I'm particularly interested in knowing the order of subjects one should study to become good in mathematics. In my not so great knowledge of the subject I would say that the best place to start could be Arithmetics, is this correct?
If you could also suggest some books or good free resources related to each subject I would really appreciate it.
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This site as well as wolfram|alpha are both excellent resources for teaching yourself math!
I'd recommend looking at the KhanAcademy: it's online, free to sign-up and participate, progresses in level of difficulty, and is self-paced. It is good for reviewing the "basics" and takes you through calculus and a bit beyond.
See also this site for mathematics-related on-line video tutorials - literally hundreds of them!
Another helpful resource for learning algebra-precalculus, calculus, linear algebra and differential equations Paul's Online Math Notes and tutorials.
Then perhaps you'd like to explore MIT's Open Courseware - Mathematics for access to classes and topics of interest in more advanced topics. From there, you'll find a list of classes with resources available, and will also learn which texts are used for the available classes. Often, course notes, videos of lectures, and exams are available to assist learners, all free of charge.
For learning how to read and write proofs, including proof by induction, a wonderful text to read and work through is How to Prove It: A Structured Approach by Daniel Velleman.
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You might want to look at the following mathematics classification guides to see the scope of courses and areas that are available:
$\bullet$ The Mathematical Atlas
$\bullet$ American Mathematical Society's Mathematics Subject Classification
$\bullet$ Engineering and Physical Sciences Research Council - Mathematical sciences
$\bullet$ Mathematics arXiv - Categories within Mathematics
This makes answering your question difficult because it is not entirely clear what your end goals are.
You certainly should learn proof techniques using such books as
From all of the above, you should see if you can refine your goals and the wonderful MSE Community can provide more guidance.
Regards
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For advice spanning from the junior high years to college training, here's a very good PF post: http://www.physicsforums.com/showthread.php?t=122924. There are many textbook recommendations and advice from multiple authors, though it may take some effort to filter out the gems of advice in the forum format.
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Work your way through the Khan Academy https://www.khanacademy.org/ Reference Wolfram Alpha https://www.wolframalpha.com/ Ask lots of questions on Stack Exchange. Surround yourself with people that love math and stay focused.
I think the "How to become a pure mathematician" website is what you're looking for. It offers a well structured approach from very basic mathematics all the way up to graduate level, with links to useful resources and books.
Also, any university curriculum can be useful to see in what order university students study different subjects. An example is Cambridge (the famous mathematical tripos), where they have pdf's outlining every part of the course:
You will also find past exam papers and a bunch of other resources. Very useful.