A Mathematical Tonic

NATURE|Vol 443|14 September 2006 BOOKS & ARTS

reader nonplussed. For instance, he discusses a problem in which a mouse runs at constant A mathematical tonic speed round a circle, and a cat, starting at the centre of the circle, runs at constant speed Dr Euler’s Fabulous Formula: Cures Many mathematician. This is mostly a good thing: directly towards the mouse at all times. Does Mathematical Ills not many mathematicians know much about the cat catch the mouse? Yes, if, and only if, it by Paul Nahin signal processing or the importance of complex runs faster than the mouse. In the middle of Princeton University Press: 2006. 404 pp. numbers to engineers, and it is fascinating to the discussion Nahin suddenly abandons the $29.95, £18.95 find out about these subjects. Furthermore, it mathematics and puts a discrete approximation is likely that a mathematician would have got into his computer to generate some diagrams. Timothy Gowers hung up on details that Nahin cheerfully, and His justification is that an analytic formula for One of the effects of the remarkable success rightly, ignores. However, his background has the cat’s position is hard to obtain, but actu- of Stephen Hawking’s A Brief History Of Time the occasional disadvantage as well, such as ally the problem can be neatly solved without (Bantam, 1988) was a burgeoning of the mar- when he discusses the matrix formulation of such a formula (although it is good to have ket for popular science writing. It did not take de Moivre’s theorem, the fundamental result his diagrams as well). long for mathematics to get in on the A question Nahin does not engage act, especially when Andrew Wiles with is whether complex numbers captured the public imagination by are indispensable to his arguments. proving Fermat’s last theorem. Now Such is his enthusiasm for complex IMAGES AKG scarcely a month goes by without numbers that he writes as though a new popular mathematics book. they always are. But there is a big Some are about famous unsolved difference between an argument problems, others are about notable that uses both addition and multi- numbers such as zero, Ț or e, and still plication of complex numbers and others, including Dr Euler’s Fabulous one that uses just addition (in which Formula by Paul Nahin, are about case the complex numbers are just important equations. a convenient notation for the real Mathematics is not easy to explain, plane). For example, he uses them and many authors writing popular in his analysis of the cat-and-mouse mathematics go out of their way not to problem above, but he needn’t have lose their readers, perhaps bearing in done. Elsewhere he explicitly says mind Hawking’s famous dictum that that he has used complex numbers in each equation included in a book will a fundamental way, when he hasn’t. halve the sales. But what if, like most For instance, to demonstrate that a readers of Nature, you already know certain quadratic equation has no some mathematics? Then a book that real solution, he points out that the avoids mathematical symbolism is familiar formula for the roots shows likely to be frustratingly vague. For- that they are both complex. This is tunately, however, among the books not a true use of complex numbers, that have appeared recently, there are because to derive the formula for the some that might be called ‘semi-pop- roots one must complete the square, ular’: they are still rather informal but and then the true reason for the lack are aimed at a more mathematically of real roots emerges: the square of a sophisticated audience. Dr Euler’s real number is never negative. Fabulous Formula is a welcome addi- On the whole these defects are not tion to this category. serious, however. Indeed, mathema- The hero of the book is Leonhard Leonhard Euler linked some fundamental mathematical constants. ticians may well find them thought- Euler (1707–83), who was one of the provoking and others will not notice most prolific mathematicians ever, continuing that (cosθ+isinθ)nǃcos(nθ)+isin(nθ). This them. On the positive side, Nahin includes to publish large numbers of papers even after is a one-line consequence of the addition gems from all over mathematics, ranging going blind in 1766. The “fabulous formula” of formulae for sines and cosines, but his proof from engineering applications to beautiful the title is eiπǃǁ1, together with the more gen- takes up an incredible six pages and uses, quite pure-mathematical identities. Most of his eral eiθǃcosθ+isinθ. These equations astonish unnecessarily, the Cayley–Hamilton theorem, topics lie just beyond the periphery of a typi- each new generation of mathematicians that which he proves in two dimensions by brute cal mathematics course: they are facts, such as encounters them: without generous helpings force. Even then the proof is not complete: at the irrationality of π, that you may have heard of hindsight, who would ever have suspected one point he says he will “refer you to any good of but never had explained in detail. It would there was such a succinct equation involving book on linear algebra for a formal proof”. The be good to have more books like this. ■ e, i and π? Nahin is not particularly concerned less experienced reader will emerge from these Timothy Gowers is at the Centre for with the philosophical problems associated pages thinking that de Moivre’s theorem is a Mathematical Sciences, University of Cambridge, with complex numbers, which are discussed dauntingly difficult result, which is a pity, as Wilberforce Road, Cambridge CB3 0WB, UK. in his earlier, less technical book An Imaginary it isn’t. ǁ Tale: the Story of √ 1 (Princeton University Sometimes Nahin reveals his engineer’s Correction Press, 1998). Rather, he takes Euler’s formula instincts by dispensing with formal proof In the review of the book Soils and Societies: as his starting point and demonstrates the wide and using numerical simulations instead. In Perspectives from Environmental Histories (Nature range of uses to which it has been put. places this works well — if a result is plausible 442, 985; 2006), the name of one of the editors As an engineer, Nahin’s perspective on the a proof isn’t always needed in a book like this was misspelled. The editors were J. R. McNeill formula is different from that of a typical — but sometimes it can leave a mathematical and Verena Winiwarter.