I am a high school student and would like to pursue a career in mathematics and I am hoping to find a serious explanatory book on math (geometry, algebra, calculus, functions and trigonometry) for further advanced studies in math. Does anyone know any very good textbooks that will truly make me a better mathematician?

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Before you read anything else, attack this one: How to Prove It: A Structured Approach, by D.J. Velleman. That book is the best mathematical tool to someone in your position. –  Git Gud Oct 6 '13 at 19:54
I have removed the "logic" tag; the question is not really about mathematical logic. (Proof techniques like the ones in "How to prove it" are not mathematical logic in any genuine sense, they are just a part of basic mathematics). –  Carl Mummert Oct 6 '13 at 20:03
I recommend checking out artofproblemsolving.com. –  littleO Dec 26 '13 at 19:15

There is a fantastic book, which you may have seen already - *What is Mathematics" by Courant and Robbins, see http://en.wikipedia.org/wiki/What_Is_Mathematics%3F. The point is, before you will become a "better mathematician", you should become a mathematician at all. The book shows what this means, and what are the possibilities.

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I would recommend learning number theory, in particular reading "An Introduction to the Theory of Numbers" by Hardy and Wright. Number theory has the advantage of requiring no prerequisite knowledge (other than of elementary arithmetic) and leads naturally into the study of (abstract) algebra, which has tremendous applications to later topics like geometry, topology, etc.

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Maybe better called "elementary number theory". –  paul garrett Oct 6 '13 at 20:10

The suggestion to work through Spivak's Calculus is good: rigorous analysis was one of the first subjects that really opened up mathematics to me and showed me how beautiful it can be (That said, there are lots of people who can't stand $\varepsilon$s and $\delta$s). Spivak's book is quite big and long though: for an introduction to Analysis, I prefer R. P. Burn's Numbers and Functions (CUP), which is shorter and leads you through the subject in a sequence of well chosen questions.

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Describing your current knowledge of mathematics would help. This question has also been answered many times before. A standard answer is that you should read and work through Spivak's Calculus.

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Thank you Git Gud and Newb. –  George Oct 6 '13 at 20:15
I am currently learning Winding Functions (just starting). I know my geometry and algebra quite well. Im a little shaggy but not too bad at trigonometry (I understand cosine, sine, tan, special triangles etc.) however I am not very advanced in calculus (limits, integrals etc.) –  George Oct 6 '13 at 20:19