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Exploration in Modern Mathematics

Mike Janssen

Section 8.2 Complex Numbers and Fractals

Motivating Questions.

In this section, we will explore the following questions.
  1. What is a complex number, and how do we represent them, both geometrically and arithmetically?
  2. What is a fractal? What are some examples of fractals?
  3. What is “sensitive dependence on initial conditions”?
In this section, we will explore other types of dynamical systems. We will see that even though these systems are deterministic, they can still generate surprising even chaotic behavior.
To begin, we review the basics of complex numbers.

Subsection 8.2.1 The Arithmetic and Geometry of Complex Numbers

Definition 8.2.1.

Let \(a\) and \(b\) be real numbers. The complex number with real part \(a\) and imaginary part \(b\) is the expression of the form \(a+bi\text{,}\) where \(i^2 = -1\text{.}\)
It is straightforward to add, subtract, and multiply complex numbers according to the following rules: given complex numbers \(a+bi\) and \(c+di\text{,}\) we define:
\begin{align*} (a+bi) \pm (c+di) \amp = (a+c) \pm (b+d)i\\ (a+bi) \cdot (c+di) \amp = (ac-bd) + (ad+bc)i \end{align*}

Activity 8.2.2.

Perform the following operations.
  1. \(\displaystyle (3 + 4i) + (1 + 2i)\)
  2. \(\displaystyle (5 + 7i) - (2 + 3i)\)
  3. \(\displaystyle (2+3i)\cdot (2-3i)\)
  4. \(\displaystyle (-1 + 6i) + (3 - 2i)\)
  5. \(\displaystyle (2 + 3i)\cdot (4 + i)\)
Since complex numbers are defined using two real numbers \(a\) and \(b\text{,}\) there is a natural correspondence with the point \((a,b)\) in the plane. We typically describe the horizontal axis as the real axis the vertical axis as the imaginary axis. See Figure 8.2.3 for some examples.
Figure 8.2.3.

Activity 8.2.4.

One interpretation of the absolute value of a real number is that it gives the “size” of the number by calculating its positive distance from 0.
Consider the complex numbers shown in Figure 8.2.3. How might we calculate their distance from 0 (that is, the origin in the plane)? Give a formula for a general complex number \(a+bi\text{.}\)

Subsection 8.2.2 Mandelbrot and Julia Sets

We next take our knowledge of complex numbers and explore what happens when they are run through a simple repeated process. Our goal will be to understand whether or not the process “blows up” and heads off to infinity that is, attains a large distance from 0.

Activity 8.2.5.

For each complex number below, compute the first five powers, and plot them in the plane. What do you notice? What do you wonder?
  1. \(\displaystyle 2+3i\)
  2. \(\displaystyle 0.5 + 0.8i\)
  3. \(\displaystyle 0.6+0.8i\)

Building the \(a+bi\) Julia Set.

We now consider the following process.
  1. Fix a number \(c=a+bi\text{.}\)
  2. Choose any complex number \(z_0\text{,}\) square it, and add \(c\text{.}\)
  3. Call the new number \(z_1\text{.}\) That is, \(z_1 = z_0^2 + c\text{.}\)
  4. Produce a new number, \(z_2 = z_1^2 + c\text{.}\)
  5. Repeat.

Activity 8.2.6.

Begin with the complex number \(a +bi = -1 + 0i\text{.}\) Carry out the above process using the numbers from Activity 8.2.5 to at least the fifth iteration. Check your work using the Sage cell below.
The process described above makes reference to a “Julia set” but does not actually define it. A rigorous definition is beyond the scope of this text, but an intuitive one is within our grasp! To that end, let \(c = a+bi\) be a fixed complex number. The Julia set of \(c\), denoted \(J_c\text{,}\) is the set of complex numbers \(z\) for which the iterative process above, repeated indefinitely, does not go off to infinity. That is, if we choose a number \(z\) and the above process does not go off to infinity, then \(z\) goes in the Julia set of \(c\text{.}\)
As we have seen, we can plot complex numbers in the plane. Thus, we can fill in the points of the plane that belong to a specific Julia set. When we do, we see something remarkable.
Figure 8.2.7. The Julia set associated to \(f(z) = z^2 - 1\text{.}\)

Exploration 8.2.8.

Use the Sage cell below to generate Julia sets for the following complex numbers. Then pick a few of your own to try. What do you notice?
  1. \(\displaystyle c=-0.4+0.6i\)
  2. \(\displaystyle c=-0.70176-0.3842i\)
  3. \(\displaystyle c=-0.835 - 0.2321i\)
  4. \(\displaystyle c=0.8i\)
  5. \(\displaystyle c=0.4 + 0.4i\)
We are now ready to explore the Mandelbrot set, which captures information about every Julia set all at once. The process for constructing the Mandelbrot set is as follows.
  1. Choose a complex number \(a+bi\text{.}\)
  2. Construct the Julia set of \(a+bi\text{.}\)
  3. If the resulting Julia set consists of just one piece, then \(a+bi\) goes in the Mandelbrot set. If the Julia set consists of more than one piece, then \(a+bi\) does not go in the Mandelbrot set.
  4. Repeat for every complex number \(a+bi\text{.}\)

Exploration 8.2.9. Exploring the Mandelbrot set.

Use the Sage interact in Figure 8.2.10 to explore the Julia sets of various complex numbers and their corresponding membership (or lack thereof) in the Mandelbrot set.
Figure 8.2.10.
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