There are many articles and books that introduce you to advanced topics in mathematics in a friendly way. The Scientific American articles written by Martin Gardner and Ian Stewart are particularly recommended. These articles have been collected together in a number of books, some of which can usually be found in your local bookshops and libraries.
"What is Mathematics?", by R Courant and H Robbins, Oxford University Press, 1996 is perhaps still the best book to convey an overall impression of the nature of university mathematics. It covers many topics from sets to soap-bubbles and has become a classic.
A big difference between pre-university and university Mathematics lies in the idea of a "proof", that is, an argument that shows that a conclusion is correct, rather than simply plausible. The texts "Numbers and Proof", by R B J T Allenby, Edward Arnold, 1997, Elementary Number Theory", by GA & J M Jones, Springer Undergraduate Mathematics Series, 1998, give a nice introduction to proofs by considering some elementary problems in number theory.
Another approach is based on an investigation of elementary set theory and logic. This viewpoint can be found in "Elements of Logic via Sets and Numbers" by D L Johnson and "Introductory Mathematics: Algebra and Analysis" by G Smith, both in the Springer Undergraduate Mathematics Series.
Two other excellent books, which will help you to prepare for the different style of mathematics that you will study here, are "A Concise Introduction to Pure Mathematics" by Martin Liebeck and "How to Think Like a Mathematician: A Companion to Undergraduate Mathematics" by Kevin Houston.
You may also consider Lara Alcock's book "How to Study for a Mathematics Degree", a practical book discussing the differences between school and university mathematics.
The following more technical texts are among those recommended for our first-year courses. They will give you more insight into the nature of university mathematics, but you may find them a bit heavy if you read them before coming to university. There is no need to purchase any of them before you arrive at college!
- "For all Practical Purposes", various authors, COMAP, W H Freeman, 1988
- "Mathematical Analysis", K G Binmore, Cambridge University Press, 1977
- "Linear Algebra", R B J T Allenby, Edward Arnold, 1995
- "Rings, Fields and Groups", R B J T Allenby, Edward Arnold, 1991
- "A First Course in Probability", Sheldon Ross, Collier Macmillan, 1988
- "Analytical Mechanics", G R Fowles, Saunders College, 1993
If you are taking a gap year, it is imperative to keep in practice by studying one or more of these books, or by revising your A-level texts.