Copyright © 2011 Pearson, Inc. 6.6 De Moivre’s Theorem and nth Roots.

Copyright © 2011 Pearson, Inc.

6.6De Moivre’s Theorem and

nth Roots

Copyright © 2011 Pearson, Inc. Slide 6.1 - 2

What you’ll learn about

The Complex Plane Trigonometric Form of Complex Numbers Multiplication and Division of Complex Numbers Powers of Complex Numbers Roots of Complex Numbers

… and whyThe material extends your equation-solving technique to include equations of the form zn = c, n is an integer and c is a complex number.


Complex Plane


Absolute Value (Modulus) of a Complex Number

2 2

The or of a complex number

is | | | | .

In the complex plane, | | is the distance of

from the origin.

z a bi z a bi a b

a bi a bi

absolute value modulus


Graph of z = a + bi


Trigonometric Form of a Complex Number

The trigonometric form of the complex number

z a bi is

z r cos isin where a r cos , b r sin , r a2 b2 ,

and tan b / a. The number r is the absolute

value or modulus of z, and is an argument of z.


Example Finding Trigonometric Form

Find the trigonometric form with 0 2 for the

complex number 1 3i.


Example Finding Trigonometric Form

Find r: r |1 3i | 12 3 2 2.

Find : tan 3

1 so

3

.

Therefore, 1 3i2 cos3

isin3

.

Find the trigonometric form with 0 2 for the

complex number 1 3i.


Product and Quotient of Complex Numbers

Let z1r

1cos

1 isin

1 and z2r

2cos

2 isin

2 .Then

1. z1z

2r

1r

2cos

1

2 isin 1

2 .

2. z

1

z2

r1

r2

cos 1

2 isin 1

2 , r20.


Example Multiplying Complex Numbers

Express the product of z1 and z

2 in standard form.

z14 cos

4

isin4

, z2 2 cos

6

isin6


Example Multiplying Complex Numbers

z1z

2r

1r

2cos

1

2 isin 1

2

4 2 cos4

6

isin

4

6

4 2 cos512

isin

512

4 2 0.259 i0.966 1.464 5.464i

Express the product of z1 and z

2 in standard form.

z14 cos

4

isin4

, z2 2 cos

6

isin6


A Geometric Interpretation of z2


De Moivre’s Theorem

Let z r cos isin and let n be a positive integer.

Then

zn r cos isin n

r n cos n isin n .


Example Using De Moivre’s Theorem

Find 3

2 i

1

2

4

using De Moivre's theorem.



The argument of z 3

2 i

1

2 is

76

,

and its modulus 3

2 i

1

2

3

4

1

41.

Hence,

z 2cos76

isin76

Find 3

2 i

1

2

4




z4 cos 476

isin 4

76

cos14

3

isin

143

cos23

isin

23

1

2 i

3

2

Find 3

2 i

1

2

4



nth Root of a Complex Number

A complex number v a bi is an nth root of z if

vn z.

If z 1, the v is an nth root of unity.


Finding nth Roots of a Complex Number

If z r cos isin , then the n distinct

complex numbers

rn cos 2k

n isin

2k

n

,

where k 0,1,2,..,n 1,

are the nth roots of the complex number z.


Example Finding Cube Roots

Find the cube roots of 1.


Example Finding Cube Roots

Write 1 in complex form: z 10i cos0 isin0

The third roots of 1 are the complex numbers

cos0 2k

3 isin

0 2k

3 for k 0,1,2.

z1cos0 isin0 1

z2cos

23

isin23

1

2

3

2i

z3cos

43

isin43

1

2

3

2i

Find the cube roots of 1.

Copyright © 2011 Pearson, Inc. 6.6 De Moivre’s Theorem and nth Roots.

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