How can high-school students learn mathematics on their own? I am a ninth grader and I would like to learn mathematics on my own. I have already learned algebra, geometry, trigonometry, some precalculus, number theory and tried to understand some calculus. Apart from those I learnt a bit from other areas of mathematics but not enough to be worth mentioning. 
I have learned several things from books but those  couldn't answer all questions so I had to turn to the internet (sometimes I can't be sure if something is correct or not). Besides I like to have proof for everything which is not always given in books. (I want understand it properly not just use a procedure).
I have tried using online lectures but the ones I have found according to my level were either going too slowly or didn't have complicated problems. Besides most didn't have the proofs. I use Khan Academy sometimes although it too lacks the difficult problems.
Could you please tell me what I can do to learn further mathematics(eg.  More number theory, proof writing, calculus and... maybe analysis altough I suppose I am not prepared well enough for that)?  Do you know any books I could read (normal high school /college /university books included just give me the name please) or lectures I could watch? Any other things I could do? I would like a good understanding of the subject as I would like to become a mathematician. 
 A: D. Coxeter: Introduction To Geometry.
D. Coxeter : Geometry Re-Visited.
Courant & Robbins: What Is Mathematics?.
P. Suppes : Axiomatic Set Theory.
Vilenkin: Stories About Sets.
I.Bromwich : Infinite Sequences And Series. (Different editions have slightly different titles.)
H. Dorrie :101 Great Problems Of Elementary Mathematics.
L. Hogben : Mathematics For The Million.
G. Polya : Mathematical Discovery:On Understanding, Learning, and Teaching Problem Solving (two volumes).
Jemeny, Snell, & Thonpson: Finite Mathematics.
U. Dudley : A Budget Of Trisectors. (For fun. A mathematician's story of his close encounters with amateur crackpots  in the field of math. Different editions have slightly different titles. )
Dover Publications (formerly Dover Press) is a good source of very cheap re-prints of older books on math, & on science in general.
At some point you will need to learn the logical foundations of the "real" numbers $\Bbb R$ and the basic consequences of it, as calculus cannot really be understood otherwise. (E.g. the Q "Why is there no positive number that's less than all positive rationals" is meaningless unless you define "number".   There's no positive member of $\Bbb R$  that's less than all positive rationals as a consequence of the $definition$ of $\Bbb R$.) And the logical base and elementary properties of complex numbers.
Also find some algebra (groups, ring, fields,vector-spaces, linear algebra). And some Statistics.
The Preface or Introduction to an introductory book should state what level of audience it is for. 
On writing: Write math in complete, grammatical sentences and do not omit punctuation, just as you would write an essay or a story. And $never$ omit $\implies$ or $\iff$ nor any other justification or explanation of how or why one assertion or formula is related to the next. 
Take a  look at American Mathematical Monthly. It is for students and teachers (i.e. not a research journal). It had a more elementary companion Mathematics Magazine. I dk whether it still does.
A: Some free texts (which I think can be read roughly in this order) that look good to me (some I've read in part; some I am assessing based on the author):


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*Eric Lehman, F Thomson Leighton, Albert R Meyer, Mathematics for Computer Science. This is my go-to introduction to mathematics for nerds. It starts out as an introduction to proofs (but is rather fun and eclectic at that, as opposed to more systematic texts that can be rather dull; but if you do need a slower or systematic treatment, check out Hammack or Day or Levin). Then it gets to basic number theory, combinatorics and statistics. Almost all of this is useful for every mathematician, whether or not they want to specialize in computer science.

*Kenneth P. Bogart, Combinatorics Through Guided Discovery. (Another version, not sure if any newer.) This is an introduction to (mostly enumerative) combinatorics, written as a collection of exercises with ample hints and motivation. (Yes, there is a version with solutions in the tar.gz source archive.)

*Irena Swanson, Introduction to Analysis. Does what it says, and appears to be rigorous at that. (Work in progress; if you find errors, tell the author!)

*Neil Strickland, Linear Mathematics with Applications. A rigorous matrix-based linear algebra text.

*Jim Hefferon, Linear Algebra. The other approach to linear algebra, via vector spaces.

*Samir Siksek, Introduction to Abstract Algebra. This is perhaps the most amusing introduction to this subject I've seen. Linear algebra is being used.

*William Stein, Elementary Number Theory.
A: I like this book by Weissman, especially after answering dozens of related questions on this site. 
In short, it is beneficial to know some quadratic forms as a preview of algebraic number theory. For that matter, a good feel for integer quadratic forms would be helpful for Lie Algebras.
