Numbers: A Very Short Introduction (Very Short Introductions) [Paperback]

Fantastic! I was looking for something that didn't just talk about maths, but really explained it - and this was perfect. Peter Higgins writes in such an accessible style that the reader is carefully guided through the maths in a way that leaves you informed and entertained. For example, the chapter on cryptography was a revelation. For a "very short introduction" you come away with enough knowledge to impress even your most scientific friends - a great buy.

[Update to review:

I fear that this long (and in places technical) review may give the impression that the book itself is a somewhat heavy and technical, which it isn't. If this review is too wordy or convoluted, be assured that the book does not suffer from the same faults. A glance at the Amazon preview facility should reassure any nervous prospective reader that the book is in fact friendly and inviting. In the (slightly edited) review that follows, I have necessarily concentrated on points which gave me some difficulty, which may give a skewed impression of the book as a whole.

If that's not enough, let me add that, having not been able to so much as look at a book on mathematics for nearly two years, I found that this book very gently and unthreateningly got me interested in reading and doing mathematics again.

It is not at all in the class of apparently 'popular', but actually difficult and technical, books such as Roger Penrose's huge and intimidating "The Road to Reality: A Complete Guide to the Laws of the Universe" (probably very readable and interesting, but only with considerable effort and time - you actually have to do tensor calculus, and so on!), or Avner Ash and Robert Gross's "Fearless Symmetry: Exposing the Hidden Patterns of Numbers" (which I think is utterly incomprehensible, for anyone other than Andrew Wiles himself!).]

This would be a good book to give someone who has already shown some interest in the subject, and might be thinking of studying a mathematical subject at university. On the other hand, the proverbial intelligent general reader, nervous of mathematics, and wanting only to be told something about it, rather than actually having to do it, might find their commitment challenged. The author has clearly thought about this potential problem, and in order not to put off the second kind of reader, has not set any explicit exercises, but instead scatters the text with occasional remarks like "an elementary calculation shows that ...", implicitly inviting the first kind of reader to pick up his or her pen and see for themselves, and the second kind of reader to take the conclusion on trust.

Having had a mathematical education, but also being old and rusty, I fell in between these two kinds of reader. I found the book to be a very pleasant invitation to renew my acquaintance with numbers, and to try out some forgotten skills.

Other readers who already know some mathematics may, like me, find some things here which are new to them. New to me were: in Chapter 1, the algorithms for testing numbers in decimal notation for divisibility by primes other than 2 and 5 (in fact, in spite of the impression given on page 14, there are algorithms of this type for all candidate divisors which are not multiples of 2 or 5, and the usual test for divisibility by 9 is a special case of this); in Chapter 3, the conjecture (due to Catalan) about aliquot sequences, and the definition of semiperfect numbers; in Chapter 6, the 'greedy' algorithm for expressing fractions as sums of reciprocals of positive integers; and in Chapter 7, the palindromic property of representations of irrational square roots as continued fractions.

The discussion of public key cryptography in Chapter 4, although not new to me, was a very good refresher, and would probably be a good "very short" introduction to the subject. The author seems to have taken particular care over it. (This would also have been a good place to insert an explanation of the divisibility algorithms from Chapter 1, because the necessary number theory is developed here. He already takes the opportunity to fill in a gap in the explanation of unique factorization into primes in Chapter 2.)

Criticisms:

Only the last of the eight chapters does not seem to me to be successful. It feels rushed, as if the author was obliged to make ruthless cuts in order to fit within the constraints of the Very Short Introductions format. I'm sure he would have liked to be more expansive. This was the one chapter where I had no opportunity to get out pen and paper and calculate.

There were a number of typographical errors (as is apparently usual with the VSI series), but all without exception were minor: there were none of those maddening ones which sometimes garble equations, and can cause real confusion. (The only misprinted equation or formula, I think, was "119 = 7 x 19" on page 17, and that will hardly confuse anybody!)

There was also what appeared to be one logical error (as opposed to a typographical error), in Chapter 7, in the section headed "Cantor's Middle Third Set". He correctly describes this set, C, as being "nowhere dense", but he doesn't explain what this term means, and I had to remind myself from elsewhere. In a non-technical book, that's fair enough, even if slightly confusing for the reader, who must imagine that he's missing something. What's worse, unless I'm simply misreading the text, is that he does /seem/ to go on to give an explanation, only, what he actually describes is the property that C is closed (which indeed it is), not that it is nowhere dense. In fact, given that C is closed, the property that it is nowhere dense is equivalent to the property that its complement is dense, i.e. the set C is "riddled with holes", wherever you look, and however closely you look. This could very easily have been explained, and even proved, geometrically, using his very clear construction (with accompanying diagram) of the set C as the intersection of a nest of closed sets, the {n}th of which consists of 2^n disjoint closed intervals, each of length 3^{-n}. (Equivalently, the complement of C is a union of infinitely many pairwise disjoint open intervals, which obviously 'almost fill up' the unit interval, i.e. the complement of C is dense, as well as open.) For this reader, at least, such an explanation would also have been easier to understand than the ones given in terms of infinite ternary expansions, which I must admit tend to make my eyes glaze over.

The text seems to hint at an unnecessarily complicated proof of the easy half of Wilson's Theorem in Chapter 2, on page 24 (but even then it is still straightforward, and I may have misread the hint).

I'm not at all sure that the repeated statement in Chapter 6 (on pages 76-7, and again on page 83) that the ancient Greeks had a conception of irrational /number/ (as opposed to irrational ratios of /magnitudes/) is correct. This is a scholarly question, on which I'm not qualified to pronounce, and the author may well be right. Still, in a book devoted to the conception of number, this interesting conceptual question could probably at least have been mentioned explicitly (especially as at least one entire book, Hart's "Multidimensional Analysis", has been devoted to the consequences of distinguishing between numbers and dimensioned quantities).

On the whole, this is a very enjoyable, and even quite easy, read, whatever your level of interest in mathematics. It does make demands on the reader's concentration, and some of the explanations are quite condensed (although also clear and illuminating). This makes me wonder what a reader with little mathematical training or self-confidence would make of it, but I would still recommend even a reader nervous of mathematics to give it a try. I, for one, was nervous when I started, but not when I finished!

.... unless said general reader also has quite a substantial background in modern mathematics. You have to be prepared to cope with a lot of algebraic manipulations, and facts about numbers which are mainly of interest to mathematicians. I wasn't quite sure who would benefit from this book, but for their sake I give it three stars.

The VSI series is impressive, but the accessibility does seem to vary. The one on the elements I remember enjoying despite my lack of knowledge of chemistry, but the one on the laws of thermodynamics was in the same category as this one. I wonder what remit is given to the authors? I also wonder who acts as reader - obviously the specialist knowledge must be checked, but perhaps OUP should consider using someone not expert in the subject to advise also.