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
Find the fourth roots of
Use the new formula, with r = 5, n = 4, k = 0 – 3,
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
Find the fourth roots of
Use the new formula, with r = 5, n = 4, k = 0 – 3,
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
Find the fourth roots of
Use the new formula, with r = 5, n = 4, k = 0 – 3,
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.
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,
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.
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,
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???
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,
1
1 3
2 2z i 2 1 0z i 3
1 3
2 2z i
The cube roots of –1
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