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Self study topics and material for mathematics

  1. Sep 11, 2011 #1
    First of all, i would like to appologise if there is already a topic related to this on which i could have posted. I was not able to find it but also only skimmed through the search results.

    Mathematics has not been one of strong points. In middle school i struggled with it - i didn't spend any time at it. In highschool i was forced into it because i needed it to further my studies and go to University later on. However i memorised it. It was for the most part without understanding, and while it got me into university i did not grasp it. Somehow, though, because i did so much of it i started to become curious; i slowly became more interested in it for, well just the concepts it brings. The problem is that i have many loopholes and gaps in my knowledge of it. Some parts i grasp, some i don't.

    As a result i have decided to go back and fill in those gaps. The best way i could think of doing this was just to revisit the topics covered from middle school up until university, and eventually beyond. For this ofcourse i picked out some textbooks and decided upon a general plan. Knowing that in the US the topics are covered roughly in this order: Algebra I, Geometry, Algebra II with Trigonometry, Pre-calculus, Calculus (up until University), that is the order in which i think it is best to tackle them. My plan is to go through the respective textbooks and if i find an area i already understand then i will fast forward through it, but hopefully that will also allow me to focus on the parts which i dont grasp - the gaps i mentioned.

    Here are the textbooks i've decided upon after some browsing on the internet:

    Algebra I: Expressions, Equations and Applications, by Foerster
    Geometry, by Jacobs
    Algebra and Trigonometry: Functions and Applications, Foerster

    This should bring me up to speed with Pre-calculus level (which i also understand is similar to Algebra II and Trigonometry, except that it might bring some additional depth and also introduce limits). The dilemma i face is which Pre-calculus textbook to choose? The only one i have considered so far is Larson's Precalculus with Limits, 2nd Edition, but i was wondering if anyone might suggest better ones? I've seen 'Principles of Mathematics' , by Allendoerfer mentioned a few times and read very pleasant commentaries on it. Would this cover the same subject matter or beyond it?


    Going past the pre-calculus level, i was thinking of one of these for the Calculus.




    Would you have other recommandations on this or on picking one of the two in particular? Also, while we're at the Calculus matter i would like to ask: Would a whole Calculus textbook cover Calculus I, II and III aswell, and perhaps even go beyond?
    Suggestions and recommendations in this would be very welcome.

    Going Beyond Calculus
    Assuming i go beyond the Calculus topics (when and if), what subject matter should i next pursue? What would follow next in the educational pattern? This is an area i am particularly blurry about since i didn't go beyond Calculus when i entered University, so i don't really have any idea what lies beyond. Suggestions, recommendations and such would be most welcome in this area. I would like to outline that i am not interested in one area that would be particularly useful to this career or that career, rather i am looking to broaden my knowledge on mathematics with this.

    Mathematical reasoning and logic

    I've seen this mentioned quite a few times and (even to someone as unknowing as me) it should be very interesting. What textbooks or material would you recommend for this? Perhaps something at an introductory level at first (as i am totally new to this field), and then further recommendations to build up upon it and perhaps reach more advanced levels.

    I would like to emphasise the fact that i'm interested in learning and understanding the mathmatical concepts involved, not just to say i've done them. I've also read a little bit and will continue to read the 'who wants to be a mathematician' sticky in this forum and it seems very interesting, especially since it contains some unique reading and information.

    That would be it. I guess what i'm asking for is some indications about how i should tackle the task at hand, preferably with suggestions of materials (textbooks) on the mentioned topics.

    Thank you for reading :)
    Last edited by a moderator: May 5, 2017
  2. jcsd
  3. Sep 11, 2011 #2
    Well, I'm an engineering student currently in a differential equations class (think the next step beyond calculus), and I've never taken a geometry course. Occasionally I miss it, but most of the axioms are just that: common sense. I think you could safely skip geometry and not miss anything. I also think you'll do fine with just algebra I/II and precalc/trig.

    Depends on what field you're going into. If physics/engineering, the next step is ordinary differential equations. I've only ever delved into one textbook on ODE's, but I've found Boyce and Diprima's Elementary Differential Equations to be quite satisfactory. It was good enough that I was able to get a head start in my class over the summer. You don't need the newest edition, so you can find a very cheap copy online. I also recommend going to Khan Academy. He has a way of explaining what's going on behind the scenes that really helps with understanding the material.
  4. Sep 11, 2011 #3
    Thank you for the reply. I've edited the initial text in the hopes that my situation is better expressed in this way. I'm not sure on a particular field at the moment, as i've said i'd just like to understand more about it at this point.
  5. Sep 11, 2011 #4
    hi for going beyond calculus i suggest first taking a course in linear algebra as it is used in many areas of higher mathematics and it is good to be acquainted with its ideas. if you haven't already you definitely should learn multi-variable and vector calculus (calculus III) along with linear algebra as they go hand in hand. Once you learned these subjects which are usually courses a first year mathematics major would take you can really go dive into many other subjects.

    If you are interested in the more applied side of math you can try ordinary differential equations. The book by boyce and diprima is not the best but it should give you a decent grounding in the techniques. however if you want true understanding and not just memorizing techniques you may want to pick up another book such as vladimir arnold's ordinary differential equations.

    if you want to go more on the pure side, you can go into abstract algebra, real analysis, or topology to name a few. these are proof based classes however so if you are not familiar with reading, understanding, and writing formal proofs it may be good to take a proofs class which is usually called introduction to mathematical reasoning or something similar to that. if you have the proper foundations and a willingness to work hard then these courses are definitely within your reach.
  6. Sep 11, 2011 #5
    Yes well, the problem is i don't have the option to take such courses. Hence i rely on books/textbooks explaining these. :) Could you point me to some explaining mathematical reasoning?

    And yes i am interested in actually understanding and not merely memorizing. I would prefer if suggested reading/materials and textbooks would favour a proper understanding.
  7. Sep 17, 2011 #6
    well one choice for multivariable calc and linear algebra is "Linear Algebra, Vector Calculus, and Differential Forms" by John Hubbard. it gives a unified approach to these concepts and is very rigorous and focused on understanding rather than simple computation.
  8. Sep 18, 2011 #7


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    Personally I'm not a fan of Foerster:
    - The explanations aren't as clear as I liked.
    - I find a disconnect between the instruction in the book and the harder problems.
    - There's no color in their diagrams/photos/illustrations.
    - I don't like the method presented in teaching word problems.

    If I were you, I would instead get the Algebra books by Lial:
    - Introductory Algebra, 9th ed. (ISBN-13: 9780321557131)
    - Intermediate Algebra, 9th ed. (ISBN-13: 9780321574978)
    These are "self-teaching" texts -- the teaching is in the examples. I have these, and if I wasn't too late with the textbook forms at my high school, I would have adopted them in my Algebra classes.
  9. Sep 18, 2011 #8
    Why not go for the decent math texts?? Try "basic mathematics" by Lang. It covers everything in algebra and geometry. And it's a text that isn't dumbed down at all.
  10. Sep 18, 2011 #9
    Well there seem to be some diverging opinion - one vouching strongly for Lial and the other for Lang. I have managed to actually see a preview on amazon of Lang's book and it seems pretty good. He seemed to explain in a good manner, as in he tried to make you understand why this notation is this or why this is the way it is, not just slap it in front of you. Cheapest one i could find was about 25$.

    About the Lial ones - i found some of the seventh edition for only 5$. I think they should be the same or nearly the same content-wise?
  11. Sep 18, 2011 #10
    I don't understand why you are worried about materials for advanced classes, when you are still learning the basics.

    It will probably take you at least a couple of years before you are ready for, say, differential equations. If you waste time looking for a book on that now, not only will you be taking time away from study, but you might end up buying something that you won't use, because something better will have come along by then.

    For example, look at this course for self-study in calculus:

    That's probably the best course you can find today for first-semester calculus. It's free, it's from MIT, it is written especially for home study, it has all the video lectures and class notes and homework and exams and solutions. It's perfect for you. And it didn't exist two years ago. Who knows what will be available two years from now?

    So just take your classes one at a time, and then see what's best for your next class when you're ready for it.

    And to answer your question about precalculus books, I like this one:

    Pretty hard to beat for ten bucks (used).
    Last edited by a moderator: May 5, 2017
  12. Sep 18, 2011 #11
    As i said, i have gaps - some material i cover quickly as i already grasp, some (where the gaps are) i cover slower. Thanks for the MIT link aswell the pre-calculus recommandation! I think you are correct, however, there will be awhile, atleast months, until i start calculus. Thanks for the advice.
  13. Sep 18, 2011 #12
    I too recommend Precalculus - Mathematics for Calculus by James Stweart. It is a great book. Older editions should be available quite cheap second-hand.
  14. Sep 25, 2011 #13
    While I agree with the second half of your post, I do not agree with the Geometry half.
    A rigorous study of Geometry will not only help develop "mathematical maturity" (I always find this phrase a little funny), but will also give you tools in identifying symmetries and structure in Physics and Engineering. Geometry is a great way to understand many of the key topics in physics (perhaps the most obvious being GR). Geometrical intuition is something that everyone could use if they plan on having a deeper understand of the Physics they are working on (though I'm less familiar with Engineering topics, I'm sure it couldn't hurt).
    While Geometry isn't *necessary* to be come good at the calculation based work (Pre-Algebra through Diff Eqs), it's necessary for (what my professor consider) the first "real" courses, like (Linear Algebra upwards).

    **I'm not arguing that there isn't real math in Diff Eqs or Calculus, it's not my place to, but I think it's safe to say that there is more substance to the higher-level, abstract material**

    Just my thoughts,
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