---
tag: 080T2TXVQV
kind: definition
name: Irrational number
msc: [11J72]
defines: 0DZXAXEJQ9
uses: [08J2M1JYAG, 0NCJ5WXN6Z]
sources:
  - url: https://en.wikipedia.org/wiki/Irrational_number
    locator: lead section
---

## Definition

A [[real number|08J2M1JYAG]] $x$ is irrational when $x$ is not a [[rational number|0NCJ5WXN6Z]].
That is, there are no integers $a$ and $b$ with $b \neq 0$ and $x = a / b$.
The set of irrational numbers is $\mathbb{R} \setminus \mathbb{Q}$.

## Remarks

The number $\sqrt{2}$ is [[irrational|0DZGSDHD92]]. The numbers $\pi$ and $e$ are also irrational.
The decimal expansion of a real number is eventually periodic exactly when the number is rational. Thus the decimal expansion of an irrational number never repeats.
The set $\mathbb{Q}$ is countable and $\mathbb{R}$ is not, thus almost all real numbers are irrational.

## Explanations

### L1 human

An irrational number is a number on the number line that you cannot write as a fraction.
The length of the diagonal of a square with side $1$ is such a number: it is $\sqrt{2} = 1.41421\dots$.