Page 4-1
Chapter 4
Calculations with complex numbers
This chapter shows examples of calculations and application of functions to
complex numbers.
Definitions
A
complex number z
is a number written as
z = x + iy
, where
x
and
y
are
real numbers, and
i
is the
imaginary unit
defined by
i
2
= -
1. The complex
number
x+iy
has a
real par
t,
x = Re(z),
and an
imaginary par
t,
y = Im(z).
We can think of a complex number as a point
P(x,y)
in the x-y plane, with the
x-axis referred to as the
real axi
s, and the y-axis referred to as the
imaginary
axi
s. Thus, a complex number represented in the form
x+iy
is said to be in its
Cartesian representatio
n. An alternative Cartesian representation is the
ordered pair
z = (x,y)
. A complex number can also be represented in polar
coordinates
(polar representatio
n) as
z = re
i
θ
= r
⋅
cos
θ
+ i r
⋅
sin
θ,
where
r =
|z|
=
2
2
y
x
+
is the
magnitude
of the complex number z, and
θ
= Arg(z) =
arctan(y/x)
is the
argument
of the complex number z. The relationship
between the Cartesian and polar representation of complex numbers is given
by the
Euler formula
:
e
i
θ
= cos
θ
+ i sin
θ.
The
complex conjugate
of a
complex number
z = x + iy = re
i
θ
, is
z = x – iy = re
-i
θ
. The complex
conjugate of
i
can be thought of as the reflection of z about the real (
x
) axis.
Similarly, the
negative
of z,
–z = -x-iy = - re
i
θ
,
can be thought of as the
reflection of
z
about the origin.
Setting the calculator to COMPLEX mode
When working with complex numbers it is a good idea to set the calculator to
complex mode, use the following keystrokes:
H
)@@CAS@
2˜˜™
@
@CHK@
The COMPLEX mode will be selected if the CAS MODES screen shows the
option
_Complex
checked off, i.e.,
