4.5 Complex NumbersObjectives:1. Write complex numbers in standard form.2. Perform arithmetic operations on complex numbers.3. Find the conjugate of a complex number.4. Simplify square roots of negative numbers.5. Find all solutions of polynomial equations.

Imaginary & Complex NumbersThe imaginary unit is defined as Imaginary numbers can be written in the

form bi where b is a real number.A complex number is a sum of a real and

imaginary number written in the form a + bi.Any real number can be written as a complex

number:

Example: 2 = 2 + 0i , −3 = −3 + 0i

i 1

Example #1Equating Two Complex NumbersFind x and y.

4x 5i 8 6yi

To solve make two equations equating the real parts and imaginary parts separately.

6

5

6

5

65

2

84

i

iy

yii

x

x

Example #2Adding, Subtracting, & Multiplying Complex NumbersPerform the indicated operation and write the

result in the form a + bi.A.

B.

(2 3i) (1 i)

(2 3i) (4 2i)

i

ii

23

312

Combine like terms.

i

ii

ii

52

2342

2432

Distribute the (-) and then combine like terms.

Example #2Adding, Subtracting, & Multiplying Complex NumbersPerform the indicated operation and write the

result in the form a + bi.

C.

D.

10i 5 15

i

(3 i)(2 5i)

i

i

ii

502

1250

250 2

Distribute & Simplify.

Remember:

1

12

i

i

i

i

iii

1311

15136

52156 2

Use FOIL, substitute and combine like terms.

Example #3Products & Powers of Complex NumbersPerform the indicated operation and write the

result in the form a + bi.A.

B.

(4 5i)(4 5i)

(4 i)2

41

12516

25202016 2

iii Since these groups are the same but with opposite signs they are conjugates of each other. The middle terms always cancel with conjugates.

ii

iiiii

8151816

441644 2

Powers of i

This pattern of {i, −1, −i, 1} will continue for even higher patterns.A shortcut to evaluating higher powers requires you to memorize this pattern, but it is not necessary to evaluate them.

Example #4Powers of iFind the following:A. i73

ii

i

ii

ii

1

1 36

362

72

Method 1:

1. If the exponent is odd, first “break off” an i from the original term.

2. Rewrite the even exponent as a power of i2 (divide it by 2).

3. Replace the i2 with −1.

4. Evaluate the power on −1. Even exponents make it positive and odd exponents keep it negative.

5. Multiply what is left back together.

Example #4Powers of iFind the following:A. i73

Method 2:

1. Use long division and divide the exponent by 4. Always use 4 which is the four possible values of the pattern {i, −1, −i, 1}.

2. The remainder represents the term number in the sequence.

1

32

33

4

18734

For this problem the remainder is 1 which means the answer is the first term in the sequence {i, −1, −i, 1} which is i.

Example #4Powers of iFind the following:B. i64

1

1 32

322

i

This time it isn’t necessary to “break off” any i because the exponent is already even.

0

24

24

4

16644

If the remainder is 0 this indicates that the value is the 4th term in the sequence {i, −1, −i, 1} since you can’t have a remainder of 4 when dividing by 4.

Therefore, the answer is 1.

Example #4Powers of iFind the following:C.

ii

i

ii

ii

·1

·1

·

·

25

252

50

This time it is necessary to “break off” an i because the exponent is odd.

3

8

11

4

12514

If the remainder is 3 this indicates that the value is the 3rd term in the sequence {i, −1, −i, 1}.

Therefore, the answer is −i.

51i

Example #4Powers of iFind the following:D.

1

1 15

152

i

2

28

7304

If the remainder is 2 this indicates that the value is the 2nd term in the sequence {i, −1, −i, 1}.

Therefore, the answer is −1.

30i

Example #5Quotients of Two Complex NumbersExpress each quotient in standard form.

A.

2 5i1 2i

i

i

ii

iii

i

i

i

i

5

1

5

125

12

141

110241

10542

21

21

21

52

2

2

Multiply the top and bottom by the conjugate of the denominator, FOIL, and simplify.

Write your final answer as a complex number of the form a + bi.

Example #5Quotients of Two Complex NumbersExpress each quotient in standard form.

B. 4 i 3 3i

i

i

ii

iii

i

i

i

i

6

5

2

118

159

199

13151299

331212

33

33

33

4

2

2

Example #6Square Roots of Negative NumbersWrite each of the following as a complex

number.

A.

B.

5

2 115

5

51

51

i

After removing the i, make sure to place it out front as this can be confusing:

ii 55

i

i

5

11

5

2

5

112

This time with the i off to

the side there is no confusion.

Example #6Square Roots of Negative NumbersWrite each of the following as a complex

number.

C. 1 16

5 36

i

i

iii

ii

ii

1429

124145

242065

6541

365161

2

Be sure to remove the i from each radical first!

Example #7Complex Solutions to a Quadratic EquationFind all solutions to the following:

3x 2 5x 13 0

i

x

6

131

6

5

6

1315

32

133455 2

Sum & Difference of Cubes

a 3 b 3

a b a 2 ab b 2

a 3 b 3

a b a 2 ab b 2

Example #8Zeros of UnityFind all solutions to the following:

A.

x 3 64

4

04

0)164)(4(

04

064

2

33

3

x

x

xxx

x

x

322

2

344

2

484

12

161444

0164

2

2

i

i

x

xx

Example #8Zeros of UnityFind all solutions to the following:

B. x 4 625 0

5,5

02555

025252

22

x

xxx

xx

ix

x

x

x

5

25

25

0252

2