Having Fun with Complex Numbers
mathwonder.org
mathwonder.org
A series of videos named "Imaginary Numbers are Real" by Welch Labs - https://www.youtube.com/playlist?list=PLiaHhY2iBX9g6KIvZ_703...
You can buy book versions (used to be free earlier) at - https://www.welchlabs.com/resources
Even though the book is designed for kids, it is also recommended for curious readers of all ages who want fresh ideas.
https://www.eoas.ubc.ca/courses/atsc113/snow/met_concepts/07...
His other research
had involved discovering the mechanism of ICE CRYSTAL GROWTH HABIT CHANGE, an outstanding problem for more than 50 years in cloud physics that is closely related to the “thousands’ variations” in snowflakes.
https://youtu.be/sehioJvr_eo?t=10m40s
(There's also a video in that channel that's made for elementary teachers, but it's not as pretty as you might hope)
Sounds grand. A bit too grand, perhaps. Does anyone know what he's alluding to? ELI am a physicist.
Is wrong
But an easy way to define the complex plane is to postulate you want multiplication of vectors in polar form to multiply distance to origin and add angles. No mystery number squares going negative here, just simple and useful geometry !
There’s a whole book about it, Visual Complex Numbers by Tristan Needham. This author is the real boss of the game
But Iirc he doesn't really cover this perspective much (excepting those parts where he hints at hypercomplex numbers)
As far as I understand, he essentially defines $i$ through a π/2 rotation. But this is exactly what $i^2=-1$ is. So in a sense, I do not think it is quack, but overblown in terms of novelty. Personally, I always liked such kinds of geometric approach to complex numbers, because it makes a lot of stuff more intuitive, even just for reals (eg you can see multiplication by -1 as rotation by π). If he makes a good dissemination of the complex numbers to kids, it could be worth it, but no idea without any sample from the book.
Then the imaginary unit becomes, not just rotation by pi/2 but a "basis vector" for rotation.
Putting on my physicst/engineer hat. this identifies rotations with the axis of rotation, which points outside the plane. (Disclaimer: this is not exactly how the author thinks about it.)
(In contrast. The basis vector of "stretches", btw which include 180-degree rotations, stay in the plane:)
The math is not novel but the perspective is.
Now this can be generalized to 3D rotations, whence you think of (the unit) quaternions as 3 independent axes of rotations.
(Euler angle and Euler formula become muddled :)
There's also the "rotational derivative" (angular velocity) bit which is THE THING worth mulling over. I think is the really novel bit (again. perspective, not math-- but I have not worked out his [degree] arithmetic )
(He calls it the fundamental equation in the video)
The physicist gets reminded of Legendre transforms (think <p,q> (- H)), where p here means angular momentum :)
It will be most cool if he can use this style to explain the "Feynman belt trick" without symbols or animation :)