Books for a beginner

Question

I am a student of 11th grade and I have completed the syllabus 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 curiosity has now increased to learn maths at a deeper level. As of now i am reading "Linear algebra done right" by Axler. That's 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, that's a great text, but for a beginner, it's 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 you may suggest may be for undergraduate level course as these topics are for undergraduate level courses, that's not a problem at all. I need an 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! :))

Answer 1

You ask a very fair question, and everyone's suggestions are pretty decent. However, I will explain the biggest jump from high school math and college math, and why you might want to consider not jumping the gun on abstract algebra, analysis, or topology.

The biggest difference between high school and college math is that in high school, each lesson is packaged nicely, and the teacher leaves you mostly to "plug and chug" different values to evaluate several mathematical statements. Sure, you might have to think twice about what you're integrating or apply the Taylor's theorem correctly, and such problems can be tricky. However, in college, you're going to spend much more time "proving" things, and freshman year geometry's proofs don't go all the way in prepping you.

Thus, my recommendation is that you start with a book that teaches logic, set theory, discrete math, and general proof strategies. Having these skills will save you so much more time in the future because you'll be used to thinking abstractly, no matter what topics your courses cover. I've read several such books, but the one I keep coming back to is Daniel Valleman's How to Prove It. You'll find yourself thinking like a mathematician, and impressing your professors on day 1.

My high school, despite ranking #1 in AP Calculus AB & BC scores throughout the entire midwest, did not prepare students for abstract thinking- and I felt that pain when jumping into linear algebra my freshman year of college. Now, each high school is different, so I don't want to be unfair toward your high school. However, I know most high school curricula don't delve into abstract thinking too much. Check the table of contents of such a book and ensure you have all those skills down before advancing.

Best of luck.

Answer 2

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).

Answer 3

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.

Answer 4

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!