roots of complex numbers sec. 6.6c hw: p. 558 39-59 odd

Roots ofRoots ofComplex NumbersComplex Numbers

Sec. 6.6cSec. 6.6cHW: p. 558 39-59 oddHW: p. 558 39-59 odd

From last class:

31 3 8i

8

2 21

2 2i

1 3i2 2

2 2i

The complex number

is a third root of –8

The complex number

is an eighth root of 1

DefinitionA complex number v = a + bi is an nth root of z if

v = zn

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

Finding nth Roots of a Complex Number

cosθ sin θz r i

θ 2π θ 2πcos sinn k k

r in n

If ,

then the n distinct complex numbers

where k = 0, 1, 2,…, n – 1, are the nth roots ofthe complex number z.

Let’s now do an example…π π

5 cos sin3 3

z i

Find the fourth roots of

Use the new formula, with r = 5, n = 4, k = 0 – 3,

41

3 2 0 3 2 05 cos sin

4 4z i

3

41 5 cos sin

12 12z i

k = 0:

fourth root continued…π π

5 cos sin3 3

z i



42

3 2 1 3 2 15 cos sin

4 4z i

3

42

7 75 cos sin

12 12z i

k = 1:

fourth root continued…π π

5 cos sin3 3

z i



43

3 2 2 3 2 25 cos sin

4 4z i

3

43

13 135 cos sin

12 12z i

k = 2:

fourth root done!π π

5 cos sin3 3

z i



44

3 2 3 3 2 35 cos sin

4 4z i

3

44

19 195 cos sin

12 12z i

k = 3:

How would we How would we verifyverify these algebraically??? these algebraically???

A new example…Find the cube roots of –1 and plot them.

1z First, rewrite the complex number in trig. form:

1 0z i cos sini Use the new formula, with r = 1, n = 3, k = 0 – 2,

31

2 0 2 01 cos sin

3 3z i

cos sin3 3i

1 3

2 2i

third root continued…Find the cube roots of –1 and plot them.



32

2 1 2 11 cos sin

3 3z i

cos sini 1 0i

third root continued…Find the cube roots of –1 and plot them.



33

2 2 2 21 cos sin

3 3z i

5 5

cos sin3 3

i

1 3

2 2i

third root done!Find the cube roots of –1 and plot them.

Now, how do we sketch the graph???Now, how do we sketch the graph???



1

1 3

2 2z i 2 1 0z i 3

1 3

2 2z i

The cube roots of –1

roots of complex numbers sec. 6.6c hw: p. 558 39-59 odd

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