The Infinite Book by John D Barrow

At first glance, John Barrow's latest book looks like a derivative attempt to cash in on the success of such recent books as Peter Beckmann's A History of Pi and his own The Book of Nothing. But don't be fooled. Whatever the author's motive, the end product is a delight. I have read all Barrow's books, and this is the one I have enjoyed the most. Others are more erudite, or more comprehensive, or superior in their own special ways. But this one is definitely the most readable, and the most fun.

Barrow ought to know a bit about infinity, since he is a Professor of Mathematical Sciences at Cambridge. The conceit around which the book is constructed is to look at the nature of infinity in many different guises. There's the mathematicians' infinity, where an infinite number of decimal numbers exist between any two decimal numbers you choose; the physicists' infinity, where the idea of infinite space is counterbalanced by the idea of infinite subdivision of matter; the philosophers' infinity, where everything recurs infinitely many times and in infinitely many places in an infinite universe; and more besides.

One of my favourites (an old chestnut to mathematicians, but probably a new delight to most of Barrow's readers) is the Hotel Infinity. Even though all of its infinite number of rooms are occupied, space can always be found for an infinite number of new guests, and then another infinite number of new guests, and so on for ever. There's the idea that some infinities are bigger than other infinities, an explanation of why the sky is dark at night, and some serious stuff suggesting there must be something fundamentally wrong with theories of physics (including Einstein's general theory of relativity) that allow for the existence of so-called singularities inside black holes and at the birth of the universe.

When he wants to, Barrow has a light touch, and he wants to throughout this book. He offers a proof that the existence of non-zero interest rates is evidence that time travel does not happen, discusses the possibility of carrying out an infinite number of tasks in a finite time, and speculates that our universe may have the peculiar properties it has because it has been created as an experiment by a slightly more advanced civilisation than our own. Or maybe our universe is a computer simulation running in someone else's universe - which could explain why the laws of physics don't always quite make sense, as a simulation is never going to be quite perfect. In other words, "god" may be a supergeek with an advanced laptop and no real friends.

The same light touch is applied to topics that have formed the subject matter of whole books, with such success that it makes you wonder whether less might not really be more when it comes to putting across deep scientific ideas. String theory is a breeze: just two pages to explain what many physicists believe to be the fundamental structure of matter. Cosmology is a bit harder - three pages plus half a paragraph. Special relativity, in a nod to the Einstein centenary, has three and a half pages. If ever proof were needed of the validity of the old adage"keep it simple", this is it.

You may have noticed, though, that none of this is new. That is not the object of the exercise, nor does it matter. Rather than a boring re-hash of tired old ideas, Barrow offers a fresh view of things you may have half-understood, seen from a different perspective.

Of course, it isn't perfect; nothing is. The biggest flaw with The Infinite Book is the same thing that infuriates me about all of Barrow's books. For some bizarre reason, he feels a need to throw in every remotely relevant quotation he possibly can. It's like an insecure undergraduate trying to impress his tutor with how much he has read, and breaks up the flow of the story. In the words of Ralph Waldo Emerson, "I hate quotations, tell me what you know".

As faults go, this one has the great advantage that it can be completely ignored by skipping all the quotes and concentrating on the real stuff, from Barrow himself. Popular science doesn't come much better than this.

John Gribbin's latest book is 'Deep Simplicity' (Penguin)