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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.

Rose2357
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    Probably mostly a duplicate of As a high school student, how can I get ahead and teach myself math?, or of probably around 100+ other questions. I'm not sure it specifically addresses 9th graders, but no doubt the information in all these questions is already sufficient. Either way this question is too broad. – rschwieb Jun 28 '19 at 16:39
  • Although this is a probably a duplicate, I would suggest finding yourself a good textbook and supplement that with internet resources. Also, I would make sure you absolutely know precalculus and algebra inside out before moving onto number theory, calculus, etc. – N. Bar Jun 28 '19 at 16:43
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    If you are interested in a good crash course on calculus, linear algebra, and complex numbers, I would recommend the MIT open course lectures by Herb Gross available here as a playlist on youtube. – JMoravitz Jun 28 '19 at 16:43
  • "Besides I like to have proof for everything which is not always given in books" It sounds like it might help to find better textbooks. You might try looking at Calculus by Spivak, though there are many good books to read in different styles. Number theory is a fun subject. If you can manage to find other students who also love math and are at your mathematical level, you can learn a lot from them. Consider attending math summer programs (such as the Ross program) or joining a local math circle, taking courses from Art of Problem Solving, or doing contest math. Find mentors if you can. – littleO Jun 28 '19 at 16:50
  • I don't know how deep you want to go but there is the 3-volume "Principia Mathematica" by Alfred North Whitehead and Bertrand Russell. It's pricey, even in paperback on amazon, and some say hard to read but it beats Isaac Newton's version which is just a lot of examples of how he solved problems. – poetasis Jun 28 '19 at 17:13
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    Generally, progress is done in mathematics by understanding the one or the other specific structure. (Then join structures together, but this is an advanced target.) Separate knowledge from solving, become good in both. (It is as in music, chess, go, physics, and programming. You can learn a lot of theory from all possible corners of the specific field, but then there will always be problems when facing a specific issue, in the examples to play violin, when you know piano, to play and win a game, to design and make an experiment work, to write clean functional code...) Just do it! – dan_fulea Jun 28 '19 at 18:32
  • I was in your position a few years ago (now in 11th grade), and using only YouTube, Wikipedia, and Desmos I built a fairly solid intuition on a lot of (what I believe to be) advanced topics. The way I got myself to remember and understand stuff was just by deriving everything, and making sure that I understood how to derive it. When I got stuck, I would look up a proof and skim it until I found a hint, then finish it off. I don't think you need to actually buy any resources, unless you want to go into something which completely deviates from what you learn in school (like topology). – uhhhhidk Jun 28 '19 at 22:47

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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.

DanielWainfleet
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  • Yes, they still have Mathematics Magazine. I get the Monthly in print, the others are available online with my annual dues. – Will Jagy Jun 29 '19 at 01:10
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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):

darij grinberg
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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.

Will Jagy
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