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I am a junior at high school (16.1 years old); although I perform at the top of my class and have some competitive achievements in the nation, I think I am both very far from the IMO and college-level mathematics. I am struggling to move on with mathematics since there isn't any clear pathway I could find. For the past year, I wanted to learn (this halt in my learning process is the case for the past 2-3 years where I didn't learn anything new in school, but also I didn't manage my time and didn't learn anything on my own time) calculus but kept on procrastinating... I don't think I have great planning skills, but I do think I have adequate mathematical skills... I really want to pursue mathematics in the future. What should I learn, and how? Do you have any suggestions? Is it too late for me to become a "good" mathematician in the future? I don't want to be in constant depression, thinking how I won't become a "good" mathematician since I am not intelligent enough and I don't learn enough; honestly, I feel like this cycle is not only prevented me from learning but also psychologically consumed me. Although this probably not the right place to ask these questions, I wanted to give it a try.

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    $\begingroup$ start with analysis and linear algebra. My recommendation for an starting point is Understanding analysis of Robert Abbott $\endgroup$
    – Masacroso
    Apr 3 at 14:22
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    $\begingroup$ about "too late": I started to study math for fun around 36 years old or so. Forget about become "good" or "bad", if you want to know something just do it. This is all. Its like: if you like pizza you will try to eat it regularly, and you are not thinking about "becoming good at eating pizza". This is the same. $\endgroup$
    – Masacroso
    Apr 3 at 14:47
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    $\begingroup$ @PrinceM regarding “wanting to learn calculus,” I have searched for resources, tried to read Spivak Calculus, watched some videos from 3b1b, wrote code for taking the integral and derivative of $x^n$ type of functions by monte carlo method, and also with trapezoids. I took an AoPS class on Precalc and finished a Precalc book. But as I said before, I don't think it is hard to learn; at this point, I just lack the motivation and the organizational skills to learn. But hopefully, I can carry on, as some answers have motivated me :) $\endgroup$ Apr 3 at 16:04
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    $\begingroup$ Sounds good. Best of luck in your mathematical pursuits! $\endgroup$
    – Prince M
    Apr 3 at 16:16
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    $\begingroup$ "16.1" years old. Sigh. I remember when fractions of years were significant to age. What does how smart you are have to do with it? If you find math interesting, learn what you can. So what if some people are better at it, and some things escape your grasp. That is true of practically everyone, including many brilliant mathematicians. Don't worry about comparing yourself to other people. Rejoice in their brilliance and what you can learn from it, but their existence does not in any way make you less or invalidate your own accomplishments. $\endgroup$ Apr 4 at 0:13
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It sounds like you have forgotten why you want to learn math. Presumably, you like math and find it fun, or at least you used to.

Imagine that you were exempt from math. For some reason, you will never have to use math ever again in school or work. You are totally free to never touch math or think about it ever again. How does that make you feel?

If you didn't have to do math ever, would you choose to do it, even a little bit, just for fun? If so, what math would you choose? I don't mean like a big commitment to study something advanced. I mean like something you could learn in $15$ minutes or less. Is there anything you actually want to know, for fun?

If you truly would be happy if you never touched math again, then stop trying to think of math as a career. Do something else. Anything else.

But if you can think of anything in math that you are curious about, and want to know...go learn it. The internet has endless possibilities for you. Watch a 3blue1brown video on YouTube. Visit Khan Academy. Go to a puzzle site.

Do it for fun. Remind yourself why you like it.

I have long wished that someone would make a MMORPG called "World of Mathcraft" or something like that. If people will willingly sit there for fun shooting frost-bolts at boars for hours on end to finish quests, imagine if they could level up with arithmetic, and could gain power-ups by learning new math tricks!

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"Real" mathematics and the mathematics you see in school are worlds apart, I've found. And "real" mathematics are indescribably better; more creative, mind-opening, beautiful.

The reason I say this is to assuage your fear: you do not need to be Ramanujan or Euler to be fulfilled. The trials and challenges you will set yourself and overcome, the beauty of this universe that you shall discover, all of it will feed your mind and soul. Perhaps the following texts should give you some perspective as to what a career in mathematics would be like; and what role you can find for yourself.

https://zenpencils.tumblr.com/post/67594550223/richard-feynman-the-beauty-of-a-flower https://www.maa.org/sites/default/files/pdf/devlin/LockhartsLament.pdf
https://www.ams.org/notices/200902/rtx090200212p.pdf

I, personally, would absolutely advocate for a career in mathematics. But do understand that there's a lot more to mathematics than pure mathematics.

Here's a little about me to give you an idea of why I say this; after all, I am not a professional mathematician per se, but a computer scientist (software engineer, R&D).

I first studied History and Political Science. I found that historians were very unsure of the validity of their methodology so would bicker about minute details constantly (at least, they had a good amount of rigor because of this profound self-doubt; differentiating honestly and pretty competently between what was certainly true, what was probable, what was improbable, and what was certainly false). Political scientists (sociologists) were much, much worse: they were instilled with a sense of institutional pride in their methods (which were rarely scientific, or done well enough to be considered scientific), and thus constantly defended biased things with false authority.

This made me want to go back to something more "solid". So I took a gap year where I read some math textbooks. That changed me. Doing math on my own was NOTHING like how it was taught in schools (and I was in a "nationally recognized" high school, supposedly a "temple of mathematics", though I can't really agree). Lockhart's Lament, linked above (which I suggest you read completely, but the two first pages and beginning of the third should suffice to give you an idea) should give you a sense for what I mean. Mathematics became my religion, so to speak. I thus applied to a combined mathematics and computer science degree, thinking "well, CS will probably useful to find a job where I can do math". I started to code a month before that, and I saw that computers were the instruments of music to mathematics' solfeggio. They allowed me to enact mathematics in the physical world. I never looked back; I have been incredibly happy with the intersection of math and CS in which I live.

Here is the truth that most people will never learn, because schools steal it from them: Mathematics is the road to awe. It is the language in which, not only our universe, but ALL possible universes, are written. Everything you see from music to traffic jams have some form of math underlying them, and being able to discern that underlying math is like adding a kaleidoscope to your vision; making every detail of the world more interesting and beautiful... Coding a video game is literally playing the role of God: defining a universe and its physics; the people that inhabit it; the poetic arcs of their lives; the way entities and matter interact to build up into emergent patterns. Every facet of CS and math is about finding a concrete way to push your baboon mind further than its biology should allow, and explore the set of both all that is absolute, and all that is possible. It is unbelievably meaningful, soul-fulfilling, spiritual ecstasy.

Plus, it pays pretty damn well ! But really, that's just a bonus. I'd still be craving and doing some form of math in a jail cell, or at the most elite gala you could imagine; or anywhere else, honestly.

So... "Where do you start ?" Well, I am, in Dyson's words (third link above), a "bird" (Poincaré used to refer to "frogs" as "logicians" and "birds" as "intuitives", by the way), so I will give you the point of view of a "bird"; which is to seek for increased vision, intuition, and perspective.

I would set as your first objective to build a "map of mathematics" in your mind. The go-to tool for that, in my opinion, is category theory. But its depth and relations with the whole of mathematics are difficult to grasp without some view of that whole. So I would start with linear algebra, since it serves as a gateway to the rest of higher mathematics; and many of its constructions can serve as introductory examples in category theory. Alongside it, I would try to build a good understanding for the "common core" of mathematics (the building blocks of math up to category theory): set theory, (boolean/propositional) logic, abstract algebra (algebraic structures: monoids, groups, rings, fields, vector spaces, algebras, modules, etc), and finally category theory. The vocabulary of these fields will be present in all the domains you explore; so a good mastery of it and its interrelations is essential. These fields, and their concepts, form the common language to all mathematics you will learn, and act as a "spinal cord", allowing you to relate, manipulate and merge all mathematics you learn within a common framework, rather than as isolated islands.

While building this "vision", your taste will become more refined, and you'll recognize the mathematics for which you have an affinity.

Additionally, I would recommend that you learn to code as well. Raw solfeggio, without the capacity to play an instrument, can be fulfilling if you're Mozart, but generally, it is even more fulfilling to play the music yourself. But how does one learn to code, not just as a skill, but as a way of thinking ? Well, programming is mostly like solving problems: you divide it into subproblems and build up a coherent solution to the whole. And a good knowledge of computer programming is a good understanding of computing paradigms: imperative, object-oriented, dynamic, functional... Many languages implement different aspects of these paradigms: for example, C is a very "imperative" programming language, close to the machine, and will teach you its inner workings; Haskell is very (a bit too much) functional, it's almost pure math; OCaml is a decent mix of functional and object-oriented; TypeScript is the same, but slightly more on the object-oriented side; Ada and Rust are excellent halfway points between imperative and functional. Out of those, the most important for a mathematician is functional programming, since almost all modern functional programming languages are based on the typed lambda calculus, and its links to category theory (and thus the rest of mathematics) are very profound. [As a side note, GPU programming is heavily related to linear algebra and its extensions; but it is a very specific kind of programming, though very mathematical and advanced.]

That said, I know that I certainly wasn't confident that I would like coding, or ever be good at it, when I started. So being taken by the hand with a non-typed language like JavaScript on Khan Academy can also be a good starting point: there's immediate feedback and a helpful debugger. However, replacing as much of the tedious task of debugging with a strongly typed compiler as possible is a decision to which all good software developers arrive someday.

Do note that I don't know which country you're from, but in mine (France), university was a soul-crushing endeavor (much like schools in general described in Lockhart's Lament). If mathematics had not already become my religion, I would not have pursued it, given how poorly it was taught in university. I say that and I was probably among the top 3 students in my year overall. So don't be demoralized if the form of learning you deal with isn't topnotch.

That's why I'd suggest you choose for your goal to quickly reach a position where you can be paid to learn more, and in a way where you can continuously broaden your intellectual horizons. With a good handle on multivariable calculus and programming, you'll find plenty of engineering, statistics or industrial research gigs. For example, fields like biostatistics are in exceedingly high demand of people with just an average handle of mathematics and computer science, for example.

I hope this helps your introspection and feeds you what modicum of wisdom I can provide. I'll repeat that mathematics are a vast domain of knowledge (and almost all domains of knowledge rely on mathematics to some extent; or are improved by some mastery of mathematics), so different profiles, interests, engagements, etc, abound. Your career possibilities are much larger than you think if you learn mathematics.

Don't hesitate to ask any question you might have in the comments.

PS: if you're interested, I wrote (and am still writing) and introduction to higher mathematics for late high schoolers, specifically with the idea of providing the building blocks and insight necessary for set theory, abstract algebra and category theory. It's far from finished, but it should give you something to sate your curiosity and accompany whatever linear algebra (and possibly analysis) resource you choose to begin with. https://github.com/Fulguritude/Mathophilie

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    $\begingroup$ A lot of text, but I wouldn't say that "real" mathematics is that different from school mathematics. The formalism is more formal and as you progress your toolbox contains more and more abstract tools which might obscure where they came from. But often, the underlying ideas are as playful as ever, just more formally done and with tools that make the idea actually work and useful. That being said, grasping the difference between "properties" and "structure" was, for me, one of the defining realizations at undergraduate level that have initially not been taught explicitly. $\endgroup$ Apr 4 at 5:28
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    $\begingroup$ The reason why I say mathematics in schools is not "real" mathematics is because of the emphasis put on rote calculation techniques, the pre-chewed guiding of problems so that student's results can be artificially inflated (to the detriment of students' actual problem exploration and solving skills). The idea that there's "one right answer, and it's at the end of the book". I often say that I learned more about problem-solving playing Zelda than in school for this reason. $\endgroup$ Apr 4 at 13:33
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    $\begingroup$ But most importantly, the hiding of anything interesting under the pretext that it is "too complex" is a huge issue. Complex ideas are excellent motivators and "teasers". The idea that "you shouldn't be able to listen to Bach until you've learned your scales" is absurd. The idea that "Eminem isn't real music" is absurd. I have never in my life seen anyone learn more when being taught less: especially when that means being presented with bastardized, butchered, limited information. Show that degree 2 polynomials are the reason why Mario can jump, and 80% your students will be more interested. $\endgroup$ Apr 4 at 13:41
  • $\begingroup$ Lol +2 for the category theory plug $\endgroup$
    – Prince M
    Apr 9 at 6:22
  • $\begingroup$ I needed to print your answer to reread it in the future because I really love what you say. Thank you, it somehow helped me. $\endgroup$
    – Amelian
    Apr 28 at 23:54
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I suggest you to start studying calculus it is very good subject and you will find it very

fun, Introduction to proofs also very good, Introduction To Number theory it

is very kind+fun subject in Mathematics , linear algebra also very good ,differential

equations are also very good after them partial differential equations ...after these

you may be capable of studying abstract algebra and real analysis ...after these you will

choose by yourself how to continue(I chose some courses I love them very much) .

I suggest the following books:

1)Calculus Early Transcendentals Ninth Edition by James Stewart, Daniel K. Clegg, Saleem Watson

2)Book of Proof by Richard Hammack Third Edition ...it is very great book in my opinion. https://open.umn.edu/opentextbooks/textbooks/7

3)A Friendly Introduction to Number Theory.....this book is also very great book...as its name it is friendly book.

4)Elementary Linear Algebra Stephen_Francis_Andrilli Fourth Edition...it is very good.

5)william-e-boyce-richard-c-diprima-elementary-differential-equations-wiley-2012

6)Partial Differential Equations- Theory and Completely Solved Problems. this book is very good.

7)Contemporary Abstract Algebra by Joseph A Gallian ninth edition.

and book may be easier than it is Abstract Algebra Schaum's outlines it is also very good.

8)introduction-to-real-analysis-4th-ed-r-bartle-d-sherbert-wiley-2011 this is very good.

some of these may be available free at internet.

In my opinion to be great mathematician you need to :

1)love and enjoy studying Mathematics very much and you should love and enjoy its

difficulties...I always say If mathematics is not difficult I don't like it...so you

should love and enjoy facing difficulties of mathematics...so saying I give up is not

something you say .

2)The vitality is very important...and you should hate laziness because it destroys dreams

if you let it occupy you.

  1. should try to solve problems to practice subjects and strengthen yourself ...solve as

much as you can ...and make balance between practicing and learning something new.

this is my opinion.

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I'd say there's no rush. Not only do you have plenty of time, but you are ahead of the curve.

So I'd try to pick up skills that one would otherwise have to pick up "on the go", and skills that make it easier for you to experiment.

For the first part I'd say it's both interesting and helpful to take a closer look at the tricks in a typical toolbox of a mathematician.

A book that goes into this line of thinking would e.g. be "Exploring Mathematics" by Daniel Grieser.

For the second part, there's little that's more helpful than being familiar with a CAS. Often, to even find a hypothesis, you need to test your way through sufficiently many examples. So, when you get to the point that using a CAS doesn't feel like it'd distract you from the problem, it really eases and speeds up your exploration process.

Here, I'd probably go with Mathematica. It is powerful, has a big community, and for every part of math there are functions to make life easier for you.
Unluckily, I don't have a real recommendation for a book here.

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