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Teaching myself mathematics

  1. May 6, 2009 #1
    Hello.

    I am going to be a freshman in high school, and I am a very advanced math student. My school goes at a pace that's way too slow for me, so I decided I would teach myself some mathematics. So far, I've got basic linear algebra, and even basic-er modern algebra, and I'm working on calculus. (I've done all the prerequisites of course.) So, what I want to know is, what comes next?
    I want to learn more advanced math, and I need some recommendations for subjects and textbooks. I'm interested in pure mathematics, not applied. I'm interested in some higher level linear algebra textbooks and good modern algebra textbooks, but I would be glad to hear any recommendations you may have. Thank you for your help.

    *Edited*
    Since there appears to be some debate over whether I really know what I'm talking about, let me clarify: I do know the distinction between basic algebra and linear/modern algebra. I am thirteen, but I will be fourteen very soon. (My birthday is May 20th) Here is a list of some of the stuff I know:
    Basic set theory, functions, binary operations, relations, basic finite group theory, permutation groups, abelian groups, and a little bit of field theory. I also know vectors in Rn, vector algebra, matrices and determinents, and some linear transformations. Thanks for your help!
     
    Last edited: May 7, 2009
  2. jcsd
  3. May 6, 2009 #2
    If you're into Algebra, you may also like any of the following:
    - Set Theory
    - Mathematical Logic
    - Theoretical CS

    There are some alright online "textbooks" in each of these categories. Maybe like...
    http://www.math.uchicago.edu/~mileti/teaching/math278/settheory.pdf [Broken]
    http://www.math.psu.edu/simpson/courses/math557/logic.pdf [Broken]
    http://www.nada.kth.se/~johanh/complexitylecturenotes.pdf

    Tons of other books are available in these areas, both online and in print. Here are what may or may not be some more advanced books:
    http://www.liafa.jussieu.fr/~jep/PDF/MPRI/MPRI.pdf
    model / proof theory... (couldn't find anything great, but there are some smallish papers floating around)
    http://www.math.uu.nl/people/jvoosten/syllabi/catsmoeder.pdf
     
    Last edited by a moderator: May 4, 2017
  4. May 6, 2009 #3
    Are you suggesting that you are 13 and already have the rudiments of linear and modern algebra down?

    What I suggest before delving into modern math is that you get a real understanding of the math you are learning - not the mickey mouse stuff you guys do in school. For that purpose, focus on the theory books of this thread: https://www.physicsforums.com/showthread.php?t=307797 . Did you prove all the trigonometric identies, etc? Are you strong in geometry, and analytic geometry?

    If you insist on moving ahead, with a weak background, you should definately complete calculus before moving ahead. Once that is done, try Friedman for Linear Algebra and Dummit and Foote for Abstract Algebra. After calculus, do analysis with Pughs book or Rudin. This is what is usually done in a 4 year undergrad.
     
  5. May 6, 2009 #4

    Landau

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    Do you mean https://www.amazon.com/Linear-Algeb...=sr_1_6?ie=UTF8&s=books&qid=1241634423&sr=1-6?
    You may also consider https://www.amazon.com/Linear-Algeb...=sr_1_5?ie=UTF8&s=books&qid=1241634423&sr=1-5

    You say you're working at calculus. If you really want a good understanding of calculus, try the great book https://www.amazon.com/Calculus-Mic...=sr_1_1?ie=UTF8&s=books&qid=1241634600&sr=1-1. After that, move on to Rudin or Pugh, as Howers already mentioned.
     
    Last edited by a moderator: May 4, 2017
  6. May 6, 2009 #5
    Useful links. Thanks.
     
    Last edited by a moderator: May 4, 2017
  7. May 6, 2009 #6
    Give the kid a break. He says he wants some Algebra related material, give it to him. You don't have to be condescending because he makes you feel insecure. Who cares if he's really 13?
     
  8. May 6, 2009 #7
    As far as other subjects you can learn, there's always discrete math such as combinatorics, graph theory and number theory.
     
  9. May 6, 2009 #8
    Seconded.
     
  10. May 6, 2009 #9

    Landau

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    I'm not following you...I don't see anyone being condenscending?
     
  11. May 6, 2009 #10
    Seconded. It was an honest question; lots of people come here asking for study advice and make mistakes about what they know (like the person who though linear algebra and advanced algebra were the same topic) so it's normal to ask this sort of question.

    Isn't there a thread where we can make a big list of all the good beginner textbooks in the various math branches? It'd cut down on these sorts of threads.
     
  12. May 6, 2009 #11
    "Are you suggesting that you are 13 and already have the rudiments of linear and modern algebra down?

    What I suggest before delving into modern math is that you get a real understanding of the math you are learning - not the mickey mouse stuff you guys do in school."

    That's rude. You can pretend it's not, but if you wouldn't talk to the guy sitting in on your PhD dissertation like that, you shouldn't talk to a complete stranger like that.
     
  13. May 6, 2009 #12

    Pengwuino

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    There really should be a thread that dictates exactly what each level of mathematics is all about and have an example of what kind of material is taught in such a subject. I think people can easily get confused on what exactly they know. I personally wouldn't have known the difference between "algebra" and "linear algebra" at that age. Maybe this person is confused as to the terms and it wouldn't be helpful to throw a complex analysis textbook at him. Then again maybe he is at htat level and thank god I don't have to take my PGRE against him.
     
  14. May 6, 2009 #13
    It may have been rude but Hower's point is valid. It's hard to judge exactly how well of an understanding you have if you've studied something on your own. I know I've made the mistake of overestimating how well I know something many times. It's possible that the OP does have good background and is ready to study LA but she/he should question whether they do.
     
  15. May 6, 2009 #14
    Granted, but it was still condescending, and that's all I was trying to point out. When I was called out on it, I specified what I meant. I don't think anybody disagrees that Hower could have been more tactful.

    And I can only assume that people who ask for something know what they're getting into. The books I linked to assume a level of mathematical maturity on par with an understanding of linear algebra and possible some abstract algebra as well. The OP can probably judge for him/herself based on the level of understanding of that material how much they really know about math.

    Oh well. I just think it's a little pretentious to offer advice where it isn't asked for. The OP didn't ask "do I know what I'm talking about". And he didn't make any claims beyond that he had been doing some self-study... certainly nothing to argue with. I don't disagree with suggesting that self-study may not be the most effectual method of doing things, and that making sure you have a strong grasp of the fundamentals is important, but... anyway, I think you guys see what I'm saying.
     
  16. May 6, 2009 #15
    The guys on my PhD dissertation are probably masters of their fields and have many many years of experience and know-how. The people on this forum are in all shapes and sizes and there is no guarantee that they know algebra from their blowhole, so questioning someone's knowledge and judgment is very permissible and necessary. I would rather someone had critically questioned what I thought I know so that I could get the proper advice I came here for rather than get something far too difficult for my level chucked at me (which I would try, fail to understand, and then get discouraged).

    The only reason I see not to question someone's competence is vanity and anyone who's coming here for the sake of showing off deserves to be affronted.
     
  17. May 6, 2009 #16
    I'm in a similar situation. I've got a whole year till high school and i honestly feel that the way they teach math in school is way too slow. I guess only 5% the stuff i learn in school might be new but other than that it's all review. So i decided to teach myself math, and so far it's pretty good. I've just wrapped up algebra (dealing with factoring, linear equations and applications as well as quadratic equations and applications). Now i'm onto geometry then trigonometry and then calculus. Is this the right path to go so far? And yes honestly i think someone should put up a thread on the logic order of math and it's content.
     
  18. May 6, 2009 #17
    The OP could've been less boastful. I doubt he would speak like he writes; and if he did, then I doubt you would consider Hower's response condescending.
     
  19. May 6, 2009 #18
    Kevin,
    Well, It's very impressive you're just going to be a freshman in a high school. How much Linear Algebra have you learned?
    If you already got some "basic" LA, then I recommend you to use Paul Halmos' Finite-Dimensional Vector Space, actually it's the textbook we're using in honors abstract linear algebra course for advanced undergraduates. Warning, the textbook is written in a very concise and so-called mature way and thus it's hard to understand. If you find it hard, then you probably like another text, Hefferon's Linear Algebra, http://joshua.smcvt.edu/linearalgebra/ it's free and it's a superb text.
    Actually, since you are so far ahead, I do NOT suggest you work on Calculus. You may start learning Real Analysis instead, since you want pure math.
     
  20. May 6, 2009 #19
    "The guys on my PhD dissertation are probably masters of their fields and have many many years of experience and know-how."
    The way I was raised, respect is not conditional on what I perceive of other people. I don't think it's being entirely respectful to talk that way to people. Yes, the OP did say he was an advanced student. He certainly is, if he's studying Modern Algebra in the 9th grade. There's a difference between arrogance and being frank.

    "The people on this forum are in all shapes and sizes and there is no guarantee that they know algebra from their blowhole, so questioning someone's knowledge and judgment is very permissible and necessary."
    Well, reasonable people can have different opinions. I find it a little pretentious, whereas you think it's to be preferred. I guess variety really is the spice of life.

    "I would rather someone had critically questioned what I thought I know so that I could get the proper advice I came here for rather than get something far too difficult for my level chucked at me (which I would try, fail to understand, and then get discouraged)."
    Again, I guess we'll just chalk this off to a difference in opinion. Who knows how the OP feels about the comment? I would have found it offensive, which is why I piped up.

    "The only reason I see not to question someone's competence is vanity and anyone who's coming here for the sake of showing off deserves to be affronted."
    Well, the only reason not to punch strangers in the face is weakness, but people don't run around assuming people can take it.
     
  21. May 6, 2009 #20
    "Actually, since you are so far ahead, I do NOT suggest you work on Calculus. You may start learning Real Analysis instead, since you want pure math."

    I would respectfully disagree. IMHO, a little knowledge of the plug-and-chug mechanical get-an-answer calculus is useful for introducing some terminology, if nothing else.
     
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