Modeling with Complex Numbers

Steve Phelps, Jun 23, 2015

This is a GeoGebraBook for the Summer 2015 MAT Math Modeling Class.

Table of Contents

  1. Introduction
    1. Nonagon Problem: Introduction
  2. Some History
    1. Impossible Quadratics
    2. Friendly Cubics
  3. Complex Numbers via Geometry
    1. Complex Number Definition
    2. Polar Form of a Complex Number
  4. Basic Operations of Complex Numbers: What's My Rule?
    1. Complex Numbers: What's My Rule?
    2. Complex Numbers: What's My Rule?
    3. Complex Numbers: What's My Rule?
    4. Complex Numbers: What's My Rule?
    5. Complex Numbers: What's My Rule?
    6. Complex Numbers: What's My Rule?
    7. Complex Numbers: What's My Rule?
    8. Complex Numbers: What's My Rule?
    9. Complex Numbers: What's My Rule?
  5. Using Complex Numbers to Perform Transformations
    1. Complex Numbers and Transformations
    2. Complex Numbers and Transformations
    3. Complex Numbers and Transformations
    4. Complex Numbers and Transformations
    5. Inversion
    6. Inversion
    7. Inversion
  6. Complex Number Self Checks!
    1. Activity 2
  7. Real-Valued Functions with Complex Roots
    1. Complex Roots and Quadriatic Graphs
    2. Visualizing the Complex Roots of a Quadratic
    3. Complex Roots and Cubic Graphs
    4. Complex Roots and Quartic Graphs

1.

Introduction

  • 1. Nonagon Problem: Introduction

1.1.

Nonagon Problem: Introduction

Consider the following regular nonagon inscribed in a unit circle. Use the measuring tool to find the lengths of all the green segments. What is the product of these lengths? How would you begin to explain (prove) this?
– Steve Phelps

2.

Some History

  • 1. Impossible Quadratics
  • 2. Friendly Cubics

2.1.

Impossible Quadratics

It is known that the equation [math]x^2+1=0[/math] has no real number solution. Finding solutions is equivalent to finding the x-intercepts of the graph of [math]f(x)=x^2+1[/math]. No intersection points, no solutions. You should be able to grab the graph and move it around to create equations that do have solutions.
– Steve Phelps

2.2.

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3.

Complex Numbers via Geometry

  • 1. Complex Number Definition
  • 2. Polar Form of a Complex Number

3.1.

Complex Number Definition

A complex number z is a point in the plane where the x-axis is measured in real units and the y-axis is measured[br]in imaginary units. Notice x=Re(z) and y=Im(z). [br][br]What is the locus of points that satisfy Re(z)=Im(z)?[br]What is the locus of points that satisfy Re(z)=3?[br]What is the locus of points that satisfy Im(z)=2?
Complex Number Definition
– Steve Phelps

3.2.

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4.

Basic Operations of Complex Numbers: What's My Rule?

  • 1. Complex Numbers: What's My Rule?
  • 2. Complex Numbers: What's My Rule?
  • 3. Complex Numbers: What's My Rule?
  • 4. Complex Numbers: What's My Rule?
  • 5. Complex Numbers: What's My Rule?
  • 6. Complex Numbers: What's My Rule?
  • 7. Complex Numbers: What's My Rule?
  • 8. Complex Numbers: What's My Rule?
  • 9. Complex Numbers: What's My Rule?

4.1.

Complex Numbers: What's My Rule?

Points A and C are complex numbers. Drag point A. Point C moves in response. What is the rule that defines points C? Try to describe it geometrically and algebraically. Is it possible to move A without moving C? Is it possible to move A so point C is at the origin? What happens to C when A moves along the Real axis? What happens to C when A moves along the Imaginary axis? Is it possible to move A so point C moves along the Real axis? Is it possible to move A so point C moves along the Imaginary axis?
– Steve Phelps

4.2.

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4.3.

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4.4.

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4.5.

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4.6.

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4.7.

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4.8.

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4.9.

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5.

Using Complex Numbers to Perform Transformations

  • 1. Complex Numbers and Transformations
  • 2. Complex Numbers and Transformations
  • 3. Complex Numbers and Transformations
  • 4. Complex Numbers and Transformations
  • 5. Inversion
  • 6. Inversion
  • 7. Inversion

5.1.

Complex Numbers and Transformations

In the figure below, point A and the vertices of the triangle are complex numbers. Every point on the triangle is being added with point A. This induces a Transformation. Drag Point A around. Under what conditions is the transformation a dilation? Under what conditions is the transformation a translation? Under what conditions is the transformation a rotation? Under what conditions is the transformation a reflection? Under what conditions is the transformation a composition of a rotation and a dilation?
– Steve Phelps

5.2.

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5.3.

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5.4.

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5.5.

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5.6.

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5.7.

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6.

Complex Number Self Checks!

  • 1. Activity 2

6.1.

Activity 2

How well do you understand the transformation caused by multiplying by a Complex Number?[br][br]Test yourself! Try this until you get 5 correct in a row! A message will appear when you are correct!
– Steve Phelps

7.

Real-Valued Functions with Complex Roots

  • 1. Complex Roots and Quadriatic Graphs
  • 2. Visualizing the Complex Roots of a Quadratic
  • 3. Complex Roots and Cubic Graphs
  • 4. Complex Roots and Quartic Graphs

7.1.

Complex Roots and Quadriatic Graphs

Drag the BLUE point (which is one of the complex conjugate roots) Notice how the quadratic graph changes.
– Steve Phelps

7.2.

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7.4.

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