You may be familiar with the fractal in the image below. The boundary of this shape exhibits quasi-self-similarity, in that portions look very similar to the whole. This object is called the Mandelbrot set and is generated by iterating a simple recursive rule using complex numbers.
In this lesson, you will first learn about the arithmetic of complex numbers so you can understand how a fractal like the Mandelbrot set is generated.
Learning Objectives
The learning objectives for this lesson include:
Identify the difference between an imaginary number and a complex number
Identify the real and imaginary parts of a complex number
Plot a complex number on the complex plane
Perform arithmetic operations on complex numbers
Graph physical representations of arithmetic operations on complex numbers as scaling or rotation
Generate several terms of a recursive relation
Determine whether a complex number is part of the set of numbers that make up the Mandelbrot set
Arithmetic with Complex Numbers
[footnote]Portions of this section are remixed from Precalculus: An Investigation of Functions by David Lippman and Melonie Rasmussen. CC-BY-SA[/footnote]
The numbers you are most familiar with are called real numbers. These include numbers like 4, 275, -200, 10.7, ½, π, and so forth. All these real numbers can be plotted on a number line. For example, if we wanted to show the number 3, we plot a point:
To solve certain problems like x2=–4, it became necessary to introduce imaginary numbers.
Imaginary Number i
The imaginary number i is defined to be i=−1.
Any real multiple of i, like 5i, is also an imaginary number.
Example
Simplify −9.
We can separate −9 as 9−1. We can take the square root of 9, and write the square root of −1 as i.
−9=9−1=3i
A complex number is the sum of a real number and an imaginary number.
Complex Number
A complex number is a number z=a+bi, where
a and b are real numbers
a is the real part of the complex number
b is the imaginary part of the complex number
To plot a complex number like 3−4i, we need more than just a number line since there are two components to the number. To plot this number, we need two number lines, crossed to form a complex plane.
Complex Plane
In the complex plane, the horizontal axis is the real axis and the vertical axis is the imaginary axis.
Example
Plot the number 3−4i on the complex plane.
The real part of this number is 3, and the imaginary part is −4. To plot this, we draw a point 3 units to the right of the origin in the horizontal direction and 4 units down in the vertical direction.
Try It Now
In the following video, we present more worked examples of arithmetic with complex numbers.
https://youtu.be/XJXDcybM84U
When we add complex numbers, we can visualize the addition as a shift, or translation, of a point in the complex plane.
Example
Visualize the addition 3−4i and −1+5i.
The initial point is 3−4i. When we add −1+3i, we add −1 to the real part, moving the point 1 units to the left, and we add 5 to the imaginary part, moving the point 5 units vertically. This shifts the point 3−4i to 2+1i.
Try It Now
To understand the effect of multiplication visually, we’ll explore three examples.
Example
Visualize the product 2(1+2i).
Multiplying we’d get
The following video presents more examples of how to visualize the results of arithmetic on complex numbers.
In the following video, we present more worked examples of arithmetic with complex numbers.
https://youtu.be/vPZAW7Lhh1E
Generating Fractals with Complex Numbers
Complex Recursive Sequences
Some fractals are generated with complex numbers. The Mandlebrot set, which we introduced briefly at the beginning of this module, is generated using complex numbers with a recursive sequence. Before we can see how to generate the Mandelbrot set, we need to understand what a recursive sequence is.
Recursive Sequence
A recursive relationship is a formula which relates the next value, zn+1, in a sequence to the previous value, zn. In addition to the formula, we need an initial value, z0.
The sequence of values produced is the recursive sequence.
Example
Given the recursive relationship zn+1=zn+2,z0=4, generate several terms of the recursive sequence.
We are given the starting value, z0=4. The recursive formula holds for any value of n, so if n=0, then zn+1=zn+2 would tell us z0+1=z0+2, or more simply, z1=z0+2.
Notice this defines z1 in terms of the known z0, so we can compute the value:
z1=z0+2=4+2=6.
Now letting n=1, the formula tells us z1+1=z1+2, or z2=z1+2. Again, the formula gives the next value in the sequence in terms of the previous value.
z2=z1+2=6+2=8
Continuing,
z3=z2+2=8+2=10z4=z3+2=10+2=12
Try It Now
In the following video we show more worked examples of how to generate the terms of a recursive, complex sequence.
https://youtu.be/lOyusyTsLTs
Mandelbrot Set
The Mandelbrot Set is a set of numbers defined based on recursive sequences.
Mandelbrot Set
For any complex number c, define the sequence zn+1=zn2+c,z0=0
If this sequence always stays close to the origin (within 2 units), then the number c is part of the Mandelbrot Set. If the sequence gets far from the origin, then the number c is not part of the set.
Example
Determine if c=1+i is part of the Mandelbrot set.
We start with z0=0. We continue, omitting some detail of the calculations
If all complex numbers are tested, and we plot each number that is in the Mandelbrot set on the complex plane, we obtain the shape to the right.[footnote]http://en.wikipedia.org/wiki/File:Mandelset_hires.png[/footnote]
The boundary of this shape exhibits quasi-self-similarity, in that portions look very similar to the whole.
Watch the following video for more examples of how to determine whether a complex number is a member of the Mandelbrot set.
https://youtu.be/ORqk5jAFpWg
In addition to coloring the Mandelbrot set itself black, it is common to the color the points in the complex plane surrounding the set. To create a meaningful coloring, often people count the number of iterations of the recursive sequence that are required for a point to get further than 2 units away from the origin. For example, using c=1+i above, the sequence was distance 2 from the origin after only two recursions.
For some other numbers, it may take tens or hundreds of iterations for the sequence to get far from the origin. Numbers that get big fast are colored one shade, while colors that are slow to grow are colored another shade. For example, in the image below[footnote]This series was generated using Scott’s Mandelbrot Set Explorer[/footnote], light blue is used for numbers that get large quickly, while darker shades are used for numbers that grow more slowly. Greens, reds, and purples can be seen when we zoom in—those are used for numbers that grow very slowly.
The Mandelbrot set, for having such a simple definition, exhibits immense complexity. Zooming in on other portions of the set yields fascinating swirling shapes.
Additional Resources
A much more extensive coverage of fractals can be found on the Fractal Geometry site. This site includes links to several Java software programs for exploring fractals.
If you are impressed with the Mandelbrot set, check out this TED talk from 2010 given by Benoit Mandelbrot on fractals and the art of roughness.
Licenses & Attributions
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Revision and Adaptation.Provided by: Lumen LearningLicense: CC BY: Attribution.
Question ID 131223, 131218.Authored by: Lumen Learning.License: CC BY: Attribution.
CC licensed content, Shared previously
Math in Society.Authored by: Lippman, David.Located at: http://www.opentextbookstore.com/mathinsociety/.License: CC BY-SA: Attribution-ShareAlike.