---
tag: 0N10YRNNWV
kind: statement
role: theorem
name: Irrationality of square roots of non-squares
msc: [11J72]
uses: [084HH1NDCB]
sources:
  - url: https://en.wikipedia.org/wiki/Quadratic_irrational_number
    locator: Square root of non-square is irrational
---

## Statement

Let $n$ be a natural number.
Then $\sqrt{n}$ is irrational if and only if $n$ is not a perfect square.

## Remarks

If $n = m^2$, then $\sqrt{n} = m$ is rational. The content is the other direction.
A proof: if $\sqrt{n} = a/b$, then $a^2 = n b^2$. By [[unique factorization|084HH1NDCB]], each prime has an even exponent in $a^2$ and in $b^2$, so also in $n$. Then $n$ is a square.
For example, $\sqrt{2}$, $\sqrt{3}$ and $\sqrt{12}$ are irrational.
The same argument shows that the $k$-th root of $n$ is irrational unless $n$ is a $k$-th power.

## Explanations

### L1 human

The square root of a whole number is either a whole number or an irrational number.
It is never a fraction such as $\frac{3}{2}$.