Imaginary Numbers and Euler's Formulas Review

 

A lot of people seem to freak out when they see an i in math or j in electrical engineering. So hopefully this will help.

The first thing we want to go over is what i and j even are.

The other aspect of imaginary numbers that will help us understand how they are used in electrical engineering (and systems/signaling in general) is that they can be graphed on a complex plane.

NOTE: You may have also noticed that when we were graphing z = (x,y) on the complex plane we formed a right triangle. So:

Also:

Looking at imaginary numbers like this, we can also see how they can be added, subtracted, and multiplied like any other numbers.

There are also a couple of new operations:

You may have noticed that I did not include division, and multiplication is a little difficult as well. For this, we will use Euler’s formulas.

The four underlined formulas are the common forms of Euler’s formula

Multiplying and dividing exponentials is a lot easier than multiplying and dividing complex numbers

If we were asked to change that back into a complex number, then:

These are a few more useful forms of euler’s formulas:

These techniques are most useful in electrical engineering when dealing with AC circuits, or in other words, circuits with sinusoidal inputs