Slide 7.5 - 1 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

OBJECTIVES

Polar Form of Complex Numbers; DeMoivre’s Theorem

Learn geometric representation of complex numbers.Learn to find the absolute value of complex numbers.Learn the geometric representation of the sum of complex numbers. Learn the polar form of a complex number.Learn to find the product and quotient of two complex numbers in polar form.

SECTION 7.5

1

2

3

5

4

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

OBJECTIVES

Polar Form of Complex Numbers; DeMoivre’s Theorem

Learn DeMoivre’s Theorem.Learn to use DeMoivre’s Theorem to find the nth roots of a complex number.

SECTION 7.5

6

7

Slide 7.5 - 4 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

GEOMETRIC REPRESENTATION OFCOMPLEX NUMBERS

The geometric representation of the complex number a + bi is the point P(a, b) in a rectangular coordinate system. When a rectangular coordinate system is used to represent complex numbers, the plane is called the complex plane or the Argand plane. The x-axis is also called the real axis, because the real part of a complex number is plotted along the x-axis. Similarly, the y-axis is also called the imaginary axis.

Slide 7.5 - 5 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

GEOMETRIC REPRESENTATION OFCOMPLEX NUMBERS

A complex number a + bi may be viewed as a position vector with initial point (0, 0) and terminal point (a, b).

Slide 7.5 - 6 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

EXAMPLE 1 Plotting Complex Nubmers

Plot each number in the complex plane.A: 1 + 3i B: –2 + 2i C: –3 D: –2i E: 3 – i

Solution

Slide 7.5 - 7 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

ABSOLUTE VALUE OFA COMPLEX NUMBER

The absolute value of a complex numberz = a + bi is:

z a bi a2 b2

Slide 7.5 - 8 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

EXAMPLE 2Finding the Absolute Value of a Complex Number

Find the absolute value of each complex number.a. 4 + 3i b. 2 – 3i c. –4 + i d. –2 – 2i e. –3i

Solutiona. 4 3i 42 32 16 9 25 5

b. 2 3i 22 3 2 4 9 13

c. 4 i 4 1i 4 2 12 16 1 17

d. 2 2i 2 2 2 2 4 4 4 2 2 2

e. 3i 0 3i 02 3 2 0 9 9 3

Slide 7.5 - 9 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

GEOMETRIC REPRESENTATION OFTHE SUM OF COMPLEX NUMBERS

Let points P, Q, and R represent the complex numbers a + bi, c + di, and (a + c) + (b + d)i, then R is the diagonal of the parallelogram OPRQ.

a bi c di a c b d i

Slide 7.5 - 10 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

EXAMPLE 3 Adding Complex Numbers Geometrically

Add the complex numbers 1 + 3i and –4 + 2i geometrically.

Solution

Locate P: 1 + 3i and Q: –4 + 2i.

3 5i 1 3i 4 2i .

Draw line segments OP and OQ.

Draw the parallelogram.

R represents –3 + 5i where

Slide 7.5 - 11 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

POLAR FORM OF A COMPLEX NUMBER

z r cos i sin ,

The complex number z = a + bi can be written in polar form

where

and tan b

a.

When a nonzero complex numbers is written in polar form, the positive number r is the modulus or absolute value of z. The angle q in the representation is called the argument of z (written q = arg z).

x r cos, y r sin, r a2 b2 ,

Slide 7.5 - 12 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

EXAMPLE 4 Writing a Complex Number in Polar Form

Write in polar form. Express the argument in degrees 0º ≤ ≤ 360º.

r a2 b2Find r. 12 3 2

z 1 3i

SolutionFor , a = 1 and b =z 1 3i 3.

1 3 2

tan b

a

3

1 3

tan 60º 3. Thus, 360º 60º300º .

z 1 3i r cos i sin 2 cos 300º i sin 300º

Find .

Slide 7.5 - 13 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

PRODUCT AND QUOTIENT RULES FOR TWO COMPLEX NUMBERS IN POLAR FORM

z1 r1 cos1 i sin1 and z2 r2 cos2 i sin2 Let

be two complex numbers in polar form. Then

z1z2 r1r2 cos 1 2 i sin 1 2

and z1

z2

r1r2

cos 1 2 i sin 1 2 , z2 0.

Slide 7.5 - 14 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

EXAMPLE 6Finding the Product and Quotient of Two Complex Numbers

Leave the answers in polar form.

Let z1 3 cos65º i sin 65º and

z2 4 cos15º i sin15º . Find z1z2 and z1

z2

.

Solution

z1z2 3 cos65º i sin 65º 4 cos15º i sin15º

z1z2 12 cos80º i sin 80º Multiply moduli Add arguments

z1z2 34 cos 65º 15º i sin 65º 15º

Slide 7.5 - 15 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

EXAMPLE 6Finding the Product and Quotient of Two Complex Numbers

Solution continued

z1

z2

3 cos65º i sin 65º 4 cos15º i sin15º

z1

z2

3

4cos50º i sin 50º

Divide moduli Subtract arguments

z1

z2

3

4cos 65º 15º i sin 65º 15º

Slide 7.5 - 16 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

DEMOIVRE’S THEOREM

Let z = r(cos + i sin) be a complex number in polar form. Then, for any integer n,

zn rn cosn i sin n .

Slide 7.5 - 17 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

EXAMPLE 8 Finding the Power of a Complex Number

Let z = 1 + i. Use DeMoivre’s Theorem to find each power of z. Write answers in rectangular form.

a. z16 b. z 10

SolutionConvert z to polar form. Find r and .

r a2 b2 12 12 2

tan b

a

1

11 so

4

z 1 i 2 cos4

i sin4

Slide 7.5 - 18 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

EXAMPLE 8 Finding the Power of a Complex Number

a. z16Solution continued

z16 2 16cos 16

4

i sin 16

4

z 2 cos4

i sin4

z16 28 cos 4 i sin 4

z16 256 1 i 0 z16 256

Slide 7.5 - 19 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

EXAMPLE 8 Finding the Power of a Complex Number

b. z 10

Solution continued

z 10 2 10cos 10

4

i sin 10

4

z 2 cos4

i sin4

z 10 2 5 cos 52

i sin

52

z 10 1

320 i 1

1

32i

Slide 7.5 - 20 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

DEMOIVRE’S THEOREMFOR FINDING COMPLEX ROOTS

The nth roots of a complex number w = r(cos + i sin), r > 0, in degrees are given by

If is in radians, replace 360º by 2π in zk.

zk r1 n cos 360º k

n

i sin

360º k

n

,

k = 0, 1, 2, …, n – 1.

Slide 7.5 - 21 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

EXAMPLE 9 Finding the Roots of a Complex Number

Find the three cube roots of 1 + i in polar form, with the argument in degrees.

SolutionIn the previous example, we showed that

1 i 2 cos4

i sin4

1 i 2 cos 45º i sin 45º

zk 21 3

cos45º 360º k

3

i sin

45º 360º k

3

,

k = 0, 1, 2.

Slide 7.5 - 22 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Solution continued

z0 21 3

cos45º 360º0

3

i sin

45º 360º03

EXAMPLE 9 Finding the Roots of a Complex Number

z1 21 3

cos45º 360º1

3

i sin

45º 360º13

z0 21

6 cos15ºi sin15º

z1 21

6 cos135º i sin135º

Slide 7.5 - 23 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Solution continued

z2 21 3

cos45º 360º2

3

i sin

45º 360º23

EXAMPLE 9 Finding the Roots of a Complex Number

z2 21

6 cos255º i sin 255º

z1 21

6 cos135º i sin135º

z0 21

6 cos15º i sin15º

z2 21

6 cos255º i sin 255º

The three cube roots of 1 + i are: