Starting out:
The Number Devil by Hans Magnus Enzensberger  Twelveyearold Robert hates his maths teacher (nothing you can relate to) but then the Number Devil comes to him in his dreams and brings the subject to life. "I found this book very funny, especially when Robert outwits the Number Devil, and I learned more from the Enzebergian devil who beguiled Robert and I, into a dream world of ideas. I would really recommend this book for anyone of any age – who has ever been mystified by maths."  review by Luxmi, year 8.
Pure Maths:
Professor Stewart's Cabinet of Mathematical Curiosities (and anything else by Ian Stewart)
"All the books are engaging. The cabinet books are made up of puzzles, and Incredible Numbers touches on many different areas of Mathematics"
How to study for a maths degree by Laura Alcock. Apparently it is "rather good" and does what it says on the tin.
The Birth of a Theorem by Cedric Villani  apparently impenetrable but Villani is an interesting speaker  watch him here.
Prime Obsession by John Derbyshire. It is about the Riemann Hypothesis and supposed to be brilliant.
The Music of the Primes by Marcus du Sautoy. You can get a taste for his writing on his website here.
Applied Maths:
How long is a piece of string? by Rob Eastaway. "A fun book explaining the maths behind everyday life, from taxis to lifts. Also some weird fractal tricks."
History of Maths:
Fermat's Last Theorem by Simon Singh
"It doesn't actually explain the proof of Fermat's Last Theorem. But it does explain why it matters and how it lasted for centuries unsolved."
Infinity by Brian Clegg.
"Partly explaining the history of the concept, and partly explaining the maths that involves infinity. Completely fun."
Physics:
The Quantum Universe: Why everything that can happen, does happen by Brian Cox and Jeff Forshaw
"An absolutely brilliant book, working upwards from the basics of electrons to explain the periodic table, computers, and then the formation of white dwarfs. What marks this book out from other popular science books is that it explains the calculations to show how the maximum mass of a white dwarf can be calculated using thought experiments (and a calculator)."
Six Easy Pieces (and the sequel Six NotsoEasy Pieces) by Richard Feynman. The first book begins fairly basic so you might want to skip to the second.
The Big Questions: Physics by Michael Brooks.
"He's also written 13 Things that Don't Make Sense which is an absolutely amazing book. They both read a bit like unfinished detective stories, and give surprisingly good depth for books that cover such a wide range of topics."
The Number Devil by Hans Magnus Enzensberger  Twelveyearold Robert hates his maths teacher (nothing you can relate to) but then the Number Devil comes to him in his dreams and brings the subject to life. "I found this book very funny, especially when Robert outwits the Number Devil, and I learned more from the Enzebergian devil who beguiled Robert and I, into a dream world of ideas. I would really recommend this book for anyone of any age – who has ever been mystified by maths."  review by Luxmi, year 8.
Pure Maths:
Professor Stewart's Cabinet of Mathematical Curiosities (and anything else by Ian Stewart)
"All the books are engaging. The cabinet books are made up of puzzles, and Incredible Numbers touches on many different areas of Mathematics"
How to study for a maths degree by Laura Alcock. Apparently it is "rather good" and does what it says on the tin.
The Birth of a Theorem by Cedric Villani  apparently impenetrable but Villani is an interesting speaker  watch him here.
Prime Obsession by John Derbyshire. It is about the Riemann Hypothesis and supposed to be brilliant.
The Music of the Primes by Marcus du Sautoy. You can get a taste for his writing on his website here.
Applied Maths:
How long is a piece of string? by Rob Eastaway. "A fun book explaining the maths behind everyday life, from taxis to lifts. Also some weird fractal tricks."
History of Maths:
Fermat's Last Theorem by Simon Singh
"It doesn't actually explain the proof of Fermat's Last Theorem. But it does explain why it matters and how it lasted for centuries unsolved."
Infinity by Brian Clegg.
"Partly explaining the history of the concept, and partly explaining the maths that involves infinity. Completely fun."
Physics:
The Quantum Universe: Why everything that can happen, does happen by Brian Cox and Jeff Forshaw
"An absolutely brilliant book, working upwards from the basics of electrons to explain the periodic table, computers, and then the formation of white dwarfs. What marks this book out from other popular science books is that it explains the calculations to show how the maximum mass of a white dwarf can be calculated using thought experiments (and a calculator)."
Six Easy Pieces (and the sequel Six NotsoEasy Pieces) by Richard Feynman. The first book begins fairly basic so you might want to skip to the second.
The Big Questions: Physics by Michael Brooks.
"He's also written 13 Things that Don't Make Sense which is an absolutely amazing book. They both read a bit like unfinished detective stories, and give surprisingly good depth for books that cover such a wide range of topics."
All of these books are available in the School Library to borrow.
General Mathematics
Recreational Mathematics

Physics/Engineering
 How long is a piece of string? Rob Eastaway. "A fun book explaining the maths behind everyday life, from taxis to lifts. Also some weird fractal tricks."
 The Quantum Universe: Why everything that can happen, does happen. Brian Cox and Jeff Forshaw
"An absolutely brilliant book, working upwards from the basics of electrons to explain the periodic table, computers, and then the formation of white dwarfs. What marks this book out from other popular science books is that it explains the calculations to show how the maximum mass of a white dwarf can be calculated using thought experiments (and a calculator)."  Six Easy Pieces (and the sequel Six NotsoEasy Pieces, Richard Feynman. The first book begins fairly basic so you might want to skip to the second.
 The Big Questions: Physics, Michael Brooks.
"He's also written 13 Things that Don't Make Sense which is an absolutely amazing book. They both read a bit like unfinished detective stories, and give surprisingly good depth for books that cover such a wide range of topics."
Biographies of Mathematicians
 The Man Who Loved Only Numbers, Paul Hoffman
 The Man Who Knew Infinity: A Life of the Genius Ramanujan, Robert Kanigel
 Alan Turing: The Enigma, Andrew Hodges
History of Mathematics
 Fermat's Last Theorem, Simon Singh
Slightly Mad
 The Thrilling Adventures of Lovelace and Babbage: The (Mostly) True Story of the First Computer by Sydney Padua
Sixth Form and Above
BlogsTerence Tao  A strongly recommended collection of advice from Terence Tao to young mathematicians:
https://terrytao.wordpress.com/careeradvice/ 
Books for ChallengesThe Mathematical Experience
Philip J Davis and Reuben Hersh A varied selection of articles both long and short and of varying technical difficulty. They cover mainly historical and philosophical aspects of the subject. The Problems of Mathematics Ian Stewart A survey of some of the central problems in maths, both past and present. Ian Stewart has written a number of “popular” books on maths (ie accessible), you might also like to try Nature’s Numbers or his most wellknown book Does God Play Dice? His style is clear and fairly straightforward. The Pleasures of Counting T W Körner Examples of how maths has been used in different situations, some of them quite unlikely. Read the preface to get an idea of the author’s style. Some of the maths on display is difficult – don’t be put off if you don’t get it even after a prolonged period of thought, you should still be able to understand the gist of the explanation. What is Mathematics, Really? Reuben Hersh Relatively accessible philosophy of maths with a historical slant. The first part is more general; the second part contains a fair few sentences of the type “But he doesn’t discover an ontology to go with his fallibilist epistemology”, which you may choose to avoid. There are some helpful notes about maths (rather than philosophy) at the back. The Mathematical Brain & The Maths Gene Brian Butterworth, Keith Devlin These both deal with the psychology of mathematics. Devlin is interested in the notion of maths as a language and how as a species we ended up doing maths in the way we do. Butterworth is a cognitive neuroscientist and includes several interesting case studies both of experimental work and of people with dysfunction when it comes to arithmetic. These books will probably be of most interest to anyone thinking of doing psychology, medicine etc.. They are not about how to do mathematical problems! 