What (Two Variable Function Pump)

What is the Two Variable Function Pump Activity?

College Algebra Related Pages: 103-104 "Complex Numbers"
College Algebra and Trigonometry Related Pages: 103-104 "Complex Numbers"
Precalculus Related Pages: 21-22 "Complex Numbers"

This activity allows you to see an applicaiton of complex numbers. The standard form of a complex number is (a+bi). Complex numbers can also be thought of as ordered pairs, instead of the standard form (a+bi) use the form (a,b) where the x-coordinate is the real part and the y-coordinate is the imaginary part.

This activity is a precursor to ultimately understanding the Mandelbrot Set, the worlds most famous fractal. In this activity you will use the ordered pair form of a complex number to determine whether that number is a "prisoner" or an "escapee" of the formula f(z)=z² + (1/2, 1/2), where z is the ordered pair. The idea of the two-variable function pump is to choose a complex number and plug it into the formula to get a new point, and then plugging that point back into the function to get another new point, and so on (this is known as an iterative process). If the point becomes smaller and smaller, appraoching 0, the point is called an "prisoner." If the point becomes larger and larger, approaching infinity, the point is called an "escapee." (Keep reading, this will all make sense soon!)

Definitions:

prisonervalue of z in the formula f(z)=z² + (1/2, 1/2) where at each iteration the resulting value becomes smaller and smaller, approaching zero
escapeevalue of z in the formula f(z)=z² + (1/2, 1/2) where at each iteration the resulting value becomes larger and larger, approaching infinity.

iterationrepeating a set of rules or steps over and over

Basic Operations are revised for this new situation as:

Addition of two pairs (+)	Component-wise, i.e., (a,b) + (c,d) = (a+c,b+d)
Subtraction of two pairs (-)	Component-wise, i.e., (a,b) - (c,d) = (a-c,b-d)
Multiplication by a number (*)	Component-wise, i.e., x * (a,b) = (xa,xb)
Division by a number (/)	Component-wise, i.e., (a,b) / x = (a/x,b/x)
Squaring a number (^2)	(a,b) ^2 = (a^2 - b^2, 2ab)

Let's choose a starting point of (0, 0) to plug into the formula f(z)=z² + (1/2,1/2) and see what happens. To reiterate (ha ha - get it, reiterate?) we are interested in choosing a starting point, plugging it into the function to get a new point, and then plugging that point back into the function to get another new point, and so on:

f(0,0) = (0, 0)² + (1/2, 1/2) = (0, 0) + (1/2, 1/2) = (0.5, 0.5)


	f(0.5, 0.5) = (0.5, 0.5)² + (0.5, 0.5) = (0, 0.5) + (0.5, 0.5) = (0.5, 1)
  f(0.5, 1) = (0.5, 1)² + (0.5, 0.5) = (-0.75, 1) + (0.5, 0.5) = (-0.25, 1.5)
  f(-0.75, 1.5) = (-0.75, 1.5)² + (0.5, 0.5) = (-2.1875, -0.75) + (0.5, 0.5) = 
      (-1.6875, -0.25)
  f(-1.6875, -0.25) = (-1.6875, -0.25)² + (0.5, 0.5) = (2.7852, 0.84375) + (0.5, 0.5)
      = (3.2852, 1.34375)

Here is the graph:

The point jumps around, getting farther away from (0,0), and then finally leaves the gray circle of radius 2. We can try other starting points and look at what happens in each situation. There are three things that can happen. The points can:

Get closer and closer to 0 -- we say they approach 0. These are the prisoners.
Get larger and larger -- we say they approach infinity. These are the escapees. We know that they are escapees if they further away from the origin than 2 units.
Get trapped -- these points are the boundary points between the prisoners and escapees. The set of all of these points for any formula of the form f(z)=z² + c, where both z and c are complex numbers, is called a Julia Set.

Julia sets are the main building blocks for the Mandelbrot set. We find the Julia sets by looking for the numbers which are neither prisoners nor escapees.