The Complex Plane


The intention in this section is to provide a brief introduction to some of the

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 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


(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.