Lesson 10E Complex Polar Planes Solidify Understanding
Represent complex numbers using polar coordinates.
Multiply complex numbers written in polar form.
We have plotted complex numbers of the form
Open Up the Math: Launch, Explore, Discuss
Alyce, Javier, and Veronica have two different ways of recording the location of artifacts at the archeological dig: one way is to use rectangular coordinates
Javier thinks the location of the point plotted on the polar grid is given by the polar coordinates
Alyce thinks the location of the point is
Veronica thinks the location of the point is
What do you think? Who has named the location of the point correctly? Explain why.
What are the rectangular coordinates of the plotted point?
Alyce and Veronica recall that they have learned how to represent complex numbers as points or vectors on a complex plane by letting the
In the complex plane, the point
The arithmetic of complex numbers from a polar perspective
Multiplying complex numbers:
Alyce and Veronica have learned that when complex numbers are written in complex form,
“You just multiply the
Javier writes out in symbols what Veronica has claimed:
Javier doesn’t understand how Veronica’s claim can be true; that is,
So, he decides to try out her rule for a specific example.
Pick two complex numbers written in the form
Rewrite both of the complex numbers in polar form.
Multiply the polar forms of the two complex numbers together using Veronica’s rule.
Convert the product from polar form back to
Did you get the same result using Veronica’s rule as you got in part a?
Javier is more convinced, but would like some proof that Veronica’s rule will work all of the time, and not just for the few examples he tried. Veronica says, “As I recall, you have to use the sum and difference identities for sine and cosine to prove it.” Javier decides to try to prove Veronica’s rule.
Rewrite the results using the sum and difference identities for sine and cosine that you wrote in the task Double Identity.
Manipulate your final expression until it matches Veronica’s claim.
Powers and roots of complex numbers:
Javier has a new insight of his own as he thinks about Veronica’s rule for multiplying complex numbers in polar form. “Since raising something to the
Finish Javier’s rule for
Javier is wondering what happens to complex numbers as they are raised to higher and higher powers. He starts with the complex number
Use Javier’s rule to raise
Plot each of the previous complex numbers on the following complex plane. That is, treat the horizontal axis as a real number axis, and the vertical axis as an imaginary number axis. A complex number
Javier was surprised to see that
Javier decides to do some of his own searching on the internet, to see if he can find out more about complex roots of real numbers. Here are some of the results of his search:
Idea #1: If
Idea #2: The modulus, or magnitude of the vectors representing the roots, are all the same.
Idea #3: The
Based on these ideas, plot the cube roots of
What are the three cubes roots of
Based on these ideas, plot the sixth roots of
What are the six sixth roots of
Ready for More?
In the real number system,
By solving the equation
Using the method described in problems 10–12.
Verify these two methods give the same results.
The complex number
The polar form of complex numbers makes computation easier using the following rules:
Raise to the
Find all the
In this lesson, we learned how to write complex numbers in polar form. We used the polar form to multiply and divide complex numbers and to raise complex numbers to powers. We also learned that every complex number has
Apply the rule
Use the definition of