers$ .

$24 \times 360 = 8640$ , and $72 \times 120 = 8640$ .

Planned interactiverecommended

Prime Factors, HCF and LCM Explorer

Split numbers into prime factors and combine shared and unshared powers.

Visualise factor trees, divisibility and HCF/LCM rather than rely on memorised recipes.

Inputs & controls

two or three integers
factor split
method selector

Learner actions

build factor trees
place factors
verify products

Visible outputs

prime factorisation
factor Venn or exponent table
HCF
LCM
product check

Implementation note

The catalogue number-theory entry has no asset; keep planned and use modest IGCSE-sized integers.

9.Rational & Irrational Numbers

Definitions

Rational Number ( $Q$ ): A number that can be written as a fraction $\frac{a}{b}$ where $a$ , $b$ are integers and $b \neq = 0$ . Includes all terminating and recurring decimals.
Examples: $\frac{3}{4}$ , -2, 0.6, 0. $\overline{3}$
Irrational Number ( $I$ ): A number that cannot be written as a fraction. The decimal never terminates or repeats.
Examples: $π$ , $2$ , $3$ , $5$

Not All Square Roots Are Irrational!

$4 = 2$ (rational), $9 = 3$ (rational). Only square roots of non-perfect-squares are irrational.

A real length that is irrational

Irrational numbers are not just abstract — $2$ is a length you could draw. It is the diagonal of a square with side $1$ : by Pythagoras, the diagonal $d$ satisfies $d^{2} = 1^{2} + 1^{2} = 2$ , so $d = 2$ . A perfectly real, measurable line segment whose exact length can never be written as a fraction, however many decimal places you use — this is essentially how the ancient Greeks first discovered that irrational numbers had to exist.

Reciprocal

Reciprocal: The reciprocal of $n$ is $\frac{1}{n}$ . The reciprocal of $\frac{a}{b}$ is $\frac{b}{a}$ . A number times its reciprocal is 1.
Example: the reciprocal of 5 is $\frac{1}{5}$ ; the reciprocal of $\frac{2}{3}$ is $\frac{3}{2}$ .

10.Challenge Questions

Classify numbers and apply your knowledge.

Challenge 1 — the imposter prime

Is 91 a prime number? Show your reasoning.

1
91 is odd, and it is not a multiple of 3 ( $9 + 1 = 10$ ) or 5, so a quick glance might suggest it is prime.
2
But testing the next prime, 7: $91 \div 7 = 13$ exactly.
3
So $91 = 7 \times 13$ , which has more than two factors — 91 is not prime.

Lesson: test divisibility by every prime up to $n$ (here, up to $91 \approx 9.5$ , so 2, 3, 5 and 7) before concluding a number is prime.

Challenge 2 — working backwards from HCF and LCM

Two numbers have HCF 8 and LCM 240. One of the numbers is 48. Find the other.

1
Use $HCF \times LCM = product of the two numbers$ : $8 \times 240 = 1920$ .
2
Other number $= 1920 \div 48 = 40$ .
3
Check: $48 = 2^{4} \times 3$ and $40 = 2^{3} \times 5$ , so $HCF = 2^{3} = 8$ ✓ and $LCM = 2^{4} \times 3 \times 5 = 240$ ✓.

Challenge 3 — factor and multiple together

A number is both a factor of 60 and a multiple of 4. List every number it could be.

1
Factors of 60: $1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60$ .
2
From that list, keep only the multiples of 4: $4, 12, 20, 60$ .

11.Set Notation

Essential Set Notation

Symbol	Meaning	Example
{ } or $\emptyset$	Empty set (no elements)	Set of even primes greater than 2:
$E$	Universal set	All elements under consideration
$\in$	Is a member of	$3 \in {1, 2, 3}$
$\in /$	Is not a member of	$4 \in / {1, 2, 3}$
n(A)	Number of elements in A	$n (a, b, c) = 3$
$A \cup B$	Union: elements in A or B (or both)	${1, 2} \cup {2, 3} = {1, 2, 3}$
$A \cap B$	Intersection: elements in A and B	${1, 2, 3} \cap {2, 3, 4} = {2, 3}$
A'	Complement: elements NOT in A	If $E = {1, 2, 3, 4, 5}$ and $A = {1, 2}$ , then $A^{'} = {3, 4, 5}$ .
$A \subseteq B$	A is a subset of B	${1, 2} \subseteq {1, 2, 3}$
A $\subset B$	A is a proper subset of B	${1, 2} \subset {1, 2, 3}$

Union vs Intersection

Think:

\cup

looks like a cup (holds everything) = OR.

\cap

= AND (the overlap).

12.Describing Sets

Sets can be described by:

Listing

A = {2, 4, 6, 8, 10}

— the set of even numbers from 1 to 10.

Set Builder Notation

A = {x : x (is an even number), 1 < x \leq 10}

— Read as: "A is the set of all x such that x is an even number, where x is greater than 1 and at most 10."

Example

1
$B = {x : x (is a prime number), x < 20}$
2
Listing: $B = {2, 3, 5, 7, 11, 13, 17, 19}$
3
$n (B) = 8$ ,

13.Venn Diagrams

Venn diagrams show sets as overlapping circles within a rectangle (the universal set

E

Key Venn Diagram Formula

n (A \cup B) = n (A) + n (B) - n (A \cap B)

Worked Example

In a class of 30 students, 18 study French, 15 study Spanish, and 8 study both. Find: $n (F \cap S)$ , $n (F \cup S)$ , $Number studying neither$

1
$n (F \cap S) = 8$ (given)
2
$n (F \cup S) = 18 + 15 - 8 = 25$
3
$Neither = 30 - 25 = 5$

Explore membership in two-set Venn diagramsOpen full screen

14.Shading Venn Diagrams

You must be able to identify and shade regions in Venn diagrams.

Three rectangles, each with two overlapping circles A and B. The first shades only the overlap (A ∩ B), the second shades both circles (A ∪ B), and the third shades everything outside circle A (A′). — Figure 1: Match each symbol to its shading: ∩ is the overlap, ∪ is everything inside either circle, and ′ is everything outside.

Planned interactivecore

Sets, Venn Diagrams and Set Notation

Translate membership and expressions into two- and three-set regions.

Connect union, intersection, complement, subset and cardinality to the picture.

Inputs & controls

two or three sets
draggable elements
set expression
region shading

Learner actions

move elements
shade region
enter notation
check

Visible outputs

membership
shaded region
region cardinalities
expression match
reasoned feedback

Implementation note

Use patterns and labels as well as colour; include universe and neither regions.

Match shaded Venn regions to set notationOpen full screen

15.Three-Set Problems

A rectangle for 30 students with three overlapping circles A, B and C. The seven regions inside hold 4, 6, 3, 5, 2, 4 and 3 students, and 3 students sit outside all three circles. — Figure 2: The regions never overlap, so adding those inside a circle counts each student once, for example $n(A) = 4 + 5 + 2 + 3 = 14$.

Three sets create eight distinct membership regions

1
Single sets
A only, B only, C only · 3 regions
2
Pair overlaps
A∩B only · A∩C only, B∩C only
3
Triple overlap
A∩B∩C · belongs to all three
4
Outside
in universal set but in none

Label every region before placing values; “only” excludes the third set.

Worked Example (3 Sets)

In a school of 100 students, 50 play football, 35 play cricket, and 30 play tennis. 15 play Football and Cricket, 10 play Football and Tennis, 12 play Cricket and Tennis, and 5 play all three.

Find how many students play none of the three sports.

1
Start from the centre: $n (F \cap C \cap T) = 5$
2
$F \cap C = 15 - 5 = 10$
$F \cap T = 10 - 5 = 5$

$C \cap T = 12 - 5 = 7$
3
$F = 50 - 10 - 5 - 5 = 30$
$C = 35 - 10 - 7 - 5 = 13$

$T = 30 - 5 - 7 - 5 = 13$
4
$T o t a l in s i d e = 30 + 13 + 13 + 10 + 5 + 7 + 5 = 83$
5
$None = 100 - 83$ = 17

Always Start From the Centre!

When filling in a 3-set Venn diagram, always start with the triple intersection, then work outwards through the pair intersections, then the "only" regions.

16.Subsets

Subsets

Subset $A \subseteq B$: Every element of A is also in B. A can equal B.
Proper Subset $A \subset B$: Every element of A is in B, but $A \neq = B$ (B has extra elements).

Number of Subsets

A set with n elements has 2^{n} subsets

Example

1
$A = 1, 2, 3$
2
Subsets: $\emptyset$ , ${1}$ , ${2}$ , ${3}$ , ${1, 2}$ , ${1, 3}$ , ${2, 3}$ , ${1, 2, 3}$
3
Total: $2^{3} = 8$ subsets

17.Set Language, Notation and Venn Diagrams

Sets and elements

A set is a collection of distinct objects (called elements). Sets are written using { } braces.

Example: $A = 2, 4, 6, 8$ — the set of the first four even numbers.

Example — using set builder

1
$A = {x : x is a natural number and x < 6} = {1, 2, 3, 4, 5}$ .
2
$B = {x : x is an even number and x < 10} = {2, 4, 6, 8}$ .
3
$A \cap B = {2, 4}$ , $A \cup B = {1, 2, 3, 4, 5, 6, 8}$ , $n (A \cup B) = 7$ .

Venn diagrams

Use overlapping circles inside a rectangle (the universal set). Shade regions to represent unions, intersections and complements.

Exam tip

When you see

\cup

think "OR" (more elements). When you see

\cap

think "AND" (fewer elements).

18.Sets — Challenge Questions

Three stepping-stone questions on set notation — from a direct application of the counting formula to spotting a shortcut for an intersection of multiples.

Challenge 1 — working from the outside in

In a survey of 50 people, 28 like tea, 30 like coffee, and 5 like neither. How many like both?

1
If 5 like neither, then $n (T \cup C) = 50 - 5 = 45$ .
2
$n (T \cap C) = n (T) + n (C) - n (T \cup C) = 28 + 30 - 45 = 13$ .

The formula was rearranged rather than applied directly — recognising which piece is missing is the real skill being tested.

Challenge 2 — counting an intersection without listing it

$E$ is the numbers 1 to 40. $A = {multiples of 4}$ and $B = {multiples of 6}$ . Find $n (A \cap B)$ without listing every element of $A$ and $B$ .

1
$A \cap B$ is the set of numbers that are multiples of both 4 and 6 — that is, multiples of $LCM (4, 6) = 12$ .
2
Multiples of 12 up to 40: $12, 24, 36$ .
3
$n (A \cap B) = 3$ .

This shortcut — the intersection of two sets of multiples is the set of multiples of their LCM — turns a listing question into a one-line calculation.

Challenge 3 — combining two classifications

$E = {1, 2, \dots, 20}$ . $P = {prime numbers}$ and $E v = {even numbers}$ . Find $n (P \cap E v)$ and $n ((P \cup E v)^{'})$ .

1
$P = {2, 3, 5, 7, 11, 13, 17, 19}$ , so $n (P) = 8$ . $E v = {2, 4, 6, \dots, 20}$ , so $n (E v) = 10$ .
2
$P \cap E v = {2}$ — the only even prime — so $n (P \cap E v) = 1$ .
3
$n (P \cup E v) = 8 + 10 - 1 = 17$ , so $n ((P \cup E v)^{'}) = 20 - 17 = 3$ .
4
Those 3 elements are the odd non-primes in the set: ${1, 9, 15}$ .

19.Exam-Style Worked Examples

Finding the HCF and LCM

1
Write as prime factors
Express each number as a product of prime factors, using index form.
2
For the HCF
Take the lowest power of each prime that appears in both.
3
For the LCM
Take the highest power of every prime that appears in either.
4
Check
$HCF \times LCM$ should equal the product of the two numbers.

Worked example 1 — HCF and LCM (4 marks)

Question. Find the HCF and LCM of 84 and 120, showing your method.

1
Write each as a product of prime factors: $84 = 2^{2} \times 3 \times 7$ and $120 = 2^{3} \times 3 \times 5$ .
2
HCF — take the lowest power of each common prime: $2^{2} \times 3 = 12$ .
3
LCM — take the highest power of every prime that appears: $2^{3} \times 3 \times 5 \times 7 = 840$ .
4
Check: $HCF \times LCM = 12 \times 840 = 10080 = 84 \times 120$ , as it must be.

Marking. 1 mark per point. The prime factorisation earns a mark on its own, so write it out even if you can spot the answer.

Worked example 2 — set notation (4 marks)

Question. $E = {1, 2, 3, \dots, 12}$ , $A = {multiples of 3}$ and $B = {even numbers}$ . List (a) $A \cap B$ , (b) $A \cup B$ , (c) $A^{'}$ , and (d) state $n (A \cap B)$ .

1
$A = {3, 6, 9, 12}$ and $B = {2, 4, 6, 8, 10, 12}$ .
2
(a) $A \cap B$ is what is in both: ${6, 12}$ .
3
(b) $A \cup B$ is what is in either: ${2, 3, 4, 6, 8, 9, 10, 12}$ — list each element once.
4
(c) $A^{'}$ is everything in $E$ not in $A$ : ${1, 2, 4, 5, 7, 8, 10, 11}$ . (d) $n (A \cap B) = 2$ .

Marking. 1 mark per point. $n ()$ asks for how many elements, not the elements themselves.

Worked example 3 — classifying numbers (3 marks)

Question. From the list $- 4, 0, \frac{3}{4}, 9, 2, π, 17$ , write down (a) the integers, (b) the irrational numbers, (c) the prime numbers.

1
(a) Integers: $- 4$ , $0$ , $9$ and $17$ — note $9 = 3$ , so it is an integer.
2
(b) Irrational: $2$ and $π$ — they cannot be written as a fraction and their decimals never terminate or repeat.
3
(c) Prime: $17$ only. $9 = 3$ is prime too if the simplified value is accepted; $0$ and $- 4$ are not prime.

Marking. 1 mark per point. Simplify surds first — $9$ looks irrational but is not.

20.Exam Tips & Common Misconceptions

Exam Tips

Read questions carefully — prime vs square vs cube numbers are commonly confused under time pressure.
For HCF and LCM, write each number as a product of primes first; circle common primes for HCF, list the highest power of every prime for LCM.
Set notation marks: $A \cup B =$ union ("or"); $A \cap B =$ intersection ("and"); $A^{'} =$ complement ("not in A").
On Venn diagrams, fill the intersection first, then work outwards — it's the most efficient strategy and prevents double-counting.
1 is not prime and 0 is not a natural number on the Cambridge syllabus — both are common trick answers.

Common Misconceptions

"2 is not prime because it's even" — 2 is prime; it's the only even prime number.
Writing HCF when LCM is asked — re-read the question; $H C F \leq e i t h er$ number, $L C M \geq e i t h er$ number.
Treating ∅ as 0 — the empty set has no elements; |∅| = 0 but ∅ ≠ 0.

21.Worked method — Classify each number in the smallest useful set

The sets are nested: every integer is rational and every rational is real; irrationals are real but not rational.

Transfer the method

Reasoning prompt. A question gives totals for two overlapping groups. How can you avoid counting learners in both groups twice?

1
Draw the universal set and the two circles before entering any numbers.
2
Place the intersection first, because it belongs to both sets.
3
Fill the non-overlapping parts next and total the required region only.

Self-check

The total of all disjoint regions must equal the stated size of the universal set.

22.Classification and counting without ambiguity

Boundary cases worth testing

Zero

Integer and rational

$0 = \frac{0}{1}$ , but it has no reciprocal because division by zero is undefined.

One

Neither prime nor composite

It has one positive factor, so it does not meet the definition of a prime number.

49

Rational

$49 = 7$ ; a square-root symbol does not automatically make a value irrational.

50

Irrational

$50 = 52$ , which cannot be expressed as a ratio of integers.

Worked set-counting example

Let the universal set be the natural numbers from 1 to 30. Set $A$ contains multiples of 2 and set $B$ contains multiples of 3. How many elements are in neither set?

$n (A) = 15$ and $n (B) = 10$ . The intersection contains the multiples of 6, so $n (A \cap B) = 5$ .

$n (A \cup B) = 15 + 10 - 5 = 20$ . Therefore the number in neither set is $30 - 20 = 10$ .

Why the subtraction is essential

The five multiples of 6 were included once in

n (A)

and once in

n (B)

. Subtracting the intersection once leaves each element counted exactly once.

23.Summary

Key Points

Natural numbers: 1, 2, 3, … Integers: …, −2, −1, 0, 1, 2, …
Prime: exactly 2 factors. 1 is NOT prime. 2 is the only even prime.
Memorise squares to $1 5^{2}$ and cubes of 1–5 and 10
HCF: lowest powers of common primes. LCM: highest powers of all primes
Rational: expressible as a/b. Irrational: non-terminating, non-repeating decimals
Reciprocal of $n = 1/ n$ . Number × $r ec i p r oc a l = 1$