The Mandelbrot set is the set of all complex numbers c 
such that iterating z <= z^2 + c does not ascend to infinity, 
starting with z=0.

// A class for complex numbers.
class complexNumber {

  private double a, b;                                               // a and b from the form a + bi     

  public complexNumber(double realNumber, double imaginaryNumber) {  // Class constructor.  
    a = realNumber; b = imaginaryNumber;
  }

  public double realPart() {                                         // Returns real value a.
    return a;
  }

  public double imaginaryPart() {                                    // Returns imaginary value b.
    return b;
  }

  public double absoluteValue() {                                    // Returns vector magnitude, sqrt(a^2 + b^2).
    return (sqrt(exp(a,2)) + exp(b,2)));
  }

}

checkMandelbrotElement(int N, complexNumber c) {

  for (int i=N, complexNumber z=0; i>=0; i--) {
    if (i == 0) {
      inTheSet(c);
    } else if (z.absoluteValue() > 2) {
      notInTheSet(z);
      break;
    } else {
      z = z^2 + c;
    }
  }

}

// True monitors do not have the same width as height, but we will assume so to make computation easier.
#define monitorWidth 800;
#define monitorHeight 800;

generateMandelbrot(complexNumber c, int N) {

  int pixels[i][j];

  for (i=0; i<=monitorWidth; i++) {
    for (j=0; j<=height, j++) {

    // Note 4 explains subtracting (i,j) by 400 and dividing by 200
    complexNumber z = new complexNumber( ((i-400)/200), ((j-400)/200) );
    checkMandelbrotElement(N, z, c);
    }
  } 

n	z <= zn + c	z <= z-barn + c
2
3
4

Update: Looks like I have lost the aformentioned file. More precisely, inst.eecs.berkeley.edu erased it. But here is a much better one, produced by Thomas Marsh and Jan Hubicka of Xaos. Includes many zoom-ins. The clip cannot be fast forwarded, so just watch the whole thing patiently.

• Note 1: Complex Numbers
  
  Complex numbers are numbers expressed in the form a + bi, where a and b are real numbers, and i is the imaginary number, the square root of -1. In fractal geometry, we graph points against two axes, a horizontal real axis and a vertical imaginary axis; this plane is often called the complex plane (as opposed to the xy plane, where x and y are both real). Hence, 4 + 5i can be graphed on an complex plane as corresponding to point (4,5). The absolute value of a complex number is simply sqrt(a^2 + b^2), which corresponds to the hypotenuese of the triangle formed by the real and imaginary parts on the complex plane.
  Return to Note 1 context
  
  
• Note 1: z=0 Initialization
  
  We start with z equal to zero because it is the critical point of z^2 + c (from introductory calculus: a point where d/dz (z^2 + c) = 0). If the recursive function was z = (z-1)^2 + c, we would start with z = 1 (0 = d/dz [(z-1)^2 + c] = [d/dz (z-1)^2] + 0 = 2(z-1) -> 2z = 2 -> z = 1). In some cases there may be multiple critical values, so they should all be tested.
  
  Mathematician Pierre Fatou (1878-1929) showed that every attracting cycle for a polynomial or rational function attracts at least one critical point. Thus, testing the critical point shows if there is any stable attractive cycle. Return to Note 2 context
  
  
• Note 3: Diverging Condition
  
  In the coding of the checkMandelbrotElement() function, we say that if |z| is ever greater than 2, we can be assured that the z sequence corresponding to parameter c will diverge, and c is thus not in the set. Proof: If |z| > 2, then |z^2 + c| >= |z^2| - |c| > 2|z| - |c|. If |z| >= |c|, then 2|z| - |c| > |z|. So, if |z| > 2 and |z| >= c, then |z^2 + c| > |z|, and thus the sequence is increasing. (It takes a little more work to prove that the sequence is unbounded and diverges.)
  
  From this line of reasoning, it follows that the bounds of the Mandelbrot set are within |c| <= 2. In the first iteration, z = 0^2 + c = c, so if |c| > 2, the sequence diverges. Return to Note 3 context
  
  
• Note 4: Computer screen scaling
  
  The explanation for subtracting 200 from ( i , j ) and then dividing by 400 stems from the proof given in Note 3. Since the bounds of the Mandelbrot set are |c| <= 2, we want the dimensions of our graph to be [-2,2] (real axis) by [-2,2] (imaginary axis) - a larger graphing space would be a waste of space. However, the pixels on a computer monitor are indexed on a [0,monitorWidth] by [0,monitorHeight] cross product set, with (0,0) being the upper-left corner. Hence we have to do some arithmetic operations on a pixel's indices prior to making a useful complex number corresponding to that pixel. To shift the origin to the center of the screen, we subtract half of monitorWidth from i and then divide this difference by a fourth of monitorWidth, and subtract half of monitorHeight from j and then divide this difference by a fourth monitorHeight:
  
  

  complexNumber z = new complexNumber( (i-(monitorWidth/2))/(monitorWidth/4) , (j-(monitorHeight/2))/(monitorHeight/4) );
  

  So, if monitorWidth = monitorHeight = 800, then i=0 -> (0-400)/(200) = -2; i=800 -> (800-400)/(200) = 2; j=0 -> (0-400)/(200) = -2; j=0 -> (800-400)/(200) = 2. Thus we produce only complex numbers in the cross product of [-2, 2] and [2, 2]. Return to Note 4 context