## Definitions

### General

The square root of a number x, is a number r so that r^{2} = x. In modern notation square root is written as \(\sqrt{x}\) or x^{1/2} . Since r^{2} = (-r)^{2} , any positive number has in principle two square roots, one positive and an opposite negative one. These are usually denoted as \(\pm\sqrt{x}\). Depending on the application, only one of the two square roots may be associated with a physical meaning. The square root of a positive number is always a real number. The square root of 0 is 0.

The graph of the positive square root function of non-negative numbers is shown in the figure below. It is a monotonic function with a parabolic form

### Properties

The derivative of the positive square root function (x>0) is:

\[ \left(\sqrt x\right)' = \frac{1}{2\sqrt{x}} \]

The integral of the positive square root function is given by:

\[ \int \sqrt x\, \mathrm{d}x = \frac{2x^{3/2}}{3} +C \]

### Identities

The following formulas are valid for positive square root function:

\[ \begin{split} &\sqrt {xy} &= \sqrt{x}\sqrt{y} \quad & x\ge 0, y\ge 0\ \\ \\ & \sqrt {\frac{x}{y}} &= \frac{\sqrt{x}}{\sqrt{y}} \quad & x\ge 0, y>0\ \\ \\ &\sqrt {x^n} &= \left(\sqrt{x}\right)^n \quad & x\ge 0\ \\ \\ \end{split} \]

### Negative numbers

Since the square of any real number (positive or negative) is always positive, it is impossible to find a real square root of a negative number. Nevertheless, an expansion of the square root to negative numbers is meaningful for many applications. A trick is necessary in that case, employing an imaginary unit number*i* with the unique property:

\[ i^2 = -1 \]

Therefore, the square root of a negative number is an imaginary number:

\[ \sqrt{-x} = i\sqrt{x} \]