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Trying to read my way through math & physics

  1. Jun 20, 2007 #1
    I am 21 and studying to be a psychologist. I am NOT studying to be a mathematician, or a physicist, or a scientist, but I am very interested in all those subjects. I find that I generally learn much better from reading (+ writing over what I've read) than I do from listening to a lecture. So I don't like going to class when I don't have to, and I don't really have much time to go to regular math/physics classes anyway.

    Recently I've been looking at the possibility of studying mathematics on my own just from books, with as few actual classes as possible - maybe none at all if I can get away with it. My goal is to be able to understand advanced fundamental physics.

    I would like to get people's suggestions for what are the very best books in each subject that I need to study in order to make it to that goal.

    First let me make clear two points related to how I have been going about studying this stuff:

    1) I am well aware that this usually takes 8 years or so of intensive undergrad / graduate study to master. I don't mind if it'll take me years and years and years of free-time study to get through everything.

    2) I do not like doing enormous numbers of practice problems. First of all, doing all those problems tends to get very boring very fast, which dilutes my interest in the subject (I once heard it called "Drill and Kill" - great name). Secondly, I find that it isn't really necessary, at least for me: If the book is good enough then I can get a good understanding of what's going on just from reading it; writing it over in my own words and doing a few practice problems will give me a bit of practice in using the new concepts; and then going on to the next subject gives me yet more practice, since the new concepts almost always reuse the previous concepts. So when I look for a book, I generally try to shy away from the big (and very expensive) textbooks which are almost entirely full of practice problems, and instead I just try to go for the most clear and concise book that I can find. (Of course, if the only clear book is a big expensive textbook then I'll go for that too, but I suspect that most of the time I'll probably be able to find a more concise book with equal or better clarity.)

    So far I have reviewed Algebra I and II (which I've forgotten in the time since I first took those classes in school), and have started reviewing Precalculus. I have been using the CliffsQuickReview math series to re-study Algebra I and II, and I've been very impressed with their clarity and ease of use, though the frequent typing errors can make things a bit difficult. I am considering getting the entire set, which includes Algebra I, Algebra II, Geometry, Trigonometry, Precalculus, Calculus, Statistics, Linear Algebra, and Differential Equations.

    That should hopefully take me through the basics. For intermediate and advanced stuff, I found a list of books on the following website: http://math.ucr.edu/home/baez/books.html

    Does anybody have any additions or comments to that list, or any comments on the choice of CliffsQuickReview to study the basics? Or any other suggestions for that matter?

  2. jcsd
  3. Jun 20, 2007 #2
    being exposed and knowing of fundamental physics is one thing but understanding comes from being able to apply what you know and that comes from doing problems, lots of em. theres no way around it. its just like anything else. someone can tell you how to hit a baseball or shoot a basketball but until you do it yourself and get a feel for what actually happens personally and work through all the weird things that can happen you don't know how to do either thing. you will also not be able to understand advanced fundamental topics like special relativity and quantum mechanics and even E&M without being extremely comfortable with the math.
  4. Jun 20, 2007 #3
    You're probably right. But I find that I learn differently than most people do, so I'd prefer to test it and find out myself. Anyway, if I find you're right it'll be very easy to switch to doing all the problems.

    But what about that list (which was the real point of my post)? Are there any really good books that should be added? Are there any books there that aren't really all that great, and that have better substitutes? Is it a comprehensive list? Can it in fact take you all the way through intermediate and advanced mathematics / physics?

    And what about the CliffsQuickReview series (the other point of my post)? Are they comprehensive enough to be used for a complete course in basic mathematics (Algebra I through Differential Equations)?
  5. Jun 20, 2007 #4
    no cliffs are not comprehensive. baez's list is more than comprehensive. i guarantee you will not get far without doing problems
  6. Jun 20, 2007 #5
    First, my background is math (enough math for a B.S.), I know very little about physics, and personally I don't care about physics.

    You have to do problems, but I agree with you that doing the same thing over and over is very boring. Unfortunately, the subjects you have listed usually have a billion exercises in them and emphasizes that kind of drill and kill style. When going through those basic subjects you should probably skim the exercises doing enough to feel comfortable (and if you can't figure something out, post your question here). You will, though, need to make sure that you understand everything! Math is one of those subjects that layers on, and so you need a solid foundation.

    Personally, I would avoid those boring subjects you listed, and try something a little more interesting and fun. First pick up a book on math reasoning (logic and proofs, see below). Then look at a basic book on probability, or graph theory, or combinatorics, and maybe later a book on abstract algebra (sometimes called modern algebra). In my opinion the ideas in these books are much cooler than the silly stuff they teach you in algebra 1,2, calc, trig, etc.

    The only prereq for the subjects I listed are time and some knowledge of math reasoning. You should probably know some basic manipulation rules like how to multiply, divide, fractions, etc. which I assume you already know.

    I would recommend the book: An Introduction to Mathematical Reasoning

    This is a nice ease into math reasoning at a reasonable pace. The first 100 pages are about logic, proofs, and fuctions which are essential to almost all the math theory you will see. The rest of the book is combinatorics and number theory (mostly number theory) which you now have the background for (wow, in just a few weeks you are doing some cool math). You will notice, and I think like, that there are much fewer exercises in books like these (math theory books that is). Instead of have 50 problems per section, there are more like 5, and these problems often make you think (instead of doing the same drill and kill nonsense). The exercises in this book and a mix of theory and application. Meaning that some are proofs and some are just computations. If you like this you should then check out a book on probability (which at your level, will probably be more computations then proofs), graph theory (a fun mix of both), combinatorics (a little more toward the computational side), or abstract algebra (mostly proofs).

    However, you will (eventually) need to go through the subjects you listed, it just can't be avoided (especially if you want to go into physics). You can avoid differential equations if you don't want to go into physics, but everything else will probably be needed at some point. I would take a look at the book I recommended, and at the same time go through the stuff you already have. You might be thinking why you would want to go through math reasoning book and I have the following reasons: 1. It is fun stuff and you don't really need any background for it, and it also gives you a taste of what you will see later on. 2. You will need it if you want to study higher math and hence higher physics. Topology, Algebra, Real Analysis are all theory and you need to have the basic reasoning skills for these classes. You may not need this stuff for physics until you get to senior level or lower level grad classes, but if you go that far you will probably need it (or so I've heard).
    Last edited: Jun 20, 2007
  7. Jun 21, 2007 #6


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    Even if you read a book and think you understand everything perfectly, when you sit down to do some problems for the first time you'll find that you don't even know where to start.
  8. Jun 21, 2007 #7


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    I agree whole-heartedly. I can't count the number of times where I think I've understood a chapter completely along with all its examples, and even know to solve the questions in the examples, without looking, only to get stuck when I start doing the problems.

    EDIT: Could someone sticky a thread for textbook recommendations? I find myself having to search for threads on specific textbook recommendations for specific topics each time I want to check if the library has them.
    Last edited: Jun 21, 2007
  9. Jun 21, 2007 #8
    I second mattmns's recommendation of the book "An Introduction to Mathematical Reasoning." Its very clear and it has all the answers to the exercises. However, it doesn't have answers to the problems.
  10. Jun 21, 2007 #9


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    I'm not from the States, and am not really sure what Algebra I or II consists of, but I agree with Mattmns, An Introduction to Mathematical Reasoning is good for practically everyone who wishes to be amused by mathematics in his free time. It's not very big a book, but it'll probably refresh your memory quite a bit if you've taken most secondary school mathematics courses.

    I also recommend Alan Jeffrey's Mathematics for Scientists and Engineers. It's quite a volume, but it's a very clear and well presented reference, and good to have around. There are some problems at the end of each chapter, and if my memory serves me right, (a recommended reading list as well,) but you wouln't really need to go through them to understand the materials. The book goes through the most fundamental aspects of differential and integral calculus, and takes you all the way (albeit step by step) to partial derivatives and (i think) multiple integrals as well. It also covers complex numbers, differential equations, polynomials, linear algebra, and other mathematical curiosities. It generally follows the English tradition of being an applied mathematics text (as opposed to pure maths), but it was written for engineering and science students. That book will probably cover all the materials of a first or second year of a ordinary degree in most universities in the UK, roughly equivilent to the first two to three years of an undergraduate degree in North America (I believe).

    For some pure mathematical studies, I suggest you go to the library, and borrow books on specific subjects, say chaos and fractals (which are already applied, though not as widely as differential equations or things like that). There is also this practically useless but very interesting mathematical concept of an intrinsic equation which you would be able to find in GCE Advanced Level textbook for Further Pure Mathematics. (You'd be hard pressed to find it elsewhere).

    In fear of getting too specific, I'll stop right here. Just, well, Alan Jeffrey's book and the library are probably your best bet.
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