What Advanced Math Should I Teach Myself Next?

[brainy kevin]

Hello.

I am going to be a freshman in high school, and I am a very advanced math student. My school goes at a pace that's way too slow for me, so I decided I would teach myself some mathematics. So far, I've got basic linear algebra, and even basic-er modern algebra, and I'm working on calculus. (I've done all the prerequisites of course.) So, what I want to know is, what comes next?
I want to learn more advanced math, and I need some recommendations for subjects and textbooks. I'm interested in pure mathematics, not applied. I'm interested in some higher level linear algebra textbooks and good modern algebra textbooks, but I would be glad to hear any recommendations you may have. Thank you for your help.

*Edited*
Since there appears to be some debate over whether I really know what I'm talking about, let me clarify: I do know the distinction between basic algebra and linear/modern algebra. I am thirteen, but I will be fourteen very soon. (My birthday is May 20th) Here is a list of some of the stuff I know:
Basic set theory, functions, binary operations, relations, basic finite group theory, permutation groups, abelian groups, and a little bit of field theory. I also know vectors in Rn, vector algebra, matrices and determinents, and some linear transformations. Thanks for your help!

 

[AUMathTutor]

If you're into Algebra, you may also like any of the following:
- Set Theory
- Mathematical Logic
- Theoretical CS

There are some alright online "textbooks" in each of these categories. Maybe like...
http://www.math.uchicago.edu/~mileti/teaching/math278/settheory.pdf
http://www.math.psu.edu/simpson/courses/math557/logic.pdf
http://www.nada.kth.se/~johanh/complexitylecturenotes.pdf

Tons of other books are available in these areas, both online and in print. Here are what may or may not be some more advanced books:
http://www.liafa.jussieu.fr/~jep/PDF/MPRI/MPRI.pdf
model / proof theory... (couldn't find anything great, but there are some smallish papers floating around)
http://www.math.uu.nl/people/jvoosten/syllabi/catsmoeder.pdf

 

[Howers]

Are you suggesting that you are 13 and already have the rudiments of linear and modern algebra down?

What I suggest before delving into modern math is that you get a real understanding of the math you are learning - not the mickey mouse stuff you guys do in school. For that purpose, focus on the theory books of this thread: https://www.physicsforums.com/showthread.php?t=307797 . Did you prove all the trigonometric identies, etc? Are you strong in geometry, and analytic geometry?

If you insist on moving ahead, with a weak background, you should definitely complete calculus before moving ahead. Once that is done, try Friedman for Linear Algebra and Dummit and Foote for Abstract Algebra. After calculus, do analysis with Pughs book or Rudin. This is what is usually done in a 4 year undergrad.

 

[Landau]

Once that is done, try Friedman for Linear Algebra

Do you mean https://www.amazon.com/dp/0130084514/?tag=pfamazon01-20?
You may also consider https://www.amazon.com/dp/0387982582/?tag=pfamazon01-20

You say you're working at calculus. If you really want a good understanding of calculus, try the great book https://www.amazon.com/dp/0914098918/?tag=pfamazon01-20. After that, move on to Rudin or Pugh, as Howers already mentioned.

 

[DukeofDuke]

If you're into Algebra, you may also like any of the following:
- Set Theory
- Mathematical Logic
- Theoretical CS

There are some alright online "textbooks" in each of these categories. Maybe like...
http://www.math.uchicago.edu/~mileti/teaching/math278/settheory.pdf
http://www.math.psu.edu/simpson/courses/math557/logic.pdf
http://www.nada.kth.se/~johanh/complexitylecturenotes.pdf

Tons of other books are available in these areas, both online and in print. Here are what may or may not be some more advanced books:
http://www.liafa.jussieu.fr/~jep/PDF/MPRI/MPRI.pdf
model / proof theory... (couldn't find anything great, but there are some smallish papers floating around)
http://www.math.uu.nl/people/jvoosten/syllabi/catsmoeder.pdf

Useful links. Thanks.

 

[AUMathTutor]

Give the kid a break. He says he wants some Algebra related material, give it to him. You don't have to be condescending because he makes you feel insecure. Who cares if he's really 13?

 

[qntty]

As far as other subjects you can learn, there's always discrete math such as combinatorics, graph theory and number theory.

 

[DrummingAtom]

Give the kid a break. He says he wants some Algebra related material, give it to him. You don't have to be condescending because he makes you feel insecure. Who cares if he's really 13?

Seconded.

 

[Landau]

Give the kid a break. He says he wants some Algebra related material, give it to him. You don't have to be condescending because he makes you feel insecure. Who cares if he's really 13?

I'm not following you...I don't see anyone being condenscending?

 

[MissSilvy]

I'm not following you...I don't see anyone being condenscending?

Seconded. It was an honest question; lots of people come here asking for study advice and make mistakes about what they know (like the person who though linear algebra and advanced algebra were the same topic) so it's normal to ask this sort of question.

Isn't there a thread where we can make a big list of all the good beginner textbooks in the various math branches? It'd cut down on these sorts of threads.

 

[AUMathTutor]

"Are you suggesting that you are 13 and already have the rudiments of linear and modern algebra down?

What I suggest before delving into modern math is that you get a real understanding of the math you are learning - not the mickey mouse stuff you guys do in school."

That's rude. You can pretend it's not, but if you wouldn't talk to the guy sitting in on your PhD dissertation like that, you shouldn't talk to a complete stranger like that.

 

[Pengwuino]

There really should be a thread that dictates exactly what each level of mathematics is all about and have an example of what kind of material is taught in such a subject. I think people can easily get confused on what exactly they know. I personally wouldn't have known the difference between "algebra" and "linear algebra" at that age. Maybe this person is confused as to the terms and it wouldn't be helpful to throw a complex analysis textbook at him. Then again maybe he is at htat level and thank god I don't have to take my PGRE against him.

 

[qntty]

It may have been rude but Hower's point is valid. It's hard to judge exactly how well of an understanding you have if you've studied something on your own. I know I've made the mistake of overestimating how well I know something many times. It's possible that the OP does have good background and is ready to study LA but she/he should question whether they do.

 

[AUMathTutor]

Granted, but it was still condescending, and that's all I was trying to point out. When I was called out on it, I specified what I meant. I don't think anybody disagrees that Hower could have been more tactful.

And I can only assume that people who ask for something know what they're getting into. The books I linked to assume a level of mathematical maturity on par with an understanding of linear algebra and possible some abstract algebra as well. The OP can probably judge for him/herself based on the level of understanding of that material how much they really know about math.

Oh well. I just think it's a little pretentious to offer advice where it isn't asked for. The OP didn't ask "do I know what I'm talking about". And he didn't make any claims beyond that he had been doing some self-study... certainly nothing to argue with. I don't disagree with suggesting that self-study may not be the most effectual method of doing things, and that making sure you have a strong grasp of the fundamentals is important, but... anyway, I think you guys see what I'm saying.

 

[MissSilvy]

That's rude. You can pretend it's not, but if you wouldn't talk to the guy sitting in on your PhD dissertation like that, you shouldn't talk to a complete stranger like that.

The guys on my PhD dissertation are probably masters of their fields and have many many years of experience and know-how. The people on this forum are in all shapes and sizes and there is no guarantee that they know algebra from their blowhole, so questioning someone's knowledge and judgment is very permissible and necessary. I would rather someone had critically questioned what I thought I know so that I could get the proper advice I came here for rather than get something far too difficult for my level chucked at me (which I would try, fail to understand, and then get discouraged).

The only reason I see not to question someone's competence is vanity and anyone who's coming here for the sake of showing off deserves to be affronted.

 

[T.O.E Dream]

I'm in a similar situation. I've got a whole year till high school and i honestly feel that the way they teach math in school is way too slow. I guess only 5% the stuff i learn in school might be new but other than that it's all review. So i decided to teach myself math, and so far it's pretty good. I've just wrapped up algebra (dealing with factoring, linear equations and applications as well as quadratic equations and applications). Now I'm onto geometry then trigonometry and then calculus. Is this the right path to go so far? And yes honestly i think someone should put up a thread on the logic order of math and it's content.

 

[General_Sax]

The OP could've been less boastful. I doubt he would speak like he writes; and if he did, then I doubt you would consider Hower's response condescending.

 

[jin8]

Hello.

I am going to be a freshman in high school, and I am a very advanced math student. My school goes at a pace that's way too slow for me, so I decided I would teach myself some mathematics. So far, I've got basic linear algebra, and even basic-er modern algebra, and I'm working on calculus. (I've done all the prerequisites of course.) So, what I want to know is, what comes next?
I want to learn more advanced math, and I need some recommendations for subjects and textbooks. I'm interested in pure mathematics, not applied. I'm interested in some higher level linear algebra textbooks and good modern algebra textbooks, but I would be glad to hear any recommendations you may have. Thank you for your help.

Kevin,
Well, It's very impressive you're just going to be a freshman in a high school. How much Linear Algebra have you learned?
If you already got some "basic" LA, then I recommend you to use Paul Halmos' Finite-Dimensional Vector Space, actually it's the textbook we're using in honors abstract linear algebra course for advanced undergraduates. Warning, the textbook is written in a very concise and so-called mature way and thus it's hard to understand. If you find it hard, then you probably like another text, Hefferon's Linear Algebra, http://joshua.smcvt.edu/linearalgebra/ it's free and it's a superb text.
Actually, since you are so far ahead, I do NOT suggest you work on Calculus. You may start learning Real Analysis instead, since you want pure math.

 

[AUMathTutor]

"The guys on my PhD dissertation are probably masters of their fields and have many many years of experience and know-how."
The way I was raised, respect is not conditional on what I perceive of other people. I don't think it's being entirely respectful to talk that way to people. Yes, the OP did say he was an advanced student. He certainly is, if he's studying Modern Algebra in the 9th grade. There's a difference between arrogance and being frank.

"The people on this forum are in all shapes and sizes and there is no guarantee that they know algebra from their blowhole, so questioning someone's knowledge and judgment is very permissible and necessary."
Well, reasonable people can have different opinions. I find it a little pretentious, whereas you think it's to be preferred. I guess variety really is the spice of life.

"I would rather someone had critically questioned what I thought I know so that I could get the proper advice I came here for rather than get something far too difficult for my level chucked at me (which I would try, fail to understand, and then get discouraged)."
Again, I guess we'll just chalk this off to a difference in opinion. Who knows how the OP feels about the comment? I would have found it offensive, which is why I piped up.

"The only reason I see not to question someone's competence is vanity and anyone who's coming here for the sake of showing off deserves to be affronted."
Well, the only reason not to punch strangers in the face is weakness, but people don't run around assuming people can take it.

 

[AUMathTutor]

"Actually, since you are so far ahead, I do NOT suggest you work on Calculus. You may start learning Real Analysis instead, since you want pure math."

I would respectfully disagree. IMHO, a little knowledge of the plug-and-chug mechanical get-an-answer calculus is useful for introducing some terminology, if nothing else.

 

[qntty]

Alright so from what I gather from the OP's post

(1) he/she has completed a precalculus course

(2) Linear Algebra was the first stop for the OP when they got bored of their school math curriculum

In that case, the OP certainly is very advanced but starting with real analysis is a big mistake. Students need a context before they begin to rigorously build up the basis of calculus. Even some math majors struggle in analysis courses.

 

[jin8]

Alright so from what I gather from the OP's post

(1) he/she has completed a precalculus course

(2) Linear Algebra was the first stop for the OP when they got bored of their school math curriculum

In that case, the OP certainly is very advanced but starting with real analysis is a big mistake. Students need a context before they begin to rigorously build up the basis of calculus. Even some math majors struggle in analysis courses.

Well, actually in China all college students starting with real analysis. (i mean math and science majors)
Maybe I'm biased, I feel extremely bad when I first learn calculus 'cause I don't like just memorize and plug-in and play, I want proves.
I think the guy posted this thread should be very smart, and he say he love pure math so I assume he will enjoy analysis.
Maybe he should start with an easier real analysis text?

 

[Werg22]

Well, actually in China all college students starting with real analysis. (i mean math and science majors)
Maybe I'm biased, I feel extremely bad when I first learn calculus 'cause I don't like just memorize and plug-in and play, I want proves.
I think the guy posted this thread should be very smart, and he say he love pure math so I assume he will enjoy analysis.
Maybe he should start with an easier real analysis text?

​

What makes you think there are no proofs in calculus? As far as I know, calculus refers to the study of limits, continuity, differentiation, Riemann integration and sequences in one dimension. Real and complex analysis refer to the study of those same notions over R^n, C^n, general metric spaces and manifolds.

 

[snipez90]

Well educational standards are very different here. Also, this is a guess, but even though college students in China start with real analysis, surely some, if not the vast majority of them have had previous exposure to calculus? There are good calculus books out there that do cover the proofs. Even when I took Calc BC my senior year, I made sure that the theorems and claims were reasonable (often through physical intuition), even if I didn't seek proofs for everything (my calc text omitted many proofs or referred to a more advanced text). So proof or no proof, memorization does not need to take priority.

Many easier "real analysis" texts devote a good portion, if not all of the book to rigorously proving the theorems that you typically encounter in basic calculus. Spivak is a very good choice because the OP can't go wrong if he likes pure math and wants to tackle calculus. Just by reading the text, one can learn basic calculus very well and pick up many introductory concepts in real analysis. By doing many of the problems, the reader will have a very solid foundation for a more advanced analysis course. Spivak actually pulls some problems from Baby Rudin, while coming up with his own challenging exercises.

Another good text that I'm working through now is Advanced Calculus by Fitzpatrick. Although Spivak gives more motivation, Fitzpatrick succinctly guides the reader through the concepts without skimping on proofs. Fitzpatrick also introduces many fundamental concepts in analysis early on, whereas Spivak builds calculus up from the basics, easing the reader into greater abstraction. But judging from your experiences with linear algebra, you can't go wrong with Spivak.

 

[chiro]

I'm in a similar situation. I've got a whole year till high school and i honestly feel that the way they teach math in school is way too slow. I guess only 5% the stuff i learn in school might be new but other than that it's all review. So i decided to teach myself math, and so far it's pretty good. I've just wrapped up algebra (dealing with factoring, linear equations and applications as well as quadratic equations and applications). Now I'm onto geometry then trigonometry and then calculus. Is this the right path to go so far? And yes honestly i think someone should put up a thread on the logic order of math and it's content.

You realize that even high school math represents hundreds and hundreds of years of mathematics and by no means is that kind of math "solved". Relationships between for example e^x cos^x sin^x (apart from the standard Euler equation) are still waiting to be solved. Trigonometry is not completely a solved problem either. There are still relationships waiting to be proven and solved and lots of people just take it for granted that a ratio defined by an infinite series justs "exists" out of thin air.

If you can explain why trigonometry is the way it is, then you should start writing a paper on it. If you can explain trigonometry you'd probably understand why physics uses rotations and understand all the rotational symmetries involved in particle physics. You see even your standard sin(x) tan(x) we all take for granted. But if you sit back ask yourself "Why is it the way it is?" it is not so simple because it just "works" for us.

Anyway still I wish you good luck with your pursuits.

 

[brainy kevin]

Thanks for all your help, it's really nice of you.I've added some information at the beginning for clarity.

 

[brainy kevin]

Are you suggesting that you are 13 and already have the rudiments of linear and modern algebra down?

What I suggest before delving into modern math is that you get a real understanding of the math you are learning - not the mickey mouse stuff you guys do in school. For that purpose, focus on the theory books of this thread: https://www.physicsforums.com/showthread.php?t=307797 . Did you prove all the trigonometric identies, etc? Are you strong in geometry, and analytic geometry?

If you insist on moving ahead, with a weak background, you should definitely complete calculus before moving ahead. Once that is done, try Friedman for Linear Algebra and Dummit and Foote for Abstract Algebra. After calculus, do analysis with Pughs book or Rudin. This is what is usually done in a 4 year undergrad.

Thank you so much for that link by the way. It's got some great books on it. Your advice is really good too.

 

[ice109]

Thank you so much for that link by the way. It's got some great books on it. Your advice is really good too.

those are all pretty low level books in that link.

for calculus look for richard courant's or tom apostol's book.

for linear algebra gilbert strang or the hoffman kunze book.

for abstract algebra dummit & foote.

for real analysis serge lang's book or w/e. they're all the same.

for complex analysis start with churhill and brown for computational practice then go to any of the good authors like serge lange, courant, rudin etc. again they're all the same

for topology munkres

hmm that about covers and undergrad pure degree. for some applied stuff:

for odes boyce and di prima

for pdes i have a good book some where but can't remember right now but i'll come back when i find it.

note all these books can be found in various places online for free

http://hk.mathphy.googlepages.com/puremath.htm#1calculus

that's a good list

 

[Fragment]

I second ice109's list, covers a lot of good books. Spivak for calculus is also recommended(Usually for more advanced students). As requested, here's a list of subjects according to logical order:

Basic algebra & Geometry
Trigonometry (Precal as sometimes is called)
[A good book on proofs and mathematical logic would be useful here)
Calculus
Linear algebra
More advanced calculus/Analysis
Follow ice109's from here I suppose


I personally found that the optimum way to self-study was to get a book that was mostly theory, and then follow it up by a more practical and hands on book.
For example, Gelfand has a nice series of books (Pretty advanced for the starting student), a lot of them are proof based and cover algebra-->Calculus. I paired Gelfand's books with Sullivan for Trig&Algebra, then went through Spivak and Apostol...Another useful thing, which I suggest to others who self-study, is to get at least two books (e-book or just books) on the particular subject. This way when something isn't presented in a way that one understands, one can refer to the other copy, so on and so forth.

Hope I helped

 

[brainy kevin]

Do you know if Jacob's books are any good?

 

[Fragment]

If you can find an older edition(1&2, but they are usually expensive) of Jacobs, then they are great. The newer versions seem more 'watered' down. If you cannot find those, then you should look for Larson or Sullivan, and Gelfand.

 

[Howers]

Granted, but it was still condescending, and that's all I was trying to point out. When I was called out on it, I specified what I meant. I don't think anybody disagrees that Hower could have been more tactful.

And I can only assume that people who ask for something know what they're getting into. The books I linked to assume a level of mathematical maturity on par with an understanding of linear algebra and possible some abstract algebra as well. The OP can probably judge for him/herself based on the level of understanding of that material how much they really know about math.

Oh well. I just think it's a little pretentious to offer advice where it isn't asked for. The OP didn't ask "do I know what I'm talking about". And he didn't make any claims beyond that he had been doing some self-study... certainly nothing to argue with. I don't disagree with suggesting that self-study may not be the most effectual method of doing things, and that making sure you have a strong grasp of the fundamentals is important, but... anyway, I think you guys see what I'm saying.

I see your point, but most students feel they have mastered elementary math in school without realizing what curriculum they are getting it from. Hence, I suggest real books to make sure that is indeed the case. If they have, nothing is lost by looking over free references.

I didn't mean anything by my remark about his age other than surprise. I know gifted people exist... in fact, that's almost all I post about. Its not my intention to slow him down at all, which is why I listed the modern books anyway in case he gets bored. I just think its safe for him to spend a couple of months getting up to speed if it is needed. Lastly, don't project overtones of your own insecurity onto me.

 

[kingdt]

Most of the pure math majors at my school took many college courses when they were in junior high and high school(or international equivalent). I would REALLY recommend trying to take classes from a local university, that way you will really learn the material well and get credit for your work (I mean on transcripts and resumes etc) instead of wasting your time in high school.