http://www.ck12.org Chapter 11. Complex Numbers
The three points are equally spaced around a circle of radius 2. Only one of the points, 2+ 0 i, is made up of only
real numbers. The other two points have both a real and an imaginary component which is why they are off of thex
axis.
As you become more comfortable with roots, you can just determine the number of points that need to be evenly
spaced around a certain radius circle and find the first point. The rest is just logic.
Example C
What are the fourth roots of 16 cis 48◦?
Solution: There will be 4 points, each 90◦apart with the first point at 2 cis( 12 ◦).
2 cis( 12 ◦),2 cis( 102 ◦),2 cis( 192 ◦),2 cis( 282 ◦)
Concept Problem Revisited
Squaring a complex number produces a new complex number. The angle gets doubled and the magnitude gets
squared, so geometrically you see a rotation.
Vocabulary
Thenthroots of unityrefer to thenthroots of the number 1.
Thereal axisis thexaxis and the imaginary axisis theyaxis. Together they make the complex coordinate plane.