Chaos by James Gleick

Chaos is what happens when the behaviour of a system gets too complicated to predict; the most familiar example is the weather, which apparently cannot be forecast accurately more than five days ahead. This book tells the story so far in the study of this new field of Physics.

Mathematics and the Physical World by Morris Kline

A stimulating account of development of basic mathematics from arithmetic, algebra, geometry and trigonometry, to calculus, differential equations and non-Euclidean geometries. Also describes how maths is used in optics, astronomy, motion under the law of gravitation, acoustics, electromagnetism, and other aspects of physics.

The ‘Uncle Albert’ Series by Russell Stannard

A best selling science/adventure series, beginning with ‘The Time and Space of Uncle Albert’. Uncle Albert and his intrepid niece, Gedanken, enter the dangerous and unknown world of a thought bubble. Their mission: to unlock the deep mysteries of Time and Space... Discover why you can't break the ultimate speed barrier, how to become older than your mother, how to put on weight without getting fat, and how to live forever without even knowing it. Other books in the series include: ‘Black Holes and Uncle Albert’ and ‘Uncle Albert and the Quantum Quest’.

The Code Book is a history of man's urge to uncover the secrets of codes, from Egyptian puzzles to modern day computer encryptions. As in Fermat's Last Theorem, Simon Singh brings life to an astonishing story of puzzles, codes, languages and riddles that reveals man's continual pursuit to disguise and uncover, and to work out the secret languages of others. Codes have influenced events throughout history, both in the stories of those who make them and those who break them. The betrayal of Mary Queen of Scots and the cracking of the enigma code that helped the Allies in World War II are major episodes in a continuing history of cryptography. In addition to stories of intrigue and warfare, Simon Singh also investigates other codes, the unravelling of genes and the rediscovery of ancient languages and most tantalisingly, the Beale ciphers, an unbroken code that could hold the key to a 20 million dollar treasure.

Look up 1729 to see why it is ‘among the most famous of all numbers’. Look up 0.7404 ($= \frac{\pi}{18}$) to discover that this is the density of closely packed identical spheres in what is believed by many mathematicians (though it was at that time an unproven hypothesis) and is known by all physicists and greengrocers to be the optimal packing. Look up Graham’s number (the last one in the book), which is inconceivably big: even written as a tower of powers ($9 ^{9^ {9 ^\cdots}}$) it would take up far more ink than could be made from all the atoms in the universe. It is an upper bound for a quantity in Ramsey theory whose actual value is believed to be about 6. A book to be dipped into at leisure.

The Art of the Infinite by Robert and Ellen Kaplan

This book unlocks the secrets of maths - revealing it to be our lost, native language, as much a part of us as the words we use every day. Number and form are the essence of our world: from the patterns of the stars to the pulses of the market, from the beats of our hearts to catching a ball or tying our shoelaces. Drawing on science, literature, history and philosophy, this book makes the rich patterns of maths brilliantly clear.

This book tells the story of one of the biggest adventures in mathematics: the search for symmetry. This is the story of how humankind has come to its understanding of the bizarre world of symmetry - a subject of fundamental significance to the way we interpret the world around us. Our eyes and minds are drawn to symmetrical objects, from the sphere to the swastika, from the pyramid to the pentagon. 'Symmetry' is all-pervasive: in chemistry the concept of symmetry explains the structure of crystals; in evolutionary biology, the natural world exploits symmetry in the fight for survival; symmetry and the breaking of symmetry are central to ideas in art, architecture and music; the mathematics of symmetry is even exploited in industry, for example to find efficient ways to store more music on a CD or to keep your mobile phone conversation from cracking up through interference.

Flatterland by Ian Stewart

In 1884, Edwin A. Abbott published “Flatland”; a brilliant novel about mathematics and philosophy that charmed and fascinated all of England. Now, Ian Stewart has written a fascinating, modern sequel to Abbott's book. Through larger-than-life characters and an inspired story line, "Flatterland" explores our present understanding of the shape and origins of the universe, the nature of space, time, and matter, as well as modern geometries and their applications.

Euclid's Window: The Story of Geometry from Parallel Lines to Hyperspace by Leonard Mlodinow

Anyone who thought geometry was boring or dry should prepare to be amazed. Despite its worthy cover this book is exactly what its title says - a story - and the plot of this story involves life, death and revolutions of understanding and belief. It stars the some of the most famous names in history, from Euclid who laid the logical foundations, to Albert Einstein, who united space and time in a single non-Euclidean geometry. It offers an alternative history of mathematics, revealing how simple questions anyone might ask about space - in the living room or in some other galaxy - have been the hidden engines of the highest achievements in science and technology.

How can one predict when the next prime number will occur? Is there a formula which could generate primes? These apparently simple questions have confounded mathematicians ever since the Ancient Greeks. In 1859, the brilliant German mathematician Bernhard Riemann put forward a hypothesis which finally seemed to reveal a magical harmony at work in the numerical landscape. The promise that these eternal, unchanging numbers would finally reveal their secret thrilled mathematicians around the world. Yet Riemann never publicly provided a proof for his hypothesis and his housekeeper burned most of his personal papers on his death. Whoever cracks Riemann's hypothesis will go down in history, for it has implications far beyond mathematics. In business, it plays a central role in security and e-commerce. In science, it brings together vastly different areas, with critical ramifications in Quantum Mechanics, Chaos Theory and the future of computing. Pioneers in each of these fields are racing to crack the code and a prize of $1 million has been offered to the winner. As yet, it remains unsolved.

The story of the solving of a puzzle that has confounded mathematicians since the 17th century. In 1963, a schoolboy browsing in his local library stumbled across the world's greatest mathematical problem: Fermat's Last Theorem, a puzzle that every child can understand but which has baffled mathematicians for over 300 years. Aged just ten, Andrew Wiles dreamed that he would crack it. Wiles's lifelong obsession with a seemingly simple challenge set by a long-dead Frenchman is an emotional tale of sacrifice and extraordinary determination. In the end, Wiles was forced to work in secrecy and isolation for seven years, harnessing all the power of modern maths to achieve his childhood dream. Many before him had tried and failed, including a 18-century philanderer who was killed in a duel. An 18-century Frenchwoman made a major breakthrough in solving the riddle, but she had to attend maths lectures at the Ecole Polytechnique disguised as a man since women were forbidden entry to the school.

The purpose of this book is to introduce the basic ideas of mathematical proof to students. The emphasis is on helping the reader in understanding and constructing proofs and writing clear mathematics. This is achieved by exploring set theory, combinatorics and number theory, topics which include many fundamental ideas which are part of the tool kit of any mathematician. This material illustrates how familiar ideas can be formulated rigorously, provides examples demonstrating a wide range of basic methods of proof, and includes some of the classic proofs.

Mathematics: A Very Short Introduction by Timothy Gowers

Tim 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.