Having Fun with Complex Numbers(mathwonder.org)
mathwonder.org
Having Fun with Complex Numbers
https://mathwonder.org/Having-Fun-with-Complex-Numbers/
13 comments
As a parent it would be nice to have a better sense of what the book is like — I'm always interested in different ways of explaining concepts to people for the first time, especially kids, but without seeing the content it's impossible to know what to expect. Unfortunately there's not a physical copy to peruse to get a sense of the main ideas and aesthetics.
You might find this resource useful - https://news.ycombinator.com/item?id=46030609
When it comes to learning/understanding Complex Numbers there is one pre-eminent source which everybody should lookup;
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
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
The research of the book's author has redeveloped the imaginary unit and complex number theory from the ground-up based on first principles, without using i = sqrt(-1) or i^2 = -1. This makes it accessible across broader educational levels—including elementary schools.
Even though the book is designed for kids, it is also recommended for curious readers of all ages who want fresh ideas.
Even though the book is designed for kids, it is also recommended for curious readers of all ages who want fresh ideas.
Looks like a "visual thinker":
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://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.
The author has a version for adults (or older students)
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)
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)
The video for elementary teachers, that is prob closer to the book, for anybody interested https://www.youtube.com/watch?v=R43XpPZ6PKg
> Dr. Qiujiang Lu is an independent researcher and software developer in Silicon Valley whose work has re-created the imaginary unit in real life from first principles, which remained unknown for 500 years since the inception.
Sounds grand. A bit too grand, perhaps. Does anyone know what he's alluding to? ELI am a physicist.
Sounds grand. A bit too grand, perhaps. Does anyone know what he's alluding to? ELI am a physicist.
“which remained unknown for 500 years since the inception”
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
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
Needham is one of my favourite books..
But Iirc he doesn't really cover this perspective much (excepting those parts where he hints at hypercomplex numbers)
But Iirc he doesn't really cover this perspective much (excepting those parts where he hints at hypercomplex numbers)
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.
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.
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/Plate_trick#The_belt_trick
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/Plate_trick#The_belt_trick
LOL :D somebody has high opinion about himself