Definition:Square Root

Positive Reals
Let $x \in \R: x \ge 0$ be a positive real number.

Then from the definition of root, we have that $\exists y \in \R: x = y^2$, and we write $y = \sqrt x$.

From Even Powers are Positive, we have that $y^2 = x \iff \left({-y}\right)^2 = x$ and so we can also write $y = \pm \sqrt x$.

The number $y = + \sqrt x$ is called the positive or principal square root of $x$, and $y = - \sqrt x$ is the negative square root of $x$.

Frequently, when written just as "$\sqrt x$", the positive one is being referred to by default.

Note also that square roots are so much more commonly used in mathematics than any other sort of root, $\sqrt x$ is frequently just called "root $x$"

Negative Reals
Let $x \in \R$ be a real number.

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

$\sqrt x = \begin{cases} +\sqrt x & : x \ge 0 \\ i \left( {+\sqrt {\left( {-x} \right)}}\right) & : x < 0 \end{cases}$

where $i$ is the imaginary unit and $i^2 = -1$.

Hence we have $\sqrt{-1} = i$.

Square roots of primes
The square root of any prime is irrational, so can not be expressed precisely by a rational fraction.

The decimal expansions of the first few primes are as follows:

Also see

 * One Equals Minus One