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DeMoivre's Theorem. Lesson 5.3. Using Trig Representation. Recall that a complex number can be represented as Then it follows that What about z 3 ? . DeMoivre's Theorem. In general (a + b i ) n is Apply to Try . Using DeMoivre to Find Roots. Again, starting with a + b i =

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DeMoivre's Theorem

Lesson 5.3


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Using Trig Representation

  • Recall that a complex number can be represented as

  • Then it follows that

  • What about z3 ?


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DeMoivre's Theorem

  • In general (a + bi)n is

  • Apply to

  • Try


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Using DeMoivre to Find Roots

  • Again, starting with a + bi =

  • also works when n is a fraction

    • Thus we can take a root of a complex number


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Using DeMoivre to Find Roots

  • Note that there will be n such roots

    • One each for k = 0, k = 1, … k = n – 1

  • Find the two square roots of

    • Represent as z = r cis θ

    • What is r?

    • What is θ?


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Graphical Interpretation of Roots

  • Solutions are:

Roots will be equally spaced around a circle with radius r1/2


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Graphical Interpretation of Roots

  • Consider cube root of 27

  • Using DeMoivre's Theorem

Roots will be equally spaced around a circle with radius r1/3


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Roots of Equations

  • Recall that one method of solving polynomials involves taking roots of both sides

    • x4 + 16 = 0x4 = - 64

  • Now we can determine the roots(they are all complex)

Try out spreadsheet for complex roots


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Assignment

  • Lesson 5.3

  • Page 354

  • Exercises 1 – 41 EOO


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