Why Do Imaginary Numbers Exist?

Date: 02/17/2001 at 00:43:21
From: Cara
Subject: Imaginary numbers

Why are there imaginary numbers? I don't understand why we need to 
have imaginary numbers or what their purpose is.

Date: 02/17/2001 at 06:01:52
From: Doctor Mitteldorf
Subject: Re: Imaginary numbers

Dear Cara,

I remember thinking just the way you do when I first came on the more 
abstract forms of math. And the name "imaginary" just adds to the 
sense that they don't really exist; they're just ideas.  
  
So there are two sorts of answers to your question. One is that 
mathematicians are just like children at Christmas. We love new toys, 
and we want to try this and that to see what happens. The mathematics 
of imaginary numbers exists just because you can do so much with it 
that's fun - perhaps the same reason that people love to play chess.
  
Did you know that even though i was invented as a solution to 
equations like x^2 = -1, you could pick any other equation that 
doesn't have a real solution and define the solution to that as our 
new "i" and end up with the same mathematics?  You could start, for 
example, with the equation sin(x) = 2, and define the number x that 
solves this equation as the fundamental imaginary number "i," and by 
the time you'd finished trying to solve all the other equations, the 
math you'd end up with is the same as what we've got with the 
standard "i."

You'd think that once you started inventing numbers to solve 
equations you'd have to keep going and going. We'll let i be the 
solution of x^2 = -1. But then, what's the square root of -i?  We'll 
call that j. And what's the square root of -j? We'll call that k....

Well, it turns out you don't have to do that. There's a combination 
of real and imaginary numbers that satisfies the equation x^2 = -i. In 
fact, there are two such combinations - just as every positive real 
number has two square roots, one the opposite of the other, every 
negative and imaginary number also has two square roots. Can you 
think of how you would go about finding the square root of -i?

I promised you there were "two sorts" of answers, but I haven't 
gotten to the second one yet. The second is that there are areas of 
science where imaginary numbers turn out to be enormously useful. 
That may be hard to imagine, because it's obvious that when you ask 
questions like "how far does the ball bounce" or "how much energy 
does it have," etc., the answer must always be a real number. But 
there are situations where, even though you start with real physical 
quantities and what you are trying to compute is another real 
physical quantity, the shortest, easiest way to compute that real 
number takes you through imaginary numbers along the way. The reason 
for this has to do with the fact that so many trig identities are 
complicated and unexpected. Exponential equations are comparatively 
simple, and obey simple laws. When you get deep into imaginaryland, 
you find a close, unexpected relation between trig and exponentials. 
It turns out that an angle is a lot like an imaginary logarithm, and a 
logarithm is like an imaginary angle. Until you do the math, I'm sure 
it's hard to even imagine what that could mean. But it's an exciting 
chapter for you to look forward to.

So my advice is to keep playing with those imaginary toys. And you 
can read more, if you like, at our FAQ on imaginary numbers:

   http://mathforum.org/dr.math/faq/faq.imag.num.html   

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