Complex Numbers

Date: 04/11/2001 at 03:56:59
From: Rosie
Subject: Complex numbers

What exactly is the complex number system comprised of?

Thank you.

Date: 04/11/2001 at 18:01:53
From: Doctor Ian
Subject: Re: Complex numbers

Hi Rosie,

Let's start with the square root function.  The most 'natural' square 
roots are the square roots of perfect squares, e.g., 

  2 = sqrt(2*2) = sqrt(4)
  3 = sqrt(3*3) = sqrt(9)

and so on. 

For any positive number x, we can write

  sqrt(x) = ?

and we can compute either an exact value or an approximation for ? . 

Once a notation exists, there is always somebody who will try to use 
it for something it wasn't intended to handle. For example, what do we 
do with 

  sqrt(-4) = ?

The answer can't be 2, since 2^2 = 4, not -4.  And it can't be -2, 
since (-2)^2 = 4, not -4.  For a long time, the answer was that the 
operation simply didn't make sense, much like dividing by zero. 

But it turns out that if you invent a single number, i, which is 
defined such that

  i*i = -1

then suddenly you can write things like

  sqrt(-4) = sqrt(4) * sqrt(-1)
           = 2 * i  

or just '2i'.  

Now, what if we want to do something like 

  3 + sqrt(-4) = ?

Well, we can make some progress:

  3 + sqrt(-4) = 3 + 2i

but now we're stuck.  We can't add a multiple of i to something that 
isn't a multiple of i, any more than we can add fractions with 
different denominators.  What we _can_ do is consider 

  3 + 2i 

the reduced form of a new kind of number, called a 'complex' number.

We can plot the real numbers on a line,

  <------------------+---------------------->
             -2  -1  0   1   2   3

But we have to plot the complex numbers on a plane: 

                     |       *  2 + 3i
                     |
       -2 + i *      |
  <------------------+------------------------>
             -2  -1  0   1   2   3
                     |
                     |   * 1 - 2i
                     |
                     |   * 1 - 4i

This plane is called the 'complex plane', and to locate the complex 
number 

  a + bi

you move a units along the real (horizontal) axis, and b units along 
the complex (vertical) axis.  That is, plotting the point 

  a + bi

on the complex plane is just like plotting the point (a,b) on a 
regular x-y plane. 

Note that we can think of the real numbers as the 'shadows' of the 
complex numbers. For every real number, like 2.5, there is an entire 
line of complex numbers, 

  2.5 + 68.92232i
  2.5 + 2i
  2.5 + 1i
  2.5 + 0i
  2.5 - 1i
  2.5 - pi*i

If you think of a light shining down from above the line, the line 
would cast a shadow on the real number line at 2.5.  

So, to answer your question, the complex numbers system is composed of 
the complex plane, the rules for locating points of the form a + bi on 
that plane, and the definition that i^2 = -1.  

Does this help?  Write back if you'd like to talk about this some
more, or if you have any other questions.

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

Date: 04/12/2001 at 04:20:02
From: Rosie
Subject: Re: Complex numbers

Thank you very much for the help. I now understand complex numbers!

Rosie