This is a book about the octonions, a bigger and better version of the complex numbers, albeit with some subtle properties. Bigger, because there are more square roots of $-1$. Better, because an octonionic formalism provides natural explanations for several intriguing results in both mathematics and physics. Subtle, because the rules are more complicated; order matters.

Some readers may be familiar with the quaternions, which lie halfway between the complex numbers and the octonions. Originally developed more than 100 years ago to be the language of electromagnetism, an effort that ultimately lost out to the use of vector analysis ultimately, the quaternions have been reborn as a useful tool for applications as diverse as aeronautical engineering, computer graphics, and robotics. What will the octonions be good for? This author believes that the octonions will ultimately be seen as the key to a unified field theory in physics. But that is a topic for another day, although hints of this vision can be found here.

This book is intended as an introduction to the octonions. It is not a mathematics text; theorems and proofs (and references!) are few and far between. Nonetheless, the presentation is reasonably complete, with most results supported by at least the outline of the underlying computations.

The only true prerequisite for reading this book is the ability to multiply matrices, and a willingness to follow computational arguments. Familiarity with linear algebra is a plus, up to the level of finding eigenvalues and eigenvectors. And of course comfort with the complex numbers is a must, or rather a willingness to become comfortable with them.

The book is divided into three parts. Part I discusses several different number systems, emphasizing the octonions. Part II is the heart of the book, taking a detailed look at a particular collection of symmetry groups, including orthogonal, unitary, symplectic, and Lorentz groups. As we demonstrate, octonions provide the language to describe the so-called exceptional Lie groups. Finally, Part III contains a rather eclectic collection of applications of the octonions, in both mathematics and physics.

A companion website for the book is available at
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