# Definition:Square Root

## 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 and denoted as:

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

### Negative Square Root

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

- $\sqrt x = i \paren {\pm \sqrt {-x} }$

where $i$ is the imaginary unit:

- $i^2 = -1$

## Complex Numbers

Let $z \in \C$ be a complex number expressed in polar form as $\polar {r, \theta} = r \paren {\cos \theta + i \sin \theta}$.

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

\(\ds z^{1/2}\) | \(=\) | \(\ds \set {\sqrt r \paren {\map \cos {\frac {\theta + 2 k \pi} 2} + i \map \sin {\frac {\theta + 2 k \pi} 2} }: k \in \set {0, 1} }\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \set {\sqrt r \paren {\map \cos {\frac \theta 2 + k \pi} + i \map \sin {\frac \theta 2 + k \pi} }: k \in \set {0, 1} }\) |

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 principal branch of the $2$nd power of $w$.

Hence, by the conventional definition of the principal branch of the natural logarithm of $z$, it is the element $w$ of $z^{1/2}$ such that:

- $-\dfrac \pi 2 < \arg w \le \dfrac \pi 2$

## 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 $54$)

## Examples

The decimal expansions of the square roots of various numbers are as follows:

### Square Root of $2$

- $\sqrt 2 \approx 1 \cdotp 41421 \, 35623 \, 73095 \, 04880 \, 16887 \, 24209 \, 69807 \, 85697 \ldots$

### Square Root of $3$

- $\sqrt 3 \approx 1 \cdotp 73205 \, 08075 \, 68877 \, 2935 \ldots$

### Square Root of $5$

- $\sqrt 5 \approx 2 \cdotp 23606 \, 79774 \, 99789 \, 6964 \ldots$

### Square Root of $10$

- $\sqrt 10 \approx 3 \cdotp 16227 \, 76601 \, 68379 \, 33199 \, 88935 \, 44432 \, 71853 \, 3719 \ldots$

### Square Root of $e$

The decimal expansion of Euler's Number $e$ starts:

- $\sqrt e \approx 1 \cdotp 64872 \, 12707 \, 00128 \, 1468 \ldots$

## Also see

## 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 RenĂ© Descartes, but there is evidence that it may have been around a lot earlier than that.

## Sources

- 2008: Ian Stewart:
*Taming the Infinite*... (previous) ... (next): Chapter $8$: The System of the World: Calculus - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**square root**