I am a high school student but I would like to self-learn higher level maths so is there a correct order to do that? I have learnt pre-calculus, calculus, algebra, series and sequences, combinatorics, complex numbers, polynomials and geometry all at high school level. Where should I go from here? Some people recommended that I learn how to prove things properly, is that a good idea? What textbooks do you recommend?

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    $\begingroup$ There are many routes that lead to many places. Calculus is a useful analytical tool, but it is not a "crowning achievement". $\endgroup$ – ncmathsadist May 3 '13 at 1:03
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    $\begingroup$ Given that your name is "Alexander Jones", it is hard to resist recommending that you study knots and/or polynomials. $\endgroup$ – Pete L. Clark May 3 '13 at 2:27
  • $\begingroup$ What do you mean? $\endgroup$ – please delete me May 3 '13 at 2:32
  • $\begingroup$ $@$Alexander: It was (supposed to be) a joke. See en.wikipedia.org/wiki/Alexander_polynomial and en.wikipedia.org/wiki/Jones_polynomial. $\endgroup$ – Pete L. Clark May 3 '13 at 3:17
  • $\begingroup$ I don't know that many maths-related jokes haha :) $\endgroup$ – please delete me May 3 '13 at 3:22

I'm not mathematician. I hold degree in management both bachelor's and master's degree. Anyway, to my knowledge, if you want to learn math from scratch, the order will be

college algebra, algebra & trigonometry, precalculus, calculus, linear algebra, differential equation, intro to analysis (or advanced calculus), functional analysis, real analysis, probability theory

There're some books I recommend

  • College Algebra by Henry Burchard Fine
  • Fundamentals of Algebra & Trigonometry by Earl Swokowski
  • Precalculus by Sheldon Axler
  • Calculus vol.1&2 by Tom M. Apostol
  • Introduction to Linear Algebra by Gilbert Strang
  • Mathematical Analysis by Tom M. Apostol
  • Introductory Functional Analysis with Applications by Erwin Kreyszig

Supplement: How to Prove It by Daniel J. Velleman. You need this book to learn how to do rigorous proofs for mathematical theorems.


Quite often the transition to higher, pure math is real analysis. Here proofs really become relevant. I would suggest this free set of down-loadable notes from a class given at Berkeley by Fields medal winner (math analog of Noble Prize) Vaughan Jones.


They are virtually verbatim and complete as a text. They build gradually so you can get a good base. The material is Prof. Jones's own treatment and the proofs are quite accessible and beautiful.

You might just give it a try and see if it works for you.


One general approach is to select a college, and start working through topics that an undergraduate mathematician would see.

It is a good idea to know proper techniques of proof, but that can also be picked up by reading lots of "good" proofs. If you feel comfortable with proving some basic things (e.g. the sum of two odd numbers is even) on your own, then I'd suggest just picking up proper methods of proof by reading other people's more advanced proofs.

From looking at what you've done, it seems that Linear Algebra could be a good next step, or perhaps a multivariable calculus course (if you haven't done that already).

  • $\begingroup$ What does "pick a college at random" mean? Not to randomly choose an undergraduate institution in which to enroll, I hope?? $\endgroup$ – Pete L. Clark May 3 '13 at 2:25
  • $\begingroup$ I will see if I can find the undergrad course structure for uni of sydney, that is the best uni in sydney and I will most likely go there (although it is ranked 50th in the world lol). $\endgroup$ – please delete me May 3 '13 at 3:25
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    $\begingroup$ @PeteL.Clark Sorry. bad choice of words. I meant to pick some college (preferably a good one) to get an idea of what's required for a math degree (e.g. Abstract algebra, real analysis, etc), and also to get an idea of in what order they are typically taught. I figure that it doesn't really matter what school you model your self-teaching after, so long as it's a "real" school. Of course, you should pick the college that you enroll en carefully. $\endgroup$ – apnorton May 3 '13 at 13:34

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