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Precalculus | Packer Collegiate Institute

Complex Roots

Warm Up:

1. Draw a perfect nonagon:

2. Solve:

(a) Solve 3 1x

(b) Solve 5 1x

(c) Use your calculator.

Let 1 3

2 2z i . Find 3z :

Let 1

2 2

3w i . Find 3w :

Also, find out how far each of these complex numbers are from the origin. distance of z : distance of w :

(d) Use your calculator. [Split this up among your group]

Let 0.309 0.951a i . Find 5a :

Let 0.809 0.588b i . Find 5b :

Let 0.809 0.588c i . Find 5c :

Let 0.309 0.951d i . Find 5d : Also, find out how far each of these complex numbers are from the origin. distance of a :

distance of b : distance of c :

distance of d :

Section 1: Graphing Roots of Unity

(e) We know that 1x is a solution to 3 1x . However we have discovered two more. Plot them carefully on the complex plane.

(a) We know that 1x is one solution to 5 1x . However we have discovered four more. Plot them carefully on the complex plane.

Whoa!!! Seriously? DO YOU SEE WHAT I SEE?

If you do, try to plot all the solutions to 8 1x in the complex plane.

Write out what some of these solutions are, exactly:

Section 2: From Visualizing to Algebra

The solutions to 3 1x are visualized below. Write down each of the solutions exactly in complex polar form:

Solution 1:

Solution 2:

Solution 3:

The solutions to 5 1x are visualized below. Write down each of the solutions exactly in complex polar form:

Solution 1:

Solution 2:

Solution 3:

Solution 4:

Solution 5:

Section 3: A perfect nonagon

If I asked you to give the me coordinates of a regular nonagon in the regular coordinate plane, you might have hesitated.

A NONAGON???

However, solve 9 1x , and use your answers to give me coordinates of a regular nonagon…