# Chapter 4: Polar Equations and Complex Numbers

## Introduction

When you first began graphing mathematical equations and values, you probably used a Cartesian graph, also known as an x, y or rectangular graph. In this chapter you will be graphing quite a bit on a polar graph, or circular graph. Polar equations and polar graphs can be a bit intimidating, particularly at first. With practice, however, you will likely come to appreciate a number of situations where a polar graph is easier or just makes more sense than a rectangular graph.

Another major topic in this chapter is imaginary numbers. You may be thinking, "If the numbers are imaginary anyway, why should I need to learn about them?" One reason is that imaginary numbers can *become* real numbers when multiplied together. It seems strange, but imaginary numbers actually 'convert' to real numbers all the time in mathematics, and *every* number you have ever seen can be written as a "complex number," which is the combination of a real and an imaginary number!

Another interesting section in this chapter is the lesson on using the quadratic formula. You probably remember using the formula in the past, and perhaps even remember the "discriminant," which can be used to identify the number of real solutions to a quadratic equation. Now you will get to learn why all of your prior lessons always specified "real" answers, when asking you to find the roots!

## Chapter Outline

- 4.1. Polar Coordinates
- 4.2. Polar and Cartesian Transformation
- 4.3. Systems of Polar Equations
- 4.4. Imaginary Numbers
- 4.5. Complex Numbers
- 4.6. Quadratic Formula and Complex Sums
- 4.7. Products and Quotients of Complex Numbers
- 4.8. Polar Form of Complex Numbers
- 4.9. Product and Quotient Theorems
- 4.10. Powers and Roots of Complex Numbers

### Chapter Summary

## Summary

This chapter focuses on the various uses of cartesian (x, y) graph and polar graph methods.

Through the material in this chapter, students should become proficient at:

- Converting values and graphs between polar and cartesian methods.
- Identifying and graphing imaginary and complex numbers.
- Using the Quadratic Formula to find the imaginary roots of quadratic equations.
- Multiplying and dividing complex numbers and graphing the solutions on polar and cartesian graphs.