Forms both a:
- division algebra if thought of with complex multiplication as the bilinear map of the algebra
- field
Rather, we should group students by subject of interest; e.g. natural sciences, social sciences, a sport, etc., just like in any working adult organization!
This way, younger students can actually actively learn from and collaborate with older students about, see notably Jacques Monod's you can learn more from older students than from faculty.
This becomes even more natural when you try to give students must have a flexible choice of what to learn.
This age distinction should be abolished at all stages of the system, not only within K-12, but also across K-12, undergraduate education and postgraduate education.
This idea is part of the ideal that the learning environment should be more like a dojo environment (AKA peer tutoring, see also dojo learning model), rather than an amorphous checkbox ticking exercise in bureaucracy so that "everyone is educated".
Perhaps, even more importantly, is that we should put much more emphasis on grouping students with other students online, where we can select similar interest amongst the entire population and not just on a per-local-neighbourhood basis.
Constructs the quaternions from complex numbers, octonions from quaternions, and keeps doubling like this indefinitely.
Unlike the quaternions, it is non-associative.
TODO what is the point of them? Why not just sum over every index that appears twice, regardless of where it is, as mentioned at: www.maths.cam.ac.uk/postgrad/part-iii/files/misc/index-notation.pdf.
Those in indices on bottom are called contravariant vectors.
It is possible to change between them by Raising and lowering indices.
The values are different only when the metric signature matrix is different from the identity matrix.
How to teach Publish your material even if it is not perfect by 
Ciro Santilli 40 Updated 2025-07-16
Ciro Santilli 40 Updated 2025-07-16Just make it very clear what you've tried, what you observed, and what you don't understand if anything at all.
This will already open up room for others to come and expand on your attempt, and you are more likely to learn the answers to your questions as they do.
And there's a good chance someone who knows more than you will come along and correct or teach you something new about the subject. For example, this has happened countless times to Ciro Santilli when doing Ciro Santilli's Stack Overflow contributions.
Perfect is the enemy of good.
Examples of famous fails:
- QED and the men who made it: Dyson, Feynman, Schwinger, and Tomonaga by Silvan Schweber (1994) chapter 7.11 "Epilogue" mentions how Julian Schwinger has lots of unpublished notes, or that his collaborators had to write most of the stuff down themselves in the end because he felt they were not perfect enough
In some systems, e.g. including Metamath, modus ponens alone tends to be enough, everything else can be defined based on it.
Pinned article: Introduction to the OurBigBook Project
Welcome to the OurBigBook Project! Our goal is to create the perfect publishing platform for STEM subjects, and get university-level students to write the best free STEM tutorials ever.
Everyone is welcome to create an account and play with the site: ourbigbook.com/go/register. We belive that students themselves can write amazing tutorials, but teachers are welcome too. You can write about anything you want, it doesn't have to be STEM or even educational. Silly test content is very welcome and you won't be penalized in any way. Just keep it legal!
Intro to OurBigBook
. Source. We have two killer features:
- topics: topics group articles by different users with the same title, e.g. here is the topic for the "Fundamental Theorem of Calculus" ourbigbook.com/go/topic/fundamental-theorem-of-calculusArticles of different users are sorted by upvote within each article page. This feature is a bit like:
- a Wikipedia where each user can have their own version of each article
- a Q&A website like Stack Overflow, where multiple people can give their views on a given topic, and the best ones are sorted by upvote. Except you don't need to wait for someone to ask first, and any topic goes, no matter how narrow or broad
This feature makes it possible for readers to find better explanations of any topic created by other writers. And it allows writers to create an explanation in a place that readers might actually find it.
Figure 1. Screenshot of the "Derivative" topic page. View it live at: ourbigbook.com/go/topic/derivativeVideo 2. OurBigBook Web topics demo. Source. - local editing: you can store all your personal knowledge base content locally in a plaintext markup format that can be edited locally and published either:This way you can be sure that even if OurBigBook.com were to go down one day (which we have no plans to do as it is quite cheap to host!), your content will still be perfectly readable as a static site.
- to OurBigBook.com to get awesome multi-user features like topics and likes
- as HTML files to a static website, which you can host yourself for free on many external providers like GitHub Pages, and remain in full control
Figure 3. Visual Studio Code extension installation.
Figure 4. Visual Studio Code extension tree navigation.
Figure 5. Web editor. You can also edit articles on the Web editor without installing anything locally.Video 3. Edit locally and publish demo. Source. This shows editing OurBigBook Markup and publishing it using the Visual Studio Code extension.Video 4. OurBigBook Visual Studio Code extension editing and navigation demo. Source. 
- Infinitely deep tables of contents:
All our software is open source and hosted at: github.com/ourbigbook/ourbigbook
Further documentation can be found at: docs.ourbigbook.com
Feel free to reach our to us for any help or suggestions: docs.ourbigbook.com/#contact