The square root of a number x, is a number r so that r2= x. In modern notation square root is written as or x1/2. Since r2= (-r)2, any positive number has in principle two square roots, one positive and an opposite negative one. These are usually denoted as . 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
The derivative of the positive square root function (x>0) is:
The integral of the positive square root function is given by:
The following formulas are valid for positive square root function:
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 numberiwith the unique property:
Therefore, the square root of a negative number is an imaginary number: