The Square Root of 2: A Dialogue Concerning a Number and a Sequence [Hardcover]

An outstanding book by a master teacher, but with serious editing deficiencies.

This is a relatively short and interesting book covering some substantial topics in mathematics. For the most part, it can be read without the need for pencil and paper. It is only five chapters long, and although easy to read, it does require concentration.

This book is written as a discussion between a master teacher and an interested student. The teacher's words are shown in a boldface font, and the student's in an indented normal font. The selection of topics, the book's organization, and the dialog between teacher and student, help guide the reader in the appropriate direction. The book is clearly the work of an excellent teacher.

The first chapter quickly displays the strengths and weaknesses of this book. A major strength is the author's ability to grab and hold your attention, making this book like a well-written mystery story, setting the stage to draw you in and to stay until a solution is revealed. It also displays the book's major weakness, poor editing.

Unfortunately, there are numerous grammatical and substantive errors, some particularly serious for the reader new to the mathematics presented here. These appear as early as the first chapter. Some examples are: On page 11 the term "perfect number" is used incorrectly when "perfect square" is meant. On page 24 the word "is" is left out of a sentence. The illustration on page 23 represents the location of the "square root of two" differently, and incorrectly, as compared to the text discussion. In the sequence of square roots on page 30, the square root of six is inappropriately missing.

Chapter 1. "Asking the Right Questions" shows how the square root of two can arise in the simplest of contexts, as the diagonal of a unit square (i.e., a square one unit on each side). It goes on to show how it is possible to get closer and closer to this square root's true value using integer fractions, but its does not yet prove that this value cannot be exactly represented this way.

Chapter 2 introduces us to the proof of the irrationality of the square root of two and its consequences. After first presenting the proof in English, Dr Flannery shows how it can be concisely presented in mathematical notation. This Chapter explains the connection between the square root of two and the European A-Series paper sizes. It touches on Pell numbers as well as decimal expansions. The term "mixed decimal" as described in this Chapter is incorrect.

Chapter 3, using more algebra than earlier, extends the previous material. Considering that the author assumes minimal mathematical sophistication from the reader, even explaining the term inverse, the material on pages 83-94 seems inappropriately demanding. That material would clearly benefit from a gentler presentation.

The final two Chapters, 4 and 5, present some additional mathematical odds and ends, including the continued fraction expansion to approximate the square root of two, and some concepts connected to Gauss and Ramanujan.

In summary, if the seriously deficient editing, the occasionally inappropriate definitions, and the slightly roller coaster requirements for mathematical maturity were corrected, this book could serve as an exemplar of the best teaching methods, i.e., focused questions that direct the student to find and confirm the right answers.

There have been plenty of mathematics books written about specific numbers, like pi, zero, e, the golden ratio, etc. All of these books typically go into a detailed the history of the number, give examples of where the number shows up, and how work with the number has affected the world. I bought this book because I was expecting it to show the square root of 2 along those same lines. Unfortunately, that wasn't the case at all.

This book is written in the form of a continuous dialogue between a teacher and student. Throughout the book, they play around with different equations and uncommon series related to the square root of two. They also not only tackle a few proofs of the number's irrationality, but discuss the nature of proof itself. The only real world examples they show are the diagonal of a square (and as usual this leads into some discussion of the ancient Greeks and the discovery of irrational numbers in general), and how the square root of 2 relates to the business standard A4 paper and why it was created (a great example that I wasn't aware of).

Again, this wasn't what I was expecting at all. But I'll give it 4 stars because it is a nice introduction to mathematical thinking, and how mathematicians go about examining and solving problems, and the reasoning behind the approaches.

This is a charming book with lots of information (see other reviews) presented in the form of a clever teacher/student dialog. However there are serious mathematical and typographical errors. For example, pi is defined to be "the ratio of the circumference of a circle to its diagonal." Flannery meant diameter; circles don't have diagonals. In another place he interchanges the numbers 48 and 84 to get the incorrect result that 36/48 = 3/7. The problem with these errors is that people read this book to get information, and the information needs to be correct.