Complex Numbers
Work with imaginary and complex numbers and the complex plane.
The Square Root of -1
Imaginary Numbers
Complex Numbers
Plotting Complex Numbers
Identifying Complex Points
Combining Complex Numbers
Adding Complex Numbers
Subtracting Complex Numbers
Absolute Value
Complex Separation
Scalar Multiplication
Scaling Magnitudes
Simplifying Expressions
Multiplying by i
Rotation by Multiplication
Multiple Rotations
Scaling and Rotating
General Multiplication
Complex Pairs
Finding the Complex Conjugate
Adding Conjugates
Multiplying Conjugates
Functions Re and Im
Sum of Two Conjugates
Dividing by i
Complex Fractions
General Division
Transforming Points
Magnitude and Argument
Principal Arguments
Computing Arguments
Using Polar Form
Adding to Multiply
Sine of a Sum
Cosine of a Sum
Comparing Expressions
Applying Argument Addition
Multiplying in Polar Form
Transforming to Multiply
Division in Polar Form
Transforming to Divide
Polar Conjugates
The Product of Conjugates
Raising to any Power
Powers of Conjugates
The Principal Root
The Square Root of i
Both Square Roots
Multiple Roots
Roots of 1
Course description
In this course you'll understand how complex numbers arise naturally as solutions to equations. You'll learn to manipulate them according to the usual rules of algebra and derive geometric insights from the complex plane.
Topics covered
- Complex numbers as solutions to quadratic equations.
- Addition and subtraction of complex numbers.
- Absolute value and scaling of complex numbers.
- Multiplying complex numbers algebraically.
- The geometric interpretation of complex multiplication.
- Complex conjugates.
- Dividing complex numbers.
- Complex numbers in polar form.
- Powers and roots.
Prerequisites and next steps
Assumes comfort with the cartesian and polar coordinate plane, quadratic equations and basic trigonometry.