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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.

Interactive
quizzes

80

Concepts and
exercises

550+
  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.

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      It's Not A Real Number

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

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      Imaginary Powers

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

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      Real and Imaginary Parts

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

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      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.

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      Functions Warm-up

      Review the key principles of mathematical functions!

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      Domain & Range

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

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      Transforming Functions

      Practice applying and visualizing function transformations.

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      Transforming Functions Practice

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

  4. 4

    Polynomials

    It's the Fundamental Theorem of Algebra.

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      A Golden Polynomial

      Review the mathematics of polynomials and discover the golden ratio.

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      Finding Roots

      Learn and practice several polynomial root-finding techniques.

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      How Many Roots?

      Apply the Factor Theorem while studying roots and multiplicity.

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      Graphs of Polynomials

      What do polynomials look like?

  5. 5

    Exponential Equations

    A doomsday forecast if left unchecked.

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      Exponents Warmup

      Review the foundational skills for working with exponents.

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      Defining Exponents

      Strengthen your intuition for how exponents behave.

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      Laws of Exponents

      Explore and practice applying the algebraic laws of exponents.

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      The Growth Rate of Exponential Sequences

      How quickly do exponential functions grow?

  6. 6

    Logarithms

    Is it all about the base?

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      Defining Logarithms

      Understand the role of each component of a logarithmic expression.

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      Log Scales

      Use exponential scaling to visualize data in a useful way.

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      Understanding Log Arithmetic

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

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      Log Arithmetic Practice

      Practice applying the arithmetic log rules you recently derived.

  7. 7

    Trigonometry

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

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      Sine and Cosine

      Get acquainted with the two fundamental functions of trigonometry.

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      Conceptual Foundations

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

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      The Unit Circle

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

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      Trigonometry Graphs

      What do the graphs of sine and cosine look like?

  8. 8

    Polar Coordinates

    The coordinate plane as a spider sees it.

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      Conversion From Cartesian

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

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      Simpler in Polar Form

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

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      Polar Transformations

      Modify and visually transform graphs in polar form.

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      More Transformations

      Now rotate, translate, and reflect polar graphs!

  9. 9

    Vector Space Transformations

    First functions, but now let's warp space.

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      Introduction to Transformations

      What is a transformation function?

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      Introduction to Vectors

      What is a vector?

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      Translation and Scaling

      Apply transformations that move, grow, and shrink vectors.

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      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.

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      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.

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      Algebraically Manipulating the Formula

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

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      How to Approach the Trigonometry

      Study the sine and cosine terms that are on one side of Euler's Formula.

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      How to Approach Complex Exponents

      Study the complex exponent that's on the other side of Euler's Formula.