Makers of Mathematics S. Hollingdale (Penguin, 1989)
There are not many books on the history of mathematics which are pitched at a suitable level. Hollingdale gives a biographical approach which is both readable and mathematical. You might also try E.T. Bell Men of Mathematics (Touchstone Books, Simon and Schuster, 1986). Historians of mathematics have a lot to say about this (very little of it complimentary) but it is full of good stories which have inspired generations of mathematicians.
Alan Turing, the Enigma A. Hodges (Vintage, 1992)
A great biography of Alan Turing, a pioneer of modern computing. The title has a double meaning: the man was an enigma, committing suicide in 1954 by eating a poisoned apple, and the German code that he was instrumental in cracking was generated by the Enigma machine. The book is largely nonmathematical, but there are no holds barred when it comes to describing his major achievement, now called a Turing machine, with which he demonstrated that a famous conjecture by Hilbert is false.
The Man Who Knew Infinity R. Kanigel (Abacus, 1992)
The life of Ramanujan, the self-taught mathematical prodigy from a village near Madras. He sent Hardy samples of his work from India, which included rediscoveries of theorems already well known in the West and other results which completely baffled Hardy. Some of his estimates for the number of ways a large integer can be expressed as the sum of integers are extraordinarily accurate, but seem to have been plucked out of thin air.
Fermat’s Last Theorem Simon Singh
You must read this story of Andrew Wiles’s proof of Fermat’s Last Theorem, including all sorts of mathematical ideas and anecdotes; there is no better introduction to the world of research mathematics. You must also see the associated BBC Horizon documentary if you get the chance. Singh’s later The Code Book (Fourth Estate) is not so interesting mathematically, but is still a very good read.
The Colossal Book of Mathematics M. Gardner (Norton 2004)
Over 700 pages of Gardner for under 20 pounds is an astonishing bargain. You will be hooked by the very first topic in the book if you haven’t seen it before (and probably even if you have): a diophantine problem involving a monkey and some coconuts — can’t say more without writing a spoiler. At the beginning, about 60 other books by Martin Gardner are listed, none of which will disappoint.
To Infinity and Beyond Eli Maor (Princeton, 1991)
Not much hard mathematics here, but lots of interesting mathematical ideas (prime numbers, irrationals, the continuum hypothesis, Olber’s paradox (why is the sky dark at night?) and the expanding universe to name but a few), fascinating history and lavish illustrations. The same author has also written a whole book about one number (e The Story of a Number), also published by Princeton (1994), but not yet out in paperback.
How to Think like a Mathematician Kevin Houston (CUP, 2009)
This sounds like the sort of book that elderly people think that young people should read. However, there is lots of good mathematics in it (including many interesting exercises) as well as lots of good advice. How can you resist a book the first words of which (relating to the need for accurate expression) are: Question: How many months have 28 days? Mathematician’s answer: All of them.
Mathematics: a very short introduction Timothy Gowers (CUP, 2002)
Gowers is a Fields Medalist (the Fields medal is the mathematical equivalent of the Nobel prize), so it is not at all surprising that what he writes is worth reading. What is surprising is the ease and charm of his writing. He touches lightly many areas of mathematics, some that will be familiar (Pythagoras) and some that may not be (manifolds) and has something illuminating to say about all of them. The book is small and thin: it will fit in your pocket. You should get it.
The Pleasures of Counting T.W. K¨orner (CUP, 1996)
A brilliant book. There is something here for anyone interested in mathematics and even the most erudite professional mathematicians will learn something new. Some of the chapters involve very little technical mathematics (the discussion of cholera outbreaks which begins the book, for example) while others require the techniques of a first or second year undergraduate course. However, you can skip through the technical bits and still have an idea what is going on. You will enjoy the account of Braess’s paradox (a mathematical demonstration of the result, which we all know to be correct, that building more roads can increase journey times), the explanation of why we should all be called Smith, and the account of the Enigma code–breaking. These are just a few of the topics K¨orner explains with enviable clarity and humour.
Beyond Numeracy J. A. Paulos (Penguin, 1991)
Bite-sized essays on fractals, game-theory, countability, convergence and much more. It is a sequel to his equally entertaining, but less technical, Numeracy.
The New Quantum Universe T. Hey & P. Walters (CUP, 2003)
All you ever wanted to know about quantum mechanics, from fusion to fission, from Feynman diagrams to super-fluids, and from Higgs particles to Hawking radiation. With potted biographies, historical background, and packed with wonderful illustrations and photographs (including an electron microscope image of a midge). This is an excellent and unusual introduction to the subject. The same authors also wrote a splendid book on relativity (Einstein’s Mirror).
Was Einstein Right? C.M. Will (Basic Books, 1988)
Einstein’s theory of General Relativity is a theory of gravitation which supersedes Newton’s theory and is consistent with Special Relativity. The basic idea is that space-time is curved and you feel gravitational forces when you go round a curve in space, in the same way as you feel centrifugal force when your car goes round a bend. This book is about observational tests of the theory, all of which have been passed with flying colours. In particular, there is a binary pulsar which loses mass by gravitational radiation and, as a result, its period of rotation increases by 76 ± 2 millionths of a second per year; General Relativity predicts 75. There is much to be learnt here about physics, cosmology and astronomy as well as about Einstein and his theory.
The Accidental Universe P.C.W. Davies (CUP, 1982)
All the buzz-words are here: cosmic dynamics; galactic structure; entropy of the Universe; black holes; many worlds interpretation of quantum mechanics, but this is not another journalistic pot-boiler. It is a careful and accurate account by one of the best writers of popular science.
Advanced Problems in Mathematics S.T.C. Siklos (1996 and 2003)
These are selections of STEP–like problems complete with discussion and full solutions. (STEP is the examination normally used as a basis for conditional offers to Cambridge.) The problems are different from most A-level questions, being much longer (‘multi-step’ is the current terminology) and sometimes covering material from apparently unconnected areas of mathematics. They are more like the sort of problems that you encounter in a university mathematics course, although they are based on the syllabuses of school mathematics. Working through one or both of these booklets would be an excellent way of getting your mathematics up to speed again after the summer break. The 2003 booklet (Advanced Problems in Core Mathematics) is in a sense a prequel, since it is based on a less advanced syllabus (basically the A-level core plus some mechanics and probability). Both these booklets can be downloaded by clicking on “STEP documents at: http://admissionstestingservice.org/for-test-taker...
Mathematical Methods for Physics and Engineering K F Riley, M P Hobson & S J Bence (Cambridge University Press 1998)
Most of A-level pure mathematics consists of what could be called ‘mathematical methods’ — i.e. techniques you can use in other areas (such as mechanics and statistics). The continuation of this material forms a basic part of every university course (and would count as applied mathematics!).
We are currently developing our very own Maths Library in the department to give our pupils more access to the titles on our reading list.
Check back soon for more information!