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### Graphing Complex Functions

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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/
```
Associated Topics:
College Imaginary/Complex Numbers
High School Imaginary/Complex Numbers

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