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I am a student of 11th grade and i have completed the syallabus of both 11th and 12th grade maths with complete understanding and it was possible coz of the love for this subject that i have. I don't want to sound "larger than life" here, but my curiousity has now increased to learn maths at a deeper level. As of now i am reading "Linear algebra done right" by Axler. Thats an amazing piece of work and now i truly understand what "matrix" is all about. I also want to extend my knowledge on the following topics :

1) Algebra (groups,subgroups,homomorphisms etc)

2) Analysis

3) Geometry/Topology

Why i am here is because i wanted you guys to recommend a book on each of the above topic that would be appropriate for a beginner like me (I must mention here I have been working on Apostol's volume 1 calculus, thats a great text, but for a beginner, its best if the concept is explained in a broad manner and in as simple and easy words as possible) . I just want a book which explains the concept broadly rather than coming to the conclusion directly (which is not a great sight for a beginner like me) .

Having said that, no book is complete i understand. Thats why i am asking your recommendation as you guys are aware which text would be the best to start with for a beginner. I have searched on the net about this, but there are dozens of works available, and out of them all , i want the one which is the best (approximation) for a beginner .

Please don't misunderstand me in any way. I am just confused which text to go for, and i understand the books which u may suggest may be for undergraduate level course as these topics are for undergraduate level courses, thats not a problem at all . I need a easy to learn (i mean easy in terms of "broad" explanation) book. I hope you guys don't mind me questioning such a question on this forum . Thanks for all the help you guys have been providing me on this forum . Maths Maths Maths... the world is beautiful coz of u! :))

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When I am looking for books that introduce topics, the first thing I do is, on mathSE, make the search: "[reference-request] TOPICHERE" and sometimes add beginner, introduction introductory, etc. to the search. –  Jordan Mahar Jun 18 '13 at 20:15
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@under-root My suggestion is: stop whatever it is you're reading and don't try to tackle the standard undergrad curriculum before reading this amazing book: How to Prove It: A Structured Approach, by D.J. Velleman. –  Git Gud Jun 18 '13 at 20:16
    
For Analysis, I think Terence Tao's book would be a good choice. –  math Jun 18 '13 at 20:26
    
Thanks all..and @Git Gud,thanks a lot..ordered it just now! :)) –  under-root Jun 18 '13 at 20:52
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@under-root I just catapulted your math growth by about 2 years ^_^ –  Git Gud Jun 18 '13 at 20:55
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3 Answers

Spivak's Calculus could really be considered an introduction to analysis. Indeed, the author has said as much (I believe in the preface to the 3rd edition). I don't think I'm going out on a limb in calling it a great text and quite readable, with challenging exercises.

It also has an answer book (hardcover or spiral bound).

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One book I strongly recommend that I don't think is that well-known outside of Canada is Introduction to Abstract Algebra by Nicholson (http://www.amazon.com/Introduction-Abstract-Algebra-Keith-Nicholson/dp/1118135350/ref=dp_ob_title_bk). This is the book I learned group theory and ring theory with, and I think it is particularly useful because there is a fairly in-depth Chapter 0 on how reason and write mathematical proofs, followed by a Chapter 1 which describes (concretely) many topics with which you are probably already familiar with (such as prime numbers and modular arithmetic) as motivating examples for further abstractions. Oh, another benefit... there's a solutions manual which you can purchase, which contains worked out solutions to a large number of exercises!

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A Survey of Modern Algebra by Birkhoff and MacLane is a great introduction to algebra.

If you can read Axler, you should be fine with this book. It is neither overly concise nor is it overly verbose.

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