Which of the following is a prime number?
Prime numbers have exactly 2 factors.
7 has only factors 1 and 7.
Types of Numbers
Numbers can be classified into different types such as natural numbers, integers, prime numbers and square numbers. Recognising these categories is essential for understanding number properties in GCSE Maths.
Overview
Not all numbers are the same. In maths, numbers are grouped into different types depending on their properties.
Examples: \( 4,\ -3,\ \frac{1}{2},\ \sqrt{2},\ 17,\ 25 \)
A single number can belong to more than one group.
For example, \(5\) is a natural number, a whole number, an integer, and also a prime number.
What you should understand after this topic
- Understand how common number sets are defined
- Understand how one number can belong to several types
- Understand the difference between rational and irrational numbers
- Recognise primes, squares, cubes and multiples
- Understand how classification questions appear in exams
Key Definitions
Natural Numbers
Counting numbers: \(1, 2, 3, 4, \dots\).
Whole Numbers
\(0\) and the positive integers: \(0, 1, 2, 3, \dots\).
Integers
Negative and positive whole numbers, including \(0\).
Prime Numbers
Numbers greater than 1 with exactly two factors.
Square Numbers
Numbers of the form \(n^2\), such as \(1, 4, 9, 16\).
Cube Numbers
Numbers of the form \(n^3\), such as \(1, 8, 27, 64\).
Rational Numbers
Numbers that can be written as a fraction \(\frac{a}{b}\), where \(b \neq 0\).
Irrational Numbers
Numbers that cannot be written exactly as a fraction, such as \(\sqrt{2}\) or \(\pi\).
Even Numbers
Numbers divisible by 2.
Odd Numbers
Numbers not divisible by 2.
Key Rules
Prime numbers
Have exactly 2 factors.
Even numbers
End in \(0, 2, 4, 6,\) or \(8\).
Rational numbers
Terminate or recur as decimals.
Irrational numbers
Never terminate or recur.
Quick Reminder
\(2\) is prime
It is the only even prime number.
\(1\) is not prime
It only has one factor.
Square roots are not always irrational
\( \sqrt{9} = 3 \), which is rational.
Fractions are rational
\( \frac{3}{4} \), \( -\frac{5}{2} \), and \( 7 \) are all rational.
How to Solve
Step 1: Understand number types
Numbers can belong to different groups depending on their properties.
Exam tip: One number can belong to more than one group.
Step 2: Natural numbers, whole numbers and integers
Natural numbers
\(1, 2, 3, 4, \dots\)
Whole numbers
\(0, 1, 2, 3, \dots\)
Integers
\(\dots, -2, -1, 0, 1, 2, \dots\)
Step 3: Prime and composite numbers
Prime numbers have exactly two factors: 1 and themselves.
\(2, 3, 5, 7, 11, 13, 17, \dots\)
\(2\) is the only even prime number.
\(1\) is not prime.
Composite numbers have more than two factors.
Step 4: Square and cube numbers
Square numbers
\(1, 4, 9, 16, 25, 36, \dots\)
Cube numbers
\(1, 8, 27, 64, 125, \dots\)
Step 5: Even and odd numbers
Even numbers
Divisible by 2.
Odd numbers
Not divisible by 2.
Step 6: Rational and irrational numbers
Rational: \(5, -2, \frac{3}{4}, 0.25, 0.\dot{3}\)
Irrational: \(\sqrt{2}, \sqrt{3}, \pi\)
Rational numbers
Can be written exactly as a fraction.
Irrational numbers
Cannot be written exactly as a fraction.
Step 7: Classify numbers carefully
Example: \(9\) is natural, whole, integer, square and rational.
Exam thinking: Always simplify first if the number is written as a root or expression.
Step 8: Exam method summary
See powers and roots for square and cube numbers.
- Simplify the number if possible.
- Check if it is whole, integer, even or odd.
- Check if it is prime, square or cube.
- Decide whether it is rational or irrational.
- List all groups that apply.
Example Questions
Edexcel
Exam-style questions inspired by Edexcel GCSE Mathematics.
Edexcel
State whether each of the following numbers is prime or composite: 11, 15, 17.
Edexcel
Write down the first five square numbers.
Edexcel
Write down the first three cube numbers.
Edexcel
Which of the following numbers are multiples of 3? 12, 14, 18, 25
Edexcel
Which of the following numbers are factors of \( 36 \)? 3, 5, 6, 9
AQA
Exam-style questions based on the AQA GCSE Mathematics specification, focusing on classification and reasoning.
AQA
State whether each number is rational or irrational: \( \sqrt{2} \), 0.75, \( \pi \), 1.2.
AQA
Which of the following numbers are integers? \( -3,\; 0,\; 4.5,\; 7 \)
AQA
Which of the following numbers are natural numbers? \( 1,\; 0,\; 5,\; -2 \)
AQA
Explain why \( \sqrt{49} \) is rational but \( \sqrt{5} \) is irrational.
AQA
Give an example of a number that is both a square number and a cube number.
OCR
Exam-style questions aligned with OCR GCSE Mathematics, emphasising reasoning and number classification using Venn diagrams.
OCR
List all the prime numbers less than 20.
OCR
Place the numbers \( -4,\; 0,\; 3,\; 0.5,\; \sqrt{3} \) into the correct sets: integers, rational numbers, and irrational numbers.
OCR
Write down all the factors of 28.
OCR
Which of the following are real numbers? \( \sqrt{9},\; -5,\; \frac{2}{3},\; \pi \)
OCR
Explain why every prime number greater than 2 is odd.
Exam Checklist
Step 1
Check whether the number is positive, negative, whole or decimal.
Step 2
Test special properties such as prime, square, cube, even or odd.
Step 3
Decide whether it is rational or irrational.
Step 4
Remember that one number may belong to more than one type.
Most common exam mistakes
Prime mistake
Saying that 1 is prime.
Rational mistake
Forgetting that integers can be written as fractions.
Root mistake
Thinking every square root is irrational.
Classification mistake
Giving only one type when a number fits several groups.
Common Mistakes
These are common mistakes students make when classifying numbers in GCSE Maths.
Thinking 1 is prime
Incorrect
A student includes 1 as a prime number.
Correct
A prime number has exactly two factors. The number 1 has only one factor, so it is not prime.
Forgetting that 2 is prime
Incorrect
A student assumes all prime numbers are odd.
Correct
2 is the only even prime number because it has exactly two factors: 1 and 2.
Assuming all square roots are irrational
Incorrect
A student treats every square root as irrational.
Correct
Square roots of perfect squares are integers. For example, \(\sqrt{16} = 4\), which is rational.
Forgetting integers are rational
Incorrect
A student separates integers from rational numbers.
Correct
All integers are rational numbers because they can be written as fractions, such as \(3 = \frac{3}{1}\).
Assigning only one category
Incorrect
A student places a number into just one group.
Correct
Numbers can belong to multiple sets. For example, 4 is natural, whole, integer, and rational.
Try It Yourself
Practise identifying different types of numbers.
Foundation
Foundation Practice
Identify basic types of numbers such as integers, primes and squares.
Is 13 a prime number? (yes/no)
Check its factors.
13 is prime.
Which number is a square number?
Think of squares.
4ยฒ = 16.
Write the square of 6.
Multiply 6 by itself.
6ยฒ = 36.
Which number is an integer?
Integers are whole numbers including negatives.
-3 is an integer.
Is 0 an integer? (yes/no)
Integers include zero.
Yes, 0 is an integer.
Which number is a multiple of 4?
Divide by 4.
12 รท 4 = 3.
Is 15 a multiple of 5? (yes/no)
Check division.
15 รท 5 = 3.
Which number is NOT a whole number?
Whole numbers have no decimals.
2.5 is not whole.
Is 25 a square number? (yes/no)
5 ร 5.
25 = 5ยฒ.
Higher
Higher Practice
Work with rational and irrational numbers and classify numbers correctly.
Which number is irrational?
Irrational numbers cannot be written as fractions.
โ2 is irrational.
Is ฯ rational or irrational?
ฯ is non-terminating and non-repeating.
ฯ is irrational.
Which number is rational?
Rational numbers can be written as fractions.
0.25 = 1/4.
Is 0.333... rational or irrational?
Recurring decimals are fractions.
0.333... = 1/3.
Which is a cube number?
Think 3 ร 3 ร 3.
3ยณ = 27.
Write the cube of 4.
Multiply 4 three times.
4ยณ = 64.
Which number is both a square and a cube?
Think 8ยฒ and 4ยณ.
64 = 8ยฒ and 4ยณ.
Is โ9 rational or irrational?
Simplify first.
โ9 = 3, which is rational.
A student says โ4 is irrational. What is wrong?
Simplify square roots.
โ4 = 2, which is rational.
Is 7/3 rational or irrational?
It is already a fraction.
All fractions are rational.
Games
Practise this topic with interactive games.
Games coming soon.
Types of Numbers Video Tutorial
Frequently Asked Questions
What are natural numbers?
Natural numbers are positive whole numbers starting from 1, such as 1, 2, 3 and so on.
What is the difference between integers and natural numbers?
Integers include negative numbers, zero and positive numbers, while natural numbers only include positive whole numbers.
How do I identify prime numbers quickly?
Prime numbers only have two factors: 1 and itself. You can test divisibility by smaller primes like 2, 3 and 5.