As I settled to read this book, my son came running up to me, a look of mock consternation on his face: “Mommy, I have got only NINE fingers!” He proceeded to demonstrate, uncurling each in turn – and indeed, there were only nine fingers. I was called upon to locate the missing finger, which I did. But he counted again, and lo! Only nine fingers! It was all terribly confusing. He then ran off laughing at my befuddlement, shouting “Only joking, Mommy”.

This scenario has been repeated for weeks, ever since he learned about zero – or ‘zewo’ – at nursery. He happily counts forward and backwards, from and to zero – the latter often followed by “Blast-off!” He sings songs about zero, plays tricks on his poor old parents with zero, returns his plate to the kitchen when it has ‘zero food’ on it. If a 4-year-old is so comfortable with the idea of zero, why should anyone go to the trouble of writing a book about it?

Charles Seife clearly thought it worthwhile. Currently a professor of Journalism at Columbia, having written for some of the best-known science publications, he holds a Mathematics degree, and studied with Andrew Wiles, the mathematician who solved Fermat’s Last Theorem (twice!). He has also done research relating to mathematics, so one can assume that he knows what he is talking about – and has something to say.

However, to the modern reader, there seems little point in writing a book on the subject. Zero, nought, nothing, nada, are all part of the common vocabulary. Yet, as Seife points out early in the book, zero is not a common number. Most of us can go through life without thinking about it. Palaces and pyramids have been built without it, inheritances divided, taxes reckoned, routes planned. Apart from the inconveniently empty state of bank accounts at the end of the month, we might never need to think about zero at all. So, why zero, and why is it dangerous, aside from the exorbitant bank charges for letters to tell us we are out of money?

I came to this book with what I considered a fair knowledge of zero. In addition to its mathematical uses, I was aware, for example, that the Romans and Greeks did not use zero, that it was introduced via Islam from India but probably pre-dated these civilisations, that its absence in the early Christian church is the reason for the arguments over the date of the millennium and more recently over the end of the decade – in short, I knew more than most. Seife covers this and more in a mathematics-free introduction. Writing clearly and with occasional flashes of Brysonesque humour, he uncovers the beginnings of human counting in a 30,000-year-old wolf-bone tally, and talks us through the development of written mathematical notation, taking in religion, mysticism, philosophy, the calendar, art, a smidge of comparative linguistics (zero and cipher the same word!), computer programming, astronomy and both classical and quantum physics along the way – including one of the nicest lay explanations of the uncertainty principle I have ever seen. No potential kittens in thought-boxes here – just the difficulty of measuring the length of a pencil (p 170). Amazingly, this is achieved with very few equations – the first appearing on p 95, almost halfway through the book. What little the author cannot explain in words is conveyed through well-chosen diagrams and images, many derived from contemporaneous sources, but even these support rather than elucidate the text.

The part of the book that worried me most was the section on Newton’s development of calculus (pp 114-126). This was clearly signalled in early chapters, adding to the apprehension. I had a niggling memory about strange notations; and while I have had little difficulty in using and understanding calculus, I was concerned about understanding proofs of the topic from first principles. However, I was pleasantly surprised. Not only is Seife’s explanation clarity itself, I actually recognised the notation and the vocabulary of fluxions and fluents, and even Newton’s ‘dirty trick’ of expunging the infinitesimals that allowed him to develop his proofs. I also seem to have learned L’Hôpital’s rule, though not by name: clearly, my maths teacher must have passed on more information than I remember!

The following chapter, Infinity’s Twin, is, I feel, the least successful part of the book, particularly the section on projective geometry and the complex plane. I came unstuck on the discussion of Riemann spheres, unable to visualise the complex rubber universe of his imagination. However, this may yield to a closer examination of the text.

The remainder of the book deals with zero – and infinity – in what is for me the more comfortable domain of physics: thermodynamics, relativity, quantum mechanics – nothing too strenuous. However, here I found one of the great surprises of the book. I have been myopic most of my life, but was not diagnosed until I was twelve because I had developed excellent coping mechanisms, one of which was looking through a ‘pinhole’ created by pinching my fingers together. I had always been aware of faint stripes when looking through this pinhole, for which I vaguely blamed my eyelashes. Thanks to Seife, I now know these stripes are interference!

In sum, I feel I could recommend this book to almost anyone who is interested in the history of mathematics, regardless of his or her subject knowledge. The mathematical treatments are light: there are few equations to scare off the layperson, and even these are offset by some truly outstanding explanations and examples. For those with an appetite for more, there is a wide-ranging selected bibliography to chew over. The appendices, usually the repository of the esoteric and arcane, are amusing little pieces – build your own time machine; why Winston Churchill is a carrot, and so on. Seife’s real gift is telling the stories behind the mathematics: his book is worth reading just for the image of Pythagoras as some crazed antediluvian guru, railing against the infamy of beans.

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