What are complex numbers?
The Mandelbrot set is the domain of convergence of the series built up by the complex sequence defined by the recursion law: Z_n+1 = Z²_n + C, therefore understanding complex numbers is important to understand what the Mandelbrot set represents.

If you are familiar with complex numbers, you can skip this section. If you haven't encountered complex numbers during your studies, don't worry! In order to learn what they are and how to use them to build fractals you only need some simple concepts.
A complex number is made by two parts: the real part and the imaginary part. The real part is a real number (i.e. an ordinary number), the imaginary part is another real number multiplied by i, which is called the imaginary unit and is defined as the square root of -1, so that i² = -1.

Example: -6 + 9i is a complex number.

Now imagine a plane with two coordinate axes on it. We can associate each point of the plane with a complex number of the form: x_coordinate + i*y_coordinate. The picture below shows the correspondence between the coordinates of a point and the complex number that can be associated with it.

We need to define some operations to deal with complex numbers:

Adding two complex numbers is very simple. Let's take two complex numbers: Z₁=a₁+ib₁ and Z₂=a₂+ib₂. The complex number resulting from Z₁+Z₂ is defined as: a₁+a₂+i*(b₁+b₂). The real part of the result is the sum of the real parts of Z₁ and Z₂; the imaginary part of the result is the sum of the imaginary parts of Z₁ and Z₂.

Example: Z₁ = 2 + i , Z₂ = 3 - 6i , Z₁+Z₂ = 5 - 5i

In order to multiply Z₁ by Z₂ we only need to remember the distributive law and the equation i² = -1. Therefore, the multiplication is defined as follows: Z₁*Z₂ = a₁*a₂ - b₁*b₂ +i*(a₁*b₂ + a₂*b₁).

Example (with the values above): Z₁*Z₂ = 6 - (-6) + i*(-12 + 3) = 12 - 9i

In particular, the square of a complex number (i.e. the complex number multiplied by itself) is: Z² = a² - b² + 2*i*a*b
Don't forget this formula, because we'll use it to build Mandelbrot and Julia sets.

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