‘What does it mean, “to do mathematics”?’ asks the author in the opening sentence of this very readable book. The response to this rhetorical question is spread over 200 pages and 10 chapters, in which there is very little use of mathematical symbolism and only one diagram.

The dearth of illustrations naturally means that geometrical ideas do not feature prominently in these ‘stories from mathematics’. Conversely, for the purposes of engaging non-specialist readers in the art of *doing* mathematics, much of the narrative is based upon elementary number theory — or rather, simply stated problems that lead to complex areas of mathematics.

The imagination is stirred by the introduction of ‘interesting numbers’, such as 6 (a perfect number), 7 (a Mersenne prime), 8 and 9 in relation to the Catalan Conjecture, 17 (the Gauss polygon), 1729 (Hardy’s taxi), and so on. But which numbers are uninteresting? What is the smallest uninteresting number? Such is the degree of fun and latent humour in Ziegler’s pedagogical style.

Use and abuse of numerical data is discussed in one short chapter, and the use and misuse of equations in communicating ideas in another. We know that scientific activity is mainly located university research labs or in industrial establishments. But where is mathematics created? Almost anywhere according to chapter 6, but often in the most informal of circumstances.

Who creates mathematics? This question accounts for the significant amount of biographical and historical observations contained in the last three chapters. Mathematicians are seen as being all too human, and often quite idiosyncratic. For instance, we read of the nomadic Paul Erdős, the disappearance of Grothendieck and the asocial Perelman. Not just their strange worldly ways, but their wonderful achievements also.

This very enjoyable book is informative on so many levels for specialists and non-specialists alike.

Peter Ruane is not nomadic, has not (yet) disappeared, and he thinks he is not asocial. Indeed, he can be contacted at ruane.hp@blueyonder.co.uk.