10.6 Roots of Complex Numbers Notice these numerical statements. 32 2 125 5 4 16 2 These are true!
Download ReportTranscript 10.6 Roots of Complex Numbers Notice these numerical statements. 32 2 125 5 4 16 2 These are true!
Slide 1
10.6 Roots of Complex Numbers
Slide 2
Notice these numerical statements.
5
32 2
3
125 5
1
4
16 2
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.
Slide 3
Ex 1) Find the four fourth roots of 81 cis
2
3
Let w = r cis θ represent a fourth root. Then w 4 81 cis
2
3
r cis r 4 cis 4 81 cis
4
r 81
4
4
2
3
but there are lots of angles that terminate at
we must consider multiples 4
r 3
4
2
3
4
2
3
2
12
4
Four 4th roots: 3 cis
6
6
2
3
3 cis
4
2
3
8
3
4
14
3
2
3
2
3
6
3
3 cis
7
6
7
6
12
3
2 k
2
3
4
2
3
4
20
3
3 cis
2
3
5
3
5
3
18
3
Slide 4
Complex Roots Theorem
For any positive integer n and any complex number z = r cis θ,
the n distinct nth roots of z are the complex numbers
n
r cis
2k
n
for k = 0, 1, 2, …, n – 1
Now, what does this mean?? Explain what “to do” in plain words!
Ex 2) Find the cube roots of 1000i
r
0 1000 1000
2
think!
(graph in head) 2
3
1000 cis
2
2 k
3
0 + 1000i
,
4 2
6
6
5
6
,
,
6
10 cis
63
,
8 4
6
63
3
2
6
, 10 cis
5
6
, 10 cis
3
2
Slide 5
Ex 3) Graph the five fifth roots of 32.
2 cis
1
5
5
32 2
0 2 k
2 cis
4
5
5
0,
2
5
,
2
4
5
,
6
5
,
8
5
2 cis 0
2 cis
6
5
2 cis
8
5
Slide 6
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
1 cis
2 k
for k = 0, 1, 2, …, n – 1
n is nth root
n
Ex 4) Find the three cube roots of unity and locate them on complex
plane.
r = 1 and they are spaced 23 rad apart
1 cis 0
1 cis
2
1 cis
3
1
1
2
3
4
3
i
2
1
2
3
i
2
Ex 5) You can use these answers to find the cube root of 8
*multiply #4 answers by 3 8 2
2
1
3i
1
3i
Slide 7
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 r ( 2) 2 (2) 2 8
3
8 cis
4
8 8
3
1 1
2 3
3
4
1
6
8
6
1 3
2
4
3 4
4
4
3
4
3
8
3
12
6
8 cis
4
,
6
8 cis
11
12
,
6
8 cis
8
3
12
12
16
12
19
11
19
12
12
12
Slide 8
Homework
#1007
Pg 526 #1, 2, 7, 8, 12, 13, 16, 17, 19, 23,
27, 42, 43, 44
10.6 Roots of Complex Numbers
Slide 2
Notice these numerical statements.
5
32 2
3
125 5
1
4
16 2
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.
Slide 3
Ex 1) Find the four fourth roots of 81 cis
2
3
Let w = r cis θ represent a fourth root. Then w 4 81 cis
2
3
r cis r 4 cis 4 81 cis
4
r 81
4
4
2
3
but there are lots of angles that terminate at
we must consider multiples 4
r 3
4
2
3
4
2
3
2
12
4
Four 4th roots: 3 cis
6
6
2
3
3 cis
4
2
3
8
3
4
14
3
2
3
2
3
6
3
3 cis
7
6
7
6
12
3
2 k
2
3
4
2
3
4
20
3
3 cis
2
3
5
3
5
3
18
3
Slide 4
Complex Roots Theorem
For any positive integer n and any complex number z = r cis θ,
the n distinct nth roots of z are the complex numbers
n
r cis
2k
n
for k = 0, 1, 2, …, n – 1
Now, what does this mean?? Explain what “to do” in plain words!
Ex 2) Find the cube roots of 1000i
r
0 1000 1000
2
think!
(graph in head) 2
3
1000 cis
2
2 k
3
0 + 1000i
,
4 2
6
6
5
6
,
,
6
10 cis
63
,
8 4
6
63
3
2
6
, 10 cis
5
6
, 10 cis
3
2
Slide 5
Ex 3) Graph the five fifth roots of 32.
2 cis
1
5
5
32 2
0 2 k
2 cis
4
5
5
0,
2
5
,
2
4
5
,
6
5
,
8
5
2 cis 0
2 cis
6
5
2 cis
8
5
Slide 6
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
1 cis
2 k
for k = 0, 1, 2, …, n – 1
n is nth root
n
Ex 4) Find the three cube roots of unity and locate them on complex
plane.
r = 1 and they are spaced 23 rad apart
1 cis 0
1 cis
2
1 cis
3
1
1
2
3
4
3
i
2
1
2
3
i
2
Ex 5) You can use these answers to find the cube root of 8
*multiply #4 answers by 3 8 2
2
1
3i
1
3i
Slide 7
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 r ( 2) 2 (2) 2 8
3
8 cis
4
8 8
3
1 1
2 3
3
4
1
6
8
6
1 3
2
4
3 4
4
4
3
4
3
8
3
12
6
8 cis
4
,
6
8 cis
11
12
,
6
8 cis
8
3
12
12
16
12
19
11
19
12
12
12
Slide 8
Homework
#1007
Pg 526 #1, 2, 7, 8, 12, 13, 16, 17, 19, 23,
27, 42, 43, 44