What is the recommended order for self-teaching advanced mathematics?

[MrApex]

Hey there , I am a 14 year old teaching myself mathematics and I was wondering what comes next.. I have completed everything uptil Precalculus(That includes Algebra 1,2 Geometry and Triginometry) .. Id appreciate it if one you could list the order of learning mathematics , I know that math is interconnected (eg: You need Abstract alg to understand Linear Alg and you need multivariable Calc and whatnot) , say I was on a road to getting a degree in Maths . What maths will I be taught and in what order. Thanksand It'd be great if some you seniors could suggest good books on self teaching some of the more advanced Math(that which comes after Pre-cal)

 

[MarneMath]

After someone has completed Algebra I&II, Geometry, and Trig, the natural progression seems to be Calculus. I'm sure a lot of people would recommend a lot of different books, but http://ocw.mit.edu/resources/res-18-001-calculus-online-textbook-spring-2005/textbook/ this is free and online. So enjoy. After calculus, the normal progression seems like this: Calc I, II, III, Linear Algebra, Differential Equations, and then a bunch of electives with an abstract algebra and real analysis course thrown into the mix.

 

[MrApex]

Thanks , Could you also tell me what the prerequisites for number theory are? and how id go about to start learning it?​

 

[micromass]

Thanks , Could you also tell me what the prerequisites for number theory are? and how id go about to start learning it?​

Elementary number theory doesn't really have any prereqs. You have to be well-versed in proofs though. You shouldn't be mystified with proofs by contradiction or induction and you should be able to form basic mathematical arguments. Furthermore, number theory can be abstract, so you might need to some mathematical maturity.

If you go deeper into number theory, then you are going to need abstract algebra and real analysis. But for a first course, I don't think those things are strictly needed (although a knowledge of abstract algebra makes a lot of things easier to grasp).

 

[MrApex]

@Micromass

The only proofing I've done was when I was learning Geometry , Could you perhaps suggest a few books on how to write proofs ? and a few books to get me started with number theory? Thanks.

 

[dustbin]

I don't really know that you need a dedicated proof book. If you pick some books that require you to work a lot of proofs, I feel this is sufficient... especially if you have someone who is willing to critique your answers (for instance, post your work up on here). I've read a couple of proof books but, other than giving a basic familiarity with proof techniques, I really feel that having someone rip apart your work and then you go back and correct/learn from the mistakes is the fastest way to learn. This is my perspective/experience, so take it as you wish.

 

[jasonRF]

If you feel like you want a book, you can find other recommendations by searching the forums. There are also some free books available online, for example:

http://www.people.vcu.edu/~rhammack/BookOfProof/

I have no idea if it is any good, but the price is right for trying it out!

jason

 

[MrApex]

Merci Beaucoup !

 

[lucid_dream]

Most people would advise you to study Calculus. It builds intuition for analysis ("calculus done right"). However, I would recommend studying a little bit of topology and set theory as well. The earlier you learn these subjects, the better of you will be in the future. I find I learn the most when I am most confused. And at first, these subjects can be very confusing.

 

[micromass]

Most people would advise you to study Calculus. It builds intuition for analysis ("calculus done right"). However, I would recommend studying a little bit of topology and set theory as well. The earlier you learn these subjects, the better of you will be in the future. I find I learn the most when I am most confused. And at first, these subjects can be very confusing.

You're suggesting him to study topology before studying calculus. Am I understanding you right?

Topology should be very straightforward (that's not the same as saying it is trivial). If you find topology confusing, then you don't have the right prereqs or maturity.

 

[lucid_dream]

You're suggesting him to study topology before studying calculus. Am I understanding you right?

Topology should be very straightforward (that's not the same as saying it is trivial). If you find topology confusing, then you don't have the right prereqs or maturity.

In my opinion, topology is confusing. In particular, I feel that many of the proofs to theorems require a high degree of creativity, and they are not always intuitive. In your opinion, if I find topology confusing, I don't have the right prereqs or maturity. But I do have the right prereqs, as I have taken graduate level classes in analysis, number theory, algebra and set theory before topology. Perhaps in your opinion I don't have the right maturity, but many of my professors (who have received PHds from Ivy League universities and the University of Chicago) have stated they found topology (in particular, algebraic topology) confusing when they first learned it as well. This would indicate to me you may be of exceptional mathematical maturity, and not have the same perception of an average, or even above average student.

I am indeed encouraging him to study topology while he is studying calculus. I do think it may be difficult, but if he sticks with it, he will quickly build his mathematical maturity. In my opinion, mathematicians should not even bother with calculus.

 

[micromass]

In my opinion, topology is confusing. In particular, I feel that many of the proofs to theorems require a high degree of creativity, and they are not always intuitive. In your opinion, if I find topology confusing, I don't have the right prereqs or maturity. But I do have the right prereqs, as I have taken graduate level classes in analysis, number theory, algebra and set theory before topology. Perhaps in your opinion I don't have the right maturity, but many of my professors (who have received PHds from Ivy League universities and the University of Chicago) have stated they found topology (in particular, algebraic topology) confusing when they first learned it as well. This would indicate to me you may be of exceptional mathematical maturity, and not have the same perception of an average, or even above average student.

I am indeed encouraging him to study topology while he is studying calculus. I do think it may be difficult, but if he sticks with it, he will quickly build his mathematical maturity. In my opinion, mathematicians should not even bother with calculus.

I don't really understand what you would find so hard about topology. Most of the terms and definitions are straightforward generalizations of things that you already know about. If you have the right prerequisites, then you should know about the things topology is trying to generalize. And if you do, then I don't see why topology is so hard.

I don't have an exceptional maturity at all. But I don't think you need one anyway.

I do agree that topology tends to have many definitions which make things confusing. I never remember what "accumulation point", "close point", "limit point", "complete accumulation point" are for example. But I can just look up those definitions. But yes, it might be common for new people to get tangled in a web of definitions they need to know. But that's about the only problem I really see.

 

[Chaostamer]

The book "Mathematical Proofs: A Transition to Advanced Mathematics" by Chartrand et al. is a pretty good introduction to thinking like a mathematician, and the content in that book furnishes the minimum required background for studying subjects like number theory and topology. Once you've written a decent number of proper mathematical proofs, you ought to be able to work through a more abstract subject.

 

[MrApex]

In my opinion, topology is confusing. In particular, I feel that many of the proofs to theorems require a high degree of creativity, and they are not always intuitive. In your opinion, if I find topology confusing, I don't have the right prereqs or maturity. But I do have the right prereqs, as I have taken graduate level classes in analysis, number theory, algebra and set theory before topology. Perhaps in your opinion I don't have the right maturity, but many of my professors (who have received PHds from Ivy League universities and the University of Chicago) have stated they found topology (in particular, algebraic topology) confusing when they first learned it as well. This would indicate to me you may be of exceptional mathematical maturity, and not have the same perception of an average, or even above average student.

I am indeed encouraging him to study topology while he is studying calculus. I do think it may be difficult, but if he sticks with it, he will quickly build his mathematical maturity. In my opinion, mathematicians should not even bother with calculus.

IN that case could you suggest a few books on topology which you think would be a good place to start? and what are the prereqs for topology?

and thank you all for the feedback (@Micromass and @Lucid Dream)

 

[eliya]

Everything in topology has a bunch of different definitions that are all equivalent, but still can be confusing, and sometimes using one definition and not another is actually helping in understanding a problem more clearly. But it's also interesting, and is a little bit like walking through an exotic zoo -- lots of weird counterexamples.

I would suggest that you learn how to do proofs as soon as possible. Reading and writing proofs is really where the fun is in Mathematics, and it will be your gateway to all advanced mathematics. Then there are several paths you can take: Linear Algebra, Abstract Algebra, Analysis, or Topology.

I took Analysis before I took topology and that helped. I also think that most topology books assume that you've taken analysis. At the very least it helps to know what limits are rigorously before taking topology.

Abstract Algebra has very few prerequisites, mostly elementary Number Theory, which is covered in the beginning of most Algebra books.

Linear Algebra has almost no prerequisites (except for what you've done so far), and is a nice and gentle mix between computing things and proving things. So you might want to look into that first.

Analysis is very wide -- it has the computational aspect, which is very important and broad, but it also has the more theoretical part where you ``just" prove theorems. It's possible to learn one and then the other, but most people go with the computational part (i.e. Calculus) and then the theoretical part which is what people usually call Analysis. I think that I would have benefited from a mix of the two using something like Spivak's Calculus book.

To put it in one line: First learn how to read and write proofs, then pick whichever subject you're interested in, but be reasonable -- don't try to learn Algebraic Geometry first.

 

[Dead Boss]

I don't really understand what you would find so hard about topology. Most of the terms and definitions are straightforward generalizations of things that you already know about. If you have the right prerequisites, then you should know about the things topology is trying to generalize. And if you do, then I don't see why topology is so hard.

I don't have an exceptional maturity at all. But I don't think you need one anyway.

I do agree that topology tends to have many definitions which make things confusing. I never remember what "accumulation point", "close point", "limit point", "complete accumulation point" are for example. But I can just look up those definitions. But yes, it might be common for new people to get tangled in a web of definitions they need to know. But that's about the only problem I really see.

What I find most confusing about topology is that there seem to be many similar (or related) definitions and concepts and not enough examples for them. It helps a lot when I know an example that comes with definition because even when I forget parts of the definition I can always fill the blanks by considering an example case. Another thing is that there is little to be done with the knowledge you gain for some time. When you learn calculus for instance you always do something with the definitions you learn which helps to clarify things quite a bit.

 

[dustbin]

IN that case could you suggest a few books on topology which you think would be a good place to start? and what are the prereqs for topology?

There is an interesting book with very minimal prerequisites (mostly just traditional high school maths) called The Knot Book. I believe the author's last name is Adams. Really fun, VERY interesting read. I would also recommend Principles of Mathematics by Allendoerfer. The first time I attempted to go through it, I didn't really like it... but I have since come to like it a lot and have worked through the whole thing. It will cover a great deal of material assuming no more prerequisite knowledge than you already possess.

 

[MPKU]

As a 4th level student of University majoring in Pure Mathematics & Physics, I think set theory would be a good approach, of course that and calculus. You can even try studying Real Analysis while doing Calculus, infact, it is probably for the best. However, with a foundation of higher mathematics(proof writing, etc.) this will probably be a huge challenege, but you certainly seem up to that :). I always found set theory enjoyable, and not very hard to learn at all; I'm sure you've already went over some in algebra (i.e. finding domain of specific functions). As for books, I'm not so sure..

There are great videos on the web of tutorials and lessons on Advanced Mathematics, just search for them, not to say you can learn math just by watching someone else do it; if basketball players watched a video on how to play basketball, that wouldn't make them good players, so you've got to get out there and try the problems.

Something I can't stress enough is to learn the theory and implications of the topics you study, not just the application (although that is a very important part of why we do mathematics). Learning the theory of calculus and WHY we study calculus will help you so much in your studies.

 

[nlsherrill]

After elementary algebra/trigonometry/geometry/pre-calculus I think a good progression is(in order):

Calc 1
Calc 2
Calc 3
Differential Equations
Linear Algebra
Partial differential equations(not required, but I recommend)
Set Theory
Abstract Algebra
Real Analysis
...

I think there are a few main things to point out about a good math route. First of all, it doesn't need to be strict, but you surely need to take the courses I listed minus PDE's. I recommended PDE's solely because there is so much more you learn in that class(at my UNI) than ODE's that I think every mathematician should see. For example, you really get to see how linear algebra is the basis for it all. Also, I think you should put off real analysis until you have taken set theory and abstract alg. This is because the jump from set theory proofs to real analysis proofs is quite a large one, so as much practice with this beforehand would be best.

I don't really think the order of taking calc 3, linear, and ODE's really matters that much. I took linear and ODE's at the same time and that was fine, although I didn't really see their connection until PDE's. I think you can take ODE's after calc 2, just make sure you know partial derivatives first.

 

[micromass]

After elementary algebra/trigonometry/geometry/pre-calculus I think a good progression is(in order):

Calc 1
Calc 2
Calc 3
Differential Equations
Linear Algebra
Partial differential equations(not required, but I recommend)
Set Theory
Abstract Algebra
Real Analysis
...

I agree but I would place Linear algebra before Calc 3 and Diff eq. Those subjects tend to make more sense with a knowledge of linear algebra.