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1+1=?
Maths is great. I did it for A-level and loved it. It emphasises logical, orderly thinking, trying to make sense of things around you, and its technique of proof is possibly the most rock-solid method of discovery there is. It's a great stepping-stone into deeper thoughts about the nature of this world. Some books which I would highly recommend for this overview of the nature of mathematics are:- Gödel, Escher, Bach: an Eternal Golden Braid by Douglas Hofstadter - winner of the Pulitzer Prize, a fascinating look at the nature of the mind through Gödel's earth-shattering discoveries, Escher's mind-warping artwork and Bach's painstakingly-constructed music (the last being, of course, of particular interest to me). Also, by the same author (somewhat later), Metamagical Themas.
- Fermat's Last Theorem by Simon Singh. This traces the many attempts to solve the infamous theorem, leading to Andrew Wiles's eventual discovery of a solution; plenty of other interesting ideas along the way.
- The Music of the Primes by Marcus du Sautoy. A look at the Riemann Hypothesis (one of the greatest unsolved problems of our time), in a similar way to the above (but this time focusing on the primes, since the Hypothesis is chiefly to do with them).
I must say, it is this more abstract side of mathematics (proof, formal systems, number theory etc.) which interests me more than the more physical side (mechanics, statistics, etc.). However, in an attempt to gain a better grasp of that, I've bought The Road to Reality by Roger Penrose, which claims to be 'A complete guide to the laws of the universe', and certainly has some scary-looking maths to back it up. I'm on about chapter 3 so far, and it's getting fairly hairy (but good...)
In case it interests you, I've put up some of the coursework I did for Decision and Discrete Maths at A-level (mainly because that was the most interesting coursework I did) in the 'Resources' section, as well as two tongue-in-cheek 'papers' I wrote in response to certain hilarious (at the time) events in my maths class (you had to be there, I think). The first concerns a pupil who tried to simplify a fraction by cancelling out additions (bit of a faux pas there); the second is a guide to the 'Big M' method of the Simplex algorithm, written in response to a teacher who forgot one of its nit-picky rules and ended up spending several weeks trying to solve a single question (his wife claimed she'd become a 'Simplex widow').
Finally, the Simplex section is the gateway to an implementation of the Simplex algorithm (including two-stage simplex, though not Big M as yet) which I wrote whilst procrastinating from A-level revision. A previous version was used to solve the Simplex problem in my D2 coursework, which featured 28 variables and required 12 pivots!