### 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\\

It can be proven that if the function passes 2, it will go to infinity if you continually apply the function.
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).