As a science writer, whose only foray into maths has been to cover infinity – by far the sexiest and most intriguing mathematical topic – I am in awe of those who successfully popularize maths.

By comparison, science is easy. We all know from school that science can be dull, but if you go about it the right way, it is naturally fascinating, because it’s about how the universe we live in works. Admittedly maths has plenty of applications, but an awful lot of mathematics is about a universe we don’t live in. It can seem that many mathematicians spend their time doing the equivalent of arguing about the dietary habits of unicorns. Not really a proper job for a grown human being.

Probably the best of the current crop of popular maths writers is Ian Stewart. Certainly the most prolific – I don’t know how he finds the time for his day job. Stewart is decidedly variable in his books. Some of them are pure unicorn territory. I find myself turning page after page thinking ‘So what? I don’t care!’ But every now and then he gets it just right – and this is such an example.

Okay, there are occasional unicorn moments, where I had to skip through a page or two to avoid dropping off (when, for example, he gets altogether too excited about the prospect of constructing a regular 17 sided polygon using only a ruler and a pair of compasses), but they are rare indeed. Stewart takes on some of the greatest problems to face mathematicians through history – even the names are evocative, like Goldbach’s Conjecture and, of course, the Riemann Hypothesis. They sound like a Sherlock Holmes story. And Stewart makes them interesting. Which is truly wonderful.

In part the readability is because of a good smattering of stuff about the people – historical context is never more important than in popular maths – but he also pitches the mathematics itself at just the right level to keep our interest without going into mind-numbing detail, or being too summary. I am very wary of describing any book as a tour-de-force, but this one certainly comes close.

Even though Stewart does not keep things enthralling throughout – the dullest chapter is the one on Fermat’s Last Theorem, which I suspect is because Stewart focuses more on the maths here and less on the people, so excellently covered by Simon Singh – there is plenty in this book to keep the imagination alive. If you hate maths this is not going to make you a convert. But if, like me, you have a grudging admiration for maths but find a lot of it impenetrable or pointless, you should have a great time in Ian Stewart’s capable hands.

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Review by Brian Clegg

There’s been a trend for a couple of years in popular science to produce ‘n greatest ideas’ type books, the written equivalent of those interminable ’50 best musicals’ or ’100 favourite comedy moments’ or whatever shows that certain TV companies churn out. Now it has come to popular maths in the form of Ian Stewart’s 17 Equations that Changed the World.

Stewart is a prolific writer – according to the accompanying bumf he has authored more than 80 books, which is quite an oeuvre. That can’t be bad. He is also a professional mathematician – a maths professor – and that potentially is a problem. The trouble is that, much more so than science, mathematicians are not ordinary people. They get excited about things that really don’t get other people thrilled. And it takes an exceptional mathematician to be able to communicate that enthusiasm without boring the pants off you. It’s notable that the most successful maths populariser ever, Martin Gardner, wasn’t a mathematician.

So how does Ian Stewart do here? Middling well, I’d say. The equations he provides us with are wonderful, fundamental ones that even someone with an interest in science alone, who only sees maths as a means to an end, can see are fascinating. In most cases he throws in quite a lot of back story, historical context to get us interested. So the meat of the book is excellent. But all too often there comes a point in trying to explain the actual equation where he either loses the reader because he is simplifying something to the extent that the explanation isn’t an explanation, or because it’s hard to get excited about it, unless you are a mathematician.

The section on the Schrodinger equation, for example, is presented in such a way that it’s almost impossible to understand what he’s on about, throwing around terms like the Hamiltonian and eigenfunctions without ever giving enough information to follow the description of what is happening. (I also always get really irritated with knot theory, as the first thing mathematicians do is say ‘Let’s join the ends up.’ No, that’s not a knot any more, it’s a twisted or tangled loop. A knot has to be in a piece of string (or rope, or whatever) with free ends.)

Inevitably, to give the book real world interest, many of the equations are from science, and Stewart proves, if anything, better at getting across the science than he is the maths (probably because it is easier to grasp the point). The only section I’d argue a little with is the one on entropy, where he repeatedly says that entropy always increases or stays the same, where it’s more accurate to say that statistically it is very, very likely to do so. But there is always a small chance that purely randomly, say a mixture of gas molecules will partly unmix. (He also uses an unnecessarily complex argument to put down the creationist argument that uses entropy to argue for divine intervention, as it’s easiest to explain that you aren’t dealing with a closed system, something he doesn’t cover.)

Overall, then, I am not sure who will benefit from this book. There’s not enough detail to interest people studying maths or physics at university, but it becomes too obscure in a number of places for the general reader. A good attempt, but would have benefited from having a co-author who isn’t a mathematician and who could say ‘Sorry, Ian, I don’t get that. Let’s do it differently.’ Bring back Simplicio. (One for the Galileo fans.)

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Review by Brian Clegg

When I was a teenager I delighted in Martin Gardner’s books like Mathematical Puzzles and Diversions, taken from his Scientific American columns. British mathematician Ian Stewart has taken over Gardner’s role and continues to amaze and boggle the mind with the possibilities of recreational maths in his latest collection.

For me it was rather a mixed bunch. The best were great fun – the worst would only really engage the sort of person who thinks calculating pi by hand is a form of entertainment. I think to some extent Stewart has a problem because Gardner had already picked off the really entertaining, truly amazing stuff, and Stewart is left with either more of the same, or things that aren’t so engaging. Even so it’s an enjoyable read for anyone who finds mathematical puzzles fun – just be prepared to skip over one or two bits.

In a few of the sections Stewart adopts a story-telling form, and these are the weakest, as he’s not a great fiction writer and the result is too whimsical and irritating. Having said that, his three part story approach to time travel is interesting, if rather limited, but would have been so much better without the H. G. Wells pastiche.

In many of his books, Stewart is excellent at explaining obscure maths to the general reader, but for this one I think he assumes just a bit too much knowledge, and his explanations (for example of the symmetry breaking in animal gaits) can be quite confusing. This was particularly unfortunate in his ‘interrogators fallacy’ section where he tries but fails to explain why some arguments used in trials don’t hold up statistically. This chapter needs totally re-writing.

Despite these concerns, there’s much to interest the recreational maths fan. I was delighted to see a piece on what he refers to as ‘bends’ but are what normal people call knots. He has to do this because it’s a classic case of mathematicians living in their own tiny and often irrelevant worlds – according to the standard mathematical definition, a knot is in an infinitely thin line and both ends of the line are joined up. That is not a knot, guys. But this piece by Stewart deals with the maths of real knots.

A mixed bag, then, but there’s enough really good stuff in here to allow it four stars and to suggest than any recreational maths enthusiasts would be mad not to add a copy to their bookshelves.

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Review by Brian Clegg

Ian Stewart has probably done more than anyone alive to further the cause of popular maths – he’s is simply excellent at putting across mathematical concepts in an approachable way. In this little (though perfectly formed, according to the blurb) book he sets out to challenge your mathematical awareness with a series of puzzles of the sort that tells a little story and leaves you with an enigma at the end.

We can’t give this book more than three stars as it’s not true popular science/maths – but it doesn’t stop it being an excellent collection of brain teasers for anyone who likes having to get the mathematical mental muscles in action – if you are in the habit of doing sudoku, stop it at once, and try something more mind-stretching in Stewart’s delightful little collection. (If you aren’t in the habit of doing sudoku, don’t start, read this book.)

There’s a great tradition of these things stretching back at least to Victorian times. Stewart mentions Dudeney and Sam Lloyd in his introduction, and of course Lewis Carroll was not averse to the odd puzzle. For those, like me, of a certain age, it’s hard to talk about mathematical puzzles and diversions without also paying homage to the great Martin Gardner – but Stewart is a worthy successor. In all there’s 30 challenges here. Chances are they won’t all appeal – there’s something very personal about this sort of thing – but certainly some will.

A couple of the puzzles caused slight concerns. The trouble with using a story setting is once you’ve done it you expect the people in the story to behave like human beings – in one puzzle we have a set of pirates who instead act like perfect calculating automata, where you know they’d really settle the differences with less logic and more blood. And again, there’s one where the answer is actually wrong. Stewart gives us the puzzle of working out exactly when the hands of a clock in the wild west would be directly on top of each other again, after high noon. His answer is to the nearest 1/11th of a second – but the mechanical escapements in use at the time would probably only move the hand every 1 to 2 seconds. That’s the trouble when maths takes on the real world – it doesn’t quite fit.

Don’t be put off though, they’re mostly a delight. Personal favourites were the one about the card game on a train (because it’s the only one I could immediately see the right answer to as I read it, and we all need to feel clever sometimes), and the one set in the house of the “Bleeping Squaresbornes”, because Stewart’s pastiche of the Osborne dialogue was so accurate (Squozzy: “I dunno how many legs I got, Squaron, how am I expected to remember kids?”) that I suspect that Ian Stewart really is Ozzie Osborne. There’s a frightening thought.

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Review by Brian Clegg