Definition

The square root of a number is what you multiply by itself to get that number, written with the radical sign. Formally, the principal square root of $n \ge 0$ is the non-negative real number $x$ such that $x^2 = n$, written $\sqrt{n}$ or $n^{1/2}$; the square root of a non-perfect-square positive integer is irrational, and $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$, $\sqrt{a/b} = \sqrt{a}/\sqrt{b}$ for $a,b \ge 0$. As a function, $\sqrt{\cdot}: [0,\infty) \to [0,\infty)$ is continuous, strictly increasing, and concave, satisfying $\frac{d}{dx}(\sqrt{x}) = \frac{1}{2\sqrt{x}}$; in the complex numbers, every non-zero $z$ has two square roots, $\sqrt{|z|}\, e^{i\arg(z)/2}$ and its negative, with the principal root taking $\arg(z)$ in $(-\pi, \pi]$.