Name:_______________________ Date assigned:______________ Band:________

Precalculus | Packer Collegiate Institute

Complex Numbers, the Complex Plane, and Polar Coordinates

Warm Up:

Let 1 2 3 4 5 2 , 2 3 , 2 5 , 3 z i iz z ii z be complex numbers. Without a calculator…

(a) calculate 1 2 3 4z z z z (b) calculate 1 2

3 4

z z

z z (c) calculate 3

1 2 4

z

z z z

Section 1:

Let 1 4p i and 3q i .

(a) Multiply pq and write in “normal” complex number

form [that means: a bi form]…

(b) Divide p

q and write in “normal” complex number

form…

Let point (1,4)P and (3,1)Q being points on a

coordinate plate. Use degrees and round to the hundredths. (a) Convert P to polar form. (b) Convert Q to polar form.

Section 2: The Complex Plane

There is one more fancy set of graphing that you can do, that you may or may not remember from Algebra II (depending

on if you did that). It’s graphing on the complex plane. Instead of having two real axes (the x- and y-axes), we have a real

axes (the horizontal axis) and an imaginary axis (the vertical axis).

On the next page is a complex plane, for your viewing pleasure.

On it, plot the four imaginary numbers (and label them with their letter designations): , , ,p

p q pqq

Now suspend some disbelief and follow me here… The complex plane looks a lot like a regular x-y coordinate plane,

right? RIGHT? So for the next five minutes, pretend it is a regular coordinate plane. For each of the four points, find the

polar coordinates of them!

“polar coordinates” of p

“polar coordinates” of pq

“polar coordinates” of q “polar coordinates” of

p

q

Time to pay some attention… Do you see anything interesting about the “polar coordinates” of pq and p

q ? If not,

stare a bit harder. And harder… Only if you need a hint should you look at this footnote.1

Your conjecture:

When you multiply complex numbers in polar form, the angles _____________ and the distances _____________.

When you divide complex numbers in polar form, the angles _____________ and the distances _____________.

We’ll try our conjecture out on some simple complex numbers to see if it works!!!

Problem: Let 1 0z i and 2 2 0z i .

Thus in our “polar” form, 1z has an angle of ____ and a distance of ____.

2z has an angle of ____ and a distance of ____.

Plot 1z and 2z , and then using your conjecture above (and no other calculations), plot where you predict 1 2z z and

1 2/z z …

1 2z z has an angle of ____ and a distance of ____.

1 2/z z has an angle of ____ and a distance of ____.

Now calculate 1 2z z and 1 2/z z :

Did they match where you predicted, based on your conjecture?

1 HINT: Look at the angles for p and q, and then the angle measure for pq and p/q. And separately, look at the r values for p and q,

and the r values for pq and p/q.

Section 3: Generalizing our result…

It seems that it is true! But how do we know our conjecture will always work? Let’s prove it.

We know that every complex number can be written in polar form, so let’s start from there.

Assume we have two complex numbers p and q written in polar form, with some distance from the origin and some

angle away from the polar axis. They can be written:

1 1

2 2

cos sin

cos sin

ir

q r ir

p r

(1) Do you see why every complex number can be written in this way? Look at the diagram above and convince

yourself of this. (Hint: draw a triangle!)

(2) Find pq … remember to write it in ( ) ( )a b i form. And simplify it as much as possible! (Hint: Think of some

trig formulas!!!)

(3) Whoa! If you haven’t seen how this proves our conjecture… take a moment!

(4) Can you prove the result for /p q ?