A complex number, Z, is the sum of an imaginary and a real number.
Z =
a + ib where a = Re{Z}, b = Im{Z} and i is the square root of -1. Because of the two component nature
of complex numbers, it is convenient to represent complex numbers as vectors in
a two dimensional plane called the complex plane. By convention, the
horizontal axis is the real axis and the vertical axis is the imaginary
one. The cartesian coordinates of Z in the complex plane are {a,b}.
Another set of valid coordinates specifying the location of Z are the polar
coordinates {r,a} where r is the length of a vector
from the origin to Z and a is the phase angle measured in a
clockwise manner from the real axis to Z. In this phasor representation, Z
= |Z| exp( i * alpha). A purely real number has alpha = 0 or alpha = pi and a purely
imaginary number has alpha = pi/
2 or alpha = -pi/ 2.
The animation below traces a circular path in the complex plane. Z begins
as a purely real number. The radius of the circle, the length of Z,
remains constant and alpha linearly increases with
time. The graphs to the left and below the circle trace the real and
imaginary projections of Z.
Section 1a Exercises:
The real and imaginary components of Z can be found using the relation:
exp( i alpha} = cos(alpha) + i sin(alpha) .
© 2000 by Prentice-Hall, Inc. A Pearson Company