Real Life Applications of Imaginary Numbers

Date: 03/08/98 at 13:25:11
From: Kevin Shah
Subject: Imaginary Numbers

Dr. Math,

I am a sophomore in high school and I am currently studying imaginary 
numbers. As an extra assignment, my teacher asked us to find out who 
uses imaginary numbers and why. Why are they so important? I tried 
looking through your archives and found info on what they are and the 
history behind them. But I need to find out who uses them and when do 
they use them.

Thanks

Date: 03/09/98 at 07:43:43
From: Doctor Jerry
Subject: Re: Imaginary Numbers

Hi Kevin,

It would be easier to ask who doesn't use complex numbers. Since 
complex numbers are often called "imaginary numbers," they often 
become suspect, seen as mathematicians' playthings. This is far, far 
from the truth, although apart from my saying this, it is not easy to 
prove.

Complex numbers enter into studies of physical phenonomena in ways 
that most people can't imagine. There is, for example, a differential 
equation, with coefficients like the a, b, and c in the quadratic 
formula, that models how electrical circuits or forced spring/damper 
systems behave. The movement of the shock absorber of a car as it goes 
over a bump is an example of the latter. The behavior of the 
differential equations depends upon whether the roots of a certain 
quadratic are complex or real. If they are complex, then certain 
behaviors can be expected. These are often just the solutions that one 
wants.

In modeling the flow of a fluid around various obstacles, like around 
a pipe, complex analysis is very valuable for transforming the problem 
into a much simpler problem.

When everything from large structures of riveted beams to economic 
systems are analyzed for resilience, some very large matrices are used 
in the modeling. The matrices have what are called eigenvalues and 
eigenvectors. The character of the eigenvalues, whether real or 
complex, is important in the analysis of such systems.

In everyday use, industrial and university computers spend some 
fraction of their time solving polynomial equations. The roots of such 
equations are of interest, whether they are real or complex.

And complex numbers are useful in studying number theory, which is the 
study of the positive integers. The techniques in complex analysis 
are just one more tool that researchers have.

-Doctor Jerry, The Math Forum
Check out our web site http://mathforum.org/dr.math/