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Complex Numbers as Rotation Matrices

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First published at 22:38 UTC on November 18th, 2022.

#mathematics
#science
#education

Math Easy Solutions

MES

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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 Fractal Woman YouTube channel: htt…

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

Download Video Notes: https://1drv.ms/b/s!As32ynv0LoaIiaFTZDoqo7BXtv_aCQ?e=vfGRLy

View video notes on the Hive blockchain: https://peakd.com/hive-128780/@mes/complex-numbers-as-rotation-matrices

View video sections playlist: https://www.youtube.com/playlist?https://www.youtube.com/playlist?list=PLai3U8-WIK0E6x0dZEAx77Kqv7LLxYdbx

View the full Vectors and Geometry of Space video series: https://www.youtube.com/playlist?list=PLai3U8-WIK0FjJpwnxwdrOR7L8Ul8VZoZ