# Definition:Square Root

## Contents

## Definition

A **square root** of a number $n$ is a number $z$ such that $z$ squared equals $n$.

### Positive Real Numbers

Let $x \in \R_{\ge 0}$ be a positive real number.

The **square roots of $x$** are the real numbers defined as:

- $x^{\paren {1 / 2} } := \set {y \in \R: y^2 = x}$

where $x^{\paren {1 / 2} }$ is the $2$nd root of $x$.

The notation:

- $y = \pm \sqrt x$

is usually encountered.

### Positive Square Root

The **positive square root of $x$** is the number defined as:

- $+ \sqrt x := y \in \R_{>0}: y^2 = x$

### Negative Square Root

The **negative square root of $x$** is the number defined as:

- $- \sqrt x := y \in \R_{<0}: y^2 = x$

## Negative Real Numbers

Let $x \in \R_{< 0}$ be a (strictly) negative real number.

Then the **square root of $x$** is defined as:

- $\sqrt x = i \left({\pm \sqrt {-x}}\right)$

where $i$ is the imaginary unit:

- $i^2 = -1$

## Complex Numbers

Let $z \in \C$ be a complex number expressed in polar form as $\left \langle{r, \theta}\right\rangle = r \left({\cos \theta + i \sin \theta}\right)$.

The **square root of $z$** is the $2$-valued multifunction:

\(\displaystyle z^{1/2}\) | \(=\) | \(\displaystyle \left\{ {\sqrt r \left({\cos \left({\frac {\theta + 2 k \pi} 2}\right) + i \sin \left({\frac {\theta + 2 k \pi} 2}\right) }\right): k \in \left\{ {0, 1}\right\} }\right\}\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \left\{ {\sqrt r \left({\cos \left({\frac \theta 2 + k \pi}\right) + i \sin \left({\frac \theta 2 + k \pi}\right) }\right): k \in \left\{ {0, 1}\right\} }\right\}\) |

where $\sqrt r$ denotes the positive square root of $r$.

### Principal Square Root

Let $z \in \C$ be a complex number.

Let $z^{1/2} = \set {w \in \C: w^2 = z}$ be the square root of $z$.

The **principal square root** of $z$ is the element $w$ of $z^{1/2}$ such that:

- $\begin{cases} \map \Im w > 0 : & \map \Im z \ne 0 \\ \map \Re w \ge 0 : & \map \Im z = 0 \end{cases}$

## Also known as

Because **square roots** (and in particular, positive square roots) are so much more commonly encountered in mathematics than any other sort of root, $\sqrt x$ is frequently just called **root $x$**

In translations of Euclid's *The Elements*, the word **side** can be found, often in quotes to emphasise the awkward nature of the language available to the Ancient Greeks.

In the words of Euclid:

*If an area be contained by a rational straight line and the first binomial, the "side" of the area is the irrational straight line which is called binomial.*

(*The Elements*: Book $\text{X}$: Proposition $54$)

## Historical Note

It is suggested by some sources that the symbol $\surd$ (a stylised **r** for **radix**) for the **square root** may have originated with René Descartes, but there is evidence that it may have been around a lot earlier than that.

## Also see