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I fell in love with mathematics a bit too late when I've already taken decisions regarding my future, career-wise. Now I would like to learn math on my own but I'm a bit confused as where to start. My knowledge of mathematics is comparable to that of a 15-16 year old highschool freshman. I would like to know how would you (if you were in my position) start your learning adventure on your own. There are a lot of resources online, of that I'm sure but I would like to follow a path. I'm pretty sure I can learn mathematics on my own and been thinking about this decision for almost a year.

Thank you!

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You might enjoy reading this thread: mathoverflow.net/questions/7120/… –  Ryan Budney Sep 19 '10 at 23:40
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On top of the great advice already given, you should seriously consider tutoring once you are prepared. –  Tom Stephens Sep 20 '10 at 0:30
    
What is your motivation, how did you fall in love with it, what is it you actually want to learn, why do you want to get good at it? –  NikolajK Jul 9 at 12:37

11 Answers 11

Pardon me if I'm understating your current knowledge, but are you familiar with geometry, elementary algebra and pre-calculus? Those are pretty standard courses for high school students, and I'm sure most standard textbooks is fine to get the basics.

If you are already familiar with that, computational calculus and computational linear algebra is usually the next step, and books like Tom Apostol's Calculus Vol. 1-2 are a nice introduction to single and multivariable calculus with some basic linear algebra.

When you're comfortable with computational mathematics, it might be nice to move on to more theoretical mathematics, like linear algebra, abstract algebra, and real analysis. Some good texts in my opinion are Friedberg, Insel and Spence for Linear algebra, Dummit and Foote for Abstract Algebra, and Rudin for Real Analysis. Those are just some suggestions that I would pick and choose parts which I would want to learn.

I'm currently self-teaching myself as well, so I don't have as much experience as many others here, so they can probably be more helpful.

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up vote 11 down vote accepted

I hope this is useful, I am just describing what worked well for me in the past. I recommend using real physical books (for example, I used Mathematics for the Million - Lancelot Hogben at this stage and I would recommend this particular book also because I am very fond of it).

Also you can a very good grip of the basic mathematical concepts by programming them (e.g. algorithms to convert between bases, factoring quadratics, finding square roots by newtons method when you get to calculus) - if you are just programming for programmings sake you will probably not learn very much mathematics (or at least, most of that which you do learn is by accident), but if you do it as an act of detailed explanation it can be an excellent process to find much clearer understanding of whatever it is you're learning about (and as a bonus you can run them to automatically solve problems that you already know how to do!).

In finding a path it would be useful for you to make a list of the more advanced theorems or applications which fascinate you (you will have to choose your own but I mean things like why are the orbits of planets in newtons universe ellipses, why you cannot square the circle or how raytracing works etc.) - then find out which setting they live in, then study the building blocks needed to get started working on the background - it doesn't even matter if you give up after a couple months extremely hard work by discovering that you cannot possibly study this stuff yet because you will still have got a lot out of it.

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I am a work in progress myself, but I think I can give some advice. I think tackling mathematics as an older person has some advantages because you have a lot more life experience than a 15-16 year old person. You know yourself better. This can be helpful in terms of keeping a consistent work ethic. I base these observations on tutoring a wide number of age groups (at a local community college).

Now, unfortunately, your question is a bit broad and I can't really tell from your description what a 15-16 year old high school freshman might know in your country. In North America, preparation can vary greatly between school districts. So my advice has to be general.

I would set myself a goal of learning the basic topics that are needed at the high school level in the local school system: read the books and do the exercises. Ensuring that you don't have major gaps in the topics covered in your school system has the advantage of making it easier to enter the formal system if at some point later you find that you want to take a class perhaps at a local community college. Such classes are likely to be tuned to the educational background of the typical high school student. As an independent learner myself, I occasionally have the problem of weird gaps in my background.

Mathematics at the high school level is straightforward I think. If you read the book and do all the exercises, you can get through most of the curriculum.

I think it's important to have social feedback. I have a few ideas in this area:

  1. You can participate in online mathematics community such as this site. In addition to this site, I like Art of Problem Solving. I think it's worth it to interact with others and keep your enthusiasm high. I also like the Mathematics Category on Yahoo Answers. There are a large variety of questions and you can compare your answers to others. The questions are quite often in the high school to early college range.

  2. When you get to a level where you feel confident enough, you might tutor either as a volunteer or as a way to make some extra money.

  3. You can start your own blog. I haven't tried it, but a few people have reported that it's a great way to learn new things.

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A good place to sample quick starting parts of mathematics is through the books of Martin Gardner, who sadly, died recently:

http://www.york.cuny.edu/~malk/biblio/martin-gardner-biblio.html

Another suggestion is to sample the essays in the wonderful book (1034 pages):

The Princeton Companion to Mathematics (2008)

Edited by Timothy Gowers

Princeton U. Press, Princeton, NJ.

These essays cover a wide swath of mathematics and there are suggestions about learning more about areas that may catch your interest.

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I can only speculate on your background, so I would recommend only that in addition to the stuff you picked up from school, learn how to program (insert language/environment of choice here). It can greatly help to have another tool apart from pen and paper to try out whatever piece of mathematics you're trying to learn.

After that little prerequisite... it's open season for problems! Math problems don't have to be hard, advanced, or a "hot topic"; all they have to be is to be captivating and fascinating to you! Look around encyclopedias like MathWorld or Wikipedia so you can get a feel of what's out there (sometimes the "Random Entry" function is helpful in this regard if you don't know where to start), and if you find something you like, you can start from there! In addition to that, the bibliographies should give you ideas on what to type into the search engine should the encyclopedia entry prove inadequate in giving you what you want. If you find your background is less than adequate in understanding fully what you saw, that's okay too; this is one good thing about being in the Internet age, the ability to search for information without having to leave your seat.

As a further supplement, go to the nearest public library and take a look at the math section. Since not all the great books are already online, so to speak, a look at the printed pages can be helpful.

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I find the act of constructing a proof to be very similar to debugging computer software. Looking for holes in an argument is very much like debugging because you're asking yourself "okay, this procedure has this kind of allowable input, ?is that what I'm giving it?, and then it outputs this kind of data ?is that what I'm expecting?" and so on. So programming IMO builds up several tools to put away in your tool-chest if you want to be a mathematician. –  Ryan Budney Sep 21 '10 at 13:50
    
Heh, indeed, trying to break your algorithm and trying to poke holes in your proof require the exact same mindset. Building up the discipline to go over such things line-by-line is great practice. –  J. M. Sep 21 '10 at 14:09

I'm also an older person trying to teach myself math. I took math till high school and since then I have taught myself to beginning-graduate level in one area (analysis) and bits and pieces of other areas at an undergraduate level. My suggestions,

  • Time is limited, don't waste it on popular books. Study high-school and undergraduate textbooks and do the problems. It may be a good idea to go to an older generation of textbooks since they are less chatty than the current ones. The Chicago Undergradute Mathematics Bibliography is useful. But don't obsess too much on which book to read. At this level it is hard to get things very wrong. Pick one book and persevere with it.
  • Work in sprints. It takes time to get into a productive frame of mind. So trying to do math for a little while each day does not lead to any progress and results in frustration. Instead, dedicate a week of your vacation or at least a whole weekend for learning something new. Between these sprints revise what you have learned by doing more problems.
  • Get it right the first time. While studying a text if you feel that you don't understand something fully, stop and work on your difficulty. Look at other books. Ask for help on forums like this. Go and brush up on the prerequisites if need be. But never leave a gap. Don't skip more than one exercise in ten.
  • Be patient. A good undergraduate training takes 4 years of work with teachers and peers. My guess is it will take at least a dozen years to achieve the same level of mathematical maturity working part-time on your own. But at the end your understanding will be better than an undergraduate's since you will have had the benefit of greater intellectual maturity and more time to reflect.
  • Is it worth it? Math is no doubt beautiful. But the time you give to it comes at the cost of your career, personal life and other interests. Are you willing to make the necessary sacrifices? If you are then it might be a good idea to make the sacrifices upfront and go back to school to learn math. The chances of fulfillment would be much higher.
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"Pick one book and persevere with it." - Not always the best idea I think; trying to stick to a really sucky book can be damaging. And sometimes it's hard to tell what's crappy and what ain't when you're just getting started. –  J. M. Sep 20 '10 at 6:36
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Your last point is a great one especially. I think a lot of graduate students I know haven't asked themselves this questions seriously! –  BBischof Sep 20 '10 at 11:15
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I disagree with taking a dozen years instead of four. I personally learn much faster on my own than in a classroom. –  Olivier Lalonde Nov 25 '10 at 5:56

Khan Academy has excellent short videos you can take at your own pace, from basic arithmetic to advanced calculus. He has a very conversational style as well.

http://www.khanacademy.org/

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I want to point out that there are, topicwise, different paths from the most suggested standard one:

As noted by "jmoy", time is limited - the comments telling you to learn some physics or programming first or on the way seem to ignore this somewhat. Sure it is good and will pay off to have these things as background, but it is not indispensable. Apparently you already have discovered your fascination for mathematics, so you don't have to back it up with these things that surround mathematics. Physics and programming can give you good intuition, but so can doing mathematics itself.

Also, going through the whole standard curriculum with lots of calculus and coordinate geometry is by no means the only way to come into mathematics!! Certainly a basic knowledge of these things will be indispensable at some point. If you actually enjoy studying calculus, then do it, it's an excellent start! But don't just do it because it is the standard curriculum in US colleges. Often people who went through this can't imagine having learned things in a different way, but that is perfectly possible, and normal in many countries (I went through a German school and university and learned much less calculus on the way than an average US student - it didn't do me any harm).

After all you are wanting to study mathematics for having a rewarding experience, and while there will be some frustration on the way, it should feel somewhat rewarding right from the beginning.

So here are three alternative routes - in case they pique your curiosity more:

Number Theory is a field which seems perfect for self study: It is concerned with objects that you know very well already - the natural numbers - yet it is a deep and difficult field and connects to virtually every other part of mathematics. This is mathematics of an entirely different flavour than you would see following the calculus path, and it can bring you just as far. In particular it will bring you to a point where you want to study calculus because it tells you things about number theory. For this road, pick up a book on Elementary Number Theory (e.g. (1), free for download - but if it's too hard going, there are more basic ones also) and work your way through that. Accompany it with the book "Fearless Symmetry" (2) an excellent popular math book, and not a waste of time! It leads you very far into things which connect to current research (the so-called "Langlands program") but is very friendly written. It gives you a path to follow; you get intuitive ideas for example about "group theory", and whenever you get to such a point where a new concept (like "group") is introduced, you should accompany it with a more formal treatment, where you can see the basic theorems about that structure and get some exercises to solve. I suggest Michael Artin's "Algebra" (3) which contains all you will need to know for a long time on this way. It starts at a low level, which either is already ok for you, or will take only little intermediate reading to bridge the gap. (If you start wondering when the calculus comes in, pick up Serre's "Course in Arithmetic" (4) and jump to part II, but only after you got some feeling for what's in the first half of Artin's "Algebra")

Combinatorics (e.g. with the book "Proofs that really count" (5)) - this is also a topic which needs no motivation from physics or any other area about which you would have to learn things first. As in number theory you can see how one develops mathematical reasoning to answer questions that make sense immediately. And again it can lead you the point where you want to study other mathematics, e.g. calculus, because it makes sense to pursue those questions in the context of what you already know...

Logic, e.g. starting with Carnielli, Epstein: "Computability" (6) - this is a basic introduction into logic and computabiliy theory, written for philosophers, and therefore starting very gently. It also paves a way into mathematics via a motivating question - When should we call something "computable"? - and leads you all the way to a proof of Gödel's famous undecidability theorems. Again this is connected to a lot of mathematics. Maybe you would after that like to go on studying non-classical logics, which will lead you to algebra, lattice theory and topology...

Whatever route you choose (and I have to emphasize again that the calculus route is very good as well, if you feel like it), talk to people about the things you study, ask and answer questions, e.g. work as a tutor, or spend time here in the forum!

Enjoy your journey to mathematics!


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If you're not fazed by sticking to only pen/paper or a blackboard/whiteboard for doing all the manipulative work (remember, math ain't a spectator sport!), then sure, you don't even need to learn how to use a computer to help you with math. One convincing reason to learn programming (at least for me) is that you can let the computer do all the grunt work so that you have time for the finer things in life (e.g. "Hmm, that's funny..." or "There's a pattern here somewhere...") –  J. M. Sep 22 '10 at 1:32
    
Sure, for number theory that's a great reason for example! In my areas on the other hand there is simply no way to use computers (maybe one future day?). So in the end it was a detour and distraction for me to learn programming. If I had ended up in a different area where it would have been a benefit I could still have learned it. So, as with calculus, if you enjoy programming (I didn't :-), go for it - if not, don't just do it because you think it is a necessary preparation. –  Who Sep 22 '10 at 15:48

I would recommend the following book:

Mathematics: Its Content, Methods and Meaning by A. D. Aleksandrov, A. N. Kolmogorov, M. A. Lavrent’ev

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Before starting Basic Arithmetic(Pre-Arithmetic) study the definition of mathematics/Its history/influential scientists/mathematicians in it. Then the branches of mathematics and their definitions and then learn the number system and mathematical skills and number skills and think deeply about it like why it is based on ten digit then go on learning Arithmetic(all the operations and their definitions) Practice it Master it completely then memorize and mesmerize(Think every time about what your memorizing) all the arithmetical tables(Addition tables, subtraction tables, Multiplication tables) which will help you for mental calculations and remember always visualize numbers. Then when your Mental Calculation is awesome!!! then move on to PreAlgebra master it then Algebra master it and then modular arithmetic which will give you nice understanding of how mathematics work then Geo , Trig , Calc all other things then study their relations do extensive word problems and numericals master them then do research on them. Take part in IMO , discover your own theorems. Solve a millionaire problems . take part in mathematical discussions and study science.

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I wrote an essay based on my experiences learning math on my own over the past few years - you can read it here.

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