---
tag: 0DZGSDHD92
kind: statement
role: theorem
name: Irrationality of the square root of 2
msc: [11J72]
uses: [080T2TXVQV, 0VB8BFRABH]
sources:
  - url: https://en.wikipedia.org/wiki/Square_root_of_2
    locator: Proofs of irrationality
  - url: https://en.wikipedia.org/wiki/Square_root_of_2
    locator: History
---

## Statement

The [[square root|0VB8BFRABH]] $\sqrt{2}$ of $2$ is an [[irrational number|080T2TXVQV]].
That is, there are no integers $a$ and $b$ with $b \neq 0$ and $(a / b)^2 = 2$.

## Remarks

A classical proof: suppose that $a / b = \sqrt{2}$ in [[lowest terms|0R64A2CVN4]]. Then $a^2 = 2 b^2$, thus $a$ is even. Then $b$ is also even, which is a contradiction.
The theorem is a special case of the [[irrationality of the square root of a prime|0YHFMXJJAZ]].

## History

The ancient Greeks knew this result. Tradition attributes its discovery to the Pythagoreans.

## Explanations

### L1 human

No fraction squared gives exactly $2$.
Fractions such as $7/5$ and $141/100$ come close, but there is always an error.
This means that the diagonal of a square with side $1$ has a length that is not a ratio of whole numbers.