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AA Section 6-8

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Imaginary Numbers

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• AA Section 6-8

1. 1. Section 6-8 Imaginary Numbers Friday, February 13, 2009
2. 2. Warmup Simplify the following: 2 2 () ( ) 1. 2 3 2. −3 2 3. 6 • 15 4. -100 Friday, February 13, 2009
3. 3. Warmup Simplify the following: 2 2 () ( ) 1. 2 3 2. −3 2 12 3. 6 • 15 4. -100 Friday, February 13, 2009
4. 4. Warmup Simplify the following: 2 2 () ( ) 1. 2 3 2. −3 2 12 18 3. 6 • 15 4. -100 Friday, February 13, 2009
5. 5. Warmup Simplify the following: 2 2 () ( ) 1. 2 3 2. −3 2 12 18 3. 6 • 15 4. -100 ≈ 9.49 Friday, February 13, 2009
6. 6. Warmup Simplify the following: 2 2 () ( ) 1. 2 3 2. −3 2 12 18 3. 6 • 15 4. -100 ≈ 9.49 ??? Friday, February 13, 2009
7. 7. Deﬁnition Friday, February 13, 2009
8. 8. Deﬁnition When k > 0, the two solutions to x2 = k are k and − k Friday, February 13, 2009
9. 9. Big Question Friday, February 13, 2009
10. 10. What Friday, February 13, 2009
11. 11. is Friday, February 13, 2009
12. 12. −k ? Friday, February 13, 2009
13. 13. Friday, February 13, 2009
14. 14. What is −k ? Friday, February 13, 2009
15. 15. Imaginary Number Friday, February 13, 2009
16. 16. Imaginary Number i = −1 Friday, February 13, 2009
17. 17. Theorem Friday, February 13, 2009
18. 18. Theorem If k > 0, −k = −1 k = i k Friday, February 13, 2009
19. 19. Example 1 Solve. x2 = -100 Friday, February 13, 2009
20. 20. Example 1 Solve. x2 = -100 2 x = ± −100 Friday, February 13, 2009
21. 21. Example 1 Solve. x2 = -100 2 x = ± −100 x = ± −1 100 Friday, February 13, 2009
22. 22. Example 1 Solve. x2 = -100 2 x = ± −100 x = ± −1 100 x=± Friday, February 13, 2009
23. 23. Example 1 Solve. x2 = -100 2 x = ± −100 x = ± −1 100 x = ±i Friday, February 13, 2009
24. 24. Example 1 Solve. x2 = -100 2 x = ± −100 x = ± −1 100 x = ± i 10 Friday, February 13, 2009
25. 25. Example 1 Solve. x2 = -100 2 x = ± −100 x = ± −1 100 x = ± i 10 x = ±10i Friday, February 13, 2009
26. 26. Example 2 Show that i 7 is a square root of -7. Friday, February 13, 2009
27. 27. Example 2 Show that i 7 is a square root of -7. 2 (i 7 ) Friday, February 13, 2009
28. 28. Example 2 Show that i 7 is a square root of -7. 2 (i 7 ) 2 ( )( ) 2 7 =i Friday, February 13, 2009
29. 29. Example 2 Show that i 7 is a square root of -7. 2 (i 7 ) 2 ( )( 7 ) 2 =i = ( −1) ( 7 ) Friday, February 13, 2009
30. 30. Example 2 Show that i 7 is a square root of -7. 2 (i 7 ) 2 ( )( 7 ) 2 =i = ( −1) ( 7 ) = −7 Friday, February 13, 2009
31. 31. Example 3 Simplify. b. (6i)(4i) a. −4 − −49 Friday, February 13, 2009
32. 32. Example 3 Simplify. b. (6i)(4i) a. −4 − −49 = 2i Friday, February 13, 2009
33. 33. Example 3 Simplify. b. (6i)(4i) a. −4 − −49 = 2i - Friday, February 13, 2009
34. 34. Example 3 Simplify. b. (6i)(4i) a. −4 − −49 = 2i - 7i Friday, February 13, 2009
35. 35. Example 3 Simplify. b. (6i)(4i) a. −4 − −49 = 2i - 7i = -5i Friday, February 13, 2009
36. 36. Example 3 Simplify. b. (6i)(4i) a. −4 − −49 = 2i - 7i = 24i2 = -5i Friday, February 13, 2009
37. 37. Example 3 Simplify. b. (6i)(4i) a. −4 − −49 = 2i - 7i = 24i2 = -5i = 24(-1) Friday, February 13, 2009
38. 38. Example 3 Simplify. b. (6i)(4i) a. −4 − −49 = 2i - 7i = 24i2 = -5i = 24(-1) = -24 Friday, February 13, 2009
39. 39. Example 3 Simplify. −25 d. c. −32 + −2 −81 Friday, February 13, 2009
40. 40. Example 3 Simplify. −25 d. c. −32 + −2 −81 2 = −16g + −2 Friday, February 13, 2009
41. 41. Example 3 Simplify. −25 d. c. −32 + −2 −81 2 = −16g + −2 = −16 2 + −2 Friday, February 13, 2009
42. 42. Example 3 Simplify. −25 d. c. −32 + −2 −81 2 = −16g + −2 = −16 2 + −2 = 4i 2 + i 2 Friday, February 13, 2009
43. 43. Example 3 Simplify. −25 d. c. −32 + −2 −81 2 = −16g + −2 = −16 2 + −2 = 4i 2 + i 2 = 5i 2 Friday, February 13, 2009
44. 44. Example 3 Simplify. −25 d. c. −32 + −2 −81 2 = −16g + −2 5i = −16 2 + −2 = 9i = 4i 2 + i 2 = 5i 2 Friday, February 13, 2009
45. 45. Example 3 Simplify. −25 d. c. −32 + −2 −81 2 = −16g + −2 5i = −16 2 + −2 = 9i = 4i 2 + i 2 5 = 5i 2 = 9 Friday, February 13, 2009
46. 46. Example 4 Simplify. −36 −64 Friday, February 13, 2009
47. 47. Example 4 Simplify. −36 −64 = 6ig8i Friday, February 13, 2009
48. 48. Example 4 Simplify. −36 −64 = 6ig8i 2 = 48i Friday, February 13, 2009
49. 49. Example 4 Simplify. −36 −64 = 6ig8i 2 = 48i = −48 Friday, February 13, 2009
50. 50. Example 4 Simplify. −36 −64 = 6ig8i 2 = 48i = −48 **NOTE** Friday, February 13, 2009
51. 51. Example 4 Simplify. −36 −64 = 6ig8i 2 = 48i = −48 **NOTE** Do NOT combine radicals that have negatives inside! Friday, February 13, 2009
52. 52. Example 4 Simplify. −36 −64 = 6ig8i 2 = 48i = −48 **NOTE** Do NOT combine radicals that have negatives inside! −36 −64 ≠ 2304 = 48 Friday, February 13, 2009
53. 53. Homework Friday, February 13, 2009
54. 54. Homework p. 391 #1 - 29 “They can because they think they can.” - Virgil Friday, February 13, 2009