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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 part, x = Re(z), and an imaginary part, 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 axis, and the y-axis referred to as the imaginary
axis. Thus, a complex number represented in the form x+iy is said to be in its
Cartesian representation. An alternative Cartesian representation is the
ordered pair z = (x,y). A complex number can also be represented in polar
coordinates (polar representation) as z = re
i
θ
= r⋅cosθ + i r⋅sinθ, where r =
|z| =
22
yx + 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
, isz = 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.,
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