Complex Numbers Worksheet - Appendix F, Cengage Page 6
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11F6
APPENDIX F
Complex Numbers
EXAMPLE 6
Writing a Complex Number in Polar Form
Write the complex number
z
2
2 3i
in polar form.
Solution
The absolute value of z is
2
2
r
2
2 3i
2
2 3
16
4
and the angle is given by
b
tan
Imaginary
a
axis
2 3
Real
π 4
axis
−3
−2
1
2
3
−1
3.
Because
tan
3
3
and because
z
2
2 3i
lies in Quadrant III, choose
−2
to be
3
4
3.
So, the polar form is
⏐ ⏐ z = 4
−3
z
r cos
i sin
4
4
− −
z
= 2
2 3
i
−4
4 cos
i sin
.
3
3
See Figure F.3.
Figure F.3
The polar form adapts nicely to multiplication and division of complex numbers.
Suppose you are given two complex numbers
z
r
cos
i sin
and
z
r
cos
i sin
.
1
1
1
1
2
2
2
2
The product of
z
and
z
is
1
2
z
z
r
r
cos
i sin
cos
i sin
1
2
1
2
1
1
2
2
r
r
cos
cos
sin
sin
i sin
cos
cos
sin
.
1
2
1
2
1
2
1
2
1
2
Using the sum and difference formulas for cosine and sine, you can rewrite this
equation as
z
z
r
r
cos
i sin
.
1
2
1
2
1
2
1
2
This establishes the first part of the following rule. Try to establish the second part on
your own.
Product and Quotient of Two Complex Numbers
Let
z
r
cos
i sin
and
z
r
cos
i sin
be complex
1
1
1
1
2
2
2
2
numbers.
z
z
r
r
cos
i sin
Product
1
2
1
2
1
2
1
2
z
r
1
1
cos
i sin
,
z
0
Quotient
1
2
1
2
2
z
r
2
2
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