In this section we will extend iteration into an additional dimension. The image at the left is created based on the way a certain function approaches its attractor under iteration. It is a small segment of the boundary of an object called the Mandelbrot set. In the interest of full disclosure, I must confess that the image was created with the offline version of the Order program, in which we have better control over the colors and shading. The online Java applets serve to illustrate the concepts, but lack the power of the compiled native code Windows version.

The Mandelbrot set and related 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. Before we get into the details we need to lay some groundwork.

We will briefly review the concept of a "real number line" which we covered in Numbers Functions and Graphs. 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.

The image at the right illustrates a short segment of the real number line, between -20 and +20. The indicated example here is an irrational number. Click on the link for the Real Number Line interactive display from which this example was taken.

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 arrowhead 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:

This mess may be simplified.

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. You will find that the product vector has length equal to the product of the lengths of the other vectors and the angle is the sum of the other vector's angles.

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:

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.

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.

Are there any questions?