Which direction to explore, books to buy for self-learning (high school student)

[pkkm]

I'm a Polish high school student (in Poland education is like this: preschool → primary school → middle school → high school (entrance age 16) → graduate school (entrance age 19~20)). My school is very good (in top 2.2% in Poland). I don't have problems with mathematics in school, but I have the impression that focus falls more often on computation by mechanical following of memorized algorithms than on theory or problem-solving.

I recently became interested in mathemathics beyond school level (my main hobby (and probably future major) is computer science -- I program in several languages). I'd like to improve my understanding (both intuitive and rigorous) of already learned material, but also learn advanced concepts outside the curriculum. I think my proofwriting needs improvement too.

I'd like to buy some good books and start learning from them, but I don't know which direction to explore first. I'd be grateful if You suggested me 2 to 5 books (price is of small importance). Here are books suggested on the Internet to people in similar situation, divided into "directions", in no particular order:


General

• How to Solve It: A New Aspect of Mathematical Method, G. Polya
• How to Prove It: A Structured Approach, Daniel J. Velleman
• Solving Mathematical Problems: A Personal Perspective, Terence Tao
• What Is Mathematics? An Elementary Approach to Ideas and Methods, Richard Courant, Herbert Robbins
• The Princeton Companion to Mathematics, Timothy Gowers, June Barrow-Green, Imre Leader
• Problem-Solving Strategies, Arthur Engel
• The Colossal Book of Mathematics: Classic Puzzles, Paradoxes, and Problems, Martin Gardner

Geometry

• Introduction to Geometry, H. S. M. Coxeter
• Geometry, Harold R. Jacobs
• Geometry: Euclid and Beyond, Robert Hartshorne
• Geometry: Seeing, Doing, Understanding, Harold R. Jacobs

Linear Algebra

• Linear Algebra Done Right, Sheldon Axler
• Linear Algebra, Kenneth M Hoffman, Ray Kunze
• Linear Algebra: A Geometric Approach, Ted Shifrin, Malcolm Adams

Calculus

• Calculus Made Easy, Silvanus P. Thompson, Martin Gardner
• Calculus, Michael Spivak

Other

• Fifty Challenging Problems in Probability with Solutions, Frederick Mosteller
• Fundamentals of Number Theory, William J. LeVegue

 

[simpy]

Hi pkkm,
You have not made it clear about your current knowledge in the subject but I think that since you are in high school, math olympiads might be of some importance to you . I get an idea that you are mainly interested in your problem solving and proof writing abilities and if you prepare for olympiads or do that level of preparation you would be able to develop both.
If that is the case then have a look at: http://www.imomath.com/index.php?options=347&lmm=0
and http://www.imomath.com/index.php?options=257&lmm=0
Anyways, for geometry I would recommend you to first go for geometry revisited by coxeter then introduction to geometry by H.S. Coxeter . Kiselev's geometry is also a good one. A thread related to geometry book discussion can be found here: https://www.physicsforums.com/showthread.php?t=610241
As for calculus,Spivak's Calculus and Apostol's calculus are two masterpiece. apostol's calculus is good for deep understanding and covers much more topics than are there in spivak.
In number theory, I.Niven and Zuckerman's introduction to theory of numbers and 'Number theory structures, problems and theory' by Titu Andreescu are good. But, If you want more for theory then go for elements of number theory by david burton and introduction to number theory by G.H.Hardy and for problems go for,, 102 number theory problems from the training of USA IMO Team by titu andreescu.
For Algebra,go here:http://books.google.co.in/books/abo...al_Equations.html?id=dWI9bvqbbkUC&redir_esc=y
Arthur Engel's book about Functional equations and E.J. Barbeau's Polynomials
For general reading, what is mathematics by courant is a nice book .You are a beginner and really should go for it.Martin Gardner's books are interesting but I find them more suitable for a middle schooler.
To improve proof writing and stuff First I would recommend you to read George Polya's how to solve it. and then Terence Tao's as I have done.
P.S. :FIRST READ A PROBLEM SOLVING APPROACH BOOK BEFORE GOING FOR PROOF WRITING .
All the books above are suitable for a high schooler keen to study mathematics in its actual way rather than learning it by memorization of algorithms and using them again and again in same type of problem.

 

[pkkm]

Thank you for your reply. I've bought some of the books mentioned.