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### Uses of Imaginary Numbers

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Date: 03/24/97 at 19:27:20
From: Ryan Boyer
Subject: Uses of Imaginary Numbers

I have been pulling my hair out to find something, anything on real
life applications of imaginary numbers.  Can you help?
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Date: 03/25/97 at 14:33:57
From: Doctor Steven
Subject: Re: Uses of Imaginary Numbers

Imaginary numbers were discovered in the process of trying to find all
the roots of polynomial equations.  Nicolo Tartaglia had a formula for
finding the roots of a cubic equation that included the roots of -1,
which he knew wasn't a number.  He also had to add and multiply these
numbers together, and in the end these "numeri ficti" (fictitious
numbers) canceled out (since he chose his polynomial equations
with care, making sure they had all real roots).

Imaginary numbers may not have any true meaning in the real world
(i.e., it is hard to have 3i dollars, or even to be 240i degrees
east).  But in many continuous applications where the state of a model
at one point in time is dependent on the state of the model at a
previous point in time, these imaginary values can affect the value.

For instance if we look at the velocity of the tip of an airplane wing
and decide that the velocity at time n is 1/2 the velocity squared at
time n-1:

Say the velocity at time n-1 is 3 + 4i meters/sec:

v(n-1) = 3 + 4i meters/sec

We say it has velocity 3 meters/sec since the imaginary part
doesn't really have any meaning in this case.

But the velocity at time n is:

v(n) = 1/2*v(n-1)^2 = 1/2*(3 + 4i)^2 = -3.5 + 12i

And we say its velocity is -3.5 meters/sec. If we had taken only
the real part of the first velocity we would have incorrectly gotten
4.5 meters/sec as our velocity.

So you can see that imaginary numbers can affect the output of the
model later in its cycle. This example is probably not an accurate
representation of the velocity of an airplane wing's tip, but it does
get the point across.

Hope this helps.

-Doctor Steven,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
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Associated Topics:
High School Number Theory

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