6 6 De Moivres Theorem and nth Roots

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6. 6 De Moivre’s Theorem and nth Roots Copyright © 2011 Pearson, Inc.

6. 6 De Moivre’s Theorem and nth Roots Copyright © 2011 Pearson, Inc.

What you’ll learn about n n n The Complex Plane Trigonometric Form of Complex

What you’ll learn about n n n The Complex Plane Trigonometric Form of Complex Numbers Multiplication and Division of Complex Numbers Powers of Complex Numbers Roots of Complex Numbers … and why The material extends your equation-solving technique to include equations of the form zn = c, n is an integer and c is a complex number. Copyright © 2011 Pearson, Inc. Slide 6. 1 - 2

Complex Plane Copyright © 2011 Pearson, Inc. Slide 6. 1 - 3

Complex Plane Copyright © 2011 Pearson, Inc. Slide 6. 1 - 3

Absolute Value (Modulus) of a Complex Number Copyright © 2011 Pearson, Inc. Slide 6.

Absolute Value (Modulus) of a Complex Number Copyright © 2011 Pearson, Inc. Slide 6. 1 - 4

Graph of z = a + bi Copyright © 2011 Pearson, Inc. Slide 6.

Graph of z = a + bi Copyright © 2011 Pearson, Inc. Slide 6. 1 - 5

Trigonometric Form of a Complex Number Copyright © 2011 Pearson, Inc. Slide 6. 1

Trigonometric Form of a Complex Number Copyright © 2011 Pearson, Inc. Slide 6. 1 - 6

Example Finding Trigonometric Form Copyright © 2011 Pearson, Inc. Slide 6. 1 - 7

Example Finding Trigonometric Form Copyright © 2011 Pearson, Inc. Slide 6. 1 - 7

Example Finding Trigonometric Form Copyright © 2011 Pearson, Inc. Slide 6. 1 - 8

Example Finding Trigonometric Form Copyright © 2011 Pearson, Inc. Slide 6. 1 - 8

Product and Quotient of Complex Numbers Copyright © 2011 Pearson, Inc. Slide 6. 1

Product and Quotient of Complex Numbers Copyright © 2011 Pearson, Inc. Slide 6. 1 - 9

Example Multiplying Complex Numbers Copyright © 2011 Pearson, Inc. Slide 6. 1 - 10

Example Multiplying Complex Numbers Copyright © 2011 Pearson, Inc. Slide 6. 1 - 10

Example Multiplying Complex Numbers Copyright © 2011 Pearson, Inc. Slide 6. 1 - 11

Example Multiplying Complex Numbers Copyright © 2011 Pearson, Inc. Slide 6. 1 - 11

A Geometric Interpretation of z 2 Copyright © 2011 Pearson, Inc. Slide 6. 1

A Geometric Interpretation of z 2 Copyright © 2011 Pearson, Inc. Slide 6. 1 - 12

De Moivre’s Theorem Copyright © 2011 Pearson, Inc. Slide 6. 1 - 13

De Moivre’s Theorem Copyright © 2011 Pearson, Inc. Slide 6. 1 - 13

Example Using De Moivre’s Theorem Copyright © 2011 Pearson, Inc. Slide 6. 1 -

Example Using De Moivre’s Theorem Copyright © 2011 Pearson, Inc. Slide 6. 1 - 14

Example Using De Moivre’s Theorem Copyright © 2011 Pearson, Inc. Slide 6. 1 -

Example Using De Moivre’s Theorem Copyright © 2011 Pearson, Inc. Slide 6. 1 - 15

Example Using De Moivre’s Theorem Copyright © 2011 Pearson, Inc. Slide 6. 1 -

Example Using De Moivre’s Theorem Copyright © 2011 Pearson, Inc. Slide 6. 1 - 16

nth Root of a Complex Number Copyright © 2011 Pearson, Inc. Slide 6. 1

nth Root of a Complex Number Copyright © 2011 Pearson, Inc. Slide 6. 1 - 17

Finding nth Roots of a Complex Number Copyright © 2011 Pearson, Inc. Slide 6.

Finding nth Roots of a Complex Number Copyright © 2011 Pearson, Inc. Slide 6. 1 - 18

Example Finding Cube Roots Copyright © 2011 Pearson, Inc. Slide 6. 1 - 19

Example Finding Cube Roots Copyright © 2011 Pearson, Inc. Slide 6. 1 - 19

Example Finding Cube Roots Copyright © 2011 Pearson, Inc. Slide 6. 1 - 20

Example Finding Cube Roots Copyright © 2011 Pearson, Inc. Slide 6. 1 - 20

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