10.6 Roots of Complex Numbers

Notice these numerical statements.

These are true! But I would like to write them a bit differently.

32 = 25 –125 = (–5)3 16 = 24

Keeping this “change in appearance” in mind, let’s extend this to the complex plane.

3i is a fourth root of 81 because (3i)4 = 81

In general … For complex numbers r and z and for any positive integer n,

r is an nth root of z iff rn = z.

We will utilize DeMoivre’s Theorem to verify.

145 332 2 125 5 16 2

23

4 23

4 4 23

4 2 23 3

23

2 2 6 2 12 2 183 3 3 3 3 3 3

2 8 14 2012 3 3 3

2 7 56 3 6 3

2 7 56 3 6 3

81 cis

81 cis

cis cis 4 81 cis

81 4

3 4 2

4 4 4 4

4 4 4

3 cis 3 cis 3 cis 3 cis

w

r r

r

r k

Ex 1) Find the four fourth roots of

Let w = r cis θ represent a fourth root. Then

but there are lots of angles that terminate at

we must consider multiples

Four 4th roots:

2

3 2 5 36 6 2

2 40 1000 1000 , ,

6 6 3 6 35 3

, ,2 6 6 2

21000 cis 10 cis , 10 cis , 10 cis

3

r

k

Complex Roots TheoremFor any positive integer n and any complex number z = r cis θ, the n distinct nth roots of z are the complex numbers

for k = 0, 1, 2, …, n – 1 2

cis n kr

n

Now, what does this mean?? Explain what “to do” in plain words!

Ex 2) Find the cube roots of 1000i 0 + 1000i

think! (graph in head)

4

6

8

6

1532 2

0 2

52 4 6 8

0, , , ,5 5 5 5

k

Ex 3) Graph the five fifth roots of 32.

82 cis

5

2 cis 0

22 cis

5

42 cis

5

62 cis

5

Ex 5) You can use these answers to find the cube root of 8*multiply #4 answers by 3 8 2

2 1 3 1 3i i

The various nth roots of 1 are called the roots of unity.1 in polar is: 1 cis 0 so nth roots of unity are of the form

for k = 0, 1, 2, …, n – 1n is nth root

21 cis

k

n

Ex 4) Find the three cube roots of unity and locate them on complex plane. r = 1 and they are spaced rad apart2

3

2 41 cis 0 1 cis 1 cis

3 3

1 3 1 31

2 2 2 2i i

1 1 12 3 6

2 2

3 6

6 6 6

( 2) (2) 8

3 38 cis

4 41 3 2 4

8 8 8 83 4 4 4 3 4 3

3 8 3 16

12 12 12 1211 19 11 19

8 cis , 8 cis , 8 cis 4 12 12 12 12

r

The complex roots theorem provides a connection between the roots of a complex number and the zeros of a polynomial.

Ex 6) Find all solutions of the equation x3 + 2 = 2i x3 = –2 + 2i (aka find 3 roots of –2 + 2i)

polar

Homework

#1007 Pg 526 #1, 2, 7, 8, 12, 13, 16, 17, 19, 23, 27, 42, 43, 44