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.
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.