New user reporting for duty, sir!
First off, there's a massive list of books here
. They're mostly very good, but you might want to take the last section or two with a generous grain of salt since it's UK-oriented and we get to do a lot more maths before university than you.
If you want to get straight into some serious maths, then my recommendations vary depending on what you're interested in. Bryant's Yet Another Introduction to Analysis is a wonderful book for real analysis (basically the rigorous foundations of calculus) - it's not actually a textbook, so the author's free to be a bit less formal in style. If you're feeling more adventurous, Spivak's Calculus is good as well if a little harder - it covers more stuff, and it has a few amusing little detours like proving that e is transcendental. (It's also invaluable as a textbook.) If you're at all interested in group theory or abstract algebra in general, try to get hold of Fraleigh's A First Course In Abstract Algebra - it gets quite hard quite fast, but it covers a lot
- the insolubility of the quintic is proved in the last chapter. Anton's Elementary Linear Algebra is an easy introduction if you're interested in matrices, although it might be less easy with a US background - I'm not sure how much you cover.
For lighter, more general stuff, I'd highly recommend Ball's Strange Curves, Counting Rabbits and Other Mathematical Explorations. It has things like a space-filling curve, Pick's Theorem (which will make you gape in awe the first time you see it), and Stirling's approximation to n! - a very fun read. If you get one book, make it that one. Darbyshire's Prime Obsessions is probably the best book out there on the Riemann Hypothesis in that it actually goes into a little of the maths, while not actually assuming any maths background. And I second the recommendations of Godel, Escher, Bach and anything by Simon Singh. (If you're interested in the intersection between maths and computer science, I'd also recommend you pick up Dewdney's New Turing Omnibus - quite a friendly book with a very intimidating name.)
And if you're feeling really
adventurous, get Concrete Mathematics by Knuth, Graham and Patashnik. Yes, that
And finally, a piece of advice that will stand you in good stead: always always always
buy maths books second hand if you can. You'll get things for £5 instead of £40.
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