# Ch 4: Complex Numbers & the Complex Plane

### About This Chapter

## Complex Numbers & the Complex Plane - Chapter Summary

Lessons in this chapter can introduce you to the imaginary numbers used to represent negative square roots, along with the complex numbers that result when these figures are added or subtracted from a real number. Once you've got a grasp of these concepts, you can learn how to solve quadratic equations, including complex numbers and add, subtract or multiply complex numbers by combining like terms and using the correct order of operations.

Delve further into the lessons to find out how the complex conjugate can be used to divide complex numbers and determine the multiplicative inverse. You can even learn how to plot the results as vectors on the complex plane, find the distance between points, convert between polar and rectangular coordinates, and represent the solution to addition, subtraction and multiplication problems. This chapter is designed to teach you the following concepts:

- Properties of complex numbers
- Complex number operations
- Graphic representations of complex numbers

Instructors in this chapter's video lessons use sample problems and illustrations to walk you through the mathematical terms and concepts. There are also video tags that allow you to jump ahead or go back to key points if you find that you need to revisit certain problem-solving steps. If you need extra practice, there are corresponding self-assessment quizzes that include sample problems and solutions.

### 1. What is an Imaginary Number?

The imaginary number 'i' is the square root of -1. Although this number doesn't actually exist, pretending that it does allows us to do a bunch of crazy math that scientists use every day. Learn the basics of that number 'i' here!

### 2. How to Add, Subtract and Multiply Complex Numbers

Knowing that complex numbers exist is the first step. But that knowledge alone won't help you do much with them. Learn the basics of complex number addition, subtraction and multiplication here!

### 3. How to Divide Complex Numbers

While adding, subtracting and multiplying complex numbers is pretty straightforward, dividing them can be pretty tricky. It comes down to the process of multiplying by the complex conjugate. Learn about what that is, and how to do it, here.

### 4. How to Solve Quadratics with Complex Numbers as the Solution

When you solve a quadratic equation with the quadratic formula and get a negative on the inside of the square root, what do you do? The short answer is that you use an imaginary number. For the longer, more helpful answer, check out this lesson!

### 5. Modulus of a Complex Number: Definition & Examples

In this lesson, we will explore complex numbers and an important characteristic of these numbers called a modulus. We will look at what a modulus of a complex number is, how it relates to other aspects of a complex number, and how to find it.

### 6. Multiplicative Inverse of a Complex Number

In this lesson, you'll learn about the multiplicative inverse of a complex number and how to find it. The process is explained with the help of relevant examples.

### 7. How to Graph a Complex Number on the Complex Plane

Graphing complex numbers is pretty straight forward, but it's not necessarily intuitive. Check out this lesson to learn the vocabulary and the conventions that you'll need.

### 8. Representing Complex Numbers With Vectors

In this lesson, we will explore complex numbers and vectors, and we will look at how these two concepts, though seemingly unrelated, work together by representing complex numbers with vectors.

### 9. How to Convert Between Polar & Rectangular Coordinates

Symmetry in a math problem may suggest a change of coordinate systems. In this lesson we show how to convert between two common coordinate systems: polar and rectangular.

### 10. How to Add Complex Numbers in the Complex Plane

Watch this video lesson to see how you can use the complex plane to help you add your complex numbers. See how similar this is to the very familiar Cartesian coordinate plane.

### 11. How to Subtract Complex Numbers on the Complex Plane

In this lesson, we define the complex plane and then show two methods for subtracting complex numbers. These methods are analogous to the methods used for adding vectors in the Cartesian plane.

### 12. Multiplication on the Complex Plane

In this lesson we review two ways to locate a point on a plane. Then, using complex numbers, one of these two ways is used to multiply numbers on the complex plane.

### 13. Representing Distances on the Complex Plane

Read this lesson to learn the formula you need to use to help you find distances on the complex plane. You'll learn that all you need to know are the coordinates of the beginning and end of your distance line.

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### Other Chapters

Other chapters within the TExES Mathematics 7-12 (235): Practice & Study Guide course

- About the TExES Math 7-12 Exam
- Real Numbers
- Mathematical Models
- Number Theory
- Number Patterns
- Functions and Graphs
- Linear Functions
- Quadratic Functions & Polynomials
- Evaluating Piecewise & Composite Functions
- Rational and Radical Functions
- Inequalities and Absolute Values
- Exponentials & Logs
- The Unit Circle
- Trigonometric Functions
- Using a Scientific Calculator for Calculus
- Understanding Limits in Math
- Understanding Rate of Change
- Calculating Derivatives of Functions
- Derivatives and Graphs
- Optimization in Calculus
- Definite Integrals and Sums
- Integration Applications in Calculus
- Working with Measurement
- Finding Volume, Area & Perimeter
- Introduction to Proofs and Constructions
- Congruence and Similarity
- Triangles
- Circles
- Real World Shapes
- Coordinate Geometry
- Understanding Transformations in Math
- Conic Sections
- Understanding Vectors
- Measuring & Displaying Data
- Data Distribution Overview
- Sampling in Statistics
- Distribution & Inference in Statistics
- Inference About a Mean
- Regression and Correlation
- Finding Probability
- Probability Distributions and Statistical Inference
- Experiments and Surveys
- Mathematical Process & Perspectives
- Teaching Strategies & Activities for the Math Classroom
- Differentiated Instructional Strategies for the Math Classroom
- Using Student Assessments in the Math Classroom
- TExES Mathematics 7-12 Flashcards