9.3 Complex Numbers; The Complex Plane; Polar Form of Complex Numbers.
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Transcript of 9.3 Complex Numbers; The Complex Plane; Polar Form of Complex Numbers.
9.3Complex Numbers;
The Complex Plane;Polar Form of Complex Numbers
Complex Numbers are numbers in the form of a biwhere a and b are real numbers and i, the imaginary unit, is defined as follows:
2 1i 1i And the powers of i are as follows:
1i i2 1i 3 2 1i i i i i 4 1i
The value of in, where n is any number can be found by dividing n by 4 and then dealing only with the remainder. Why?
Examples:
18i ? 18 4 4 2with a remainder of 18 2i i Then from the chart on the previous slide18 2 1i i
1)
2) 27i ? 27 4 6 3with a remainder of 27 3i i27 3i i i
Then from the chart on the previous slide
In a complex number a bi
a is the real part and bi is the imaginary part.
When b=0, the complex number is a real number.
When a0, and b0, as in 5+8i, the complex number is an imaginary number.
When a=0, and b0, as in 5i, the complex number is a pure imaginary number.
Lesson Overview 9-5A
Lesson Overview 9-5B
5-Minute Check Lesson 9-6A
Real Axis
Imaginary Axis
O
z a bi
The Complex Plane
Let z a bi 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).
2 2z a b
Real Axis
Imaginary Axis
O x
y
|z|
z a bi
sinicosr
is sometimes abbreviated as
cisr
4
-3 Real Axis
Imaginary Axis
z =-3 + 4i
z = -3 + 4i is in Quadrant II
x = -3 and y = 4
4
-3
z =-3 + 4i
5r
Find the reference angle () by solving
x
ytan
3
4tan
3
4tan 1
13.53
4
-3
z =-3 + 4i
5r
Find r: 2213 r
2r
4
-3 Real Axis
Imaginary Axis
i3
3
12
Find the reference angle () by solving
x
ytan
3
1tan
30
3
12
3
11 tan
3
12
33030360 sinicosrz
3303302 sinicosz
Find the cosine of 330 and substitute the value.
Find the sine of 330 and substitute the value.
Distribute the r
6
5
6
52
sinicos
Write in standard (rectangular) form.
2
3
6
5
cos
2
1
6
5
sin
i
2
1
2
32 i 3
Lesson Overview 9-7A
Product Theorem
Lesson Overview 9-7B
Quotient Theorem
5-Minute Check Lesson 9-8A
5-Minute Check Lesson 9-8B
Powers and Roots of Complex Numbers
DeMoivre’s Theorem
81 81 3
2 2i
4326 i
What if you wanted to perform the operation below?
Lesson Overview 9-8A
Lesson Overview 9-8B
Theorem Finding Complex Roots
roots
Find the complex fifth roots of
The five complex roots are:
for k = 0, 1, 2, 3,4 .
2 2r a b
32 0i2 232 0 232 32
1 0
32tan 1 0tan 0
32 0 0cos i sin
5 0 360 0 36032
5 5 5 5
k kcos i sin
2 72 72cos k i sin k
0k 2 0 0cos i sin 2 1 0i 2
1k
2 72 72cos k i sin k
2 72 72cos i sin 62 1 90. . i
2k 2 144 144cos i sin 1 62 1 18. . i
3k 2 216 216cos i sin 1 62 1 18. . i
4k 2 288 288cos i sin 62 1 90. . i
The roots of a complex number a cyclical in nature.
That is, when the points are plotted on a polar plane or a complex plane, the points are evenly spaced around the origin
4
2
-2
-4
-5 5
Complex Plane
6
4
2
2
4
5 5 10
Polar plane
6
4
2
2
4
5 5 10
Polar plane
6
4
2
2
4
5 5 10
4
2
-2
-4
-5 5
To find the principle root, use DeMoivre’s theorem using rational exponents.
That is, to find the principle pth root of
a bi Raise it to the power1
p
Example
Find 3 i
First express as a complex number in standard form. i0 i
Then change to polar form 1r 2 2r a b
btan
a
1
0tan 1 1
0tan 90
1 90 90cos i sin
You may assume it is the principle root you are seeking unless specifically stated otherwise.
1 90 90cos i sin Since we are looking for the cube root, use DeMoivre’s Theorem and raise it to the power1
31
3 1 11 90 90
3 3cos i sin
1 30 30cos i sin
3 11
2 2i
3 1
2 2i
Example:
Find the 4th root of 2 i
1
42 i
Change to polar form 5 153 4 153 4cos . i sin .
Apply DeMoivre’s Theorem
1
41 1
5 153 4 153 44 4
cos . i sin .
1 22 38 4 38 4. cos . i sin .
96 76. . i