Definition:Square Root

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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 highlight 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

Square Root of $9$

A square root of $9$ is the integer $3$:

$3^2 = 9$


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

  • Results about square roots can be found here.


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.


Sources

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