Complex Numbers Worksheet - Appendix F, Cengage Page 5
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APPENDIX F
Complex Numbers
Polar Form of a Complex Number
Imaginary
axis
Just as real numbers can be represented by points on the real number line, you can
(3, 2)
3
represent a complex number
or
−
( 1, 3)
3 + 2i
or
2
−
z
a
bi
1 + 3
i
as the point
a, b
in a coordinate plane (the complex plane). The horizontal axis is
1
called the real axis and the vertical axis is called the imaginary axis, as shown in
Real
Figure F.1.
axis
−2
−1
1
2
3
The absolute value of a complex number
a
bi
is defined as the distance
− −
between the origin
0, 0
and the point
a, b .
( 2, 1)
or
− − i
2
The Absolute Value of a Complex Number
Figure F.1
The absolute value of the complex number
z
a
bi
is given by
2
2
a
bi
a
b
.
If the complex number
a
bi
is a real number that is, if
b
0 ,
then this defi-
nition agrees with that given for the absolute value of a real number.
2
2
a
0i
a
0
a .
To work effectively with powers and roots of complex numbers, it is helpful to
Imaginary
axis
write complex numbers in polar form. In Figure F.2, consider the nonzero complex
number
a
bi.
By letting
be the angle from the positive real axis (measured
counterclockwise) to the line segment connecting the origin and the point
a, b ,
you
can write
( , )
a b
a
r cos
and
b
r sin
r
b
2
2
where
r
a
b
.
Consequently, you have
θ
Real
a
bi
r cos
r sin
i
axis
a
from which you can obtain the polar form of a complex number.
Polar Form of a Complex Number
Figure F.2
The polar form of the complex number
z
a
bi
is given by
z
r cos
i sin
2
2
where
a
r cos , b
r sin , r
a
b
,
and
tan
b a.
The number r
is the modulus of z, and is called an argument of z.
NOTE The polar form of a complex number is also called the trigonometric form. Because
there are infinitely many choices for
,
the polar form of a complex number is not unique.
0 ≤
Normally, is restricted to the interval
< 2 ,
although on occasion it is convenient to
use
< 0.
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