Complex Numbers as Rotation Matrices

Math Easy Solutions

description

In this video I go over a very interesting concept and that is in interpreting complex or imaginary numbers as and equivalent rotational matrix form. This concept was brought to my attention by Lori Gardi from the @FractalWoman YouTube channel. Rewriting complex numbers as rotation matrices allows for looking at complex numbers from a different perspective, and this case that perspective is in simply rotating vectors. The imaginary number "i" is thus viewed as simply a 90 degree rotation when viewed as a rotation matrix. I also rewrite the famous Euler's formula into a rotation matrix form and that turns out to just be a 180 degree rotation. Since complex numbers appear often in mathematics and physics, it may be insightful to reinterpret such complex equations as simply vector rotations!

Note that while I only covered 2D rotation matrices, a similar concept can be applied to 3D rotation matrices, and which I may cover in future videos so stay tuned.

The topics covered as well as their timestamps are listed below.

Introduction: 0:00
Topics to Cover: 0:41

Fractal Woman and Other References: 1:28
Rotating a Vector: 3:01
Matrix Multiplication: 12:47
Matrix Addition: 25:29
Rotation Matrix: 27:13
Complex Numbers: 36:58
Relating the Complex Plane with the Rotation Matrix: 43:30
- Complex Numbers as Vector Rotations: 57:12
- Reinterpreting Euler's Formula: 1:10:02
- Applications to Other Fields: 1:20:51