Introduction
If $ 2 $ is a real number, and $ 2j $ is an imaginary number, then $ 2+2j $ is a complex number. Complex numbers can be written in many forms. The example described previously is a complex number in Cartesian form. While addition and subtraction are easy to perform on complex numbers in Cartesian form, multiplication and division are not. This page will show how to convert a complex number in Cartesian form to exponential one.
Procedures
It's much easier to perform multiplication and division on complex numbers in exponential form because the format of the exponential form looks like:
- $ z = re^{i\varphi}\, $
Suppose a complex number in Cartesian has the form $ x + yi $, then $ r $ is actually the magnitude of the two components of the complex number in Cartesian form, and thus can be easily found:
- $ r = \sqrt{x^2+y^2}\, $
While $ \varphi $ can be computed using $ tan^{-1} $:
- $ \varphi = tan^{-1}\left(\frac{y}{x}\right) $
One needs to be careful in determining the sign of the angle. A way to avoid mistake is to draw a Real-Imaginary plane and plot the components of a complex number.
For example, suppose one wants to convert $ 3-j4 $ to an exponential form, one can start with the magnitude:
- $ r = \sqrt{3^2+(-4)^2} = \sqrt{25} = 5\, $
and
- $ \varphi = tan^{-1}\left(\frac{4}{3}\right) = 53.1^o $
Plotting the real component of the complex number on positive Real axis, and imaginary component on the negative Imaginary axis shows that the point lies on the fourth quadrant. Therefore the sign of the angle is negative. Thus:
- $ 5e^{-j53.1^o}\, $
is the result of the conversion.
