Applications of Complex Numbers

Date: 12/06/97 at 23:23:17
From: David Stewart
Subject: Applications of complex #s in society today

Dear Dr.Math,
 
I am a sophomore in high school in Dallas, TX. In my Algebra 2 class, 
my teacher assigned a project for us in which we must find some common 
applications of complex numbers, like the square root of negative one, 
in our society today. My teacher already gave me one example, which is 
that it is used to describe electrical currents. I was wondering if 
you know of any other uses of imaginary numbers. 

Dave

Date: 12/07/97 at 08:32:26
From: Doctor Jerry
Subject: Re: Applications of complex #s in society today

Hi David,

Engineers use complex numbers in analyzing stresses and strains on 
beams and in studying resonance phenomena in structures as different 
as tall buildings and suspension bridges. The complex numbers come up 
when they look for the eigenvalues and eigenvectors of a matrix. The 
eigenvalues are roots of a certain polynomial equation associated with 
a matrix. The matrices may be quite large, maybe 10000 by 10000, and 
the associated polynomials are of very high degree.

Complex numbers are used in studying the flow of fluids around 
obstacles, such as the flow around a pipe.  

Mathematicians use complex numbers in many ways, but one way is in 
studying infinite series, like

   e^z = 1+z+z^2/2!+z^3/3!+z^4/4!+...,

where z = x+i*y is a complex variable. This is a more "natural" 
environment to study series than on the real line. You might be 
interested in a fact that comes from the above series: it is that

   e^(i*pi) = -1.

This brief equation relates 4 of the most fundamental constants in 
mathematics, e, i, pi, and 1. Your calculator may be able to handle 
complex numbers. You might be able to check that

   e^(i*t) = cos(t)+i*sin(t),

from which the earlier result follows.  Just let t = pi.

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