DeMoivre's Theorem

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SWBAT use DeMoivre's Theorem to find powers of complex numbers

Big Idea

If I know how to multiply complex numbers how can I find powers of complex numbers?

Bell Work

10 minutes

The lesson today starts with an activity that is both review and discovery. The question reviews how to multiply complex numbers in trigonometric form by asking students to find the z^2, z^3, and z^4. Students will need the meaning of a power and use yesterday's work on operations to find the answers. 

As the students work the power problems many will see the pattern and be able to find a rule for determining the power of a complex number in trigonometric form. 

I give students several minutes (3-4) to find the results for the requested powers. We put those answers on the board and verify they are correct.  

DeMoivre's Theorem

10 minutes

I now move to the discovery part of the bell work. I ask,  "How they would find z^6 or z^10?" Usually a student sees the pattern and explains that the coefficient is raised to that power and the angle is multiplied by the power.  If no one sees a relationship, I will take the answers and break them down by asking:

  • How did you find z^2?
  • How did you do z^3?
  • So you used your answer for z^2 to help find the other 2 answers can we write the coefficient in another way using multiplication or exponents?
  • The angle in your answer for z^3 is 3 times the original angle why is that?

These questions guide students in seeing the pattern. As a class we write a rule for z^n.  I explain that this equation is called DeMoivre's Theorem.  To make it interesting a ask students if they think this theorem was discovered like we have just done? I want students to see they are using mathematical practices in this discovery.

To verify students understanding I give students a problem to work. As students work I check students understanding and help those that are struggling. Some struggles will be realizing the complex number must first be rewritten in trigonometric form. I remind students about the rule and how the complex numbers looks to use the theorem. 


5 minutes

To end the class today I give students 3 problems and ask them to determine if it would be easier to use DeMoivre's Theorem to evaluate or to just multiply out the power then explain why they made that decision. 

These problems are designed to make the students think about different methods. Two of the problems are asking students to square a number the first is in standard form which most students will say to just multiply the problem out.  The last 2 problems can be done easily using either method.  Most students will use DeMoivre's Theorem since the last problem is already in trigonometric form. The second problem has a larger power which makes multiplying out using algebra techniques harder than using DeMoivre's Theorem.