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Precalculus - Unit Five

Complex Numbers


In this unit, you will explore how complex numbers can be used to generate fractals. "The existence of these patterns [fractals] challenges us to study forms that Euclid leaves aside as being formless, to investigate the morphology of the amorphous. Mathematicians have disdained this challenge, however, and have increasingly chosen to flee from nature by devising theories unrelated to anything we can see or feel."

-Benoit Mandelbrot (Polish/French mathematician; 1924 - 2010)

Click on a link to jump ahead to the following sections:

  • Lessons
  • Other Resources


Lessons

  • Introduction to Fractals

    • 5.01 Investigation - Introduction to Fractals
    • 5.02 Sierpinski - More fractal investigations
    • 5.02 Koch and Levy - More fractal investigations
    • 5.03 Cantor - Even more fractal investigations

  • Lesson 1: Introduction to Complex Numbers
    • 5.1 Investigation - Simple Arithmetic
    • 5.1 Practice - Simple Arithmetic - Practice

  • Lesson 2: Complex Graphing
    • 5.2 Investigation - Geometric Effects of Complex Arithmetic
    • 5.2 Practice - Complex Graphing - Practice
    • Blank Graphs - 4 Blank Graphs on a single page

  • Lesson 3: Complex Multiplication
    • 5.3 Investigation - The Geometric Effect of Complex Multiplication
    • 5.3 Practice - Complex Multiplication - Practice

  • Lesson 4: Distance and Midpoint Formulae
    • 5.4 Investigation - Distance on the Complex Plane
    • 5.4 Practice - Distance and Midpoint Formulae - Practice

  • Lesson 5: Polar Form and DeMoivre’s Theorem
    • 5.5 Investigation - Polar Form and DeMoivre’s Theorem
    • 5.5 Practice - DeMoivre’s Theorem - Practice

  • Lesson 6: More Complex Multiplication
    • 5.6 Investigation - Discovering the Geometric Effects of More Complex Multiplication
    • 5.6 Practice - More Complex Multiplication - Practice

  • Lesson 7: Polynomial Roots, Revised
    • 5.7 Investigation - Polynomial Roots, Revised
    • 5.7 Practice - Polynomial Roots - Practice

  • Lesson 8: The Fundamental Theorem of Algebra
    • 5.8 Investigation - Does Every Complex Number Have a Root?
    • 5.8 Practice - Fundamental Theorem - Practice

  • Lesson 9: The Binomial Theorem
    • 5.9 Investigation - The Binomial Theorem/li>
    • 5.9 Practice - Binomial Theorem - Practice https://youtu.be/iOQjV0FB9nY
    • Video - Binomial Theorem - Nancy Pi video

  • Assessments
    • UPA 1 - Complex Treasure Hunt
    • UPA 2 - Fractals








Resources - Matrices

The following are links to resources for Complex Numbers:

  • Entire Unit - Coming Soon

  • Complex Numbers - Entire unit on Khan Academy



The following are other free resources:

  • GeoGebra - Free online math tools

  • Desmos - Free online graphing calculator






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