A quaternion takes the form *ai+bj+ck+d, *with *a,b,c,d *real numbers and where *i, j, *and *k *each square to –1. The quaternions are a four-dimensional analog of the complex numbers, but they are not commutative: *ij = k = –ji. *With this multiplication, the quaternions form a division ring. Jumping from complex numbers to quaternions essentially requires negotiating two complications: non-commutativity and keeping track of the extra dimensions. Morais, Georgiev, and Sprößig’s *Real Quaternionic Calculus Handbook *works through the fundamental properties and formulas necessary for working with quaternions.

The first chapters cover elementary properties of quaternions and their various representations. The remaining chapters take fundamental mathematical constructions and develop quaternion analogs. This includes sequences and series, transcendental functions, matrices, and polynomials. The focus is computational and most proofs amount to elementary arguments with algebraic manipulations.

Aside from some brief introductory commentary before each chapter, there is little motivation. The applications of quaternions to geometry and physics are only broadly mentioned. The narrative is all business, with definitions up front followed by derivations of results. Many sections consume less than a page. The writing is terse but mostly lucid, but the book could use a more thorough index and/or a notation guide.

Although the material is elementary and accessible with just some calculus and linear algebra, it would be a tough slog without some external goal in mind. As such, I see this more as a practitioner’s or future-practitioner's handbook rather than a way to generate excitement about the topic. However, there are lots of exercises throughout (with solutions), and it could be used as a text for an early graduate or advanced undergraduate course. I like it best as a self-study guide; anyone mathematically mature enough to have a reason to study quaternions carefully will likely have little trouble with the math anyway, and this book is great for skipping around, picking your spots, and working through a bunch of examples and practice problems.

And if by chance you are writing a test suite for quaternion computation in a computer algebra system, this is definitely the book for you.

*Bill Wood (Twitter: @MathProfBill) is an assistant professor of mathematics at the **University** of **Northern Iowa**. He once briefly worked as a kernel tester for Mathematica. This is his first review for which that seemed worth mentioning.*