1. Section 7Chapter 8

2. 1 Copyright © 2012, 2008, 2004 Pearson Education, Inc. Objectives 2 6 5 3 4 Complex Numbers Simplify numbers of the form where b > 0. Recognize subsets of the complex numbers. Add and subtract complex numbers. Multiply complex numbers. Divide complex numbers. Find powers of i. 8.7

3. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Simplify numbers of the form where b > 0. Objective 1 Slide 8.7- 3

4. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Imaginary Unit i The imaginary unit i is defined as That is, i is the principal square root of –1. Slide 8.7- 4 Simplify numbers of the form where b > 0.

5. Copyright © 2012, 2008, 2004 Pearson Education, Inc. For any positive real number b, Slide 8.7- 5 Simplify numbers of the form where b > 0. It is easy to mistake for with the i under the radical. For this reason, we usually write as as in the definition of

6. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Write each number as a product of a real number and i. Slide 8.7- 6 CLASSROOM EXAMPLE 1 Simplifying Square Roots of Negative Numbers Solution:

7. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Multiply. Slide 8.7- 7 CLASSROOM EXAMPLE 2 Multiplying Square Roots of Negative Numbers Solution:

8. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Divide. Slide 8.7- 8 CLASSROOM EXAMPLE 3 Dividing Square Roots of Negative Numbers Solution:

9. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Recognize subsets of the complex numbers. Objective 2 Slide 8.7- 9

10. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Complex Number If a and b are real numbers, then any number of the form a + bi is called a complex number. In the complex number a + bi, the number a is called the real part and b is called the imaginary part. Slide 8.7- 10 Recognize subsets of the complex numbers.

11. Copyright © 2012, 2008, 2004 Pearson Education, Inc. For a complex number a + bi, if b = 0, then a + bi = a, which is a real number. Thus, the set of real numbers is a subset of the set of complex numbers. If a = 0 and b ≠ 0, the complex number is said to be a pure imaginary number. For example, 3i is a pure imaginary number. A number such as 7 + 2i is a nonreal complex number. A complex number written in the form a + bi is in standard form. Slide 8.7- 11 Recognize subsets of the complex numbers.

12. Copyright © 2012, 2008, 2004 Pearson Education, Inc. The relationships among the various sets of numbers. Slide 8.7- 12 Recognize subsets of the complex numbers.

13. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Add and subtract complex numbers. Objective 3 Slide 8.7- 13

14. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Add. Slide 8.7- 14 CLASSROOM EXAMPLE 4 Adding Complex Numbers Solution:

15. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Subtract. Slide 8.7- 15 CLASSROOM EXAMPLE 5 Subtracting Complex Numbers Solution:

16. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Multiply complex numbers. Objective 4 Slide 8.7- 16

17. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Multiply. Slide 8.7- 17 CLASSROOM EXAMPLE 6 Multiplying Complex Numbers Solution:

18. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Slide 8.7- 18 CLASSROOM EXAMPLE 6 Multiplying Complex Numbers (cont’d) Multiply. Solution:

19. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Slide 8.7- 19 CLASSROOM EXAMPLE 6 Multiplying Complex Numbers (cont’d) Multiply. Solution:

20. Copyright © 2012, 2008, 2004 Pearson Education, Inc. The product of a complex number and its conjugate is always a real number. (a + bi)(a – bi) = a 2 – b 2 ( –1) = a 2 + b 2 Slide 8.7- 20 Multiply complex numbers.

21. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Divide complex numbers. Objective 5 Slide 8.7- 21

22. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Find the quotient. Slide 8.7- 22 CLASSROOM EXAMPLE 7 Dividing Complex Numbers Solution:

23. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Slide 8.7- 23 CLASSROOM EXAMPLE 7 Dividing Complex Numbers (cont’d) Find the quotient. Solution:

24. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Find powers of i. Objective 6 Slide 8.7- 24

25. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Because i 2 = –1, we can find greater powers of i, as shown below. i 3 = i · i 2 = i · ( –1) = –i i 4 = i 2 · i 2 = ( –1) · ( –1) = 1 i 5 = i · i 4 = i · 1 = i i 6 = i 2 · i 4 = ( –1) · (1) = –1 i 7 = i 3 · i 4 = (  i) · (1) = –I i 8 = i 4 · i 4 = 1 · 1 = 1 Slide 8.7- 25 Find powers of i.

26. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Find each power of i. Slide 8.7- 26 CLASSROOM EXAMPLE 8 Simplifying Powers of i Solution: