In What order should I Learn math in?

Question

I am a 13 year old boy from Hong Kong, My dad is a math teacher which probably accounts for SOME of my interest in math. I am really interested in math. I self-taught myself (with little help form my dad when I needed help) Trigonometry, Differential and Integral Calculus (well most of it), Solid of a revolution, Taylor and Maclaurin series, and recently I have been learning some set theory.

I have a feeling that the things I learn are going all over the place, I just learn what i find interesting, and sometimes I don't understand some topics because I don't know some previous topics so I will need to go back to learn them. Should there be some order with what to learn first and what is the order?

Thanks in advance

Answer 1

I recommend finding some old math tournament problems, there's a million to chose from. Find some at about your level and learn whatever you have to do in order to be able to solve the problems.

If there is a question that interests you that you couldn't solve, figure out what you need to study in order to be able to solve it.

There's a world of difference between "knowing" an area of math, and being able to use it like a weapon.

Answer 2

You should start with linear algebra (vectors, matrices, linear transformations, vector spaces), then move on to abstract algebra, and from there there's a lot of places you could jump to. A great abstract algebra book with heavily linear algebraic approach is Michael Artin's Algebra.

Answer 3

I think, at your age, you can play more with Combinatorics, planar geometry and elementary number theory. These fields are full of interesting and exciting results and most of the time, these results have interesting history which gives information about history of mathematics.

It is always good if you study mathematics in a formal way starting from formalism of set theory. But math is not only about the precision and formalism, it is also about having vision and intuition.

I believe that starting with basic combinatorics, geometry, number theory and analysis is a very good way to develop a preliminary vision about what is mathematics and what it is going to do.

Also, solve as many problems as you can!

Answer 4

artofproblemsolving.com is a website where students learn math with an emphasis on creative problem solving, and also train for math contests. They have developed a curriculum, including books on number theory and counting and probability. The curriculum emphasizes combinatorics, planar geometry, and elementary number theory, as Arash suggested.