8.3 Graphical representation 229
numbers, withy 1 = 10 , are represented by points on the xor real axisand pure imaginary
numbers, withx 1 = 10 , lie on the yor imaginary axis. The representation is called the
Argand diagram.
The distance of point zfrom the origin, , is called the modulusor
absolute valueof zand is written
r 1 = 1 mod 1 z 1 = 1 |z| (8.14)
Complex numbers with the same modulusr 1 = 1 |z|lie on the circle of radius rin the
plane. When|z| 1 = 11 the point lies on the unit circle.
The polar representation
The position of the pointz 1 = 1 x 1 + 1 iyin the plane can be specified in terms of the polar
coordinates rand θ, as in Figure 8.1. Then
x 1 = 1 r 1 cos 1 θ, y 1 = 1 r 1 sin 1 θ
and the complex number can be written in the polar form
z 1 = 1 r(cos 1 θ 1 + 1 i 1 sin 1 θ) (8.15)
The angle θis called the argumentor angle of z,
θ 1 = 1 arg 1 z (8.16)
The angle can have any real value but, as discussed in Chapter 3, the trigonometric
functions in (8.15) are periodic functions of θso that the number zis unchanged when
a multiple of 2 πis added to θ. A unique value can be computed from xand yby
the prescription given in Section 3.5 for the transformation from cartesian to polar
coordinates; thus, given that tan 1 θ 1 = 1 y 2 x,
if x 1 > 10
if x 1 < 10
(8.17)
in which the inverse tangent has its principal value.
=
+
−tan
1y
x
π
argz
y
x
=
−tan
1rxy=+
22...........................................................................................................................................................................................................z=x+iy
θ
x(realaxis )
y(imaginary axis)
r
x
y
o
..........
...
..
...
...
..
...
...
...
...
..
...
...
..
...
...
..............
.......
.......
.................................................................................................................................................................................................................................
.........
..........
...........
.........
..........
...........
..........
.........
...........
..........
..........
..........
..........
...........
.........
..........
...........
.........
..........
...........
.........
..........
...........
.........
..........
...........
..........
.........
...........
..........
.........
...........
..........
.........
...........
..........
.........
...........
..........
..........
..........
..........
..........
..........
..........
..........
..
Figure 8.1