Slide 6-1

COMPLEX NUMBERS AND POLAR

COORDINATES

8.1 Complex Numbers8.2 Trigonometric Form for

Complex Numbers

Chapter 8

Slide 8-2

and i is the imaginary unit Numbers in the form a + bi are called complex

numbers a is the real part b is the imaginary part

If 0, then . a a i a

i2 1 i 1

8.1 Complex Numbers

Slide 8-3

Examples

a) b)

c)

d) e)

25 25 5i i 30 30i

125 125 25 5 5 5i i i

22

3 3 3 3

3

1 3

3

i i

i

98 98

49 49

982

49i i

Slide 8-4

Example: Solving Quadratic Equations

Solve x = 25 Take the square root on both sides.

The solution set is {5i}.

2

2

25

25

25

5

x

x

x i

x i

Slide 8-5

Another Example

Solve: x2 + 54 = 0

The solution set is

2

2

54 0

54

54

54 9 6

3 6

x

x

x

x i i

x i

3 6 . i

Slide 8-6

Example: Products and Quotients

Multiply: Divide: 8 8.

22

2

8 8 8 8

8 1

8

8

1

i i

i i

56.

8

8 8

56 5

5

7

6

6

8

i

i

i

Slide 8-7

Addition and Subtraction of Complex Numbers

For complex numbers a + bi and c + di,

Examples

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

a bi c di a c b d i

a bi c di a c b d i

(10 4i) (5 2i)

= (10 5) + [4 (2)]i

= 5 2i

(4 6i) + (3 + 7i)

= [4 + (3)] + [6 + 7]i

= 1 + i

Slide 8-8

Multiplication of Complex Numbers

For complex numbers a + bi and c + di,

The product of two complex numbers is found by multiplying as if the numbers were binomials and using the fact that i2 = 1.

(a bi)(c di) (ac bd ) (ad bc)i.

Slide 8-9

Examples: Multiplying

(2 4i)(3 + 5i) (7 + 3i)2

2

2(3) 2(5 ) 4 (3) 4 (5 )

6 10 12 20

6 2 20( 1)

26 2

i i i i

i i i

i

i

2 2

2

7 2(7)(3 ) (3 )

49 42 9

49 42 9( 1)

40 42

i i

i i

i

i

Slide 8-10

Powers of i

i1 = i i5 = i i9 = i

i2 = 1 i6 = 1 i10 = 1

i3 = i i7 = i i11 = i

i4 = 1 i8 = 1 i12 = 1and so on.

Slide 8-11

Simplifying Examples

i17

i4 = 1

i17 = (i4)4 • i

= 1 • i

= i

i4

4

1 11

1i

Slide 8-12

Property of Complex Conjugates

For real numbers a and b, (a + bi)(a bi) = a2 + b2.

The product of a complex number and its conjugate is always a real number.

Example

2

2

5 3

2(5 3 )(2 )

(2 )(2 )

10 5 6 3

47 11

57 11

5 5

i

ii i

i i

i i i

ii

i

Slide 8-13

We modify the familiar coordinate system by calling the horizontal axis the real axis and the vertical axis the imaginary axis.

Each complex number a + bi determines a unique position vector with initial point (0, 0) and terminal point (a, b).

8.2 Trigonometric Form for Complex Numbers

Slide 8-14

Relationships Among x, y, r, and

x r cosy r sin

r x2 y2

tan y

x, if x 0

Slide 8-15

Trigonometric (Polar) Form of a Complex Number

The expression

is called the trigonometric form or (polar form) of the complex number x + yi. The expression cos + i sin is sometimes abbreviated cis .Using this notation

(cos sin )r i

(cos sin ) is written cis .r i r

Slide 8-16

Example

Express 2(cos 120 + i sin 120) in rectangular form.

Notice that the real part is negative and the imaginary part is positive, this is consistent with 120 degrees being a quadrant II angle.

1cos120

2

3sin120

2

1 32(cos120 sin120 ) 2 ,

2 2

1 3

i i

i

Slide 8-17

Converting from Rectangular Form to Trigonometric Form Step 1 Sketch a graph of the number x + yi in

the complex plane.

Step 2 Find r by using the equation

Step 3 Find by using the equation choosing the quadrant

indicated in Step 1.

2 2 .r x y

tan , 0y

x x

Slide 8-18

Example

Example: Find trigonometric notation for 1 i. First, find r.

Thus,

2 2

2 2( 1) ( 1)

2

r a b

r

r

1 2 1 2sin cos

2 22 25

4

5 5 51 2 cos sin or 2 cis

4 4 4i i

Slide 8-19

Product Theorem

If are any two complex numbers, then

In compact form, this is written

1 1 1cos sin and r i

1 1 1 2 2 2

1 2 1 2 1 2

cos sin cos sin

cos sin .

r i r i

r r i

2 2 2cos sin ,r i

1 1 2 2 1 2 1 2 cis cis cis .r r r r

Slide 8-20

Example: Product

Find the product of

4(cos50 sin50 ) and 2(cos10 sin10 ).i i

4(cos50 isin50) 2(cos10 isin10) 42 cos(50 10) isin(50 10) 8(cos60 isin60)

81

2 i

3

2

4 4i 3

Slide 8-21

Quotient Theorem If

are any two complex numbers, where then

1 1 1cos sin and r i 2 2 2cos sinr i

1 1 1 11 2 1 2

2 2 2 2

1 1 11 2

2 2 2

cos sincos sin .

cos sin

In compact form, this is written

cis cis

cis

r i ri

r i r

r r

r r

2 2 2 2cos sin , 0,r i r

Slide 8-22

Example: Quotient

Find the quotient.

16(cos70 sin 70 ) and 4(cos40 sin 40 )i i

16(cos70 isin70)

4(cos40 isin40) =

16

4cos(70 40) isin(70 40)

4cos30 isin30

43

2

1

2i

2 3 2i

Slide 8-23

De Moivre’s Theorem

If is a complex number, and if n is any real number, then

In compact form, this is written

1 1 1cos sinr i

1 1cos sin cos sin .n nr i r n i n

cis cis .n nr r n

Slide 8-24

Example: Find (1 i)5 and express the result in rectangular form.

First, find trigonometric notation for 1 i

Theorem 1 2 cos225 sin 225i i

55

5

2 5 225 5 2

1 2 cos225 sin 225

cos( ) sin( )

4 2 cos1125 sin1125

2 24 2

2 2

25

4 4

i i

i

i

i

i

Slide 8-25

nth Roots

For a positive integer n, the complex number a + bi is an nth root of the complex number x + yi if

.n

a bi x yi

Slide 8-26

nth Root Theorem

If n is any positive integer, r is a positive real number, and is in degrees, then the nonzero complex number r(cos + i sin ) has exactly n distinct nth roots, given by

where

cos sin or cis ,n nr i r

360 360 or = , 0,1,2,..., 1.

k kk n

n n n

Slide 8-27

Example: Square Roots

Find the square roots of Trigonometric notation:

For k = 0, root is For k = 1, root is

1 3 i

1 3 i 2 cos60 isin60

2 cos60 isin60

1

2 21

2 cos60

2 k

360

2

isin60

2 k

360

2

2 cos 30 k 180 isin 30 k 180

2 cos30 isin30

2 cos210 isin210

Slide 8-28

Example: Fourth Root

Find all fourth roots of Write the roots in rectangular form.

Write in trigonometric form.

Here r = 16 and = 120. The fourth roots of this number have absolute value

8 8 3.i

8 8 3 16 cis 120i

4 16 2.

120 36030 90

4 4k

k

Slide 8-29

Example: Fourth Root continued

There are four fourth roots, let k = 0, 1, 2 and 3.

Using these angles, the fourth roots are

0 30 90 30

1 30 90 120

2 30 90 210

3 30 90

0

1

2

3 300

k

k

k

k

2 cis 30 , 2 cis 120 , 2 cis 210 , 2 cis 300

Slide 8-30

Example: Fourth Root continued

Written in rectangular form

The graphs of the roots are all on a circle that has center at the origin and radius 2.

3

1 3

3

1 3

i

i

i

i

Slide 8-31

Polar Coordinate System

The polar coordinate system is based on a point, called the pole, and a ray, called the polar axis.

Slide 8-32

Rectangular and Polar Coordinates

If a point has rectangular coordinates (x, y) and polar coordinates (r, ), then these coordinates are related as follows.

2 2r x y

cosx r siny r

tan , 0y

xx

Slide 8-33

Example

Plot the point on a polar coordinate system. Then determine the rectangular coordinates of the point.P(2, 30)r = 2 and = 30, so point P

is located 2 units from the origin in the positive direction making a 30 angle with the polar axis.

Slide 8-34

Example continued

Using the conversion formulas:

The rectangular coordinates are

cos

2cos30

32 3

2

x r

x

x

sin

2sin30

12 1

2

y r

y

y

13, .

2

Slide 8-35

Example

Convert (4, 2) to polar coordinates.

Thus (r, ) =

2 2

2 24 2

16 4

20 2 5

r x y

r

r

r

tan 2

4

1

2 26.6

2 5,26.6

Slide 8-36

Rectangular and Polar Equations

To convert a rectangular equation into a polar equation, use

r c

a cos bsin.

y r sinand

and solve for r.

you will get the polar equation

For the linear equation ax by c ,

x r cos

Slide 8-37

Example

Convert x + 2y = 10 into a polar equation. x + 2y = 10

cos sin

cos sin110

cos 2s

1

in

0

2

rba

c

r

r

Slide 8-38

Example

Graph r = 2 sin

1

1.414

2

0

-1

-1.414

r

330

315

270

180

150

135

-1.732120

-290

-1.73260

-1.414

-1

0

r

45

30

0

Slide 8-39

Example

Graph r = 2 cos 3

01.4121.4101.412r

9075604530150

Slide 8-40

Example

Convert r = 3 cos sin into a rectangular equation.

2

2 2

2 2

3cos sin

3 cos sin

3

3

r

r r r

x y x y

x x y y

Slide 8-41

Circles and Lemniscates

Slide 8-42

Limacons

Slide 8-43

Rose Curves