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I became interested in how numbers actually work recently. I want a source that covers number theory. Explaining how and why numbers react the way they do. A friend recommended Mathematical Proofs: A Transition to Advanced Mathematics.

I want to learn about the concept of numbers, how they interact etc. I am a high school student, but most college level material will be appropriate.


-- Also, in my search I have found Mathematics in 10 lessons: The Grand Tour, and Foundations and Fundamentals of Mathematics.

If anyone could confirm that those two resources are adequate, I would appreciate it, and other resources are still welcome.

P.S. I do not like the Khan Academy. His presentations do not... serve my needs.

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I presume you've looked into Rosen's book? – J. M. Dec 23 '10 at 3:02
Number theory is more closely tied to integers than "numbers" in general. It sounds like you are after a more general overview of mathematics to include analysis, numerical methods, foundations of mathematics, and abstract algebra, rather than just number theory. This question should probably be retagged. – Mitch Schwartz Dec 23 '10 at 10:29
What tag would you suggest? – user5061 Dec 23 '10 at 14:14
@user5061: Hmm, I'd say soft-question, reference-request, learning. – Mitch Schwartz Dec 23 '10 at 16:54

For an overview of the many parts of mathematics and the goals and accomplishments of various subareas within mathematics I recommend Timothy Gowers (ed.) The Princeton Companion to Mathematics. The book has pointers to more technical issues if you want to go into more depth after having a broad view.

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It seems like you're interested in seeing how mathematics is used beyond calculation in a general sort of way. Here are some recommendations -

Dr. Euler's Fabulous Formula
Paul Nahin (and check out his other work as well)

Number: The Language of Science
Tobias Dantzig

The Pleasures of Counting
T.W. Korner

Gödel, Escher, Bach
Douglas R. Hofstadter

These texts try to tease out interesting ideas in an accessible way. They're not exactly going to pave the road to learning proof (for that, I think a teacher is best), but they might help you see where you're going.

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You should try reading "Mathematics - form and function" by Saunders MacLane. The author tries to convey what Mathematics is "all about" and at the same time giving a overview over several different areas. (including manifolds, number theory, groups/fiels, algebra, geometry, etc),_Form_and_Function

The presentation may at times be too dense for someone who has not encountered the material before, but if you're prepared to read slow and careful at times, the book is excellent.

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