Fractal Explorer

Fractal Explorer

Complex numbers



Certain fractals like the Mandelbrot set fractal or the Julia set fractal are build of iterations with complex numbers.

Complex numbers consist out of two parts: the real and the imaginary part. The basic form is a + bi. Where a is the real and b the imaginary part. I is the imaginary unit which has following property: i² = -1.

At first it might be difficult to believe that a number squared can result in something negative, but as soon as you accept this fact, complex numbers will be much easier to understand.

This fact offers an enormous amount of oppertunitys, not only can unsolvable equations now be solved (i.e. x²+1=0) but the second dimension within the complex numbers allow to iterate over a second variable. Thus creating plotable results.


Basic operations

The usual laws for operating with number cannot be applied to complex numbers. They need a special treatment.


Basically you just add the real part to the real part and the imaginary part to the imaginary part.

(a+bi) + (c + di) = (a+b) + (b+d)i


The subtraction is pretty much the same as the addition:

(a+bi) - (c + di) = (a-b) + (b-d)i


Because of the fact, that i² = -1 the multiplication of complex numbers is a bit more complicated. Because of i² is -1 the multiplication of the two imaginary parts is a real number. The resulting real part is the two real parts + the two imaginary parts multiplied. The resulting imaginary part is the imaginary part of the first and the real part of the second complex number plus the imaginary part of the second and the real part of the first.

(a + bi)(c + di) = (ac + bd) (bc + ad)i


The complex plane

Because of the two parts, the real and the imaginary part, a complex number is basically a number with two dimensions. All one dimensional numbers (natural, irreal, real and so on..) can be ploted to the number line. For complex numbers one needs a plane which is called the complex plane. Usually the y-axis is used for the imaginary part and the x-axis for the real part.



The iteration with complex numbers is ued in numerous fractals. The iteration formula used in the Mandelbrot set is:

Zn+1 = (Zn)² + c

Z and C are complex numbers. The start value for Z is always 0. C is the constant part which determines the location of the iteration series in the complex plane.

Further information regarding the Mandelbrot set.