Publisher: Princeton University Press
Audience: Mathematicians, scientists, and engineers
Reviewer: Mike James
Understanding how numbers work is essential for any scientist or engineer. The irrationals are the most confusing and fascinating type of number, so a book that might make things seem easier is worth considering.
The subtitle of this book, A story of the numbers you can't count on, suggests, to me at least, that issues such as countability and orders of infinity might be at the heart of this book and this might make it particularly interesting to the programmer and computer scientist.
Unfortunately this isn't really the case. The book is more concerned with explaining in part the history of the irrationals and about proof that particular numbers are irrational and transcendental. This is a mathematician's view of irrational numbers and there isn't much here for the general reader' even if they have a technical competence in a mathematical subject.
Not only is the book math-oriented it rarely manages to say anything in a simple way that a non-math expert could decode. This starts right at the beginning in the introduction we have a "novel" definition of the irrationals -
"The set of all read numbers having different distances from all rational numbers"
After you have spend some time figuring out what this means, it is a "nice" definition and if you already know some things about the irrationals it does make you think. Howeve,r this in the introduction where you might hope for a gentle guide to the irrationals and what is coming next, not a clever novel definition that is remarkably subtle.
From here we have a history of the irrationals as those numbers that upset Pythagoras. The problem here is that the chapter is great but if you haven't a firm grasp on irrational numbers a lot of it will simply make no sense. The next few chapters follow the history up to Fermat and then go on to the use of continued fractions to explore irrationals, mostly pi and e.
By Chapter 6, which starts to consider the transcendentals, we have spent a lot of time dealing with how irrationals occur, what problems they cause, and proving that particular numbers are irrationals, but the reader hasn't been offered a good definition of an irrational, let alone any deep insights into what makes an irrational different from a rational. Yes, there is a definition that says that an irrational is not a ratio of two integers and its decimal expansion never repeats or terminates, but these aren't deep insights and they are given almost in passing. I suppose my complaint is that there is no physics or philosophy in the discussion.
In the chapter on transcendentals we once again have no real discussion of what a transcendental number is. The chapter just continues from the previous chapter with a particular problem. Chapter 7 goes on to show that pi and e are transcendental, but again the reader isn't provided with much of a clue as to what this might mean.
Chapter 8 returns to continued fractions to prove that the golden ratio is the most irrational of numbers - again more speculation on the ideas would have been welcome. Chapter 9 deals with the issue of the randomness of the decimal expansion of the irrationals. - this about the only chapter, and it is very short, that a non-mathematician is likely to get something out of. Chapter 10 finally gets to the modern theory of the irrationals, and we meet the three rigorous theories of the irrationals - Weierstrass-Heine, Cantor-Heine-Melray and Dedekind.
The final chapter is on the question "does irrationality matter?" In many ways this could have been the most interesting chapter for the general reader - after all there are many who question the whole idea of the continuum as a physical thing. But this is a math book, so any questions of the "reality" of irrationals is avoided. Whether the chapter concludes that irrationals matter or not isn't clear to me, even after a number of readings, but given that they are the subject of the book I suppose the verdict must be that they are.
The final chapter also illustrates another problem with trying to read the book - the chapters don't really address the subjects of their titles at all directly. The chapter about "do the irrationals matter" starts off with no hint that this is the question being discussed. It goes off at a tangent and discusses tables, then a collection of problems, the approximation of pi and so on. Never does it directly address the question, or discuss it, or explain what the examples have to do with it. This is fairly typical of the rest of the chapters, where it can be difficult to discover what the "plan of attack" is.
I am sad to say that I cannot recommend this book to a general audience, In places it is a very difficult read, and I'm not even sure about recommending it to the beginning mathematician unless they are interested in this particular branch of number theory and already know a lot of the background.