On Quaternions and Octonions

I really enjoyed the book at first. I liked the discussion of primes, units, and unique factorization for the natural integers, the Gaussian integers, and the Eisenstein and Kleinian numbers. I also liked the discussion of spherical geometry. But after that it became too dense with too much left out or not well explained. So then it became hit or miss on what I really understood. I think that the author assumes some prior familiarity with many of these topics. For instance, in the discussion of Hurwitz quaternions, it is not explicitly stated that the product of two Hurwitz quaternions is again a Hurwitz quaternion and that they form a ring (It is tacit in the proof of the factorization theorem though). Admittedly this is trivial to prove. The Hurwitz quaternions have coefficients that are either all whole integers or all half integers, have integer norms, and have Euclid division with a remainder having a smaller norm. Therefore any left (right) ideal is a left (right) principal ideal. Octonion multiplication is also distributive but is not associative, only alternative which is a weaker property than being associative. Octonions also form a normed division algebra with the norm of a product also equal to the product of the norms. The author extensively discusses factorization of octonion integers, which is a complicated subject. Octonions have a richer structure than quaternions. Like the Hurwitz quaternions, octonions having coefficients that are either all integers or all half integers form a ring, but there are also mixed sets that form rings. It's easy to see for the octonion ring of the first form that it has Euclid division with a remainder having a smaller norm. The additional relations (Z z^-1) z = Z and (Q z^-1) z = Q are needed to modify the corresponding Hurwitz quaternion proof. The author derives many identities involving changing how multiplied factors are to be associated. Hurwitz’s theorem on composition algebras (algebras having the norm of a product equal to the product of the norms) is proved. This explains the existence of the complex numbers, the quaternions, and the octonions with the commutative and associative properties that they have and explains why there are no others. Some knowledge of abstract algebra, number theory, and symmetry and group theory is helpful. For those for whom some topics are new, more detailed explanations would make the going easier. I found helpful the Wikipedia articles "Octonion", "Alternative algebras", and "Cayley–Dickson construction" (“eyes-right” in the author’s Appendix 1 : On Dickson Doubling Rules in Chapter 6 The Composition Algebras). For quaternions, there is an Iowa State University course handout available online titled "Quaternions" by Yan-Bin Jia that gives an introductory treatment. All in all, not easy going but well worth the effort.

While I quickly learned that quaternions are a special variant of complex numbers having one real end three imaginary parts (and octonions have one real end seven imaginary parts), the book then delved into many proofs about them (most of which I could not follow the notation). However no place in the book was I able to find any definition for arithmetic operations on them (which was really what I was looking for when I bought the book, so I could program a quaternion arithmetic library).

A very beautiful little book with some interesting takes, especially the links to algebraic integers and groups. Some statements are quite elliptical and require a good background in mathematics to follow, but generally fascinating for those who like Conway's work.

I love it. There are many fascinating, deep and unique tidbits from John Conway himself.The style is simple and lucid, assuming you are a mathematician.

Candid review by one of the masters of the subject. The text is accessible to undergraduate students, very concise and clearly written.

The authors summarized and expanded upon their referents.therir work is an appreciable advance of the field.

If you want a good laugh take a look at chapter sixin this hastily compiled piece of trash. No one whodoes not already understand the material on octonianswill be able to penetrate this unannotated formulary.The same goes for the entire second half of the book.[The first half is just a rehash of material so familiar that there is no need to see it in print for the N+1 st time. ]The only worthwhile entry here is the reference guidingreaders to John Baez's article on octonians which:a) is available free online andb) actually explains in a reader friendly way the history, math, and applications involved.The authors (not to mention editors) should be ashamedat such a sloppy treatment of this rich and historicallyinteresting episode in mathematics.

This books gives a window into the newer notation in group theory.Sometimes things that are "obvious" to Conway and his co-arthor,just aren't to the rest of us.But in contrast to that he gives concrete examplesof new approaches that are beyond classical Coexterand Cartan type approaches.If you are looking for physics applications to quantum mechanicsfor modern group theory,you might want to try another book,but if you want an idea of what a Moufang loop is or whyoctonions are not associative, then you might like this book.Some time in this century we may even geta chance to understand Freudenthal's metasymplectic geometry?This book for me is sort of a study guide towhat i should try to learn for the future?

I found this book to have been a terrible exposition of the quaternions and the octonions (especially the quaternions). The notation is poor and many symbols are written without adequate and clear explanations. The book goes into virtually no detail at all about the algebraic properties of the quaternions, does not explicitly or formally define its terms (the terms "projectively", "projective group", "rotation", and "reflection", among others, are never defined in a clear and formal way). In a lot of passages, I found myself often trying to guess or read between the lines in order to understand what the authors meant by certain sentences. Moreover, the informal conversational writing style of this book just make it even more obscure and confusing.Overall, I cannot recommend this book anybody, especially for an introduction to the quaternions. It lacks rigour, it lacks formalism, it lacks clarity and it lacks structure. It is, I find, a disastrously poor presentation and the authors simply did not deliver at all.

Another great bok by my favourite mathematician

Wer eine Einführung zum Thema Quaternionen (und Oktonionen) sucht, wird mit diesem Buch nicht unbedingt glücklich. Es setzt eine gewisse Vertrautheit mit Quaternionen voraus (mit komplexen zahlen natürlich selbstverständlich sowieso).Ist dies gegeben und ist man darüberhinaus solide mathematisch vorgebildet, bietet das Buch eine umfassende EInführung in die algebraisch-gruppentheoretischen Eigenschaften und Anwendungen der Quaternionen und insbesondere auch der Oktonionen. Manches hier Dargestellte wird man kaum anderswo finden. Aber es ist nicht alles leicht lesbar, sondern wirklich nur für "Profis".Da es ein recht schmales gebundenes Buch ist( rund 160 Seiten), von den Autoren in TeX gesetzt und nicht einmal fadengeheftet, finde ich den Preis, den A.K. Peters verlangt, allerdings unverschämt hoch.

The book explains what quaternions are. Has proofs. No problem understanding it. It goes also into the symmetries, which explains why some operations disappear.

*****