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Just starting in on higher math, at 30 help!

  1. Mar 14, 2007 #1
    I'm looking to start doing some self-study over the next few years, or however long it will take(I can imagine the smirks when many of you read "the next few years".. hehe). I have always had a bit of interest in physics and higher math, but unfortunately, was never in higher math classes in school. And so, had no chance outside of self study to persue it. Also...I'm an artist (illustration), and was never interested in anything but computers and art.

    But now, as I get older, I've really found an interest. Not to mention the explosion of new matierial for we, the non-scientific public, to mull over (everythign from Brian Greenes work, to Michio Kaku....).

    So, my question is from one in need of pity.

    How do I begin this long road of teaching myself this stuff, partially, just for the enjoyment. Plus, I can't imagine using one's brain more efficiantly would be of any harm. :) I live in New York City, and have an incredible sourse for resource materials. Any ideas?

    Thanks. Sincerely.
    Justin Cash
     
    Last edited: Mar 14, 2007
  2. jcsd
  3. Mar 15, 2007 #2
    I'm 21 and only got into higher maths and physics a few months ago. I feel old too, even though I shouldn't, but I think that's all do with societal perception and other stuff. IMO, it's never too late to start learning if you enjoy maths. If you enjoy maths, chances are you'll be fairly good at it.

    I think following a syllabus or cirriculum will help you learn, even if you don't go to college or whatever. I can't reccomend any specific books since I live in the UK, but try find whatever text books they teach at high school / college and go through em! Anything with explanations and lots of questions.
     
  4. Mar 15, 2007 #3
    Congratulations! Self-study of a field is one of the most rewarding (and most painful) things you will ever do. :biggrin:

    Having started my own self-study path back in the 90's, I think the key is to narrow down your interests or you'll wander aimlessly in the immense landscape that is mathematics.

    My own interests are in General Relativity and QFT/LQG... so that helped steer my course (which I haven't completed, yet). That path included:

    Bert Mendelson, "Topology" Dover
    Walter Rudin, "Principles of Mathematical Analysis" McGraw Hill
    Robert Bartle, "The Elements of Real Analysis" Wiley
    John Lee, "Introduction to Smooth Manifolds" Springer Verlag

    All of these are very readable books, but will require serious study to get through. Regardless of your path, these might be worth going through simply because they are now so much a part of contemporary physics that not knowing these topics may well hinder you in any physics quest you may have.

    Should your interests be different from mine, you still should find the reason you're studying all this math (although, I've been told it's not too peculiar to study it for its own sake, but the jury's still out... :rofl: ). Then I would recommend picking up a solid textbook in that field and just start reading.

    When you get stumped, figure out what's stumping you and pick up a book on *that*.

    (I suggest keeping notes on your path... I've backtracked on backtracks a few times and then wondered where I left off!)

    Much luck! And, as always, come back here (and other web resources) for advice and questions as you study.

    ZM "mathemetizing himself since 1993"

    PS Please, keep good notes as you go... you'll be referring to them frequently.
    PPS: Unlike your (well, mine, anyway) official student days, you really *should* work the homework problems :biggrin:
     
    Last edited: Mar 15, 2007
  5. Mar 15, 2007 #4

    arildno

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    Main advice:
    DON'T SKIMP ON THE MATHS!!!
    This cannot be emphasized too strongly; use your time to develop your mathematical skills, and most importantly, your understanding!
    Furthermore:
    There does not exist anything more important than the PROOFS in maths; understanding these, and be able to prove them, and exercise proofs on your own is far more important than being able to perform any one particular algorithm correctly.
    Those students who do not master maths are precisely those who have the wrong focus: Namely trying to memorize algorithms without using sufficient time to understand the logic behind its construction.
     
  6. Mar 15, 2007 #5
    Thanks to all..

    Thanks so much for all the help!

    I am in the middle of getting transfered to another design school in the city, to get my degree(and on with the rest of my life HOPEFULLY.. haha), so the next week or so will be filled with that. But once that is done, I will have more time to ingest other things other than getting my portfolio together. If you're not artistically inclined, I do wish you could be, but I also do pity those of us who are trapped within it's lovely walls.

    I sometimes feel I can empathize with mathmaticians.. in that, the passion they have to get to the final goal keeps them tied to the love of their lives. Whether we have down time or not, our goal remains.

    As for the advice, I need to sit down and work out my interest in math/science and figure out which field to look into. I'm still learning on a bystander level, watching documentaries and the like, trying to get a surface level view of what is being studied at the moment. Hopefully that might spark a wire in my brain and set me on a path.

    I've also emailed the City College of NY, where (last I heard), Michio Kaku teaches. I emailed them about the possibility of taking a course. I'm not sure of the prereq.'s, but I imagine they are pretty stout. But we'll see.

    Thanks again. :)
     
  7. Mar 26, 2007 #6
    Some books and tips

    First I feel I should give a small "warning":

    In maths, even if the book explains everything step by step as it goes along, you will have to stop and think, and some things will seem difficult at first. Even if you are very smart, you will read a maths book considerably slower than a normal pocket book.

    That's really a good thing though. It means you are being forced to think wider and cleverer. Don't let that intimidate you, just don't give up.

    I don't know how much mathematics you know from school and so on, but you are lucky in that there are plenty of EXCELLENT books on mathematics.
    I have been comparing my books with the corriculum books of a psychology student I know. While they have some good, some bad, and some horrible books, all the math books I've used and read have been fantastic. In maths there are plenty of authors with a passion, and plenty of understanding, and there is very little room for just throwing fancy terms around, or talking nonsense.

    The books in my experience differ somewhat in their focus, and in what they presume you already are familiar with. By this I mean that one book might give you lots of examples of applications in physics, and expect you to be content with getting to proofs and rigid analysis later (or not at all), while another might focus on building the theory little by little, and use purely mathematical examples.
    I tend to prefer the latter kind of books, mostly because I am frustrated by the uncertainty involved in applying formulas I don't feel I know the limitations and sources of I think. This differs from person to person though.

    When it comes to prerequisites, some books presume that the reader has done a lot of work with maths and will feel comfortable seeing familiar relations in the book, but most I have come across actually start more or less from scratch. My first book in abstract algebra, or the book I used in real analysis, for example doesn't require any knowledge beyond knowing what + means and things like that. They would be extremely difficult (impossible for most people) to get through if that is all you are familiar with though.

    One thing you should know is that calculus, which is basically a set of methods for calculating changes in formula values as their variables increase or decrease, is considered fundamental knowledge for further studies at all the higher learning institutions I have come across.

    There are literally hundreds of calculus-books, and I very much suspect there are hundreds of very good ones, but to give you an example, the book I would turn to if I were in your shoes is:

    Calculus - A Complete Course
    by Robert A. Adams

    That one starts at the basics and builds on them to cover quite a lot of ground. It will probably last you a full year, and when you've gotten through it you will be able to get further into:

    - Differantial equations
    Which roughly speaking relate speeds of changes of function values to the function values themselves, and ask you to find out what the unknown function which has the described properties can/must be. Very central in modern physics.

    - Analysis
    Which, again roughly speaking, explores the foundation for the calculus techniques, and the nature of the continuous number line.

    - Linear algebra
    Which, again very roughly speaking, explores what you can deduce from knowing that some objects (such as numbers, but we could also be talking about for example formulas) are in some constant proportion to oneanother.
    [itex]\begin{array}{cc}x + y = 3 \\ 2*y = 2\end{array}[/itex] would be a trivial example, where the "objects" as I put it are the unknown numbers x and y.

    - and a huge load of applications

    , without too much trouble. Of course, there would still be a lot to learn, but you would have a foundation to build on.

    A subject I like a lot is abstract algebra. It, again very roughly speaking, explores formulas and the manipulation of formulas, when the formulas need not represent numbers, and need not follow any of the rules that you are used to (such as [itex]a*b = b*a[/itex]), unless such rules have been defined in the particular case. A formula in abstract algebra can perfectly well decribe something like turning a cube around in space ([itex] a [/itex] might represent turning the cube upside down for example).
    Abstract algebra is usually seen as building on linear algebra, and linear algebra is usually covered after or alongside calculus (there is a little linear algebra in the calculus book by Adams), so you have kind of a natural progression there.

    Here are three books I have enjoyed a lot, which demand nothing more than knowledge of simple compulsory school mathematics:

    Linear Algebra and its Applications
    by David C. Lay

    A First Course in Abstract Algebra
    by John B. Fraleigh

    A Companion to Analysis
    by Tom W. Körner

    Of course, as you can probably tell from my previous mention of these topics, getting through these books will be a handfull if you haven't done a bit of university level maths.

    Some classics, which can be said to concern mostly quantities you can count to are:

    Concrete Mathematics
    by Ronald L. Graham, Donald E. Knuth, and Oren Patashnik

    An Introduction to the Theory of Numbers
    by G. H. Hardy and E. M. Wright
    (This one is getting a bit dated. But of course, being that this is maths, what it teaches is still true and usable, and I certainly had fun with it)

    My book selection here is affected by my interests in maths. You should find out what you are interested in. But don't miss calculus, or you'll get into trouble down the line.

    Some readers of this post might scorn my crude descriptions of the mathematical subjects I have mentioned, but I think the descriptions should be sufficient to give a bit of a feel for what the subjects are about.

    If your interest is in modern psysics, you will want to have (in addition to a thorough understanding of calculus) some familiarity with differantial equations, statistics, linear algebra, and eventually some topology.

    I go to a university, but in practice I do self study, and I have all the while (not that this is necessarily a good idea). I would advice you to follow some progression, like those described on lecture schedules for courses at many educational facilities (available online).

    It is important to work some excercises, or what you read won't stick, and you'll run into situations where you can't even figure out what you're reading about.
    Conversely, if all you do is excercises, you will get stuck and not know how to solve them.
    So get a bit of both.

    And again: Keep some progression. Don't read "every once in a while", or a full day every two weeks, or something like that, or you're not likely to get far. You should try to keep something fresh available in your mind at all times.


    The subject you are about to enter can be great fun, and really change the way you think, and expand your scope.

    Good luck!
     
    Last edited: Mar 26, 2007
  8. Mar 26, 2007 #7
    My simple suggestion is - get Maple. I'm 53 years old and just began studying calculus less than a year ago. I discovered Maple just a few months ago and never figured there was anything like that which could make even very difficult math so easy and fun.

    As far as books, if you're have an interest in, and are new to calculus, a good place to start is "Idiot's Guide To Calculus". It's an excellent book. You can make a paper cover for it if the title bothers you though - :-) Another good one, though slightly more advanced is "Calculus - An Intuitive Approach" by Kline. There is a ton of really good material around. Have a look at: http://online.math.uh.edu/ and a very good page at that site with lots of videos: http://online.math.uh.edu/HoustonACT/videocalculus/ besides the ones just mentioned, that site is full of lessons and videos.
     
  9. Mar 26, 2007 #8
    maple/matlab (or octave)/ MIT OCW open courseware website/

    Mathematics Handbook and/or Physics Handbook(STocker&Harris)

    Stewarts Calculus Text, Some Linear Algebra Text(eg. Anton or friedberg)
    Some Vector calculus Text(eg Marsden), DiffEqs, Larsons Proving Techniques Tech.
     
  10. Mar 27, 2007 #9

    mathwonk

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    you are thirty, and expect smirks from me? I am 64, and would give a lot to be 30. do whatever in the world you want to do. you have a gift that I cannot even aspire to, youth.
     
  11. Mar 27, 2007 #10

    JasonRox

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    My best advice would be took take a basic algebra textbook or intermediate high school textbook. I know everyone is saying start higher than that, but honestly, you might not remember the small stuff and the tricks to solving an equation.

    You need to revise those skills. Also, don't worry about taking longer. This only take days to read.
     
  12. Apr 2, 2007 #11
    I like Ghandi's quote: "Live as if you were to die tomorrow. Learn as if you were to live forever." Time is a illusion. It exists, but we are foolish to let it affect us.
     
  13. Apr 3, 2007 #12

    Chronos

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    Geometry, geometry, geometry. Relationships are the most fundamental aspect of both math and science [e.g., quantum gravity]. World class scientists [and chess players] reflexively think in geometrical terms, not numbers. Numbers are to scientists like colors are to artists. Unsurprisingly enough, it is extraordinarily difficult to master geometrical thinking.
     
    Last edited: Apr 3, 2007
  14. Apr 3, 2007 #13

    Hurkyl

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    It differs from person to person. e.g. some say that algebraic geometry was invented to allow geometric thinkers to do algebra -- others say that it was invented to allow algebraic thinkers to do geometry. :tongue:
     
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