Question

Are the following values of c ∈C in the Mandlebrot set? Show sufficient steps for the marker to follow your work.

(a) c = 1 /3

(b) c = 0.2 + 0.3i

Answer 1

We define the *Mandelbrot set* to be the set of all complex numbers c 
such that Or+(f,0) is bounded, where f(z) : z |--> z^2 + c.

We define a *Julia set* to be the set of all complex numbers z such 
that Or+(f,z) is bounded, where f(z) : z |--> z^2 + c, and c is some 
given complex number.

but, For what values of c is Or+(f,0) bounded?  i.e., we 
pick various values of c, and calculate f(f(f(...f(0)))), and find out 
if this stays bounded or diverges to infinity.  The rather surprising 
result is that when one plots the various values of c (*not z*, but c) 
in the complex plane, one obtains a fractal. 
And when it is bounded it yields Mandelbrot's set. 

a) c = 1/3  
   Now, f(0)           = 0 + 1/3 
        f(f(0))        = 1/3^2 + 1/3
                       = 1/9 + 1/3 = 4/9
        f(f(f(0)))     = (1/3^2 + 1/3)^2 + 1/3 = 1/3^4 + 1/3^2 + 2/3^3 + 1/3 = 43/81
       after calculating f(f(....f(0))), it tends to infinite. So it is not a M-set.

b) c = 0.2 + 0.3i
   Now, f(0)           = 0 + 0.2 + 0.3i
        f(f(0))        = (0.2 + 0.3i)^2 + 0.2 + 0.3i = 0.15 + 0.42i
        f(f(f(0)))     = (0.15 + 0.42i)^2 + 0.2 + 0.3i = 0.0461 + 0.426i
       after calculating f(f(...f(0))..) we find it is converging to a finite complex number. Hence it is a M-set.
  Note: I have used Mathematica to calculate the values. I can also provide you mathematical code to compute f(0). Let me know in the comment section. But the basic idea I have already mentioned.