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
Inverses
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
Identities
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
Rotation
Multiple Transformations
Inversion
Are You Ready for Euler's Formula?
Algebraically Manipulating the Formula
How to Approach the Trigonometry
How to Approach Complex Exponents
Adders
Multipliers
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.