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!