Review of "Dual Numbers," by W. B. Vasantha Kandasamy and Florentin Smarandache

Review of

Dual Numbers, by W. B. Vasantha Kandasamy and Florentin Smarandache, Zip Publishing, 2012. 159 pp., available for download at 

http://fs.gallup.unm.edu/eBooks-otherformats.htm. ISBN 9781599731841.

A dual number is defined as a nonzero number g such that g² = 0. This of course has the immediate consequence that gⁿ = 0 for all n ≥ 2. At first glance such a number seems unlikely until you limit the elements of discourse to integers modulo an integer. For example, if you let
g = 5 and restrict the numbers to Z₂₅, then g² = 0.

 The general tactic is to create a set of elements having a form like a + bg, where a, b ɛ Z and
g = 5 where the discourse is restricted to Z₂₅. In other cases, for example in Z₉, the two numbers 3 and 6 can be used as a g element. While the addition of elements of this form is not that much different, the squaring of the g element yielding zero significantly changes the results of the multiplication operation.

 With these conditions, algebraic systems can be constructed that are semigroups, rings, semifields and other algebraic structures. Using the a + bg forms to create polynomials, vectors and matrices significantly expands the use of dual numbers and all of these consequences are examined in this book. While some background in linear and abstract algebra is necessary to understand the material, a great deal of the proofs are based on fundamental verification of the properties of the particular algebraic structure. Many of the concepts and problems in this book will make excellent basic exercises in a course in abstract algebra.

 The book closes with a set of 116 problems over the material in the book. No solutions or hints of solutions are given.