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Complex Numbers: What and Why?


Date: 9/1/96 at 6:51:2
From: Anonymous
Subject: Complex Numbers: What and Why?

Dear Dr. Math:

What actually is a complex number? How does it work? What sort of 
problems do complex numbers solve? What are some examples?

Thank you very much!

Martin


Date: 10/19/96 at 21:45:18
From: Doctor Lynn
Subject: Re: Complex Numbers: What and Why?

Hi Martin.

Complex numbers are a fascinating extension to "normal" maths.  They 
are used a great deal at the university level.

I'm not certain of the history of the discovery of complex numbers, 
but I know that someone thought of them at least as early as the 
sixteenth century, although they suggested them more as a joke than as 
"serious" maths.

The easiest way to describe complex numbers is that they are a way of
finding the square root of a negative number.  You have probably been
told at some time that this is not possible and with normal numbers, 
it isn't.  But if we invent a new kind of number, called complex
numbers (or imaginary numbers), it is.

We start by calling the square root of -1 by the letter i (some people
use j).  We then say that the square root of any negative number is 
the square root of the positive number multiplied by i.  So the square 
root of -16 is 4i.

The second aspect of complex numbers comes from trying to answer the
question "what happens if we add a real number to a complex number?"
(a real number is the kind of number you are used to using).

The way of understanding this kind of question is generally described
with a special diagram called an Argand diagram.

Imagine drawing the number line across your page like the x axis of a
graph. Any real number can be represented as a point on this line, so
we call it the "real axis".  But i and the other complex numbers do 
not fit in between the real numbers; they are completely different 
things. So we draw another axis upwards, called the "imaginary axis", 
like this:

        ^Im
        |
       1-
        |
        |
---|----+----|----|-->Re
  -1    |    1    2
        |
      -1-         x
        |

Any number on the imaginary axis represents that number times i.
So where I have put 1 on the imaginary axis, it means i.

Now, if we try to add a real number and a complex number, the answer
doesn't fit on either axis, because it has a real part and a complex
part which won't go together.  So we mark a point on the diagram which
corresponds to the answer by measuring the real part along the real 
axis and the imaginary part on the imaginary axis.

So the place where I have put a "x" on the diagram represents the 
complex number 2 + (-1)*i, which is normally just written 2-i.

I'm going to stop there, but there is a LOT more things which complex
numbers do.  In answer to your question about what kind of problems 
they solve, the simplest answer is - complex ones!  They are used in 
engineering and in the design of computers but mainly they are used by 
mathematicians when they are working on other more complicated 
theorems.

I don't know if you have studied quadratic equations, but complex 
numbers provide the answers to some of these which would have been 
impossibe to solve before.  For example, the solutions to the 
quadratic equation x^2-6x+25=0 are 3+4i and 3-4i.  You can show that 
they work by putting them into the equation:

(3+4i)^2-6(3+4i)+25 = (9+24i+16i^2)-(18+24i)+25
                    = 9+24i+16i^2-18-24i+25
                    = 16+16i^2
but i is the square root of (-1), so i^2 = -1, so
                    = 16-16
                    = 0

You can try it yourself with (3-4i) and you should get the same 
answer.

Wow!  That's an awful lot of maths.  I hope it helped.  If you have 
any more questions, please write back.

-Doctor Lynn,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Imaginary/Complex Numbers

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