Imaginary Numbers. You CAN have a negative under the radical. You will bring out an “i“...
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![Page 1: Imaginary Numbers. You CAN have a negative under the radical. You will bring out an “i“ (imaginary).](https://reader035.fdocuments.in/reader035/viewer/2022070414/5697c01e1a28abf838cd0fa3/html5/thumbnails/1.jpg)
Imaginary
Numbers
![Page 2: Imaginary Numbers. You CAN have a negative under the radical. You will bring out an “i“ (imaginary).](https://reader035.fdocuments.in/reader035/viewer/2022070414/5697c01e1a28abf838cd0fa3/html5/thumbnails/2.jpg)
1 i 1. 16
2. 81
3. 45
4. 200
You CAN have a negative under the radical. You will
bring out an “i“ (imaginary).
4i9i3 5i10 2i
![Page 3: Imaginary Numbers. You CAN have a negative under the radical. You will bring out an “i“ (imaginary).](https://reader035.fdocuments.in/reader035/viewer/2022070414/5697c01e1a28abf838cd0fa3/html5/thumbnails/3.jpg)
Powers of i
(The i Chart)
![Page 4: Imaginary Numbers. You CAN have a negative under the radical. You will bring out an “i“ (imaginary).](https://reader035.fdocuments.in/reader035/viewer/2022070414/5697c01e1a28abf838cd0fa3/html5/thumbnails/4.jpg)
“I one, I one!!”Negatives in the middle.
1
1
i
i
1
2
3
4
i
i
i
i
.25
.5
.75
0Reference
![Page 5: Imaginary Numbers. You CAN have a negative under the radical. You will bring out an “i“ (imaginary).](https://reader035.fdocuments.in/reader035/viewer/2022070414/5697c01e1a28abf838cd0fa3/html5/thumbnails/5.jpg)
75
29
251
9536
5.
6.
7.
8.
i
i
i
i
Simplify
iii
1
![Page 6: Imaginary Numbers. You CAN have a negative under the radical. You will bring out an “i“ (imaginary).](https://reader035.fdocuments.in/reader035/viewer/2022070414/5697c01e1a28abf838cd0fa3/html5/thumbnails/6.jpg)
Complex Number
Standard Form(add this to your
notes)
a bi
![Page 7: Imaginary Numbers. You CAN have a negative under the radical. You will bring out an “i“ (imaginary).](https://reader035.fdocuments.in/reader035/viewer/2022070414/5697c01e1a28abf838cd0fa3/html5/thumbnails/7.jpg)
Add and Subtract Complex Numbers
![Page 8: Imaginary Numbers. You CAN have a negative under the radical. You will bring out an “i“ (imaginary).](https://reader035.fdocuments.in/reader035/viewer/2022070414/5697c01e1a28abf838cd0fa3/html5/thumbnails/8.jpg)
Add/Subt Complex Numbers
1.Treat the i’s like variables2.Combine like terms3.Simplify (no powers of i
higher than 1 are allowed)
4.Answer in standard form a + bi
![Page 9: Imaginary Numbers. You CAN have a negative under the radical. You will bring out an “i“ (imaginary).](https://reader035.fdocuments.in/reader035/viewer/2022070414/5697c01e1a28abf838cd0fa3/html5/thumbnails/9.jpg)
Simplify
9. (3 2 ) (7 6)i i
10 8i
![Page 10: Imaginary Numbers. You CAN have a negative under the radical. You will bring out an “i“ (imaginary).](https://reader035.fdocuments.in/reader035/viewer/2022070414/5697c01e1a28abf838cd0fa3/html5/thumbnails/10.jpg)
Simplify
10. (6 5) (1 2 )i i 6 5 1 2i i 5 7i
![Page 11: Imaginary Numbers. You CAN have a negative under the radical. You will bring out an “i“ (imaginary).](https://reader035.fdocuments.in/reader035/viewer/2022070414/5697c01e1a28abf838cd0fa3/html5/thumbnails/11.jpg)
Simplify
11. (9 4 ) (2 3 )i i
9 4 2 3i i 7 7i
![Page 12: Imaginary Numbers. You CAN have a negative under the radical. You will bring out an “i“ (imaginary).](https://reader035.fdocuments.in/reader035/viewer/2022070414/5697c01e1a28abf838cd0fa3/html5/thumbnails/12.jpg)
Simplify
12. 9 (10 2 ) 5i i
9 10 2 5i i 1 7i
![Page 13: Imaginary Numbers. You CAN have a negative under the radical. You will bring out an “i“ (imaginary).](https://reader035.fdocuments.in/reader035/viewer/2022070414/5697c01e1a28abf838cd0fa3/html5/thumbnails/13.jpg)
Simplify
4 3 4 313. (11 4 ) (2 6 )i i i i 4 3 4 311 4 2 6i i i i
9 10i
4 39 10i i 9 1 10 i
![Page 14: Imaginary Numbers. You CAN have a negative under the radical. You will bring out an “i“ (imaginary).](https://reader035.fdocuments.in/reader035/viewer/2022070414/5697c01e1a28abf838cd0fa3/html5/thumbnails/14.jpg)
Multiplying Complex Numbers
(no powers of i higher than 1 are allowed in your answer)
![Page 15: Imaginary Numbers. You CAN have a negative under the radical. You will bring out an “i“ (imaginary).](https://reader035.fdocuments.in/reader035/viewer/2022070414/5697c01e1a28abf838cd0fa3/html5/thumbnails/15.jpg)
Multiplying Complex Numbers
14. (3 )i i 23i i
3 ( 1)i 1 3i
![Page 16: Imaginary Numbers. You CAN have a negative under the radical. You will bring out an “i“ (imaginary).](https://reader035.fdocuments.in/reader035/viewer/2022070414/5697c01e1a28abf838cd0fa3/html5/thumbnails/16.jpg)
Multiplying Complex Numbers
15. (2 3 )( 6 2 )i i 212 4 18 6i i i
12 22 6( 1)i 12 22 6i 6 22i
![Page 17: Imaginary Numbers. You CAN have a negative under the radical. You will bring out an “i“ (imaginary).](https://reader035.fdocuments.in/reader035/viewer/2022070414/5697c01e1a28abf838cd0fa3/html5/thumbnails/17.jpg)
Multiplying Complex Numbers
16. 3 8 5i i 224 15 8 5i i i
24 7 5 1i 24 7 5i 29 7i
![Page 18: Imaginary Numbers. You CAN have a negative under the radical. You will bring out an “i“ (imaginary).](https://reader035.fdocuments.in/reader035/viewer/2022070414/5697c01e1a28abf838cd0fa3/html5/thumbnails/18.jpg)
Dividing Complex Numbers
(no powers of i in the denominator are allowed in your answer)
![Page 19: Imaginary Numbers. You CAN have a negative under the radical. You will bring out an “i“ (imaginary).](https://reader035.fdocuments.in/reader035/viewer/2022070414/5697c01e1a28abf838cd0fa3/html5/thumbnails/19.jpg)
What is a Conjugate?
Example: (2 – 4i)
![Page 20: Imaginary Numbers. You CAN have a negative under the radical. You will bring out an “i“ (imaginary).](https://reader035.fdocuments.in/reader035/viewer/2022070414/5697c01e1a28abf838cd0fa3/html5/thumbnails/20.jpg)
17. Dividing – Multiply top & bottom by the Conjugate
3 42 4
ii
2 4
2 4
i
i
2
2
6 12 8 164 8 8 16
i i ii i i
6 20 16
4
1
116
i
10 20
20i
10 2020 20
i
12
i
![Page 21: Imaginary Numbers. You CAN have a negative under the radical. You will bring out an “i“ (imaginary).](https://reader035.fdocuments.in/reader035/viewer/2022070414/5697c01e1a28abf838cd0fa3/html5/thumbnails/21.jpg)
18. Dividing – Multiply top & bottom by the Conjugate
ii
5 23 4
3 4
3 4
i
i
2
2
15 20 6 89 12 12 16
i i ii i i
15 26 8
9
1
116
i
7 26
25i
7 2625 25
i