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I've always loved playing with numbers, but haven't had any formal guidance in the study of advanced mathematics and number theory. Is there a book (or a few books) on mathematics that I wouldn't have learned in my high school curriculum (algebra, trig, and calculus) that could help me understand higher level maths and get me started on number theory? I know it's quite a broad question, but when I see things like this xkcd, and in trying to understand concepts I go to Wikipedia articles to find them chock full of terminology that flies over my head, it's quite frustrating.

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rutherglen.ics.mq.edu.au/wchen/lnentfolder/lnent.html is a good set of (free) notes on Number Theory. –  JavaMan Nov 27 '11 at 7:03
    
The Art of Problem Solving Volume 1 and 2 (especially 2) are nice books which go beyond the standard high school curriculum in general and venture into competition math. They got me interested in higher math. –  Shayla Nov 27 '11 at 7:08
    
If you're asking about book for self-study of number theory, you can find something in related question: math.stackexchange.com/questions/1774 (In fact, Stein and Underwood Dudley, which were suggested by Arturo, were mentioned there too.) Maybe you can have a look at questions related/linked to that one too. –  Martin Sleziak Nov 27 '11 at 7:13
    
When I was in high school and looking for extras, I stumbled upon, and immensely enjoyed, Joe Roberts' Elementary Number Theory - a Problem Oriented Approach. His book has a very unusual style. All the theory is developed as series of exercises. The first half of the book is only definitions and exercises. The second half has hints/solutions/historical remarks. It was a perfect fit for me. Warmly recommend it for self-study. –  Jyrki Lahtonen Nov 27 '11 at 9:45
    
@JyrkiLahtonen, you didn't mention the most unusual thing about the Roberts book - it's all done in calligraphy. –  Gerry Myerson Nov 27 '11 at 11:40
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up vote 3 down vote accepted

That xkcd strip is not about number theory, but about set theory (and a particularly slippery part of set theory at that; the Axiom of Choice is tricky!).

A possible first set theory book might be Halmos's Naive Set Theory, though it can be tough in parts and suffers somewhat for having few exercises.

There are several introductory Number Theory books; e.g., William Stein's Elementary Number Theory: Primes, Congruences and Secrets; Leveque's Elementary Number Theory; and Underwood Dudley's book of the same title.

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