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2. Complex Numbers. ALGEBRA 2 LESSON 5-6. 1. Simplify – –169. 2. Find |–1 + i |. –13 i. 5-6. Pages 274–276 Exercises. 1. 2 i 2. i 7 3. i 15 4. 9 i 5. 5 i 2 6. 4 i 7. 4 i 2 8. 9 i . 9. –10 i 10. 6 i 2 11. 2 + i 3

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Complex numbers

2

Complex Numbers

ALGEBRA 2 LESSON 5-6

1. Simplify – –169.

2. Find |–1 + i|

–13i

5-6


Complex numbers1

Pages 274–276 Exercises

1. 2i

2. i 7

3. i 15

4. 9i

5. 5i 2

6. 4i

7. 4i 2

8. 9i

9. –10i

10. 6i 2

11. 2 + i 3

12. 8 + 2i 2

13. 6 – 2i 7

14. 3 + 2i

15. 7 – 5i

16. 2 + i

17. –2 – 5i 2

18. 4 + 6i 2

19. 2

20. 13

21. 2 2

22. 17

23. 3 5

24. –4i

Complex Numbers

ALGEBRA 2 LESSON 5-6

5-6


Complex numbers2

25. –5 + 3i

26. –9 – i

27. 3 + 2i

28. 4 – 7i

29. 6 + 3i

30. 1 – 7i

31. 7 + 4i

32. –2 – 3i

33. 10 + 6i

34. –7 – 10i

35. 10

36. 26 – 7i

37. 9 + 58i

38. 9 – 23i

39. –36

40. 65 + 72i

41. ± 5i

42. ±

43. ±

44. ± i 7

45. ±6i

46. ±

47. –i, –1 – i, i

48. –2i, –4 – 2i, 12 + 14i

49. 1 – i, 1 – 3i, –7 – 7i

50. ± i 65

51. ±7i

8i 3

3

i 15

3

i 2

2

Complex Numbers

ALGEBRA 2 LESSON 5-6

5-6


Complex numbers3

52. ± i

53. No; the test scores were real numbers. He added the scores and divided by the number of scores. The set of real numbers is closed with respect to addition and division so he should have gotten a real number.

54. a.A: –5, B: 3 + 2i, C: 2 – i, D: 3i, E: –6 – 4i, F: –1 + 5i

b. 5, –3 – 2i, –2 + i, –3i, 6 + 4i, 1 – 5i

55. a. Check students’ work.

b. a circle with radius 10 and center at the origin

56. –5, 5

57. 288i

58. –1 + 5i

59. 10 – 4i

60. 8 – 2i

61. 11 – 5i

Complex Numbers

ALGEBRA 2 LESSON 5-6

5-6


Complex numbers4
Complex Numbers

  • You can apply all the operations of real numbers to complex numbers

  • Addition/ Additive inverse

  • Multiplication


Complex numbers5
Complex Numbers

ALGEBRA 2 LESSON 5-6

Find the additive inverse of –7 – 9i.

–7 – 9i

–(–7 – 9i) Find the opposite.

7 + 9iSimplify.

5-6


Complex numbers6
Complex Numbers

ALGEBRA 2 LESSON 5-6

Simplify (3 + 6i) – (4 – 8i).

(3 + 6i) – (4 – 8i) = 3 + (–4) + 6i + 8i Use commutative and associative properties.

= –1 + 14i Simplify.

5-6


Complex numbers7
Complex Numbers

ALGEBRA 2 LESSON 5-6

Find each product.

a. (3i)(8i)

(3i)(8i) = 24i2Multiply the real numbers.

= 24(–1)Substitute –1 for i2.

= –24 Multiply.

b. (3 – 7i)(2 – 4i)

(3 – 7i)(2 – 4i) = 6 – 14i – 12i + 28i2 Multiply the binomials.

= 6 – 26i + 28(–1) Substitute –1 for i2.

= –22 – 26iSimplify.

5-6


Complex numbers8

x= ±i 6 Find the square root of each side.

Check: 9x2 + 54 = 0

9x2 + 54 = 0

9(i 6)2 + 54 0

9(i(– 6))2 + 54 0

9(6)i2 + 54 0

9(6i2) + 54 0

54(–1) –54

54(–1) –54

54 = 54

–54 = –54

Complex Numbers

ALGEBRA 2 LESSON 5-6

Solve 9x2 + 54 = 0.

9x2 + 54 = 0

9x2 = –54 Isolate x2.

x2 = –6

5-6


Complex numbers9
Complex Numbers

  • Functions of the form f(z) = z2 + c where c is a complex number generate fractal graphs on the complex plane like the one at the left. To test if z belongs to the graph, use z as the first input value and repeatedly use each output as the next input. If the output values do not become infinitely large, then z is on the graph.

    • f(z) = z2 + a+bi


Complex numbers10

Use z = 0 as the first input value.

f(0) = 02 – 4i

= –4i

f(–4i)= (–4i)2 – 4i First output becomes second input. Evaluate for z = –4i.

= –16 – 4i

= [(–16)2 + (–16)(–4i) + (–16)(–4i) + (–4i)2] – 4i

= (256 + 128i – 16) – 4i

= 240 + 124i

Complex Numbers

ALGEBRA 2 LESSON 5-6

Find the first three output values for f(z) = z2 – 4i. Use z = 0 as the first input value.

f(–16 – 4i)= (–16 – 4i)2 – 4i Second output becomes third input. Evaluate for z = –16 – 4i.

The first three output values are –4i, –16 – 4i, 240 + 124i.

5-6


Assignment 43
Assignment 43

  • Page 274 34-48 even, 56-64 even


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