Imaginary

Numbers

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

Powers of i

(The i Chart)

“I one, I one!!”Negatives in the middle.

1

1

i

i

1

2

3

4

i

i

i

i

.25

.5

.75

0Reference

75

29

251

9536

5.

6.

7.

8.

i

i

i

i

Simplify

iii

1

Complex Number

Standard Form(add this to your

notes)

a bi

Add and Subtract Complex Numbers

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

Simplify

9. (3 2 ) (7 6)i i

10 8i

Simplify

10. (6 5) (1 2 )i i 6 5 1 2i i 5 7i

Simplify

11. (9 4 ) (2 3 )i i

9 4 2 3i i 7 7i

Simplify

12. 9 (10 2 ) 5i i

9 10 2 5i i 1 7i

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

Multiplying Complex Numbers

(no powers of i higher than 1 are allowed in your answer)

Multiplying Complex Numbers

14. (3 )i i 23i i

3 ( 1)i 1 3i

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

Multiplying Complex Numbers

16. 3 8 5i i 224 15 8 5i i i

24 7 5 1i 24 7 5i 29 7i

Dividing Complex Numbers

(no powers of i in the denominator are allowed in your answer)

What is a Conjugate?

Example: (2 – 4i)

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

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