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DeMoivre’s Theorem
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DeMoivre’s Theorem
The Complex Plane
Complex Number
A complex number z = x + yi can be interpreted geometrically as the point (x, y) in the complex plane. The x-axis is the real axis and the y-axis is the imaginary axis.
Complex Plane
Magnitude or Modulus of z
Let z = x + yi be a complex number. The magnitude or modulus of z, denoted by |z| is defined as the distance from the origin to the point (x, y). In other words
2 2z x y
Polar Form of a Complex Number
If r ≥ 0 and 0 ≤ ≤ 2, the complex number z = x + yi may be written in polar form as z = x + yi = (r cos ) + (r sin )i or z = r (cos + i sin )
The angle is called the argument of z. |z| = r
Plotting a Point in the Complex Plane and Writing it in Polar Form
Plot the point corresponding to z = 4 – 4i and write an expression for z in polar form
Plot the point 2 3 and write it in polar form.
Express the argument in degrees.
i
Plot the Point in the Complex Plane and Convert from Polar to Rectangular Form
Plot the point corresponding to
2(cos30 sin 30 )
in the complex plane, and write an
expression for in rectangular form.
o oz i
z
Write Numbers in Rectangular Form
2 (cos 120o + i sin 120o)
4 cos sin2 2i
Multiplying and Dividing Complex Numbers
1 1 1 1 2 2 2 2
1 2 1 2 1 2 1 2
2
1 11 2 1 2
2 2
Let (cos sin ) and cos sin
be two complex numbers. Then
cos sin
If 0, then
cos sin
z r i z r i
z z r r i
z
z ri
z r
Finding Products and Quotients of Complex Numbers in Polar Form
If z = 3 (cos 20o + i sin 20o) and w = 5 (cos 100o + i sin 100o), find (a) zw
(b) z/w
(c) w/z
DeMoivre’s Theorem
DeMoivre’s Theorem is a formula for raising a complex number to the power n.
If z = r (cos + i sin ) is a complex number, then
zn = rn [(cos (n) + i sin (n)] where n ≥ 1 is a positive integer.
Using DeMoivre’s Theorem
Write [2(cos 20o + i sin 20o)]3 in the standard form a + bi.
Using DeMoivre’s Theorem
= 23 [(cos (3 x 20o) + i sin (3 x 20o)] = 8 (cos 60o + i sin 60o)
1 38 4 4 3
2 2
Using DeMoivre’s Theorem
Write (1 + i)5 in standard form a + bi
First we have to change to (1 + i) to polar form
2 2
1
1 1 2
1tan
1 4
r
Using DeMoivre’s Theorem
5
5
2 cos sin4 4
2 cos 5 sin 54 4
5 54 2 cos sin
4 4
2 24 2 4 4
2 2
i
i
i
i
Finding Complex Roots
Let w = r(cos 0 + i sin 0) be a complex number and let n ≥ 2 be an integer. If w ≠ 0, there are n distinct complex roots of w, given by the formula
0 02 2cos sin
0,1,2,. . . , 1.
nk
k kz r i
n n n n
where k n
Finding Complex Roots
Find the complex fourth roots of -16i
First we have to change the number to polar form
22
1
0 16 16
16 3tan
0 2
r
Finding Complex Roots
1
4
3 316 cos sin
2 2
1 3 1 3(16) cos sin
4 2 4 2
3 2 3 22 cos sin , 0,1,2,3
8 4 8 4
3 32 cos sin
8 2 8 2
n
i
z i
k ki k
k ki
Finding Complex Roots
0
0
1
1
3 0 3 02 cos sin
8 2 8 2
3 32 cos sin
8 8
3 1 3 12 cos sin
8 2 8 2
7 72 cos sin
8 8
z i
z i
z i
z i
Finding Complex Roots
2
3
3 2 3 22 cos sin
8 2 8 2
11 112 cos sin
8 8
3 3 3 32 cos sin
8 2 8 2
15 152 cos sin
8 8
z i
i
z i
i
