SECTION 8.3
POLAR FORM AND DEMOIVRE’S THEOREM
483
8.3 POLAR FORM AND DEMOIVRE’S THEOREM
At this point you can add, subtract, multiply, and divide complex numbers. However, there
is still one basic procedure that is missing from the algebra of complex numbers. To see this,
consider the problem of finding the square root of a complex number such as
i
. When you
use the four basic operations (addition, subtraction, multiplication, and division), there
seems to be no reason to guess that
That is,
To work effectively with
powers
and
roots
of complex numbers, it is helpful to use a polar
representation for complex numbers, as shown in Figure 8.6. Specifically, if
is a
nonzero complex number, then let
u
be the angle from the positive
x
-axis to the radial line
passing through the point (
a, b
) and let
r
be the modulus of
So,
and
and you have
from which the following
polar form
of a
complex number is obtained.
a
1
bi
5
s
r
cos
d
1
s
r
sin
d
i
r
5
!
a
2
1
b
2
b
5
r
sin
,
a
5
r
cos
,
a
1
bi
.
a
1
bi
1
1
1
i
!
2
2
2
5
i
.
!
i
5
1
1
i
!
2
.
Figure 8.6
Imaginary
axis
Real
axis
Complex Number:
a
+
bi
Rectangular Form: (
a
,
b
)
Polar Form: (
r
,
)
b
(
a
,
b
)
r
a
0
θ
Definition of Polar Form
of a Complex Number
The
polar form
of the nonzero complex number
is given by
where
and
The number
r
is the
modulus
of
z
and
is the
argument
of
z
.
tan
5
b
y
a
.
a
5
r
cos
,
b
5
r
sin
,
r
5
!
a
2
1
b
2
,
z
5
r
s
cos
1
i
sin
d
z
5
a
1
bi