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?
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).