---
tag: 0VB8BFRABH
kind: definition
name: Square root
msc: [26A09]
defines: 0F7FQS6XY6
uses: [068BZBXE0C]
sources:
  - url: https://en.wikipedia.org/wiki/Square_root
    locator: lead section
  - url: https://en.wikipedia.org/wiki/Square_root
    locator: Properties and uses
---

## Definition

Let $x$ be a real number with $x \ge 0$.
The square root of $x$, written $\sqrt{x}$, is the unique real number $y$ with $y \ge 0$ and $y^2 = x$.

## Remarks

The [[existence of square roots|068BZBXE0C]] shows that exactly one such $y$ exists.
The function $x \mapsto \sqrt{x}$ on $[0, \infty)$ is increasing, and $\sqrt{a b} = \sqrt{a} \sqrt{b}$ for $a, b \ge 0$.
For each real $x$, $\sqrt{x^2} = |x|$, the [[absolute value|00S7C68D51]] of $x$.

## Explanations

### L1 human

The square root of $x$ is the side length of a square with area $x$.
The equation $y^2 = 25$ has two solutions, $5$ and $-5$. The symbol $\sqrt{25}$ means the positive one, $5$.