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Alg 3 1.10
1
1.10 Imaginary Numbers

Definition: i = 1
ex.
9

i is the “imaginary unit”

in general:
b
POWERPOINT
9  1
2


3
3i
b 2  1 
 3  1
(i 3 )2 = i2 ( 3 )2 = -3
SIMPLIFYING EXPRESSIONS WITH i
1)
81
2) 121
14
21
4) 8i  3i
3) 
5) -2i  -5i
6)
9  4
EVALUATING POWERS OF i
i1 =
i5 =
i2 =
i6 =
i3 =
i7 =
i4 =
i8 =
i 3
bi is a “pure imaginary number”
bi
i2 = -1
ex. (3i)2 = 32 i2 = -9

i9 etc.
6) i12 =
7) i17 =
8) i23 =
9) i26 =
SOLVING EQUATIONS
10) x2 + 81 = 0
11) x2 + 72 = 0
Alg 3 1.10
2
PRACTICE
12) 225
14)

15
40
13)  24
15) (4i)(-8i)
16)
i 21
17) i19
18)
x2  7  0
19) x 2  24  0
The Set of Complex Numbers
General Form: a + bi
a, b are real numbers, i is the imaginary unit
All real numbers are complex numbers (b = 0, therefore a + bi = a)
All pure imaginary numbers are complex numbers (a = 0, therefore a + bi = bi)
Alg 3 1.10
3
Equal Complex Numbers: 2 complex numbers are equal only if their real parts are equal AND
their imaginary parts are equal.
a + bi = c + di if and only if a = b and c = d
1) 5x - 3yi = 2+ 9i
Real: 5x = 2
2) 4x + 9yi = 12
; Imaginary: -3y = 9
Adding Complex Numbers: add the real parts and add the imaginary parts
3) (8 + 7i) + (-12 + 11i)
4) (-7 + 4i) + (9 - 6i)
Subtracting Complex Numbers: subtract the real parts and subtract the imaginary parts
5) (-7 + 4i) - (3 + 2i)
6) (9 - 6i) - (12 + 2i)
Multiplying Complex Numbers: FOIL
7) (8 + 5i)(2 - 3i)
8) (2 + 3i)(3 - 4i)
9) (-6 + 2i)(5 - 3i)
10) (-5 + 6i)2
Alg 3 1.10
4
1.10 Simplifying Expressions with Complex Numbers
A simplified fraction has no i in the denominator. Why?
Pure Imaginary Number in the Denominator
1)
2  8i
3i
2)
3  7i
2i
Complex Number in the Denominator
3)
1
6  3i
conjugates
complex conjugates
same terms, different sign
x- 3,x+ 3
take the product (x - 3 )(x + 3 )
x2 + x 3 - x 3 - 3
x2 - 3
6 + 3i, 6 - 3i
(6 + 3i)(6 - 3i)
36 - 18i + 18i - 9i2
45
product is real, rational number
product is real number
4)
1
9  7i
5)
2
3x 8 yi
6)
3  9i
4  2i
7)
4  5i
3  7i
Multiplicative Inverse = Reciprocal
Alg 3 1.10
5
Multiplicative Inverse = ? Don’t forget to rationalize!
8) 7 + 2i
9)
3i
2 4 i
10) 8 - 2i
Practice
7  13i
2i
11) Simplify.
5
3 i
12) Simplify.
13) Simplify.
4 i 2
i 2
14) (5 - 7i)(5 + 7i)
Alg 3 1.10
6
ALGEBRA REVIEW SHEET
1.10
Simplify each of the following please.
 81
(1)
(2)
(13) 4i
3
 3i 13

 15  10


(14) 3  4i   5  2i 



(15) 3  4i   5  2i 
(3)  3 5 3  5

(4) 
9


 25


(5) 4  12  2  3
(16) 3  4i 5  2i 

(17) 2  7i   3  4i   7  6i 
2
(18) 4  3i 
14
21
(6)

(7)
4i 2
 16
(19) 2  5 i
(8)
 i 3  i 2 
3
(20) 2  3i 
3
123
(9) i
(10) i
(11) i
(12)
57
2
 i4
i3  i5  i7

(21) 1  i 
(22)
(23)
(24)
2  4 2 
3
4  3i
5  2i
6 
2
6 
2
4  i
1 + 2i

3  i
1  3i

5i 9
Alg 3 1.10
7
ALGEBRA REVIEW SHEET ANSWERS
1.10
(1) 9i
(13) 7i
(2)  5 6 i
(14) 8 + 2i
(3) 45i
(15) 2 + 6i
(4) 15
(16) 23 + 14i
(5) 48
(17) 6  3i
(6)
6i
3
(18) 7 + 24i
(7) 32
(19) 0
(8) i
(20) 46 + 9i
(9) i
(21)
1  i
4
(10) i
(22)
14  23i
29
(11) 0
(23) 2  3
(12) i
(24)
3  i
5