Maths 1001: absolutely everything you need to know about mathematics in 1001 bite-sized explanations

By Richard Elwes

There are many places where this book could live very happily: on the shelf in with the textbooks, on a coffee table, by the bed, and if it wasn't a little heavy to hold up, it would also make good reading in the bath. The reason for its versatility is that it's a mixture between an encyclopaedia and a popular maths book. In some sense, it's a plain English encyclopaedia of maths, embellished with some examples for entertainment. So whether you're trying to get at the "true" meaning of something textbooks only define using passionless symbols, or are looking for a little diversion before going to sleep, this book can give you both.

Its author, Richard Elwes, is an old friend of Plus. He won the Plus new writers award in 2006 and has since written us several other articles, including his latest, Exotic spheres, or why 4-dimensional space is a crazy place. In these articles he writes about really hard mathematical concepts with amazing simplicity. He knows exactly what, and how much, you need to know to intuit the meaning of these ideas, and he knows just how to talk about them in ordinary English. The ability to harness intuition and strip away technicalities, while still remaining correct, is what makes a great popular maths writer. And in Maths 1001 Elwes has put this talent to good use.

The scope of the book is pretty monumental. Its chapters cover basic number theory, geometry, algebra, analysis, logic, discrete maths and probability and statistics. In addition there are chapters on mathematical physics, metamathematics and on games and recreational maths. Each chapter is made up of explanations that are literally bite-sized, two or three paragraphs at the most. Yet, between them, these bits cover a surprisingly wide range of maths, starting with the very basics in each chapter and ending with some very advanced stuff. The chapter on numbers, for example, starts with addition and ends with the Riemann hypothesis. You'll certainly find all the maths that's covered at school in this book, and it touches on pretty much everything you'll come across in a maths degree.

With so little room to play with, Elwes' talent for conciseness comes in very handy. While basic concepts, say that of a group or the idea of a limit, are defined rigorously, he does not resort to mere symbols. Using words, Elwes adds the meaning and some of the context that in ordinary textbooks often go missing. Thus, each chapter tells a story, with the harder ideas at the end following on quite naturally from the basics. Personally I enjoyed the sections exploring the very basics of an area, say number systems or the elementary laws of logic, because it's rare to find them explored in such a concise way.

The exhaustive approach also has its downsides of course. There's little room for examples or historical context. Elwes takes an unashamedly mathematical approach in which maths, rather than real-life problems, motivates maths. The idea of a dynamical system, for example isn't introduced in terms of planets orbiting the sun or rabbits spawning more rabbits, but in terms of the iteration of mathematical functions. If you like your maths to be all about puzzles, games, or practical problems, you might feel a little cheated. I for one found this approach refreshing, as I get a little tired of the "honest, it's useful and/or fun" approach to popular maths.

Having said all this, the book is much more, in terms of entertainment value, than a mere informal reference guide. Elwes and his editors have still found some room for diversions, for example we learn about the mathematician John Conway and his wife romantically quoting digits of pi at each other, and about Lewis Carrol's froggy diversions into logic. There is also the chapter on recreational maths, which collects some bits that didn't make it into the rest of the book, for example the golden section. More importantly, to my mind, Elwes' concise style, distilling the essence of ideas without getting bogged down in technicalities, brings across the beauty of the ideas and of the mathematical journey that gave rise to them. If it's the development of ideas that you're interested in, then you'll like the chapter on metamathematics, which explores mathematical reasoning and its limits.

So who is this book for? If you're a student of maths at any level, then it makes a great companion, giving you a broad overview and slightly friendlier perspective on things than an ordinary textbook. If you're just interested in maths, then it provides a quick way to get your grounding in the different areas and an idea of the kind of questions that occupy research mathematicians. It's amazing just how far Elwes' accessible explanations can take you. It's not a cover to cover read, rather something to dip in and out of, skipping any bits you find boring. I for one will be keeping this book handy because I'm sure I'll be using it — and that's not something I can say about many of the books I review.

Book details:

Maths 1001

Richard Elwes

softback — 400 pages (2010)

Quercus Publishing Plc

ISBN: 978-1848660632