Complex Numbers

Revel in the beauty of algebra as you explore complex numbers, fractals, and Euler's formula.

Algebraic Intuition

Why Complete This Course?

The Pathway to Euler's Formula

It's Not A Real Number

Imaginary Powers

Real and Imaginary Parts

The Complex Plane

The Triangle Inequality

Arithmetic Shortcuts

Multiplying Complex Numbers

Complex Exponents

The Mandelbrot Set

Julia Sets

Functions Warm-up

Domain & Range

Transforming Functions

Transforming Functions Practice

Symmetries and Isometries

Complex Transformations



Composition and Inversion

Recursion and Limits

Koch's Snowflake

A Golden Polynomial

Finding Roots

How Many Roots?

Graphs of Polynomials

Graphs and Repeated Roots

Complex Roots

Factoring with Real Coefficients

Exponents Warmup

Defining Exponents

Laws of Exponents

The Growth Rate of Exponential Sequences

Modeling Exponential Growth

Doomsday Scenarios

Finance and Exponents

Domain and Range Misconceptions

Defining Logarithms

Log Scales

Understanding Log Arithmetic

Log Arithmetic Practice

The Change of Base Formula

Graphing Logs

Log Equations

Applying Log Scales

Sine and Cosine

Conceptual Foundations

The Unit Circle

Trigonometry Graphs

Relating the Functions

Other Trigonometry Function Graphs

Trigonometry Graphs Problem Solving


Inverse Trigonometry

Conversion From Cartesian

Simpler in Polar Form

Polar Transformations

More Transformations

Rose Curves

Graphing Complex Numbers

Introduction to Transformations

Introduction to Vectors

Translation and Scaling

Identity and Reflection


Multiple Transformations


Are You Ready for Euler's Formula?

Algebraically Manipulating the Formula

How to Approach the Trigonometry

How to Approach Complex Exponents



What Is Exponentiation?

Complex Exponentiation

Trig Identities with Euler's Formula

Roots of Unity with Euler's Formula

Physics with Euler's Formula

Course description

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.

Topics covered

  • Arithmetic with Complex Numbers
  • The Complex Plane
  • Complex Exponents
  • Fractals
  • Function Transformations
  • Complex Number Transformations
  • Composition and Composition Recursion
  • Exponentials and Complex Exponentials
  • Euler's Formula
  • Roots of Unity

Prerequisites and next steps

A thorough understanding of basic algebra is an essential prerequisite for this course.


  • Functions and Quadratics
  • Pre-Calculus