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