Explanation of the Mandelbrot Set

Consider the function
f_c(z) = z^2+c\\

Where z and c are complex numbers. Complex numbers are numbers in the form of
\\ ai+b\\ $Where $i=\sqrt{-1}

Now let's check if 0.5 is in the Mandelbrot Set. To do so, start at 0
\\ f_{0.5}(0) = 0^2 + 0.5 = 0.5\\ f_{0.5}(0.5) = 0.5^2 + 0.5 = 0.75\\ f_{0.5}(0.75) = 1.0625\\ f_{0.5}(1.0625) = 1.62890625\\ f_{0.5}(1.62890625) = 3.15333557\\

Since this function has passed 2, 0.5 is not in the Mandelbrot Set. Compare this to 0.25.
\\ f_{0.25}(0) = 0^2+0.25 = 0.25\\ f_{0.25}(0.25) = 0.3125\\ f_{0.25}(0.3125) = 0.34765625\\ f_{0.25}(0.34765625) = 0.370864868\\ f_{0.25}(0.370864868) = 0.38754075\\ f_{0.25}(0.38754075) = 0.400187833\\ f_{0.25}(0.400187833) = 0.410150302\\ f_{0.25}(0.410150) = 0.418223\\ f_{0.25}(0.418223) = 0.424911\\ f_{0.25}(0.424911) = 0.430549\\ f_{0.25}(0.430549) = 0.435373\\ f_{0.25}(0.435373) = 0.439549\\ f_{0.25}(0.439549) = 0.443204\\ f_{0.25}(0.443204) = 0.446429\\ f_{0.25}(0.446429) = 0.449299\\

This will never pass 2, so 0.25 is in the Mandelbrot Set.
This process can also be done to complex numbers.

M(x) =$ the number of iterations required for $f_x$ to pass 2.$
The website is just a 2d plot of M(x).