Complex Numbers as Rotation Matrices

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
1. Fractal Woman and Other References: 1:28
2. Rotating a Vector: 3:01
3. Matrix Multiplication: 12:47
4. Matrix Addition: 25:29
5. Rotation Matrix: 27:13
6. Complex Numbers: 36:58
7. 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

Related Videos:

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In this video I rewrite and reinterpret complex numbers as being simply the rotations of a vector. This concept was first brought to my attention by the Fractal Woman YouTube channel, so make sure to check out her work.