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RE: IMAGINARY NUMBERS -Reply



There have been several answers to Jennifer Seckinger's question about useful
applications of complex numbers. One in particular hits on the point I
would like to say a little more about.

Tim Sullivan said:

"Real" and "Imaginary" are taken too literally by most students.
I have a tough time convincing them that the imaginary part of the
complex exponential is just as good a solution as the real part. "Real" and
"Imaginary" are just historical terms and don't mean "physical" and
"unphysical". ...

In my classes, especially electronics, I emphasize that the complex number
approach is taken to simplify the algebra of dealing with phase shifts, and so
you don't have to manipulate trig identities all over the place. As a
simplifying procedure, you are really solving two problems at once, 90
degrees out of phase. Whichever phase you started with on the driving
voltage, that is what you have when you get to stating the current. Viewed
in this manner, it is (or at least should be :-) ) clear that it makes no
difference whether you take the cos(theta) projection or the sin(theta)
projection. Calling them "real" and "imaginary" confuses the issue terribly,
IMHO. Unfortunately, this confusion is perpetuated in some books, as has
been mentioned.

To cite a specific case, in what is a very widely used electronics textbook
(see H&H, 2nd ed., p.31-2), let me quote a few pertinent sentences:
"Because the actual voltages and currents are real quantities that vary with
time, ..." Then two rules are given, the second of which is: "2. *Actual*
[italics in the original] voltages and currents are obtained by multiplying
their complex number representations by exp(jwt) and then taking the real
part: V(t) = Re(Ve^jwt), I(t) = Re(Ie^jwt)." And later we read "Thus, in the
general case the acutal voltages and currents are given by V(t) =
Re(Ve^jwt)..." etc.

The poor student who is trying to understand this stuff comes away with the
notion that "real" is equated with "actual," meaning what is physically
measured. Instead, you can perfectly well use *either* projection for the
measured quantities. The terms "real" and "imaginary" are most unfortunate
baggage carried over from math, and have absolutely nothing to do with
physical reality or lack thereof. Besides, when solving a.c. circuits, what you
are usually concerned with are the magnitude of the total impedance and the
phase angle between two quantities, like driving voltage and current, and you
never really have to take projections.

Sorry to get on the soap box. But this is a sensitive subject with me. The
complex number method is so powerful for a.c. circuit analysis, that students
should be taught it earlier than we presently do. And when you see its power,
it can be an exhilarating experience, though I may not go as far as James
Marsh, who said:

"Indeed, when you discover how powerful complex numbers are for the first
time, it is not unusual for the heavens to open and for choirs of angels to be
heard singing. At least that was my experience. JSM"

Rondo Jeffery
Weber State University, Ogden, UT
RJeffery@weber.edu