De Moivre's Theorem

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Objective

SWBAT use De Moivre's Theorem to raise complex numbers to a power.

Big Idea

How did mathematicians raise numbers to high powers before there were calculators?

Launch and Explore

15 minutes

Today we are going to build upon the work we did yesterday with multiplying complex numbers in trig form by raising these numbers to higher powers. I plan to start by asking the class how to calculate:

(sqrrt(3) - i)^10

I expect my students to groan and roll their eyes and that is perfect - I explain that I do not want to torture them and I suggest that we find a fast and efficient way to do this. I explain that we are going to use something called De Moivre's Theorem and that it will help us calculate this value quickly.

At this point, I always give some background about Abraham de Moivre. A fun fact from his Wikipedia page is that he supposedly predicted the day he was going to die by using an arithmetic series. It is noted, "As he grew older, he became increasingly lethargic and needed longer sleeping hours. He noted that he was sleeping an extra 15 minutes each night and correctly calculated the date of his death on the day when the additional sleep time accumulated to 24 hours, November 27, 1754."

The one hint that I will give students is that to calculate (sqrrt(3) - i)^10, students should keep in mind what we did in the previous lesson and put the complex number in trigonometric form. After this hint, I will let students loose and have them discuss this task with their table groups. Ten minutes is usually sufficient for them to discuss and come up with their own idea of what the answer should be.

Share

15 minutes

If students follow my hint, they will have put the number in trig form as:

cos330° + isin330°

Most of my students will realize that this expression can also be written as:

[2(cos330° + isin330°)]^10

and the 2 and will be raised to the tenth power and the quantity in the parentheses will be raised to the tenth power. For those who do, the new r value is easy to obtain. It has to be 2^10.

The other part of the quantity was more problematic. Many students knew right away that we could use our multiplication process from yesterday, and just do the same process over and over and that the new angle would be 330° times 10 since we are adding 330° to itself ten times. 

Other students were not convinced - a few students will say that the answer should have more than just two terms since if we raise (a + b) to the tenth power, there will be more than just two terms. It is important to be prepared for these types of questions so that you can clear up these types of issue that may arise. I explained that (a + b)(a + b) should have four terms (a^2 + ab + ab + b^2), but when when we multiplied just two quantities together yesterday, we still ended up with two terms because we simplified using the cos(A + B) and sin(A + B) formulas. The same thing will happen here every time we multiply 2(cos330° + isin330°) by something else. 

This image shows an approach that was especially powerful to my students who were not convinced that raising this complex number to the tenth power would  work out so nicely. They were confident in our conclusion from multiplying two complex numbers yesterday, so I showed them how raising this number to the tenth power could just be a sequence of multiplying two numbers together repeatedly. 

Summarize

10 minutes

After going through the first example, I give students this worksheet and I will have students do Question #1 with their groups. This is a good opportunity to see if students can generalize the process that we just did with in our in-class example. My students usually pick up on the structure of De Moivre's Theorem pretty quickly.

After discussing the general case of De Moivre's Theorem, students can work on the rest of the worksheet and finish up what they don't finish for homework. It might be helpful to show students how they can check their answer for these problems using their graphing calculator.

In the video below, I discuss two problems in the homework that may be tricky for some students.

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