# How to go from beginner to expert in mathematics?

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.

• What do you know now? Basic arithmetic? High school algebra? Commented Jan 17, 2013 at 22:23
• I think we should have a post, something like "A layman guide to Mathematics" ... of course not to offend @user
– S L
Commented Jan 17, 2013 at 22:27
• @anorton I know arithmetics, algebra, discrete mathematics, statistics and logic. My question is more for me to go back and pick up some of the basics I might have overlooked in these years :) Commented Jan 17, 2013 at 22:33
• @experimentX we could also think about turning this question into a community wiki if it could be useful for other people too Commented Jan 17, 2013 at 22:35
• well certainly, ... only that I am not expert on math!! i'll just watch along!!
– S L
Commented Jan 17, 2013 at 22:37

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.

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.

• For learning proofs, I particularly enjoyed 100% Mathematical Proof. (But try to find it at a library as there aren't very many cheap used copies available at this moment.) Commented Jul 10, 2014 at 15:56

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

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

• Oh no! Hope all is well now! MSE again? Commented May 9, 2013 at 0:39
• Just a slow day for me...and I made a couple stupid mistakes I had to quickly delete...Big sigh! ;-) Commented May 9, 2013 at 0:41
• @amWhy: I have also had my share of errors! :-) We all have those days, but are always striving at getting better and errors always happen! Commented May 9, 2013 at 0:55

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.

For probability & some statistical programming here.

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.