Lesson 10 Complex Polar Planes Solidify Understanding

Learning Focus

Represent complex numbers using polar coordinates.

Multiply complex numbers written in polar form.

We have plotted complex numbers of the form $a + b i$ on a complex plane as the point $(a, b)$ using the horizontal axis to represent the real component of the complex number and the vertical axis to represent the imaginary component. Can we also use the polar coordinates $r$ and $θ$ to represent complex numbers?

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 $(x, y)$ and the other is to use polar coordinates $(r, θ)$ . The three friends know a lot about plotting points and graphing functions on a rectangular coordinate grid and they are wondering if they can sketch graphs on a polar grid. They have found some polar graph paper in the archeological supplies and are trying to make sense of it. They have learned that angles are measured with the initial ray pointing horizontally to the right (the positive horizontal axis) and positive angles are measured counterclockwise.

Javier thinks the location of the point plotted on the polar grid is given by the polar coordinates $(6, 120 °)$ .

Alyce thinks the location of the point is $(6, - 240 °)$ .

Veronica thinks the location of the point is $(- 6, 300 °)$ .

1.

What do you think? Who has named the location of the point correctly? Explain why.

2.

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 $x$ -axis represent the real component of the complex number and the $y$ -axis represent the imaginary component. They are wondering if complex numbers can be related to Javier’s work with the polar grid. From the internet, they learn that complex numbers can be expressed in polar form and that much of the arithmetic of complex numbers can be made easier and enhanced when complex numbers are written in polar form. They are using the following problems to introduce Javier to the key ideas.

3.

The point $(2, 5)$ can be plotted on a rectangular grid. Find the polar coordinates for the same point. (Express the angle measure to the nearest tenth of a degree.)

4.

In the complex plane, the point $(2, 5)$ represents the complex number $2 + 5 i$ . In general, any point $(a, b)$ represents a corresponding complex number $a + b i$ . How are the horizontal distance $a$ and the vertical distance $b$ represented in polar form? That is, how can we use $r$ and $θ$ to describe the complex number $a + b i$ ?

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, $r \cdot \cos θ + i \cdot r \cdot \sin θ$ , the product of two complex numbers is easy to find.

“You just multiply the $r$ ’s and add the $θ$ ’s,” Veronica says excitedly.

Javier writes out in symbols what Veronica has claimed:

$If a_{1} + b_{1} i = r_{1} \cos θ_{1} + i r_{1} \sin θ_{1}$ $and$ $a_{2} + b_{2} i = r_{2} \cos θ_{2} + i r_{2} \sin θ_{2}$ ,

$\begin{aligned} then (a_{1} + b_{1} i) (a_{2} + b_{2} i) & = (r_{1} \cos θ_{1} + i r_{1} \sin θ_{1}) (r_{2} \cos θ_{2} + i r_{2} \sin θ_{2}) \\ = (r_{1} \cdot r_{2}) \cos (θ_{1} + θ_{2}) + i (r_{1} \cdot r_{2}) \sin (θ_{1} + θ_{2}) \\ = a_{3} + b_{3} i \end{aligned}$

Javier doesn’t understand how Veronica’s claim can be true; that is,

$(r_{1} \cos θ_{1} + i r_{1} \sin θ_{1}) (r_{2} \cos θ_{2} + i r_{2} \sin θ_{2}) = (r_{1} \cdot r_{2}) \cos (θ_{1} + θ_{2}) + i (r_{1} \cdot r_{2}) \sin (θ_{1} + θ_{2}) .$

So, he decides to try out her rule for a specific example.

5.

Javier’s experiment:

a.

Pick two complex numbers written in the form $a + b i$ and multiply them together algebraically, as you normally would.

b.

Rewrite both of the complex numbers in polar form.

c.

Multiply the polar forms of the two complex numbers together using Veronica’s rule.

d.

Convert the product from polar form back to $a + b i$ form.

e.

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.

6.

Javier’s proof:

a.

Multiply out $(r_{1} \cos θ_{1} + i r_{1} \sin θ_{1}) (r_{2} \cos θ_{2} + i r_{2} \sin θ_{2})$ as the product of two binomials.

b.

Rewrite the results using the sum and difference identities for sine and cosine that you wrote in the task Double Identity.

c.

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 $n$ th power is just like repeated multiplication, we can write a rule for finding powers of a complex number written in polar form, which will be much easier than multiplying out ${(a + b i)}^{n}$ . Sweet!”

7.

Finish Javier’s rule for ${(a + b i)}^{n} = (r \cdot \cos {θ + i \cdot r \sin θ)}^{n} =$

Javier is wondering what happens to complex numbers as they are raised to higher and higher powers. He starts with the complex number $1 + \sqrt{3} i = 2 \cos 60 + 2 i \sin 60$ .

8.

Use Javier’s rule to raise $1 + \sqrt{3} i$ to the following powers by using the polar form of this complex number, $2 \cos 60 + 2 i \sin 60$ , then convert the resulting complex number in polar form back to the form $a + b i$ :

a.

${(1 + \sqrt{3} i)}^{1} =$

b.

${(1 + \sqrt{3} i)}^{2} =$

c.

${(1 + \sqrt{3} i)}^{3} =$

d.

${(1 + \sqrt{3} i)}^{4} =$

e.

${(1 + \sqrt{3} i)}^{5} =$

f.

${(1 + \sqrt{3} i)}^{6} =$

9.

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 $a + b i$ is plotted as the vector from the origin to the point $(a, b)$ . How does your understanding of exponential growth show up in the diagram? In the polar form of the powers?

10.

Javier was surprised to see that ${(1 + \sqrt{3} i)}^{3} = - 8$ and ${(1 + \sqrt{3} i)}^{6} = 64$ . He knows that ${(- 2)}^{3} = - 8$ and $2^{6} = 64$ , because $\sqrt[3]{- 8} = - 2$ and $\sqrt[6]{64} = 2$ , but now he has found complex numbers that seem to do the same things.

a.

What does ${(1 + \sqrt{3} i)}^{3} = - 8$ imply about $1 + \sqrt{3} i$ ?

b.

What does ${(1 + \sqrt{3} i)}^{6} = 64$ imply about $1 + \sqrt{3} i$ ?

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 $x^{n} = a$ , we expect $n$ real or complex roots of $a$ , since the Fundamental Theorem of Algebra says the polynomial equation

$x^{n} - a = 0$ will have $n$ solutions over the complex numbers.

Idea #2: The modulus, or magnitude of the vectors representing the roots, are all the same.

Idea #3: The $n$ roots are $\frac{360 °}{n}$ apart.

11.

Based on these ideas, plot the cube roots of $- 8$ on the complex plane.

What are the three cubes roots of $- 8$ ?

12.

Based on these ideas, plot the sixth roots of $64$ on the complex plane

What are the six sixth roots of $64$ ?

Ready for More?

In the real number system, $\sqrt{- 4}$ is undefined. However, in the complex number system, the equation $x^{2} = - 4$ has two solutions. Find the two square roots of $- 4$ in two different ways:

1.

By solving the equation $x^{2} = - 4$ for its complex solutions.

2.

Using the method described in problems 10–12.

3.

Verify these two methods give the same results.

Takeaways

The complex number $a + b i$ can be written in polar form as:

The polar form of complex numbers makes computation easier using the following rules:

Multiplication:

Raise to the $n th$ power:

Find all the $n th$ roots:

Vocabulary

modulus
polar coordinates
rectangular coordinate system
Bold terms are new in this lesson.

Lesson Summary

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 $n th$ roots that are easy to find in polar form.

Retrieval

1.

Apply the rule $(x, y) \to (x - 7, y - 4)$ to point $A$ . Label as $A^{'}$ .

2.

Use the definition of $\log x$ to find the value of $x$ . Recall that $\log x$ has a base of $10$ . (NO CALCULATORS)

a.

$\log x = - 1$

b.

$\log 10^{P} = x$