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Complex Number Review
How much do you remember? (10.2)
POD
Calculate the following. Up to the board.
4
3
2
i
i
i
SAT Prep
SAT #1
SAT Prep
SAT #2
SAT Prep
SAT #3
Review the cycle
Remember what happens with successive powers of i?
12
11
10
9
8
7
6
5
4
3
2
i
i
i
i
i
i
i
i
i
i
i
Review the cycle
Remember what happens with successive powers of i?
1
1
1
1
1
1
12
11
10
9
8
7
6
5
4
3
2
i
ii
i
ii
i
ii
i
ii
i
ii
i
Review the cycle
Remember what happens with successive powers of i? Here’s a way to keep track of the pattern.
What would i23 equal? What would i101 equal?
i
-1
-i
1
Connection to radical signs
What is the definition of i?
Using that, rewrite the following.
5
75
8
25
4
Connection to radical signs
What is the definition of i?
Using that, rewrite the following.
i
i
i
i
i
55
3575
228
525
24
Graphing complex numbers
What sort of coordinate system do we use to graph complex numbers? What is on each axis?
Plot 7+11i, 5-2i, 3, -9i.
Graphing complex numbers
What connection do you see between this axis and our pattern shortcut?
i
-1
-i
1
Adding and subtracting complex numbers
Like adding polynomials, you combine like terms.
)117()25(
)117()57(
)52()117(
)52()117(
ii
ii
ii
ii
Adding and subtracting complex numbers
Like adding polynomials, you combine like terms.
iii
iii
iii
iii
1312)117()25(
614)117()57(
65)52()117(
169)52()117(
Multiplying complex numbers
Like multiplying binomials, you FOIL.
)117)(25(
)117)(57(
)52)(117(
)52)(117(
ii
ii
ii
ii
Multiplying complex numbers
Like multiplying binomials, you FOIL.
iiiiiii
iiiiiii
iiiiiii
iiiiiii
691322693522145535)117)(25(
4210455424955357749)117)(57(
136955131455223514)52)(117(
574155571455223514)52)(117(
2
2
2
2
Complex conjugates
Give the complex conjugates of the following.
i
i
i
i
i
i
24
63
7
92
43
65
Complex conjugates
Give the complex conjugates of the following.
ii
ii
ii
ii
ii
ii
2424
6363
77
9292
4343
6565
Complex conjugates
Multiply the complex conjugates. What happens?
)24)(24(
)63)(63(
)7)(7(
)92)(92(
)43)(43(
)65)(65(
ii
ii
ii
ii
ii
ii
Complex conjugates
Multiply the complex conjugates. What happens?
18216216)24)(24(
39363363)63)(63(
5014949)7)(7(
85814814)92)(92(
25169169)43)(43(
6136253625)65)(65(
2
2
2
2
2
2
iii
iii
iii
iii
iii
iii
Complex conjugates
Multiply the complex conjugates. What happens?
General rule:
So, how would you factor (x2 + 9)?
22))(( babiabia
Dividing complex numbers
Multiplying complex conjugates comes into play here so we can eliminate the complex numbers in the denominator.
i
i
i
i
23
46
25
117
Dividing complex numbers
Multiplying complex conjugates comes into play here so we can eliminate the complex numbers in the denominator.
What are the real and imaginary components?
13
2410
49
82418
)23)(23(
)23)(46(
23
46
29
6913
425
226935
)25)(25(
)25)(117(
25
117
2
2
2
2
i
i
ii
ii
ii
i
i
i
i
ii
ii
ii
i
i
Make up your own
Choose one operation– addition, subtraction, multiplication, or division– make up your own numbers, and solve.
Everyone put a problem on the board!
