Mathematicians organise numbers into sets with different properties. Understanding these sets — natural numbers, integers, rational numbers and real numbers — helps you choose the right techniques, describe results precisely and prepare for the rational and irrational number work that appears at GCSE and beyond.
What are natural numbers?
Natural numbers (also called counting numbers) are the positive whole numbers used to count objects:
1, 2, 3, 4, 5, …
The set continues without end. Most mathematicians today include zero in the natural numbers (giving 0, 1, 2, 3, …), though some older texts exclude it. Natural numbers never include negatives or fractions.
Why they matter: You use natural numbers to count — for example, the number of students in a class or the number of days in a week.
What are integers?
Integers extend the natural numbers by including zero (if not already) and all the negative whole numbers:
…, −4, −3, −2, −1, 0, 1, 2, 3, 4, …
Every natural number is an integer, but not every integer is a natural number (for example, −3 is an integer but not a natural number).
Examples of integers in context: temperature (−5 °C), floors in a building (floor −2 = two below ground), a bank overdraft (−£50).
What are rational numbers?
A rational number is any number that can be written as a fraction p/q, where p and q are both integers and q ≠ 0.
Examples of rational numbers:
| Number | As a fraction | Rational? |
|---|---|---|
| 3 | 3/1 | ✓ Yes |
| −7 | −7/1 | ✓ Yes |
| 0.5 | 1/2 | ✓ Yes |
| 0.333… (recurring) | 1/3 | ✓ Yes |
| 2.75 | 11/4 | ✓ Yes |
| √2 | Cannot be written as p/q | ✗ No |
| π | Cannot be written as p/q | ✗ No |
Every integer is a rational number (write it over 1). Every fraction with integer numerator and denominator is rational. Recurring and terminating decimals are always rational. Numbers that cannot be expressed as a fraction (like √2 and π) are called irrational — these are covered fully at GCSE.
What are real numbers?
Real numbers include everything on the number line — all rational numbers and all irrational numbers combined. The set of real numbers contains:
- Integers: …, −2, −1, 0, 1, 2, …
- Fractions: 1/3, 7/4, −5/2, …
- Irrationals: √2, √3, π, …
At KS3, nearly all the numbers you work with are real. The concept becomes important when you reach complex numbers at A-level (where numbers like √(−1) exist outside the real number line).
How do the sets nest inside each other?
Think of the sets like Russian dolls, each one inside the next:
Natural numbers ⊂ Integers ⊂ Rational numbers ⊂ Real numbers
This means:
- Every natural number is an integer.
- Every integer is a rational number.
- Every rational number is a real number.
- But not every real number is rational (π, √2 are examples).
What about other special types of numbers you meet at KS3?
Square numbers: integers that are perfect squares — 1, 4, 9, 16, 25, 36, … These are integers (and therefore also rational and real).
Cube numbers: 1, 8, 27, 64, 125, … Also integers.
Prime numbers: natural numbers greater than 1 with exactly two factors — 2, 3, 5, 7, 11, … They are a special subset of natural numbers.
Even and odd integers: integers divisible by 2 (even: …, −4, −2, 0, 2, 4, …) and integers not divisible by 2 (odd: …, −3, −1, 1, 3, …).
Frequently asked questions
Is zero a natural number?
There is a genuine historical disagreement here, but the UK National Curriculum treats zero as an integer, and most modern conventions include 0 in the natural numbers. For exam purposes at KS3, treat zero as a whole number that is also an integer and also a rational number. If a question asks for "positive integers", zero is excluded — "positive" means strictly greater than zero.
Is every fraction a rational number?
Yes, provided the numerator and denominator are both integers and the denominator is not zero. 3/4, −7/2 and 100/1 are all rational. Division by zero is not defined in standard maths, so 5/0 is not a number at all — it is undefined.
How do I know if a decimal is rational?
A decimal is rational if and only if it terminates (e.g. 0.75) or recurs (e.g. 0.333… = 1/3 or 0.142857142857… = 1/7). Non-terminating, non-recurring decimals — like π = 3.14159265… — are irrational. If you can find the fraction it equals, it is rational.
Why does it matter which type of number a result is?
Type matters for two reasons. First, precision: saying "n is an integer" tells you far more than "n is a real number". Second, technique: some operations only produce certain types of result (the product of two integers is always an integer; the quotient of two integers is not always an integer). At GCSE and A-level, proof questions often require you to state whether expressions produce integers, rationals or irrationals — knowing the definitions makes those proofs straightforward.
For guided KS3 number exploration with Professor Pi, visit AI Tutors.