Rational and irrational numbers GCSE questions ask you to sort a given list of numbers by whether each one can be written as an exact fraction of two integers. A rational number can; an irrational number, like most surds or $\pi$, has a decimal expansion that never terminates or repeats, so it cannot.
What is a rational number?
A rational number is any number that can be written as a fraction $\frac{p}{q}$, where $p$ and $q$ are both integers and $q$ is not zero. This includes:
- All whole numbers and integers, since 5 can be written as $\frac{5}{1}$.
- All terminating decimals, such as 0.75, which equals $\frac{3}{4}$.
- All recurring decimals, such as $0.\dot{3}$, which equals $\frac{1}{3}$.
- All simple and mixed fractions, such as $\frac{2}{5}$ or $3\frac{1}{2}$.
The word "rational" comes from "ratio" — a rational number is defined by being expressible as a ratio of two whole numbers.
What is an irrational number?
An irrational number cannot be written as an exact fraction of two integers. Its decimal expansion goes on forever without ever settling into a repeating pattern. Common examples tested at GCSE include:
- $\pi$ (pi), which starts 3.14159265... and never repeats.
- Square roots of numbers that are not perfect squares, such as $\sqrt{2}$, $\sqrt{3}$, $\sqrt{5}$, and $\sqrt{7}$.
- Numbers built from combining a rational number with an irrational one through multiplication or addition, such as $3 + \sqrt{2}$ or $4\sqrt{5}$.
These irrational square roots are also called surds — a surd is specifically an irrational root that is left in root form because writing it as a decimal would lose exact accuracy.
How do you tell rational and irrational numbers apart?
| Number | Rational or irrational? | Reason |
|---|---|---|
| $\frac{7}{8}$ | Rational | Already a fraction of two integers |
| $0.6$ | Rational | Terminating decimal; equals $\frac{3}{5}$ |
| $\sqrt{16}$ | Rational | Equals 4 exactly, a perfect square root |
| $\sqrt{10}$ | Irrational | 10 is not a perfect square; no exact fraction exists |
| $\pi$ | Irrational | Non-terminating, non-repeating decimal |
| $0.454545...$ | Rational | Recurring decimal; equals $\frac{5}{11}$ |
Worked example: State whether $\sqrt{49}$ and $\sqrt{50}$ are rational or irrational.
$\sqrt{49} = 7$, because 49 is a perfect square ($7 \times 7 = 49$). Since 7 is a whole number, it is rational.
$\sqrt{50}$ cannot be simplified to a whole number, because 50 is not a perfect square. Its decimal value, 7.0710678..., continues forever without repeating, so $\sqrt{50}$ is irrational.
Why are square roots of non-perfect squares always irrational?
This is a key GCSE fact you are expected to know, even though the full proof goes beyond the specification. The square roots of the perfect squares (1, 4, 9, 16, 25, 36, 49, and so on) are always whole numbers, and therefore always rational. Every other positive integer's square root has been proven to be irrational — there is no fraction of two integers that, when squared, gives exactly that number.
This is why simplifying surds at GCSE always aims to extract any perfect-square factor, leaving the smallest possible irrational part: $\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}$.
How do operations affect whether a number is rational or irrational?
Combining rational and irrational numbers follows predictable rules that GCSE questions often test directly:
- Rational + rational = rational. Adding, subtracting, multiplying, or dividing two rational numbers always gives another rational number.
- Rational + irrational = irrational, unless the rational part cancels the irrational part exactly. For example, $3 + \sqrt{2}$ is irrational, but $\sqrt{2} - \sqrt{2} = 0$ is rational.
- Irrational × irrational can be rational. This is the trickiest rule: $\sqrt{2} \times \sqrt{2} = 2$, which is rational, because the two irrational parts multiply to remove the root entirely.
Worked example: Show that $(2 + \sqrt{3})(2 - \sqrt{3})$ is rational.
Expanding using the difference of two squares: $(2 + \sqrt{3})(2 - \sqrt{3}) = 2^2 - (\sqrt{3})^2 = 4 - 3 = 1$. The irrational parts cancel completely, leaving the rational number 1.
Frequently asked questions
What is an irrational number, in simple terms?
An irrational number is a number that cannot be written exactly as a fraction of two integers. Its decimal form carries on forever without ever repeating in a fixed block, unlike a rational number, whose decimal either stops or repeats a pattern.
Are all surds irrational?
Most surds are irrational, but not all square roots count as surds. A surd is specifically an irrational root — $\sqrt{9}$ equals 3 exactly, so it is not a surd, but $\sqrt{7}$ cannot be simplified to a whole number or fraction, so it is a genuine surd and irrational.
What are some everyday examples of rational numbers?
Any whole number, terminating decimal, or fraction you meet in daily life is rational: 5 apples, £3.50, half a pizza ($\frac{1}{2}$), or a recurring decimal like $0.\dot{6}$ (which equals $\frac{2}{3}$). Rational numbers cover almost every number used outside of geometry and advanced algebra.
How do I identify an irrational number quickly in an exam?
Check for three signals: a square root (or other root) of a number that is not a perfect square or cube, the symbol $\pi$, or a decimal that is described as non-terminating and non-repeating. If a number fits none of those, and can be written as a fraction of two whole numbers, it is rational.
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