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I don’t think quaternions would be taught in a typical first course of abstract algebra. Do you know of a textbook where they are featured prominently?

by smadge 9y ago

lasagnaphil 9y ago
Probably some of the introductory level Computer Graphics books. The field uses a lot of quaternions, so they might have a good explanation about it.
dsacco 9y ago
I don't know of any book which features them "prominently", but I also don't think you'd really need one. They are taught in various abstract algebra books, they're just taught in the fashion of, "Here's an exercise that introduces a peripheral topic it's useful to know about." For example, groups and rings of quaternions show up in MacLane & Birkhoff's Algebra (62, 426; 282) and Lang's Algebra (9, 545, 723, 758). Edit: In an effort to find more applied information I put down my math books and picked up the information theoretic ones. You can find more information about the use of quaternions in the two volume Handbook of Digital Signal Processing and Salomon's Data Compression. More generally, when quaternions aren't explicitly referred to it's helpful to look up the coverage of complex rotations, especially with respect to the Discrete Fourier Transform. For a discussion of rotations with quaternions in the context of animation, this is a reasonably short paper: http://www.cs.cmu.edu/~kiranb/animation/p245-shoemake.pdf http://www.cs.cmu.edu/~kiranb/animation/p245-shoemake.pdf.
gnulinux 9y ago
When I took abstract algebra in Berkeley years ago, they were taught but they weren't the focus of the course. Basically, they're an example of skew field (division ring) so they have some interesting properties that was briefly studied. But obviously, one has to study more to understand their applications.
theoh 9y ago
https://en.wikibooks.org/wiki/Abstract_Algebra/Quaternions https://en.wikibooks.org/wiki/Abstract_Algebra/Quaternions or https://books.google.ie/books?id=ouvZKQiykf4C&pg=PA98&lpg=PA98&dq=quaternions https://books.google.ie/books?id=ouvZKQiykf4C&pg=PA98&lpg=PA... I am from Dublin, where quaternions were invented, so they get mentioned a lot by mathematicians and physicists here, maybe getting a higher billing than they do elsewhere. Computer graphics is obviously a place to go for introductions also, but it is typically going to be a more applied and less rigorous treatment.

I don’t think quaternions would be taught in a typical first course of abstract algebra. Do you know of a textbook where they are featured prominently?

by smadge 9y ago

I don’t think quaternions would be taught in a typical first course of abstract algebra. Do you know of a textbook where they are featured prominently?

lasagnaphil 9y ago

Probably some of the introductory level Computer Graphics books. The field uses a lot of quaternions, so they might have a good explanation about it.

dsacco 9y ago

I don't know of any book which features them "prominently", but I also don't think you'd really need one. They are taught in various abstract algebra books, they're just taught in the fashion of, "Here's an exercise that introduces a peripheral topic it's useful to know about." For example, groups and rings of quaternions show up in MacLane & Birkhoff's Algebra (62, 426; 282) and Lang's Algebra (9, 545, 723, 758). Edit: In an effort to find more applied information I put down my math books and picked up the information theoretic ones. You can find more information about the use of quaternions in the two volume Handbook of Digital Signal Processing and Salomon's Data Compression. More generally, when quaternions aren't explicitly referred to it's helpful to look up the coverage of complex rotations, especially with respect to the Discrete Fourier Transform. For a discussion of rotations with quaternions in the context of animation, this is a reasonably short paper: http://www.cs.cmu.edu/~kiranb/animation/p245-shoemake.pdf http://www.cs.cmu.edu/~kiranb/animation/p245-shoemake.pdf.

gnulinux 9y ago

When I took abstract algebra in Berkeley years ago, they were taught but they weren't the focus of the course. Basically, they're an example of skew field (division ring) so they have some interesting properties that was briefly studied. But obviously, one has to study more to understand their applications.

theoh 9y ago

https://en.wikibooks.org/wiki/Abstract_Algebra/Quaternions https://en.wikibooks.org/wiki/Abstract_Algebra/Quaternions or https://books.google.ie/books?id=ouvZKQiykf4C&pg=PA98&lpg=PA98&dq=quaternions https://books.google.ie/books?id=ouvZKQiykf4C&pg=PA98&lpg=PA... I am from Dublin, where quaternions were invented, so they get mentioned a lot by mathematicians and physicists here, maybe getting a higher billing than they do elsewhere. Computer graphics is obviously a place to go for introductions also, but it is typically going to be a more applied and less rigorous treatment.