Number Systems Invented to Solve the Hardest Problem - History of Rings | Ring Theory E0

As someone who never saw enough pure math to string together a full picture of these concepts and their origins, this is absolute gold. Will be very happy if there is more. :)

What a tour de force. I learnt a fantastic amount here in a very enjoyable way without being mired in detail. In this field, you truly are the Lord of the Rings .

I got a little lost on some parts, but it was definitely worth to continue watching! This was so interesting!

Really nicely presented. At 37:11 Wedderburn and Artin showed that any non-commutative algebra over the reals is a product of matrices over R, C and H. Thanks for the wonderful refresher

Sir, this is 3b1b caliber work with maybe even deeper content. I can't believe I just found your channel. I know there are other number systems but to have a complete guide with the context for why they were made and a quick explanation is mind bending. I needed this video so bad I can't even describe how I even feel about it. Thank you.

I am in awe. To be exposed to the greatest minds in math is a transcendental experience.

p-adics?!??? Also A_inf, B_dR, B_crys, B_st, Galois deformation rings, Hecke rings, and so much more!! FLT really is astounding.

This video is so good. It feels like it was made to perfectly cater to my interests and current level of knowledge. I’m so glad that I found your channel. Thank you.

Amazing video, somehow you managed to cover so much ground in this video while having it remain intuitive and understandable. I never realised how interesting rings and fields were

I'm amazed at the scope you were able to cover in less than 40 minutes. Brilliant work really (or should i say, complexly :p). Keep it up.

Beautifully done video. More, please, when you can. 🎉😊

Such a wonderful video. Please keep at it! I feel like I've just realized for what purpose those thick algebra books are so meticulously categorized!!

A much needed refresher on rings, with additional paths for further education. Bravo

The second anti-commutative 4D algebra with x² = 0 and y² = -1 is not the dual-quaternions as you said, but rather the planar-quaternions. The dual-quaternions are an 8D algebra and contains the planar-quaternions, containing an extra anti-commuting term squaring to -1. These along with several other algebras can be generated as Clifford algebras, denoted as Cl(p, q, r), where p is the number of orthogonal elements squaring the +1, q the number of such elements squaring to -1, and r the number squaring to 0. The planar-quaternions are Cl(0,1,1) and the dual-quaternions are Cl(0,2,1). As a bonus, the quaternions are Cl(0,2,0), ℂomplex numbers Cl(0,1,0), dual numbers Cl(0,0,1), hyperbolic numbers (the more descriptive name for the split-complex numbers) Cl(1,0,0), and the ℝeals are also included as Cl(0,0,0). These algebras are often very useful for describing geometric transformations in space, which is why they're often called geometric algebras. ℂomplex numbers are well known for describing 2D rotations, and the quaternions for 3D rotations. Geometric algebras extend these to higher dimensional rotations, as well as a few other things. Your third example, which is Cl(1,1,0), is often used as a simplified version of Cl(1,3,0), used for modelling a 2D slice of the 4D spacetime of Special Relativity. I loved seeing the binary rationals, not because I'm already a fan (this is actually the first time I've heard about them formally), but because I happen to be enjoy programming and computing, so I instantly recognized it as ideal fixed point and floating point numbers. It also made me consider how ℤ[1/10] would be the ring of all decimal expansions. (I'd assume finite, because otherwise it'd be indistinguishable from the ℝeals.) I was hoping for a little more time spent on modular integers, but they'll probably come up when you make the video on p-adics, because the p-adic integers with n digits of precision is equivalent to ℤ mod p^n. Again, my interest in computing makes me naturally more interested in the 2-adics specifically, and things like ℤ mod 256, ℤ mod 65536, ℤ mod 2^32, etc, since they're exactly the rings that 8-bit, 16-bit, and 32-bit integers represent. Integer "overflow" is usually treated as an error by most programmers, but it's just a natural part of doing modular arithmetic that should be completely expected.

Love how the music makes me feel like a Viking mathematical pioneer.

Sweet! This stuff is gold! Love the animation and explanations!! Well done ^^

Top tier math content right here

as someone who was struggling through some other videos about the quaternions, I am sufficiently glad this video is only 5 days old... having said that, great video!

i wish this video existed 8 years ago good job man

This was amazing. This video made me understand concepts that I have heard before but never quite understood. There were still some things I had trouble wrapping my head around espacially towards the end but overall this was a great experience. Thank you!