Introduction: Fractals Generated by Complex Numbers

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

 
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.
A shaded fractal shown on a graph.