The Complex Plane
THIS IS NOT SIMPLE...
objects commonly associated with the
topic of chaos. The Mandelbrot set and
Julia sets are prime examples of the
complexity hidden in simple systems. The
mathematics behind these objects is
straightforward. The results are
fantastically detailed and beautiful.
We will briefly review the concept of a
"real number line" which we covered in
Fundamentals. We have made use of this
sort of number line in laying out the
axes of the graphs displayed so far. This
line of numbers is generally taken to
be increasing to the right or up,
extending from minus infinity to plus
infinity. The number line is "everywhere
dense" meaning that between any two
numbers however close, there are
infinitely many other numbers. Included on the
real number line are integers, fractions
and irrational numbers. Run the Real
Number Line display to review the concept
if you feel the need.
The real numbers with which we have been
working satisfy most of the needs for
calculation and accounting in our
everyday life. There is no place on the real
number line however for a number which
when multiplied by itself gives a
negative result, for example the square
root of minus 1. Not wishing to leave
negative numbers without square roots,
mathematicians reasoned that such roots
must be off the real number line. Such
numbers are called "imaginary" numbers.
They are of course as real as any other
but imaginary remains the way in which
we refer to them.
Imaginary numbers are measured by their
distance from the real number line. Like
real numbers they form a whole continuum
from negative to positive infinity, an
imaginary number line perpendicular to
the real number line. The lines intersect
at zero which is their only common value.
The two perpendicular number lines
define a plane, a two dimensional surface,
and the question comes up, "What is
the nature of numbers which do not lie on
either of the number lines?". These
numbers are called "complex".
Complex numbers have a real part and an imaginary
Complex numbers may be written as a + b
times i, where a and b are real numbers
and i represents the square root of minus
1. This formula means go a distance a
along the real line, turn 90 degrees left
and go a distance b in the imaginary
direction. Multiplying by i has the
effect of redirecting movement in the
complex plane 90 degrees counterclockwise.
All numbers may be thought of as
complex numbers of the form described. If
b happens to be zero, the number is
all real. If a happens to be zero the
number is all imaginary.
The Complex Plane display shows a region
of the complex plane, bounded by upper
and lower limits in both the real and
imaginary dimensions. Real values are
plotted horizontally and imaginary values
are plotted vertically. What was the X
axis in previous displays is now the R (real)
axis. The former Y axis is now the
I (imaginary) axis. The limiting values
are displayed in input boxes located
near the ends of the left and bottom
boundaries of the displayed region of the
complex plane. If limit changes are
appropriate, you may enter new values in
these input boxes.
There is a series of control buttons
across the top of the display. When
required, the Action button causes the
display to do something. If the action is
continuous, the Cut button is enabled so
you can turn the action off. The
remaining buttons allow you to manipulate
the size and location of the displayed
region of the complex plane if that is
appropriate. The Near button reduces the
limits symmetrically by 25%, effectively
bringing any plotted objects nearer to
the viewer. The Far button has the
opposite effect. The Left, Right, Up and Down
buttons shift the visible region of the
complex plane one whole window in the
indicated direction. Buttons that are not
required for a particular display are
Run the Complex Plane display and
experiment with the controls.
The ordinary arithmetic operations of
adding, subtracting, multiplying and
dividing are defined for complex numbers.
We will be dealing with addition and
multiplication in this section of the
program. To add complex numbers just add
the real parts together and the imaginary
parts together to form the sum.
Plotting the numbers on the complex plane
illustrates the process. Run the
Complex Number Addition display.
An alternative way to represent a complex
number in the plane is to show it as a
directed line segment. A line of a
certain length in a given direction. This
sort of line segment is called a "vector".
Each complex number may be thought of
as a vector from the origin (0+0*i) to
the number (a+b*i). With regard to
vectors we speak of a head and a tail.
The head of a vector is located where the
line ends, the tail where the line begins.
We put an arrow head on the head end
of the vectors. To add complex numbers
then we may just place them tail to head.
The sum is a vector from the origin to
the head of the added vector. Run the
Complex Number Vector Addition display to
The previous displays illustrated the
adding of complex numbers which is a
straightforward process intuitively
similar to the adding of real numbers.
Multiplication of complex numbers is more
complicated and does not have such a
simple geometric representation in the
complex plane. To multiply two complex
numbers we must multiply them as
binomials. This means that we multiply each
part of one number by each part of the
other and add all the products. In
(a+b*i)*(c+d*i) = (a*c)+(a*d*i)+(b*i*c)+(b*i*d*i)
This mess may be simplified.
In the expression:
(a+b*i)*(c+d*i) = (a*c)+(a*d*i)+(b*i*c)+(b*i*d*i)
Consider the term (b*i*d*i). The symbol
'i' represents the square root of -1. So
i*i is just -1. This line of reasoning
reduces (b*i*d*i) to (-b*d). Now let's
collect the real parts together (a*c-b*d)
and the imaginary parts (a*d+b*c)*i
giving us the complex number in the
standard form as below:
Multiplying two complex numbers then
gives a new complex number as shown.
On the next display we see a graphical
example of complex number multiplication
using the vector representation. The
resulting vector is not as clearly related
to the two original vectors as was the
case in addition. Run the Complex Number
Multiplication display. Try to predict
where the head of the product vector will
Now that we know how to multiply complex
numbers we can raise complex numbers to
a power by multiplying it by itself the
indicated number of times. For now let's
just look at squaring the number a+b*i.
Based on the rule we demonstrated above,
that would be:
(a+b*i)*(a+b*i) = (a*a-b*b)+(a*b+b*a)*i
This simplifies to:
(a+b*i)2 = (a2-b2)+(2*a*b)*i .
With the arithmetic of complex numbers
under our belts, can functions of complex
numbers be far behind?
In answer to the question of the above
paragraph, of course not. let's consider
the function z2=z12+c, where z2, z1 and c
are all complex numbers. This says to
get z2 take the complex number z1 and
square it, then add the constant complex
number c. Given what we now know, these
are certainly do-able instructions. Next
suppose that we iterate this function,
watching what happens to the value z2 as
iterations occur. Remember that iteration
means we take the result z2 and plug
it back in the function as a new z1. Then
calculate a new z2.
Previously when we iterated a function we
were interested in whether the result
settled down to a single value, a series
of values or never settled at all
except to stay in a certain range. We
will be interested in the same questions
with regard to iterating z2+c. Within the
limited domain in which we iterated
the functions of real numbers, we always
arrived at one of the conditions
listed. In this case, iterating z2+c, we
will find another possibility. That is
that the result of iteration may grow
larger and larger without limit.
If the sequence of numbers generated by
iteration increases without limit as
suggested above, the sequence is said to
"diverge". If the sequence approaches a
single finite value it is said to "converge".
As we have seen it is possible
that a function being iterated may do
neither. It may visit a set of points or
it may dither about in a confused fashion
in some vicinity. In this section we
will be concerned with whether or not
z2+c diverges under iteration. For a
sequence of complex numbers to diverge it
is only necessary that either part,
real or imaginary, diverge.
It is likely that the result of our
iteration will depend on where we choose to
begin. Think about the real number
iteration of x2. If x starts less than 1 the
function is attracted to 0 under
iteration. If x starts greater than 1, the
function grows without limit. In the next
display you will have an opportunity
to experiment with iteration of z2+c.
Move the cursor to any point on the
display. This establishes the complex
constant c. The initial value of z is
zero. Click on the Action button to start
iterating. Each subsequent click
iterates one time.
If the absolute value of either the real
or imaginary part of z exceeds 2.0, the
function is on its way to infinity and
the process is halted by killing the
Action button. After the iteration
settles down, or blows up as the case may be,
you may move the cursor to another
location and begin again, testing that
location for the behavior of z2+c under
iteration. A colored 'x' marks the
location on the screen of each calculated
z. A trail of x's marks the progress
of each iteration. The current value of
the complex number z and the current
iteration number are shown at the top
left corner of the drawing area. Run the
Iteration of z2+c display.
We intentionally left you without much
direction on where to try iterating z2+c
on the last display. You should have
discovered that the iteration of this
simple function led to some complex
results. There it is again, that theme of
complexity out of simplicity. Back in the
section on graphing functions of real
numbers, when we plotted the function
sin(x) on the real number line, the values
of the function were constrained to lie
on the same line as the values of x.
Expanding our domain and range from one
dimension to two opens up the
possibility of much richer variety.
On the real number line, a function takes
a number and transforms it to another
point on the line. In the complex plane a
function transforms a number to
another point on the plane. On the real
number line, if a function approached
some number as a limit, the wildest it
could get was to jump back and forth
across the limit as it zeroed in. In the
complex plane there are all sorts of
paths by which a function might approach
a limit. Some of these you should have
seen on the screen as a trail of colored
x's. Also you should have found that
some points iterate out of bounds.