7. Powers and Roots of Complex Numbers

by M. Bourne

Consider the following example, which follows from basic algebra:

(5e3j)2 = 25e6j

We can generalise this example as follows:

math expression

The above expression, written in polar form, leads us to DeMoivre's Theorem.

DeMoivre's Theorem

math expression

or (r ∠ θ)n = (rn ∠ nθ)

Challenge

I'm going to challenge you here...

I have never been able to find an electronics or electrical engineer that's even heard of DeMoivre's Theorem. Certainly, none of them know how it is applied in 'real life'.

Please let me know if there is a good application.

I've always felt that while this is a nice piece of mathematics, it is rather useless.. :-)

Update: I received this reply to my challenge from user Richard Reddy:

"Much of what you're doing with complex exponentials is an extension of DeMoivre's Theorem. In general, the theorem is of practical value in transforming equations so they can be worked more easily. Often, what you see in EE are the solutions to problems in physics. There was a time, before computers, when it might take 6 months to do a tensor problem by hand. DeMoivre's theorem is a time-saving identity, easier to apply than equivalent trigonometric identities.

I like your site."

 

Example:

Find (1 - 2j)6


Answer



Let's play with this. Here is how Livemath performs DeMoivre's Theorem.

LIVEMath

 

Complex Roots

If an = x + yj then we expect n complex roots for a.

 

Example:

If a5 = 7+ 5j , then we expect 5 complex roots for a.

In general, if we are looking for the nth roots of an equation involving complex numbers, the roots will be math expressionapart i.e.

2 roots will be 180° apart

3 roots will be 120° apart

4 roots will be 90° apart

5 roots will be 72° apart etc.

 

Example:

Find the two square roots of -5 + 12j.


Answer


Let's play with this. Here is how Livemath performs this operation.

LIVEMath

Let's see what happens when we have a CUBIC type. You can also check your work with this.

LIVEMath

 

Exercises:

1. Evaluate (2 ∠ 135°)8


Answer


2. Find: math expression


Answer


3. (i) Find the first 2 fourth roots of 81(cos60° + jsin60°)

(ii) Then sketch all fourth roots of 81(cos60° + jsin60°) showing relevant values of r and θ.


Answer


4. At the beginning of this section, we expected 3 roots for

x3 = 8.

Find the roots and sketch them.


Answer




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