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Next year I'll start my bachelor's degree on mathematics, but I want to study something about it while I'm idle.

I've found this:

http://webdocs.registrar.fas.harvard.edu/courses/Mathematics.html

And this:

http://www.ufpe.br/proacad/images/cursos_ufpe/matematica_bacharelado_perfil_4904.pdf

They're syllabi from some mathematics courses, one of them is from the university near me, the other is from Harvard.

Can someone suggest a study way? Where should I start and which books I should get?

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7  
I hope the downvoter will explain himself. OP's question seems very reasonable to me. If it is because this would not be MSE-appropriate and should be redirected to somewhere else, downvoting is not the right way to say this. –  Patrick Da Silva Feb 27 '12 at 1:50
    
I didn't get. Is my question too bad? –  Vÿska Feb 27 '12 at 2:34
    
It's not, that the whole point of my comment. –  Patrick Da Silva Feb 27 '12 at 2:52
    
Oh, ok. But who said it was dumb? I'm not aware of that. –  Vÿska Feb 27 '12 at 3:17
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I still disagree with downvoting without explanation. This is bad MSE behavior, because MSE should be an open-discussion community going towards improving everyone's minds. –  Patrick Da Silva Feb 27 '12 at 4:45

3 Answers 3

up vote 6 down vote accepted

I would highly recommend you get an introduction to proofs from a good book. Other computational skills and methods can be learned easily during class, but proofs require some insight and a familiarity with a new style of mathematics when transitioning from high school. Toward this end, I would recommend, as I have to many others,

"Mathematical Proofs: A Transition to Advanced Mathematics" by Gary Chartrand, Albert D. Polimeni, and Ping Zhang.

There is an entire chapter devoted to each of the following:

  • Communicating Mathematics
  • Naive Set Theory
  • Logic
  • Direct Proof
  • Proof by Contrapositive
  • Existence and Proof by Contradiction
  • Mathematical Induction (and Strong Induction)
  • Equivalence Relations (Equivalence Classes, Congruence Modulo n, Modular arithmetic)
  • Functions (Bijective, Inverse, Permutations)
  • Set Theory (up to Schroder-Bernstein Theorem and the Continuum Hypothesis)
  • Number Theory
  • Calculus (Limits, Infinite Series, Continuity, Differentiability)
  • Group Theory (up to Isomorphic Groups)

With Three Additional Chapters online covering:

  • Ring Theory
  • Linear Algebra
  • Topology

Try to get through as much of this as you can before you start university and you will be very glad that you did!

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They told me I need college algebra and trigonometry to enter calculus. –  Vÿska Feb 27 '12 at 2:24
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This is not a calculus textbook, it is a textbook on proofs in a variety of fields of mathematics. In this book you will start to learn real mathematics, don't worry about what "prerequisites" you are supposed to have for this book. Near the end (Proofs in Calculus, Group Theory, Topology, etc.) you probably won't be able to get a good handle on until you have some experience, but the starting chapters should be highly accessible! You will get a taste of higher level mathematics in this book, so make sure you buy it and go through it! –  Samuel Reid Feb 27 '12 at 2:49
    
Ok, thanks for the advice. it seems a great book! –  Vÿska Feb 27 '12 at 3:56
    
    
Do you think these books will also be useful as the one you suggested? –  Vÿska Mar 1 '12 at 2:53

Firstly, I don't think that looking at syllabi is a necessary start.

I suspect that you are getting a math degree because you either think that math is easy or fun (or perhaps both). If I presuppose that you, as an about-to-be-undergrad math student, are exactly where I was when I was about to enter undergraduate math, then I will suppose that you know calculus, trig, and elementary probability really well, you haven't really done proofs in a while or at a sufficient collegiate level, and math is both fun and easy.

Then I think you should do something you find fun. Perhaps Barbeau's Polynomials would be a great start (the reviews say it's an extension of high school math, which is true only in the sense that a high schooler should be able to pick up the book, but I would probably learn still more from it if I were to go through it again even now). It's a problem book. And if you've a good network, you can ask others when you're stumped. Being able to ask others is important.

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Just a note, the book mixedmath mentioned is geared towards Abstract Algebra. –  Samuel Reid Feb 27 '12 at 2:01
    
I've studied a little college Algebra and trigonometry, but I don't remeber it very well, at that time (I was 16 and now i'm 22) I hated mathematics with all my forces. Some years later, i discovered that the problem was not mathematics per se - mathematics is a great intelectual journey and a tool to the unimaginable - the problem was the lack of skill my teachers had, i remember asking "How can i use this or how can i make something with this?" and receiving shallow and stupid answers. Now i kinda feel the taste of mathematics and i really want to go further. –  Vÿska Feb 27 '12 at 2:30
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Looks like a great book! Thanks for the recommendation! –  user23211 Feb 27 '12 at 3:09
    
@mixedmath I got this book. –  Vÿska Feb 29 '12 at 6:07
    
@Gustavo: Hey, that sounds great! I hope you enjoy it. –  mixedmath Feb 29 '12 at 6:23

What I suggest you is that you have someone to guide you through your studies ; talk to your teachers and find one amongst them who understands your way and is willing to spend a little time to show you the way. What I mean by that is that this teacher should be able to ask you and understand what are your interests and give you an appropriate book to begin with. If you're lucky and that this teacher has enough patience, maybe he will help you go through some incomprehensions when you have some.

My point is this : don't learn mathematics alone. Find people and books to do so.

Hope that helps,

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Yup. I'm already looking at forums, people, tools, etc. –  Vÿska Feb 27 '12 at 2:25

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