---
tag: 04J6G4H78J
kind: statement
role: theorem
name: Density of the irrationals
msc: [26A03]
uses: [0STX0WRDNG, 080T2TXVQV, 0DZGSDHD92]
sources:
  - url: https://en.wikipedia.org/wiki/Irrational_number
    locator: Set of all irrationals
---

## Statement

Let $x$ and $y$ be real numbers with $x < y$.
Then there is an [[irrational number|080T2TXVQV]] $z$ with $x < z < y$.

## Remarks

Proof: by the [[density of the rationals|0STX0WRDNG]], there is a rational $q$ with $x / \sqrt{2} < q < y / \sqrt{2}$ and $q \neq 0$. Then $z = q \sqrt{2}$ is irrational and $x < z < y$.
Thus both $\mathbb{Q}$ and $\mathbb{R} \setminus \mathbb{Q}$ are dense in $\mathbb{R}$.

## Explanations

### L1 human

Between any two different real numbers, there is also a number that is not a fraction.
So fractions and non-fractions are mixed together everywhere on the number line.