Can an Imaginary Number Be a Valid Answer?

Date: 01/12/2006 at 15:06:55
From: Greg and the group from Neelin
Subject: are imaginary numbers valid answers?

We know that the answer to the equation x^2 + 1 = 0 is root negative 
1 and the only answers to that are +/- i.  But if "i" does not exist
how can it be an answer?

Date: 01/12/2006 at 22:16:40
From: Doctor Peterson
Subject: Re: are imaginary numbers valid answers?

Hi, Greg.

Your question is just the one that bothered mathematicians when 
complex numbers were first discovered, and for centuries after.  But
the same question was asked about negative solutions; only positive
solutions to equations were considered meaningful, because it was
assumed that the solution had to represent some measurable length.

Now, if your equation arises from a problem where only positive
numbers make sense (such as a length), then the answer is that
negative or imaginary solutions are invalid--but that's due to the
problem, not the equation itself.  Likewise, there are problems in
which only a real number makes sense (such as a coordinate in space);
then positive and negative solutions are valid, but an imaginary one
is not.

But an equation itself doesn't relate to any specific real-world
problem; it only asks what numbers make it true, and complex numbers
are numbers.  So we can ask "what are all the real solutions to this
equation?", and the answer will be "there are none", only because the
problem was stated in terms of real numbers.  Without that 
restriction, imaginary solutions are perfectly valid.

After this idea was accepted, that complex numbers really are numbers
(and DO exist, in the sense that any numbers, though abstract, can be
said to exist), it was found that there ARE real problems in which
complex solutions make sense.  For example, in electronics there are
problems in which the complex solutions of an equation tell you how a
circuit will respond to certain inputs; imaginary numbers correspond
to alternating current, and real numbers to direct current (more or
less).  In this setting, complex solutions not only are valid, they 
are what you're looking for!

See this page:

  Using Imaginary Numbers
    http://mathforum.org/library/drmath/view/53879.html 

If you have any further questions, feel free to write back.


- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/