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10.7 Complex Numbers

10.7 Complex Numbers. Objective 1 . Simplify numbers of the form where b > 0 . Slide 10.7- 2. Simplify numbers of the form where b > 0. . Slide 10.7- 3. Simplify numbers of the form where b > 0. .

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10.7 Complex Numbers

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  1. 10.7 Complex Numbers

  2. Objective 1 Simplify numbers of the form where b > 0. Slide 10.7- 2

  3. Simplify numbers of the form where b > 0. Slide 10.7- 3

  4. 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 10.7- 4

  5. CLASSROOM EXAMPLE 1 Simplifying Square Roots of Negative Numbers Solution: Write each number as a product of a real number and i. Slide 10.7- 5

  6. CLASSROOM EXAMPLE 2 Multiplying Square Roots of Negative Numbers Solution: Multiply. Slide 10.7- 6

  7. CLASSROOM EXAMPLE 3 Dividing Square Roots of Negative Numbers Solution: Divide. Slide 10.7- 7

  8. Objective 2 Recognize subsets of the complex numbers. Slide 10.7- 8

  9. Recognize subsets of the complex numbers. Slide 10.7- 9

  10. Recognize subsets of the complex numbers. 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 10.7- 10

  11. Recognize subsets of the complex numbers. The relationships among the various sets of numbers. Slide 10.7- 11

  12. Objective 3 Add and subtract complex numbers. Slide 10.7- 12

  13. CLASSROOM EXAMPLE 4 Adding Complex Numbers Solution: Add. Slide 10.7- 13

  14. CLASSROOM EXAMPLE 5 Subtracting Complex Numbers Solution: Subtract. Slide 10.7- 14

  15. Objective 4 Multiply complex numbers. Slide 10.7- 15

  16. CLASSROOM EXAMPLE 6 Multiplying Complex Numbers Solution: Multiply. Slide 10.7- 16

  17. CLASSROOM EXAMPLE 6 Multiplying Complex Numbers (cont’d) Solution: Multiply. Slide 10.7- 17

  18. CLASSROOM EXAMPLE 6 Multiplying Complex Numbers (cont’d) Solution: Multiply. Slide 10.7- 18

  19. Multiply complex numbers. The product of a complex number and its conjugate is always a real number. (a + bi)(a – bi) = a2 – b2( –1) = a2 + b2 Slide 10.7- 19

  20. Objective 5 Divide complex numbers. Slide 10.7- 20

  21. CLASSROOM EXAMPLE 7 Dividing Complex Numbers Solution: Find the quotient. Slide 10.7- 21

  22. CLASSROOM EXAMPLE 7 Dividing Complex Numbers (cont’d) Solution: Find the quotient. Slide 10.7- 22

  23. Objective 6 Find powers of i. Slide 10.7- 23

  24. Find powers of i. 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 Slide 10.7- 24

  25. CLASSROOM EXAMPLE 8 Simplifying Powers of i Solution: Find each power of i. Slide 10.7- 25

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