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Precalculus: High School27 chapters | 212 lessons | 1 flashcard set

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Lesson Transcript

Instructor:
*Yuanxin (Amy) Yang Alcocer*

Amy has a master's degree in secondary education and has taught math at a public charter high school.

After watching this video lesson, you will be able to convert complex numbers from rectangular form to polar form easily by following the formulas you will see here. You will also learn how to find the power of a complex number.

Just like we make use of our imagination when we daydream, we also make use of imaginary things in math when we talk about **complex numbers**. These are our math numbers with a real and imaginary part. They are written in the form *a + bi* where *a* is the real part and *b* is the imaginary part. Just like a daydream can play out like a television show, so do our complex numbers graph out like our regular (*x, y*) points on the Cartesian plane. The only difference is that instead of an (*x, y*) point, we now have an (*a, b*) point. Our *x* axis is now our real axis and our *y* axis is now our imaginary axis. But everything else is the same.

What are some examples of complex numbers? We can have 1 + 2i, 3 - 4i, or even -2 + 7i. Turning them into coordinate points, we have (1, 2), (3, -4), and (-2, 7).

This video lesson is about turning our complex number into polar form, so let's talk about that now. Remember that in polar form, instead of a real axis and an imaginary axis or an *x*-axis and a *y*-axis, we now label the same points with a radius and an angle telling us how far in a counterclockwise direction we go from the positive *x* axis or positive real axis. The conversion of our complex number into polar form is surprisingly similar to converting a rectangle (*x, y*) point to polar form. The formulas are identical actually and so is the process. It is just that instead of *x* and *y*, we now have *a* and *b*. Our formulas are the following:

When we use these formulas, we turn a complex number, *a + bi*, into its polar form of *z = r(cos (theta) + i*sin (theta))* where *a = r*cos (theta)* and *b = r*sin (theta)*.

Be careful when you are calculating the angle. Look first into which quadrant your complex number graphs. If it is the first quadrant, then you can rely on your calculator's answer for the inverse tangent. If it lies in either the second or third quadrant, then you need to add 180 degrees or pi to your calculator's answer. Lastly, if the complex number lies in the fourth quadrant, then you need to add 360 degrees or 2pi to the answer you get from the calculator. This is because your calculator only gives you the very first answer that fits, but this answer may not put you in the right quadrant.

Let's look at an example.

*Convert 2 - 4i to polar form.*

We have our *a* = 2 and our *b* = -4. Plugging these into our *r = sqrt (a^2 + b^2)*, we get *r = sqrt (2^2 + (-4)^2) = sqrt (4 + 16) = sqrt (20) = 4.47*. Okay, we have found our radius. It is the square root of 20, roughly 4.47.

Now, we need to find our angle. We calculate *theta = tan^-1 (-4/2) = tan^-1 (-2) = -63.4 degrees*. But wait, my complex number is in the fourth quadrant, so I need to add 360 degrees to this answer to get my real angle. I get -63.4 + 360 = 296.6 degrees. Now I have my angle, too.

I can now plug these into my polar form of my complex number, *z = r (cos (theta) + i*sin (theta) = 4.47 (cos (296.6) + i*sin (296.6))*. My answer, then, is *z = 4.47 (cos (296.6) + i*sin (296.6))*. I am done!

That wasn't too bad. Just keep in mind the quadrant that your complex number is in when calculating your angle and make the proper adjustments.

Now, let's talk about **De Moivre's formula**. This formula tells us how to find the power of a complex number.

Looking at the formula, we see that it tells us that the power of a complex number is simply the arguments of the cosine and sine parts multiplied by the power. This makes our life rather easy. Let's take a look.

We will continue with our complex number in polar form, *z = 4.47 (cos (296.6) + i*sin (296.6))*.

*Find z^10.*

First, we will rewrite our number as *z = (sqrt (20)) (cos (296.6) + i*sin (296.6))*. Now, *z^10 = 20^5 (cos (296.6) + i*sin (296.6))^10*. We got 20^5 since a square root taken to the tenth power turns into a power of 5. We can go ahead and evaluate 20^5. It is 3,200,000. We now have *z^10 = 3,200,000 (cos (296.6) + i*sin (296.6))^10*. Applying De Moivre's formula, we get *z^10 = 3,200,000 (cos (10*296.6) + i*sin (10*296.6))*. This turns into *z^10 = 3,200,000 (cos (2,966) + i*sin (2,966))*. Evaluating this, we get *z^10 = 3,200,000 (0.07 + i*0.997) = 224,000 + i*3,190,400*. And we are done.

Let's review what we've learned:

We learned that **complex numbers** are our math numbers with a real and imaginary part. They are written in the form *a + bi* where *a* is the real part and *b* is the imaginary part. To convert a complex number into polar form, we use these formulas:

We are careful when calculating the inverse tangent. If our original complex number is in the second or third quadrant, then we add 180 to our answer. If our original complex number is in the fourth quadrant, then we add 360.

If we wanted to find the power of a complex number, then we use **De Moivre's formula**:

To use this, we first convert our complex number to polar form and then use De Moivre's formula to help us with the calculation of finding the power of our complex number in polar form.

You should have the ability to do the following after this lesson:

- Define complex numbers
- Identify the formulas to convert complex numbers into polar form
- Recall the special precautions that need to be taken when calculating the inverse tangent
- Describe how to use De Moivre's formula to find the power of a complex number

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Precalculus: High School27 chapters | 212 lessons | 1 flashcard set

- Graphing Functions in Polar Coordinates: Process & Examples 7:18
- Complex Numbers in Polar Form: Process & Examples 7:21
- Converting Between Parametric & Rectangular Forms 5:33
- Graphs of Parametric Equations 5:37
- Parametric Equations in Applied Contexts 5:29
- Conic Sections in Polar & Parametric Forms 6:35
- Go to Polar Coordinates and Parameterizations

- Go to Continuity

- Go to Limits

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