1. Slide 7.5 - 1 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

2. OBJECTIVES Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Polar Form of Complex Numbers; DeMoivre’s Theorem Learn geometric representation of complex numbers. Learn to find the absolute value of complex numbers. Learn the geometric representation of the sum of complex numbers. Learn the polar form of a complex number. Learn to find the product and quotient of two complex numbers in polar form. SECTION 7.5 1 2 3 5 4

3. OBJECTIVES Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Polar Form of Complex Numbers; DeMoivre’s Theorem Learn DeMoivre’s Theorem. Learn to use DeMoivre’s Theorem to find the n th roots of a complex number. SECTION 7.5 6 7

4. Slide 7.5 - 4 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley GEOMETRIC REPRESENTATION OF COMPLEX NUMBERS The geometric representation of the complex number a + bi is the point P(a, b) in a rectangular coordinate system. When a rectangular coordinate system is used to represent complex numbers, the plane is called the complex plane or the Argand plane. The x- axis is also called the real axis, because the real part of a complex number is plotted along the x- axis. Similarly, the y-axis is also called the imaginary axis.

5. Slide 7.5 - 5 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley GEOMETRIC REPRESENTATION OF COMPLEX NUMBERS A complex number a + bi may be viewed as a position vector with initial point (0, 0) and terminal point (a, b).

6. Slide 7.5 - 6 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 1 Plotting Complex Nubmers Plot each number in the complex plane. A: 1 + 3i B: –2 + 2i C: –3 D: –2i E: 3 – i Solution

7. Slide 7.5 - 7 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley ABSOLUTE VALUE OF A COMPLEX NUMBER The absolute value of a complex number z = a + bi is:

8. Slide 7.5 - 8 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 2 Finding the Absolute Value of a Complex Number Find the absolute value of each complex number. a. 4 + 3i b. 2 – 3i c. –4 + i d. –2 – 2i e. –3i Solution

9. Slide 7.5 - 9 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley GEOMETRIC REPRESENTATION OF THE SUM OF COMPLEX NUMBERS Let points P, Q, and R represent the complex numbers a + bi, c + di, and (a + c) + (b + d)i, then R is the diagonal of the parallelogram OPRQ.

10. Slide 7.5 - 10 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 3 Adding Complex Numbers Geometrically Add the complex numbers 1 + 3i and –4 + 2i geometrically. Solution Locate P: 1 + 3i and Q: –4 + 2i. Draw line segments OP and OQ. Draw the parallelogram. R represents –3 + 5i where

11. Slide 7.5 - 11 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley POLAR FORM OF A COMPLEX NUMBER The complex number z = a + bi can be written in polar form where When a nonzero complex numbers is written in polar form, the positive number r is the modulus or absolute value of z. The angle q in the representation is called the argument of z (written q = arg z).

12. Slide 7.5 - 12 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 4 Writing a Complex Number in Polar Form Write in polar form. Express the argument  in degrees 0º ≤  ≤ 360º. Find r. Solution For, a = 1 and b = Find .

13. Slide 7.5 - 13 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley PRODUCT AND QUOTIENT RULES FOR TWO COMPLEX NUMBERS IN POLAR FORM Let be two complex numbers in polar form. Then and

14. Slide 7.5 - 14 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 6 Finding the Product and Quotient of Two Complex Numbers Leave the answers in polar form. Solution Multiply moduliAdd arguments

15. Slide 7.5 - 15 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 6 Finding the Product and Quotient of Two Complex Numbers Solution continued Divide moduliSubtract arguments

16. Slide 7.5 - 16 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley DEMOIVRE’S THEOREM Let z = r(cos  + i sin  ) be a complex number in polar form. Then, for any integer n,

17. Slide 7.5 - 17 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 8 Finding the Power of a Complex Number Let z = 1 + i. Use DeMoivre’s Theorem to find each power of z. Write answers in rectangular form. Solution Convert z to polar form. Find r and .

18. Slide 7.5 - 18 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 8 Finding the Power of a Complex Number Solution continued

19. Slide 7.5 - 19 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 8 Finding the Power of a Complex Number Solution continued

20. Slide 7.5 - 20 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley DEMOIVRE’S THEOREM FOR FINDING COMPLEX ROOTS The nth roots of a complex number w = r(cos  + i sin  ), r > 0,  in degrees are given by If  is in radians, replace 360º by 2π in z k. k = 0, 1, 2, …, n – 1.

21. Slide 7.5 - 21 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 9 Finding the Roots of a Complex Number Find the three cube roots of 1 + i in polar form, with the argument in degrees. Solution In the previous example, we showed that k = 0, 1, 2.

22. Slide 7.5 - 22 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solution continued EXAMPLE 9 Finding the Roots of a Complex Number

23. Slide 7.5 - 23 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solution continued EXAMPLE 9 Finding the Roots of a Complex Number The three cube roots of 1 + i are: