Imaginary Number: 1i

ii 1

POWERS of i: Is there a pattern?

Pattern repeats every 4th power: Divide power by 4 and use remainder

Ex: 23i

2i

3i

4i 8i

7i

6i

5i

iiiii 5354354 )1()(

111)1(4

112

i 11)1(3

1

i 1

1

i 1

I ONE, I ONE! LOSERS IN THE MIDDLE

ii

12 i

ii 3

14 i

LOSERS=NEGATIVE

Example 1: Simplifying Powers of i [A] 45i

[B] [C]

[D]

35i 14i

68i

i

i

i

3

384

1

0

174

i

i[E] 73i [E] 132i

1

2

234

i

i

i

i

i

1

1114

i

i

i

1

1184

1

0

334

i

i

Example 2 Simplify Square Roots of Negative Numbers[A] 18 [B] 28

[E] 5125x[F] 1215128 yx

26i

xix 55 2 xyix 28 67

[C] 72

[D] 2367 cba

72i

acicba 1133

23i

Example 3 Multiplying Pure Imaginaries 1st: Convert all square roots into imaginary number notation[A] ii 72 [B] ii 65

30

30 2

i

i

i

108

108 3

[C] 1512 [D] 328

[E] ii 912 2 [F] 9818

14

14 2

i

i

i

i

42

421

2723

56

452

15322

i

ii

16

48

24222

i

ii

Example 4: Operations with Complex Numbers

Complex Number: binomial term of real and imaginary #

[A]

)53)(25( ii [C] )52)(24( ii

i

iii

242

104208 2

Add and Subtract: Combine Like TermsMultiply: FOIL, Distributive Property, Laws of ExponentsDivision: Rationalize with Conjugates

)23()67()35( iii [B] )1(3)32(4 iii

[D]

i

i

115

)263()375(

i

iii

1115

33128 2

i

iii

1925

1062515 2

Example 5: Simplifying Using Complex Conjugates

[D]i

i

23

2

[E]

i

i

32

35

)32(

)32(

i

i

13

91994

9615102

2

ii

iii

[A]x8

5

[B]

21

3

i[C]

yx

x363

75

Binomial Conjugate)23(

)23(

i

i

Binomial Conjugate

13

849

23462

2

ii

iii

xx

x

218

215

x

xi

4

25

7321

733

ii

i

7

7

21

73

ii

yxyx

yxx

6163

61753

xy

y

yx

yx

18

425

18

4252

Example 6: Equations with Imaginary Solutions

[A] 0483 2 x [B] 0205 2 x

ix

x

x

2

4

2052

2

Additional examples to come with quadratic formula

[C] 0814 x [D] 01282 4 x

ix

x

x

4

16

4832

2

0)9)(9( 22 xx

3

92

x

xix

x

3

92

092 x092 x 0)8)(8(

06422

4

xx

x

22

82

ix

x

22

82

x

x

PRACTICE: Equations with Imaginary Solutions [A] 22 2725 xx [B] 4372 2 x

23

18

3622

2

ix

x

x

[C] 0499 4 x [D] 164 x

62

24

7232

2

ix

x

x

0)73)(73( 22 xx

3

21,

3

21ix

0)4)(4(

01622

4

xx

x

ix 2,2