How do you study math in your spare time?

I have recently graduated from high school, and intend on continuing mathematics study in university. For the next couple of months, I don't really have anything to do, and would like to continue doing math in my spare time.

The math I've done so far (and enjoyed) is calculus and optimisation, linear algebra, vectors, vector calculus, complex numbers, and matrices.

I guess my question is, how do I do this? I've had an idea of working through problems from my high school text book, or even borrowing a university level textbook. Are either of these good ideas?

How do you study math in your spare time?

Solution 1:

I think your next move should be Algebra. Without a good understanding of groups you will probably get stuck in every branch of your studies. This notion it will be the starting point of so many developments in Mathematics that it is not even worthwhile defending my thesis since no mathematician born after 1850 would oppose it.

Personally like and would study Herstein "Algebra" or Artin. Workout all the examples and all the exercises since at this stage the mathematics is still very very fundamental and you really have to digest every step of the matter. I think a good study of these subjects will take a couple of months. When you're finished maybe you can switch to Differential Geometry.

Solution 2:

My opinion, as is the opinion of many mathematics professors, is that one should begin studying "the big three" which are Algebra (really we mean "Abstract" or "Modern" Algebra), Topology, and Real Analysis. For Topology, I'd recommend starting with the book "A Course in Point Set Topology" by John B. Conway as it starts the motivation of the subject with metric spaces and then moves on to the more general definition of topological spaces and nearness. Once one is able to have a decent background in these fields, they can move on to be able to do well as a pure and/or applied mathematician. However, there is also combinatorics and discrete geometry as well as many other subjects, and personally I've found Axiomatic Set Theory to be the most astounding. A lot of people also have fun with graph theory

Solution 3:

I'd recommend number theory. Self contained, fun, good for self study over the summer.

Self study text for Elementary Number theory