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Judging from his videos, I do not think he actually creates something new. He takes a geometric approach to constructing complex numbers, but these approaches e

Judging from his videos, I do not think he actually creates something new. He takes a geometric approach to constructing complex numbers, but these approaches exist. Not all approaches to complex numbers are algebraic (ie about extending the real field). 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.

gsf_emergency_6 11mo ago
He takes reals to be "stretch" and imaginaries to be "rotate". As in, real space, not an abstraction like "complex plane". 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 :) https://en.wikipedia.org/wiki/Tangloids https://en.wikipedia.org/wiki/Tangloids https://en.wikipedia.org/wiki/Plate_trick#The_belt_trick https://en.wikipedia.org/wiki/Plate_trick#The_belt_trick

gsf_emergency_6 11mo ago

He takes reals to be "stretch" and imaginaries to be "rotate". As in, real space, not an abstraction like "complex plane". 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 :) https://en.wikipedia.org/wiki/Tangloids https://en.wikipedia.org/wiki/Tangloids https://en.wikipedia.org/wiki/Plate_trick#The_belt_trick https://en.wikipedia.org/wiki/Plate_trick#The_belt_trick