**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
**

**part.
**

**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
**

**disabled.
**

**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
see this.
**

**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
**

**symbols:
**

**(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:
**

**(a*c-b*d)+(a*d+b*c)*i
**

**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
**

**lie.
**

**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.
**