Types of numbers
The main types of numbers are natural numbers (1, 2, 3, ...), whole numbers (the natural numbers and 0), integers (whole numbers and their negatives, such as −4), rational numbers (any number that can be written as a fraction), irrational numbers (such as √2 and π, which cannot), and real numbers, which include all of these. Each set sits inside the next. At GCSE you also need the special kinds of number: even and odd, prime, square and cube numbers, factors and multiples.
The number sets
| Type | Symbol | What it contains | Examples |
|---|---|---|---|
| Natural numbers | ℕ | The counting numbers | 1, 2, 3, 4, ... |
| Whole numbers | Natural numbers and zero | 0, 1, 2, 3, ... | |
| Integers | ℤ | Whole numbers and their negatives | −3, −1, 0, 5, 12 |
| Rational numbers | ℚ | Anything that can be written as a/b, with a and b integers and b ≠ 0 | 3/4, −2, 0.5, 0.333... |
| Irrational numbers | Real numbers that cannot be written as a fraction | √2, √5, π | |
| Real numbers | ℝ | All rational and irrational numbers: every point on the number line | −2, 1/3, √2, π |
| Complex numbers (A-level) | ℂ | Numbers a + bi, where i = √−1 | 3 + 2i, −i, 5 |
The sets are nested: every natural number is an integer, every integer is rational (5 = 5/1), and every rational and irrational number is real. So 5 is natural, whole, an integer, rational and real all at once. Every real number is also a complex number, with b = 0.
Natural numbers and whole numbers
Natural numbers are the numbers you count with: 1, 2, 3, and so on forever. Some books include 0 as a natural number and some do not, so check what your course means. Whole numbers are the natural numbers with 0 included.
Integers
Integers are whole numbers that can be positive, negative or zero, with no fraction or decimal part, such as −8, 0 and 15. Zero is an integer that is neither positive nor negative. More in the integers guide.
Rational numbers
A rational number can be written as one integer divided by another, as long as the bottom is not zero. Every terminating decimal (0.25 = 1/4) and every recurring decimal (0.333... = 1/3) is rational. See rational numbers for more.
Example 1: a recurring decimal as a fraction
Show that 0.272727... is rational.
It equals the fraction 3/11, so it is rational. Changing recurring decimals into fractions is in the AQA GCSE Maths specification (N10).
Irrational numbers
An irrational number cannot be written as a fraction. Its decimal goes on forever without repeating. Famous examples are √2 = 1.41421..., π = 3.14159... and the square root of any whole number that is not a perfect square, such as √3, √5 and √7. Square roots left in this exact form are called surds.
Real and complex numbers
Real numbers are all the rational and irrational numbers together: every point on the number line. At A-level Further Maths you meet numbers that are not on the number line at all. The square of any real number is positive or zero, so no real number squares to −1. Mathematicians define i = √−1, and numbers such as 3 + 2i are complex numbers, plotted on an Argand diagram instead of a line.
Types of number at GCSE
AQA GCSE Maths also expects you to use the vocabulary of prime numbers, factors and multiples, and to recognise powers and roots. These are the special types of number to know:
| Type | Meaning | First few |
|---|---|---|
| Even numbers | Divisible by 2 | 2, 4, 6, 8, 10 |
| Odd numbers | Not divisible by 2 | 1, 3, 5, 7, 9 |
| Prime numbers | Exactly two factors: 1 and itself | 2, 3, 5, 7, 11, 13 |
| Square numbers | n × n, for a whole number n | 1, 4, 9, 16, 25 |
| Cube numbers | n × n × n, for a whole number n | 1, 8, 27, 64, 125 |
| Triangular numbers | 1, 1 + 2, 1 + 2 + 3, ... | 1, 3, 6, 10, 15 |
| Factors of 12 | Numbers that divide into 12 exactly | 1, 2, 3, 4, 6, 12 |
| Multiples of 12 | The 12 times table | 12, 24, 36, 48 |
- 1 is not prime: it has only one factor.
- 2 is the only even prime. Every other even number has 2 as a factor.
- The reciprocal of a number is 1 divided by it: the reciprocal of 4 is 1/4. See reciprocal.
Break any whole number into primes with the Prime Factorisation Calculator, and find highest common factors and lowest common multiples with the HCF and LCM calculators. Square numbers have their own guide, perfect squares.
Example 2: sort the numbers
Which of these are integers, which are rational and which are irrational? 0.6, √16, √11, −7, 22/7, π, 0
| Number | Integer? | Rational? | Irrational? |
|---|---|---|---|
| 0.6 | No | Yes: 3/5 | No |
| √16 | Yes: it equals 4 | Yes | No |
| √11 | No | No | Yes: 11 is not a square number |
| −7 | Yes | Yes: −7/1 | No |
| 22/7 | No | Yes | No |
| π | No | No | Yes |
| 0 | Yes | Yes: 0/1 | No |
All seven are real numbers. Always simplify first: √16 looks like a surd but is the integer 4.
Check your understanding.
Common mistakes
- Calling 1 a prime number. A prime has exactly two factors; 1 has one.
- Thinking recurring decimals are irrational. They repeat, so they are rational.
- Thinking negative numbers are not integers. −5 is an integer.
- Writing that π = 22/7 or 3.14. These are approximations; π is irrational.
- Calling every square root irrational. √25 = 5 is an integer.
- Thinking a number can only be one type. 4 is natural, whole, an integer, rational, real, even and square.
Practise with a short timed paper
Number questions come up on every GCSE Maths paper. The daily GCSE Maths Foundation and GCSE Maths Higher papers are short timed papers of 8 questions, with a new one every day.
Common questions
What are the types of numbers?
What is the difference between rational and irrational numbers?
Is 0 a natural number?
Is π a rational number?
Are all integers rational?
What are real numbers?
Revision cards
Say the answer first, then flip the card to check it.
Where to go next
- Solve it
Prime Factorisation Calculator
Break any number into primes with a factor tree
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GCSE Maths Foundation
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GCSE Maths Higher
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Maths · NumberRational numbers: definition, examples and decimalsA rational number is any number you can write as a fraction of two integers. See examples, rational vs irrational numbers, and repeating decimals.