## Non-Fiction Reviews

## The Spirit of Mathematics

Algebra and all that

(2023) David Acheson, Oxford University Press, £14.99 / US$17.95, hrdbk, 178pp, ISBN 978-0-192-845508-6

There is a long tradition of writing books about recreational mathematics. See, for example, W. W. Rouse Ball’s classic

Mathematical Recreations & Essaysthat was first published in 1892 and currently in its 13th edition.The Spirit of Mathematicsis one of the most recent books in this genre, with the added bonus that it deals withhow to thinkthrough problems and simplify them.David Acheson’s objective is to take the reader through a wide variety of mathematical topics in a light-hearted informal way that is not overwhelming. He has included a plenitude of clever tricks in this book, which should appeal to anyone who likes to solve puzzles. The demands on the reader are simple: a willingness to think; and some recollection of basic school-level mathematics.

The book does of course contain equations, but their presence is frugal. One drawback to this approach is that sometimes clarity is sacrificed for conciseness. In chapter 16, The Puzzled Farmer, for example, the reader is referred to chapter 12 for explanation of a derivation. Chapter 12 then suggests that you look at chapter 11. This would be acceptable in a text book, but not in a book of this nature.

Acheson covers a lot of ground. The book is arranged into twenty nine chapters, averaging about five pages each. Most of the chapters focus on a single topic, and each topic is presented by way of problems and examples. This is exactly the right approach for a layman audience. Perhaps the most important point in this book is that you don't need to solve everything in order to answer a question. Chapter two for example presents three people who love to fill baths. Weirdly, they only work in pairs, and they each pour water at their own favoured rate. The conundrum is to work out how long it would take them if all three worked together? This can be solved using three simultaneous equations in three variables, but Acheson says don’t lose sight of the question. All you need to do is calculate the total of their three rates, then take the reciprocal. His rationale is explained, as is the answer, which is… well, go read the book!

There are numerous tips like this, which are well known by mathematicians, but are probably largely unknown to laymen. For example, chapter 11 presents one of the simple algebraic proofs of Pythagoras’ theorem. It uses the identity

(a+b)² = a² + b² + 2ab, and a simple geometric construction to prove the theorem -simple that is compared with the horribly complex proof in Euclid’s elements. The lesson here is that there is always more than one way to visualise a problem.Another good example is described in chapter 24. It details how Karl Gauss only took a minute or two to answer his teacher, when his class was asked to add up all the numbers from 1 to 100. Gauss's trick was to chalk the sum down twice like this:

1 + 2 + … + 100

100 + 99 + … + 1There are 100 columns, each of which adds up to 101, so grand total =

100 x 101 = 10,100But that includes all of the numbers from 1 to 100

twice, so the answer is10,100/2 = 5,050The beauty of Gause’s trick was that whilst his class mates slogged away at ninety nine separate additions, he thought about the problem in a new way that only needed one addition, one multiplication and one division. He was just only 7 years old at the time!

One slight criticism of

The Spirit of Mathematicsis that it only presents traditional topics. It could have reached a wider audience if some modern-day topics had been included. For instance: how advertisers arrive at claims like90% of dentists use our toothpaste; that often misused clichéwin-win situation; and government statements like:Inflation is currently 10% but the rate at which it is increasing is falling.In summary, this book is both interesting and entertaining, and it should appeal to any numerate person who has a casual interest in puzzles or mathematics. Also, most of it shouldn’t be too scary for those readers who claim to have never understood algebra at school. Indeed, it might even help them overcome their fear.

John Watkinson BSc (Hons) Mathematics

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