# What is the square root of -3?

Sep 10, 2015

$- 3$ has no Real square root.

The principal Complex square root of $- 3$, denoted $\sqrt{- 3}$ is equal to $i \sqrt{3}$, where $i$ is the imaginary unit and $\sqrt{3}$ is the positive square root of $3$.

#### Explanation:

There is no Real number that is the square root of $- 3$ since ${x}^{2} \ge 0$ for all $x \in \mathbb{R}$.

$- 3$ has two Complex square roots, $i \sqrt{3}$ and $- i \sqrt{3}$, where $i$ is the imaginary unit, approximately called 'the' square root of $- 1$. $i$ satisfies ${i}^{2} = - 1$.

$\sqrt{3}$ is the positive square root of $3$.

$- \sqrt{3}$ is also a square root of $3$, in that ${\left(- \sqrt{3}\right)}^{2} = 3$

$\sqrt{- 3} = i \sqrt{3}$ is called the principal square root of $- 3$.