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# Complex Numbers

## The beauty of Algebra through complex numbers, fractals, and Euler’s formula.

This course is for those who want to fully master Algebra with complex numbers at an advanced level.

The prize at the end will be combining your newfound Algebra skills in trigonometry and using complex variables to gain a full understanding of Euler’s identity. Euler's identity combines e, i, pi, 1, and 0 in an elegant and entirely non-obvious way and it is recognized as one of the most beautiful equations in mathematics.

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1. 1

### Introduction

Why study Euler's Formula?

1. #### Algebraic Intuition

Warm up your algebraic skills and intuition with this opening set!

2. #### Why Complete This Course?

What are complex numbers good for?

3. #### The Pathway To Euler's Formula

This is one of the most beautiful theorems in mathematics.

2. 2

### Complex Numbers

From the basics to fractals.

1. Included with

#### It's Not A Real Number

Get started with an introduction to complex numbers and complex arithmetic.

2. Included with

#### Imaginary Powers

Explore the patterns that emerge when complex numbers are squared and cubed.

3. Included with

#### Real and Imaginary Parts

Think about complex numbers as a combination of their real and imaginary components.

4. Included with

#### The Complex Plane

Graph complex numbers and build intuition for representing them as vectors.

3. 3

### Functions and Transformations

Slide it, squeeze it, stretch it, flip it.

1. Included with

#### Functions Warm-up

Review the key principles of mathematical functions!

2. Included with

#### Domain & Range

What happens if a function's input or output is restricted?

3. Included with

#### Transforming Functions

Practice applying and visualizing function transformations.

4. Included with

#### Transforming Functions Practice

Further strengthen your skill with transformation techniques by solving these challenges.

4. 4

### Polynomials

It's the Fundamental Theorem of Algebra.

1. Included with

#### A Golden Polynomial

Review the mathematics of polynomials and discover the golden ratio.

2. Included with

#### Finding Roots

Learn and practice several polynomial root-finding techniques.

3. Included with

#### How Many Roots?

Apply the Factor Theorem while studying roots and multiplicity.

4. Included with

#### Graphs of Polynomials

What do polynomials look like?

5. 5

### Exponential Equations

A doomsday forecast if left unchecked.

1. Included with

#### Exponents Warmup

Review the foundational skills for working with exponents.

2. Included with

#### Defining Exponents

Strengthen your intuition for how exponents behave.

3. Included with

#### Laws of Exponents

Explore and practice applying the algebraic laws of exponents.

4. Included with

#### The Growth Rate of Exponential Sequences

How quickly do exponential functions grow?

6. 6

### Logarithms

Is it all about the base?

1. Included with

#### Defining Logarithms

Understand the role of each component of a logarithmic expression.

2. Included with

#### Log Scales

Use exponential scaling to visualize data in a useful way.

3. Included with

#### Understanding Log Arithmetic

Leverage your experience with logarithmic scales to understand the arithmetic rules for logarithms.

4. Included with

#### Log Arithmetic Practice

Practice applying the arithmetic log rules you recently derived.

7. 7

### Trigonometry

What is a sin(gerine) / cos(gerine)?

1. Included with

#### Sine and Cosine

Get acquainted with the two fundamental functions of trigonometry.

2. Included with

#### Conceptual Foundations

Explore the connection between the sine and cosine functions and steady movement around a circle.

3. Included with

#### The Unit Circle

Develop an essential toolkit for understanding and calculating sine and cosine values.

4. Included with

#### Trigonometry Graphs

What do the graphs of sine and cosine look like?

8. 8

### Polar Coordinates

The coordinate plane as a spider sees it.

1. Included with

#### Conversion From Cartesian

What are polar coordinates and how are they related to traditional coordinates?

2. Included with

#### Simpler in Polar Form

Sometimes using polar coordinates is more efficient and elegant than using x and y.

3. Included with

#### Polar Transformations

Modify and visually transform graphs in polar form.

4. Included with

#### More Transformations

Now rotate, translate, and reflect polar graphs!

9. 9

### Vector Space Transformations

First functions, but now let's warp space.

1. Included with

#### Introduction to Transformations

What is a transformation function?

2. Included with

#### Introduction to Vectors

What is a vector?

3. Included with

#### Translation and Scaling

Apply transformations that move, grow, and shrink vectors.

4. Included with

#### Identity and Reflection

What matrix can reflect a vector across the x or y axis?

10. 10

### Euler's Formula

You have reached the summit.

1. Included with

#### Are You Ready for Euler's Formula?

Review all of the skills that you'll need to use in concert to understand Euler's Formula.

2. Included with

#### Algebraically Manipulating the Formula

Use a bit of lite algebra to manipulate the terms in Euler's Formula.

3. Included with