High school student advice

Question

I'm have a solid high school math bases and I'm a Mathematics enthusiast. I want some good books suggestion to progress even more and be able to have a scientific career.

Answer 1

I suggest you an excellent book: What is mathematics by Courant and Robbins.

Answer 2

I highly recommend Judson's Abstract Algebra: Theory and Applications. On top of being totally free, I think it does a good job of introducing you to advanced undergraduate math without assuming you've taken a bunch of other courses on linear algebra and set theory and logic.

Linear algebra is also essential to learn for pretty much any field. If you have ambitions of being a theorist, the "abstract vector space" approach is probably better than the "matrix approach" (in e.g. Strang)--though eventually it is good to see the material developed both ways. I first learned out of Friedel, Insel, and Spense and found it to be good.

If you've already done Calculus, I cannot highly enough recommend Real Mathematical Analysis by Pugh. This is a very challenging book and it may very well be too advanced for you right now. But I read a bunch of it in high school and I really credit it to forming my mathematical intuition, so if you have the background for it, it can be a good experience. A gentler introduction to real analysis is Lebl's Basic Analysis.

Ultimately, more than anything, mathematics is about solving problems. Whatever books you choose, make sure you are solving problems that are interesting and challenging to you. Challenging should not mean impossible though so make sure you pace yourself. If something is too challenging, try and "back up a few steps" to something that isn't like hitting your head against a brick wall. Best of luck and happy mathing!