Graphing Complex and Real Numbers

Date: 02/26/2003 at 20:32:45
From: Paul Lehmbeck
Subject: Graphing Complex and Real Numbers

I have been finding the zeros for polynomial functions in class and I 
have found real, imaginary, and complex numbers as the zeros. But when 
I graph the zeros I run into a problem. Since on the Cartesian plane 
you can only graph real zeros and real solutions, I wonder if you are 
truly graphing the function when you omit the complex and imaginary 
zeros and solutions. For example, for f(x) = x^2+1 there are no real 
zeros, but the imaginary zeros are -i and i.  

MAIN QUESTION - Since you can't graph the imaginary zeros, is the 
function truly being represented by the graph?   

This has led me to wonder if it is at all possible to graph the real, 
imaginary, and complex input and output solutions on the same graph 
and still remain in the 2nd or 3rd dimension without hypothesizing 
about a fourth dimension? If it is possible to represent all solutions 
to a function, including real, complex, and imaginary numbers, on a 
single graph, what would that graph be if it is not a Cartesian plane 
or Complex plane?  If it is not possible, would you then need two 
graphs for the same function - one for the real inputs and outputs, 
and another for the imaginary and complex inputs and outputs (if it is 
possible) to show all solutions?  

Thank you for your time and explanation.

Date: 02/27/2003 at 15:49:41
From: Doctor Tom
Subject: Re: Graphing Complex and Real Numbers

Hi Paul,

Great question!

At this stage, almost every function you've seen takes real values as 
inputs and gives real values as outputs, so when you plot such a 
function on the standard x-y plane, you are showing all the values of 
the function with real-valued inputs.

For a function like y = x^2 + 1 that has no real roots, you are still 
plotting all the information for real inputs. The curve never passes 
through y = 0, so there simply are no real roots. You can't plot the 
output for an input value of i, since i is imaginary and not found on 
the real line.

In our standard three-dimensional space it is impossible to plot 
functions, like polynomials, that have complex inputs and complex 
outputs. You can imagine the real-imaginary complex plane, and plot 
outputs above it, but to display all the information, you would need 
to display two output values - the real and the imaginary components.

Given this restriction, here are a few things that are normally done:  
It is possible to have one 3-D plot that shows the real part of the 
function and another 3-D plot that shows the imaginary component.  
Both would appear to be surfaces over the complex plane whose heights 
above that plane correspond to the real and imaginary components of 
the function's values.

It's a bit tricky from a pair of plots like this to find roots, 
however.  The reason is that to find a zero, the function has to be 
zero both in its real AND imaginary components. In general, functions 
will have lines or curves on the complex plane above which the real 
part is zero or above which the imaginary part is zero. Only where 
those curves cross (and where both components are zero) is where the 
function has a root.

The other method commonly uses plots in the third dimension the 
absolute value of the complex output.  If z = a+bi, the absolute value 
of z is the square root of (a^2 + b^2), and this can be zero only if 
both a and b are zero.  Any place this surface touches the complex 
plane is a root, but the plot throws away the individual real and 
imaginary components of the function's output.  Plotting all three 
will give you a lot of information to help visualize what's going on.  
Unfortunately, you need a computer to do this quickly enough and 
accurately enough that you can get useful images.

Remember that you are usually looking at a tiny fraction of the 
available polynomials. For example, could you find the roots of:

(3-2i)x^2 + (7+3i)x - (15 + 43i) = 0?

It is just a quadratic equation, and the only thing that makes it 
different from the ones you've looked at is that it has complex 
coefficients.

There is a reasonable way to extend almost all of your favorite 
functions to the complex domain, although some strange things do 
happen. For example, it makes sense to talk about sine(3-2i) or 
log(-4+3i).

You may have seen this:

e^(it) = cos(t) + i sin(t)

You have probably only used it for real values of t, but it is true 
for complex values as well, and by messing around with those, you 
could learn a lot!

I hope this helps.

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