Complex Numbers Page 3
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1
2
3Appendix B
Complex Numbers
P. Danziger
2.3
Powers
Theorem 6 (Demoivre’s Theorem)
re
= r (cos (nθ) + i sin (nθ))
Example 7
12
Find (i + i)
1 + i =
2e
.
4
So
12
12
(1 + i)
=
2e
4
12
12
=
2
e
4
6
3
= 2
e
= 64e
=
64.
Note e =
1.
2.4
Roots of Complex Numbers
th
In order to find the n
root of a complex number z = x + iy = re we use the polar form, z = re .
Since θ is an angle,
( +2
)
re = re
for any intger k. Thus
1
1
( +2
)
z
=
re
1
+2
= r
e
1
+2
+2
= r
cos
+ i sin
Taking k = 0, 1, . . . , n
1 gives the n roots.
1
Since r
0, r
always exists, even for even roots.
Example 8
Find All cube roots of 8.
1
2
2
8 = 8e
, so, 8
= 2e
.
3
3
2
4
Taking k = 0, 1, 2 gives 2, 2e
and 2e
as the three cube roots of 8.
3
3
2.5
Fundamental Theorem of Algebra
th
Note that in C all numbers have exactly n n
roots.
This leads to the Fundamental Theorem of algebra:
Every polynomial over the Complex numbers of degree n has exactly n roots
i.e. if f (z) = a
+ a
z + . . . + a z
0
1
then there exist z
, z
. . . , z
C such that
1
2
f (x) = (z
z
)(z
z
) . . . (z
z ).
1
2
That is f can be decomposed into linear factors.
3
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