Graphing Complex Functions

Date: 08/11/98 at 03:44:56
From: Michael Liddle
Subject: Complex solutions to polynomials

In my Calculus class (Paraparaumu College, New Zealand), we have just 
begun a new unit on "Complex Numbers." In our study so far, we have 
learned the significance of i and its properties and the way in which 
it behaves with operations and relations. We have also learned about 
the modulus and argument of z.

My question, however relates to the complex solutions to various 
polynomials. Up until now we have said that, for example, the quadratic 
equation (y = x^2 + 5x + 12) when y = 0 has no solutions. We have now 
been told that in effect this equation in fact has two solutions, just 
that they are both "imaginary" as opposed to "real" numbers. What I  
want to find out is where (if anywhere), these numbers lie on the graph 
of this equation (x,y Cartesian form). 

I asked my calculus teacher this question and he said that in 20 years 
of teaching he has never been asked this before.

My problem may well lie in the fact that my experience in this area is 
extremely limited, and I am simply as yet unable to comprehend (or 
accept) that in fact they do not lie anywhere on the graph. However I 
thought that maybe that there was a third "imaginary axis," or that 
the answer may lie in a "fourth dimension" that cannot be represented 
graphically.

Thank you very much for your assistance in this area.

Date: 08/14/98 at 14:17:34
From: Doctor Benway
Subject: Re: Complex solutions to polynomials

Hi Michael, 

The easy answer to your question is that yes, you do need four 
dimensions to picture where the solutions are if you want to plug 
complex variables such as 1 + 2i into your equations. The regular way 
of representing complex numbers in the plane is to let the x coordinate 
be the real part and the y coordinate be the imaginary part, so 1 + 2i 
would correspond to the point (1, 2). However, when we plot functions 
of complex numbers, both the "input" variable, the domain of the 
function, and the "output" variable, the range of the function, require 
two axes. Therefore you would need four dimensions: two dimensions to 
plot the coordinate of the point you are plugging into the equation 
and two dimensions to plot the output. For example, when we square 
(1 + 2i), our result is -3 + 4i. To plot this would require four 
dimensions, two to represent (1 + 2i) and two to represent (-3 + 4i). 

However, not all is lost! There is another way of thinking about 
functions (which should be instructive when you're only looking at 
functions of a real variable as well), and that is thinking of them as 
a mapping of points to other points. For example, imagine you are 
standing on the real line on the point corresponding to the number 3.  
Suddenly out of nowhere comes the function "x^2 - 1" which moves all 
the points around. You are swept away from 3 in a whirlwind of 
mathematical activity. When the dust finally settles, you find yourself 
standing on the point corresponding to the number 8 (3^2 - 1 = 8). 

Thinking of functions this way, you see that functions move points 
around. A little more thought shows the following: solutions to the 
equation x^2 - 1 = 0 can be thought of as the points which get moved 
to 0 when the function x^2 - 1 strikes. Had you been standing on the 
points corresponding to the numbers 1 or -1, then you would have ended 
up standing on the origin after the function hit.

Now imagine yourself standing on the complex plane instead of on the 
number line. Say you are standing on the point corresponding to the 
complex number -2 + i, which in cartesian coordinates would be the 
point (-2,1). Along comes the function x^2 + 4x + 5, which sweeps up 
all the points and sets them down in different places. When the dust 
settles, you find yourself standing at the origin, (0,0). "Aha," you 
say to yourself, "I must have been standing on a solution to 
x^2 + 4x + 5 = 0."  

One way to represent these mappings is to have one set of axes for the 
domain and another set of axes for the range. It's a little confusing 
to picture what the function is doing, but if you trace the domain 
with one finger and the range with another, you get an idea of what's 
happening. 

I hope this helps answer your question. This problem bothered me for a 
long time when I was in high school too, and I was very happy when I 
finally found a way to think about it that made sense to me. Thanks 
for writing.

- Doctor Benway, The Math Forum
Check out our web site! http://mathforum.org/dr.math/   

Date: 08/15/98 at 12:32:57
From: Doctor Sonya
Subject: Re: Complex solutions to polynomials

Hi there. My name is Dr. Sonya, and I am one of the other math 
doctors. I just wanted to point you towards a very interesting 
discussion in our archives about what the complex roots of polynomials 
have to do with the graph. The URL of the page is:

  http://mathforum.org/dr.math/problems/complex.roots.html   

and it is one of our best discussions.

- Doctor Sonya, The Math Forum
Check out our web site! http://mathforum.org/dr.math/