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Personal Mathematics Overhaul

  1. Sep 4, 2011 #1
    Hello All :smile:

    Here is the quick version of my story to bring you up to speed. I did not learn any mathematics in high school because i dropped out. All of the math that I have learned was after I turned 25. Moreover, all of the math that I learned was in an engineering curriculum in the USA. So basically I can do Algebra, Trig, Calculus and differential equations. My foundation and my fundamentals are not so great. Unfortunately, I did not discover the value of going 'beyond the syllabus' until after I completed all of my maths in college.

    I am now in a position where I, for my own personal satisfaction, want to rebuild my mathematical knowledge base, from a pure maths point of view. I am not sure how to go about this. I am not sure where to start. The expression "you don't know what you don't know" comes to mind.

    What I have decided to do as a start, is to go through Spivak's calculus and relearn calculus from a rigorous standpoint. And anything else that I encounter along the way that is unfamiliar, I learn it as I encounter it.

    I woud like to ask those of you who feel well prepared mathematically what subjects you learned in high school? Algebra, Geometry, and Calculus come to mind. Anything else?

    And can anyone recommend any texts that are rigorous in nature but also assume little prior knowledge (i.e. rigorous introductory texts)? Online/free materials would be best, but all suggestions are welcome. I would really like to learn math correctly and with rigor in the way that a mathematics major would learn it.

    Thanks for your time :smile:
  2. jcsd
  3. Sep 5, 2011 #2


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    I recommend taking a look at a mathematical methods book, and then identifying what are the areas that are problematic. Once you identify them you can look more in depth. I don't recommend to jump right into the abstract if you feel your foundation is shaky. It is better to think in terms of a gradient and let yourself in. Also, the more you go through the topics at both the mathematical methods and the subject specific book, you'll repeat and learn more.

    A last comment, DO EXERCISES. If you cannot do the exercises you must either take a step back (go to a lighter treatment) or reread the chapter from another reference to make sure you follow.

    I recommend to take a look at Mathematical Methods in the Physical Sciences by Mary Boas as your "port of departure" in to your math adventure.
  4. Sep 5, 2011 #3
    Reading calculus just for the sake of it can be really boring (at least for me), especially since textbooks are filled, for the sake of completeness, with every theorem possible. Unless you really know what you are doing, which, if you did, you wouldn't be reading calculus, it is a pretty certain way to lose your motivation.

    Something I really enjoy doing is taking something I usually use, e.g. the gradient in polar coordinates and try to find out why it is the way it is. More often than not, things like this take many hours and you learn a lot of useful things. The best part is that in the end you will be able to use the tools that you already know much better, as well as picking up a few cool stuff along the way (which are usually the topic of the next exploration :) )
  5. Sep 5, 2011 #4

    I commend your effort of learning math out of your own will.:biggrin: If you're looking to learn from a pure maths point of view, I suggest getting a proofs book-Spivak is heavy in proofs- (assuming you don't know to do them) and Spivak/Apostol (provided you've got basics of Calc 1 & 2) for a rigorous treatment. I must agree with Pyrrhus about doing ALL the exercises. Good Luck

    Link to a very good Proofs book by Daniel J. Velleman:

    Last edited by a moderator: May 5, 2017
  6. Sep 5, 2011 #5
    Hello Pyrrus :smile: Thanks for your reply. I should have mentioned that I have already taken a Mathematical Methods for Engineers graduate course. I did quite well in it and it was probably the first course I took that required any level of rigor. This is what spawned my interest in math for the sake of math and not for the sake of application. I would really like to now go through algebra and calculus from a purely math point of view. I must say that I am not terrible at either of these subjects, I actually do quite well. But as I said (admittedly, not very clearly) I wish to revisit these subjects from the mathematician's point of view.

    Hi meldraft :smile: Reading calculus for the sake of calculus is actually why I am doing it :wink: I can do calculus just fine, but I really want to get a 'from the ground up' understanding of it.

    Hi BloodyFrozen! :smile: I actually have that book! Great! I purchased it while taking the math methods course, but it was naive of me to think I would have any time to dedicate to it while taking 4 graduate courses and doing my research :yuck: Now I have the time!

    I will be going on a 6 week trip to Africa and will have plenty of quite nights to read and work problems. I will be bringing Spivak and Velleman for sure. I would really like to bring one more thing. Perhaps something on introductory set theory? I know that in the classes I have taken, I have only encountered setbuilder notation (at least that's what I think it is called) a handful of times. So when I see things like [itex]\mathbf{R}\rightarrow\mathbf{R}^2[/itex] or [itex]\{x\in \mathbf{R} : x = x^2\}[/itex] I start to get discombobulated :redface: so I think I need to work on that. I feel like a good introductory but rigorous algebra book would present that stuff and maybe set theory is overkill. Opinions?

    :smile: Thanks for all the replies so far! I think we will eventually flesh out a good track for me to follow.
    Last edited by a moderator: May 5, 2017
  7. Sep 5, 2011 #6
    The Velleman book is pretty good, but there are other roads to learning proofs depending on your interests. Spivak may be a good place to start as it is well-written and should cover ground that you are mostly familiar with. Another option, if you don't want to spend so much time on Calculus, is Abbott's "Understanding Analysis." It should pick up where your old Calculus texts left off.

    If you want to learn Linear Algebra in more depth, the Axler book is very good. If you want to delve into Abstract Algerba, Pinter's book is inexpensive and very well written for someone new to the field. For Complex Analysis, you could try Needham's "Visual Complex Analysis" - I enjoy it, but it might not be for everyone.

    Those are some of my favourite introductory ('pure') books at the moment...

    Edit: I just saw your reply above. If you are wanting basic set theory, Velleman should be good for you. He goes over most of what you would need for other subjects.
  8. Sep 5, 2011 #7

    Stephen Tashi

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    A person does not necessarily have to be clear about his goals, but some clarity abot them is certainly helpful to people who try to advise!

    Question 1: Is the main goal of this to gain a deeper understanding of calculus and calculus realated subjects (such as differential equations)? Or do you aspire to study other branches of mathematics someday, such as linear algebra, number theory and topology?

    Question 2: What timeline are you trying to follow. For example, would you be happy to study advanced calculus for the next 10 years? Is that too long a time?

    Suppose an "average bright person" sets out to study mathematics in a detailed fashion, making sure he grasps everything that is in chapter 1 before going on to chapter 2 and making sure he understands calculus completely before taking up other branches of math. In my opinion, he will usually become bogged down. The material he covers will be throroughly understood but not much material will be covered. If that doesn't bother you then by all means get a pile of books and study. However, if there is a particular type of math that you are unable to do but wish to master, then a method that works faster is to tackle that type of math directly, get confused by various things, go to more elementary books and the internet to ask about them. This is very frustrating to some people, who only want to move along in a confident manner. However, if the main purpose is to learn some particular field of math as opposed to have mental comfort, the direct attack is what I recommend.
  9. Sep 5, 2011 #8
    Enhoy your time! [itex]\{x\in \mathbf{R} : x = x^2\}[/itex] (this is set builder-notation:wink:) As someone else has already mentioned, Velleman will teach you an intro to set builder.

    You might want to bring some other texts because I think you'll complete Velleman before then but don't get too hung up on reading math! You're in Africa for God's sake! Anyways, if you want to start Abstract Algebra, I'd recommend Pinter (so far it's good, just started a few days ago).

    I'd also recommend thinking about what Stephen Tashi said.
  10. Sep 5, 2011 #9


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    Certainly if you feel confidents with the computations of the mathematics. I recommend to move to a Set Theory book, and then to a Real Analysis book. I personally like Ken Binmore's books: Logic, Sets and Numbers, Mathematical Analysis, and Topological ideas. They are at introductory level and self contained.

    Obviously from your example, you need to understand sets, relations, functions, cartesian product and other foundation concepts.
  11. Sep 5, 2011 #10
    Excellent. :smile: I have the Axler text and will bring that too. Thank you for the recommendations.

    Hi Stephen :smile: I don't see the options as being mutually exclusive. So both!

    Ten years is pushing it! But time is not so much a factor. I think you have helped me to answer what my main goal is: I believe that if I build a very strong and rigorous foundation in algebra, calculus, and some set theory, I would be in a good position to branch out in the event that I want to. That seems to be what has bothered me the most about my math background: every time I want to branch out, I find that my non rigorous background holds me back quite a bit. I have to go way back and retrace my prior maths before moving into something new. I suspect that this is not uncommon, but if I strengthen my foundations I believe I will not have to go back as far as I usually do.

    That's me! Average, not too dumb :wink: I do think that I could do calculus, algebra and set theory in a detailed fashion, mastering one topic before moving on to another, in a 'reasonable' amount of time. I would do them in parallel of course.

    Splendid then! I will enjoy myself! It would be unwise of me to leave the house at night though (I don't care to be someone's dinner :bugeye: ) so there is plenty of time to read.

    Great! Thanks again for your thoughts.

    So in addition to the suggestions you all have been kind enough to give me, I have 2 more questions:

    1) When I say that I can do Algebra, what I mean is that I can manipulate expressions and deal with exponents and solve equations (kind of weak at inequalities, but I'm working on it) and all that good stuff. I have no experience with formalities in algebra: proofs and axioms and the like.

    If I want to revisit Algebra with rigor, should I be looking for a rigorous algebra text? Or am I now into linear algebra or abstract algebra?

    2) Does anyone know of any free online sources/texts in any of these subjects? Preferably in PDF format. I would rather carry 1 iPad in Africa than 5 texts :wink: I have found many resources in PDF format, but I don't know anything about the authors. Moreover, there are umpteen different kinds of set theory. Is ZFC the type I should be delving into?

    Thanks again for all of your time!
  12. Sep 5, 2011 #11
    Have you taken a class on Linear Algebra, its part of some engineering curriculum but not all. Its very useful in engineering (among others) and generally its a good class to get into more proof based math.

    I'm taking it right now (its not part of our EE program but I'm taking it anyway) and its like half computational and half proof. It also helps to get some type of intro to proof book too, it'll help with some of the notation (stupid sets), logic, and some techniques to writing (and reading) proofs. Discrete math courses are sometimes used in the same way.
  13. Sep 5, 2011 #12


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  14. Sep 6, 2011 #13


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    I strongly suggest a book called "Creative Mathematics" by H S Wall. The book is written in the spirit of the Moore Method where you prove everything yourself. The author goes over Calculus, the elementary trigonometric functions, differential equations, linear spaces and some applications to mechanics. Despite all this, the book is pretty slim...the reason being most of the theorems have no proofs and the exercises have no answers. You have to provide those yourself :)

    It may seem weird at first but it is the best way to learn mathematics...by building it up yourself. If you have an engineering degree, it should be well within your reach. it was used for teaching freshmen I believe
  15. Sep 6, 2011 #14

    This is a good site to find some Linear Algebra Texts (free). I'd try the one by J. Heffernon

    Direct Link:


    Scroll down to the subtitle: "Here Is Linear Algebra", and there's a link for pdf download.:smile:

    There are also books for other math topics besides Linear Algebra if you follow some of the links.

    P.S. - I don't know if you like reading electronically but if you do, it's possible to download online Kindle (not sure about IPad - only certain for PC) and buy books from Amazon to your Kindle account.
  16. Sep 7, 2011 #15
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