OpenStax-CNX module: m49408
7
modulus
We use the term
to represent the absolute value of a complex number, or the distance from the
(x, y) .
r,
θ
origin to the point
The modulus, then, is the same as
the radius in polar form. We use
to indicate
the angle of direction (just as with polar coordinates). Substituting, we have
z = x + yi
z = rcos θ + (rsin θ) i
(5)
z = r (cos θ + isin θ)
Writing a complex number in polar form involves the following conver-
a general note label:
sion formulas:
x = rcos θ
(6)
y = rsin θ
2
2
r =
x
+ y
Making a direct substitution, we have
z = x + yi
(7)
z = (rcos θ) + i (rsin θ)
z = r (cos θ + isin θ)
modulus
argument
r
θ
r
θ
r (cos θ + isin θ) .
where
is the
and
is the
. We often use the abbreviation
cis
to represent
Example 4
Expressing a Complex Number Using Polar Coordinates
4i
Express the complex number
using polar coordinates.
Solution
z = 4i
z = 0 + 4i.
On the complex plane, the number
is the same as
Writing it in polar form, we
r
have to calculate
rst.
2
2
r =
x
+ y
2
2
r =
0
+ 4
(8)
r =
16
r = 4
π
x.
x = rcos θ,
x = 0,
θ =
.
Next, we look at
If
and
then
In polar coordinates, the complex
2
π
π
π
z = 0 + 4i
z = 4 cos
+ isin
4
.
number
can be written as
or
cis
See Figure 6.
2
2
2