**Fractal Theory: The Mandelbrot Set**

So that brings us to today. The Mandelbrot set is found in the complex plane. Each point on that plane represents a single complex number of the form a + bi, where a is the distance left or right from the center line (negative when left, positive when right) and b is the distance above or below the center line (negative when below, positive when above) and i is the root of -1.

With the understanding we now have of complex math, we can write out the Mandelbrot formula in a much more compact form. If we had a computer language that understood complex numbers (some do), we could even write a program in the following form:

Z = (0 + 0i)

C = (a + bi)

for (counter = 0; ABS(Z) <= 2.0 && counter < MaxIters; counter++)

Z = Z * Z + C;

That's it. The Mandelbrot set is *really* simple. All the lines of code at the beginning of this discussion were just teaching the computer how to do the single line of math above. Multiply Z by itself. Add C. The answer is the new value for Z. Repeat until the absolute value of Z is greater than two, or until our counter expires.

The only unexplained feature above is the ABS(Z) part. ABS stands for absolute value. The absolute value of a complex number is simply its distance from the center of the plane. If we think of the number as a point on the plane, we can measure the distance from the center with a ruler. Or, that distance can be calculated (because it's simple to create a right-angle triangle on a plane) by determining the square root of a squared plus b squared. If the answer is less than or equal to two, you are looking at a point in a Mandelbrot set. If the answer is greater than two, the point is outside the set.