---
tag: 0YHFMXJJAZ
kind: statement
role: theorem
name: Irrationality of the square root of a prime
msc: [11J72]
uses: [080T2TXVQV, 0VB8BFRABH]
sources:
  - url: https://en.wikipedia.org/wiki/Irrational_number
    locator: Square roots
---

## Statement

Let $p$ be a prime number.
Then $\sqrt{p}$ is an [[irrational number|080T2TXVQV]].

## Remarks

This generalizes the [[irrationality of the square root of 2|0DZGSDHD92]].
More generally, the square root of a natural number $n$ is irrational unless $n$ is a perfect square. See [[the roots of integers|0XWCKJZRYH]].

## Explanations

### L1 human

The numbers $\sqrt{2}$, $\sqrt{3}$, $\sqrt{5}$ and $\sqrt{7}$ are all irrational.
The proof for $\sqrt{2}$ works for each prime: if $a^2 = p b^2$, then $p$ divides $a$, and then $p$ also divides $b$.