Z_{n}=(Z_{(n-1)})^2+C

where Z and C are complex numbers and C is constant, the mandelbrot set is the set of numbers Z_{0} for which Z_{n} is bounded. that is, for any value n, including infinity, Z_{n} is not infinite.

If you wish to calculate which points are part of the Mandelbrot set, it is reasonable to iterate a few hundred times and check if the absolute value of the current Z is more than some arbitrary number such as 2.

It is closely related to a julia set. the variable in a julia set is C, and Z_{0} is constant for any point on a julia set. If you take a point from a representation of the Mandelbrot set, you can create a Julia set for it by using its value of Z_{0} and plotting on the complex plane of C. There are programs that do this, such as the julia program in the Xscreensaver package.

Computer-generated representations of the Mandelbrot set are generally colorful, while the technical definition of the set allows for only two states: membership and non-membership. The colors typically come from either the number of iterations it takes for Z_{n} exceed some value (indicating that the point is not a member) or the value of Z_{n} with some arbitrary n. For example, if abs(Z_{n}) is less than 2, assume it is a member and color it black. If it's between 2 and 3, color it red. 3-4, orange; 4-5, green; 5-6, blue; 6-100, indigo; 100-infinity, violet (numbers are pulled from the air and should not be taken as suggestions).