---
product_id: 14349728
title: "On Quaternions and Octonions"
brand: "john horton conway"
price: "€ 173.81"
currency: EUR
in_stock: true
reviews_count: 13
url: https://www.desertcart.it/products/14349728-on-quaternions-and-octonions
store_origin: IT
region: Italy
---

# On Quaternions and Octonions

**Brand:** john horton conway
**Price:** € 173.81
**Availability:** ✅ In Stock

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- **What is this?** On Quaternions and Octonions by john horton conway
- **How much does it cost?** € 173.81 with free shipping
- **Is it available?** Yes, in stock and ready to ship
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## Description

On Quaternions and Octonions

## Images

![On Quaternions and Octonions - Image 1](https://m.media-amazon.com/images/I/61inS0Zr8AL.jpg)

## Customer Reviews

### ⭐⭐⭐⭐ 







  
  
    Much of the later material could have more detailed explanation
  

*by D***P on Reviewed in the United States on January 31, 2020*

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.

### ⭐⭐⭐⭐ 







  
  
    Very mathematically "dense" book.
  

*by G***6 on Reviewed in the United States on January 8, 2019*

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 beautiful book
  

*by D***Y on Reviewed in the United States on May 12, 2014*

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.

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*Product available on Desertcart Italy*
*Store origin: IT*
*Last updated: 2026-03-08*