I just graduated high school having taken AP calculus and am heading off to college this fall. I really enjoy math and have a great interest in it and am wondering if anyone can reccommend me any good books on math to read this summer. I will be taking math classes at college so obviously I am not trying to learn everything in math but more looking for an overview of the world of mathematics if that makes sense. Thanks
There is also Concepts of Modern Mathematics by Ian Stewart and What is Mathematics? by Courant - Robbins - Stewart. My personal favorite is the first recommendation, though.
"Mathematics: a very short introduction" by fields medallist Timothy Gowers FRS, Rouse Ball Professor of Mathematics at Cambridge University (i.e. the biggest of big cheeses in UK Maths -- Roger Penrose held the identical chair at Oxford). The book is especially appropriate for an 18 year old about to go off to University. Website: http://www.dpmms.cam.ac.uk/~wtg10/ It's the best short overview I've encountered, though Ian Stewart is good as well. From a totally different angle (psychological) try The Number Sense: How the Mind Creates Mathematics by Stanislas Dehaene. It's full of fun examples, like how they got Dobbin to count, chimps doing arithmetic, and babies spotting disappearing puppets...
"A Pure Course in Mathematics" by Hardy. "How to Prove It" Velleman (sp?) Not really overviews, but I would think very useful indeed.
Gower's "further reading" section is superb. He recommends books to readers with different kinds of interest -- history, applicability, formality, philosophy -- and for areas he doesn't cover, e.g., probability, women in mathematics. I like his subtle "maybe not for further reading" recommendations: "Russell and Whitehead's famous Principia Mathematica (Cambridge University Press, 2nd edn., 1973) is not exactly light reading, but if you found some of my proofs of elementary facts long-winded, then for comparison you should look up their proof that 1 + 1 = 2."