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Section 7

Chapter 8. Section 7. 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.

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Section 7

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  1. Chapter 8 Section 7

  2. 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. Simplify numbers of the form where b > 0. Objective 1 Slide 8.7- 3

  4. Simplify numbers of the form where b > 0. Slide 8.7- 4

  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 Slide 8.7- 5

  6. Write each number as a product of a real number and i. CLASSROOM EXAMPLE 1 Simplifying Square Roots of Negative Numbers Solution: Slide 8.7- 6

  7. Multiply. CLASSROOM EXAMPLE 2 Multiplying Square Roots of Negative Numbers Solution: Slide 8.7- 7

  8. Divide. CLASSROOM EXAMPLE 3 Dividing Square Roots of Negative Numbers Solution: Slide 8.7- 8

  9. Recognize subsets of the complex numbers. Objective 2 Slide 8.7- 9

  10. Recognize subsets of the complex numbers. Slide 8.7- 10

  11. 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. Recognize subsets of the complex numbers. Slide 8.7- 11

  12. The relationships among the various sets of numbers. Recognize subsets of the complex numbers. Slide 8.7- 12

  13. Add and subtract complex numbers. Objective 3 Slide 8.7- 13

  14. Add. CLASSROOM EXAMPLE 4 Adding Complex Numbers Solution: Slide 8.7- 14

  15. Subtract. CLASSROOM EXAMPLE 5 Subtracting Complex Numbers Solution: Slide 8.7- 15

  16. Multiply complex numbers. Objective 4 Slide 8.7- 16

  17. Multiply. CLASSROOM EXAMPLE 6 Multiplying Complex Numbers Solution: Slide 8.7- 17

  18. Multiply. CLASSROOM EXAMPLE 6 Multiplying Complex Numbers (cont’d) Solution: Slide 8.7- 18

  19. Multiply. CLASSROOM EXAMPLE 6 Multiplying Complex Numbers (cont’d) Solution: Slide 8.7- 19

  20. The product of a complex number and its conjugate is always a real number. (a + bi)(a – bi) = a2 – b2( –1) = a2 + b2 Multiply complex numbers. Slide 8.7- 20

  21. Divide complex numbers. Objective 5 Slide 8.7- 21

  22. Find the quotient. CLASSROOM EXAMPLE 7 Dividing Complex Numbers Solution: Slide 8.7- 22

  23. Find the quotient. CLASSROOM EXAMPLE 7 Dividing Complex Numbers (cont’d) Solution: Slide 8.7- 23

  24. Find powers of i. Objective 6 Slide 8.7- 24

  25. Because i2 = –1, we can find greater powers of i, as shown below. i3 = i · i2 = i · ( –1) = –i i4 = i2· i2 = ( –1) · ( –1) = 1 i5 = i · i4 = i · 1 = i i6 = i2· i4 = ( –1) · (1) = –1 i7 = i3· i4 = (i) · (1) = –I i8 = i4· i4 = 1 · 1 = 1 Find powers of i. Slide 8.7- 25

  26. Find each power of i. CLASSROOM EXAMPLE 8 Simplifying Powers of i Solution: Slide 8.7- 26

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