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### 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/
```
Associated Topics:
College Imaginary/Complex Numbers
High School Equations, Graphs, Translations
High School Imaginary/Complex Numbers

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