---
title: "What Is an Irrational Number?"
url: https://www.learningloftinstitute.com/glossary/irrational-number
source: Learning Loft Institute
language: en
updated: 2026-08-21
section: "Glossary › What Is an Irrational Number?"
---
# What Is an Irrational Number?
> **Answer.** An irrational number cannot be written as a fraction of two integers, which means its decimal never stops and never settles into a repeating block. π, √2, √3 and e are the ones you are expected to recognise. A root sign is not the test: √9 is 3, and 3 is rational.
## Key facts
| | |
| --- | --- |
| Definition | An irrational number is a real number that cannot be written as a fraction of two integers, so its decimal runs on for ever without ever repeating. |
| Where students meet it | Grade 9 and 10, and again on GCSE Higher and IGCSE Extended papers, where surd manipulation sits in the additional content for higher-attaining pupils. |
## The short list you actually have to recognise
At school level the irrational numbers that appear are few enough to name. Everything else in a classification question will be rational, so learning this list is most of the work.
Three of the four are roots, and that is not a coincidence. The square root of any whole number that is not a perfect square is irrational, which is where almost every exam example comes from.
- π = 3.14159… — the ratio of a circle's circumference to its diameter, and the one that appears in nearly every list
- √2 = 1.41421… — the diagonal of a unit square, and historically the first number known to be irrational
- √3 = 1.73205… — and in general the root of any whole number that is not a perfect square
- e = 2.71828… — met at A level rather than GCSE, but included in lists often enough to be worth knowing
## Why the root sign is not the test
The habit students fall into is to scan a list for √ and tick whatever has one. It fails immediately: √9 is 3, √16 is 4, √(1/4) is 1/2, and every one of those is rational. What decides the matter is whether the number underneath is a perfect square.
It fails from the other direction too. π ÷ π is 1. √2 × √2 is 2. Irrational numbers can combine into rational ones, so a number is not irrational merely because an irrational number appears somewhere inside it. The only reliable question is the same one as before: does a fraction of two whole numbers equal this number?
## Surd: the school word for leaving it exact
A surd is a root left in root form because writing it as a decimal would mean rounding it. 3√2 is exact. 4.24 is a rounded stand-in that is very slightly wrong, and every subsequent line of working inherits that error.
This is what an instruction like give your answer in surd form is really asking for: the number itself, not an approximation of it. The same reasoning explains why an area is written as 12π rather than 37.7, and why a Pythagoras answer of √52 is often preferred to 7.21.
## Questions
### Is π irrational?
Yes, and it was proved to be so by Johann Heinrich Lambert in 1761. 22/7 and 3.14 are approximations used for convenience; neither equals π. This matters in practice, because a question asking for an exact answer wants the π left in place rather than replaced by a decimal.
### Is √9 irrational?
No. √9 is 3, which is an integer and therefore rational. The root sign is a way of writing a number, not a category of number. The test is whether what sits underneath is a perfect square: √9, √16 and √225 are all rational, and √2, √3 and √10 are not.
### Can an irrational number be negative?
Yes. −√2 and −π are irrational, because being expressible as a fraction of two integers has nothing to do with sign. If a number is irrational then so is its negative, since attaching a minus sign could never turn an impossible fraction into a possible one.
### Is a recurring decimal irrational?
No, and this is the most common confusion about the word. 0.333… goes on for ever but repeats, and it is exactly 1/3. An irrational decimal goes on for ever and never repeats, which is why no fraction can capture it. Forever is not the test; the absence of a repeating block is.
## Sources
1. National curriculum in England: mathematics programmes of study — Department for Education. https://www.gov.uk/government/publications/national-curriculum-in-england-mathematics-programmes-of-study/national-curriculum-in-england-mathematics-programmes-of-study (accessed 2026-08-21)
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