# What should I learn next for pure math?

I can do basic single variable calculus which is essentially all you do at A Level in the UK. I also just read "What is mathematics?" by Richard Courant which I found very good. I would like to know where to go after I finish my proof writing and basic discrete maths books I am currently reading. Should I study real analysis? What book? Linear algebra?

I would appreciate advice on where to go next and what book would be good. I am currently 14 and my end goal would to become a pure mathematician.

The reason I ask this question is because I was trying to create a structure and I thought about the Gerard t'Hooft physics one or even the Pure mathematician or statiscian plan but they don't seem particularly specific in what you are meant to do where i.e they call a topic calculus and then a later one vector calculus and then analysis but as a pure mathematician I want to go straight into analysis, of reals obviously. In conclusion, I am asking this because I am going to go by steps now and choose the next topic as it comes.

EDIT:

Would this work:

In order...

Elementary Discrete Maths, Real Analysis, Linear Algebra, ODE's, Probability, Fourier Analysis, Complex Analysis, PDE's, Graduate stuff which I will get to when needed.

I'm not sure if probability is necessary but I think it'll be interesting.

• When you say "create a structure", do you mean plan out a course for yourself? – Sharkos May 29 '13 at 15:51
• Study whatever you find interesting. If I were in your position, I would find a great undergrad analysis text (there are quite a few, just puruse your library) as well as a great undergrad linear algebra text. For linear algebra, I'de sugguest linear algebra done right, by Axler. If you're feeling more ambisious, try Hoffman and Kunze. – RougeSegwayUser May 29 '13 at 15:53
• @Sharkos By structure that is what I mean. – tridianprime May 29 '13 at 16:22
• Linear algebra is basic to essentially all other math; I would recommend it before analysis. – Brian Rushton May 29 '13 at 16:55
• I think its also worth mentioning that when learning maths, its best not to seperate what you are doing into different chunks, and that it is much better to learn different topics side-by-side - it's suprising how interlinked things can become. Also, I feel like this is a good article to read; in particular the words of Imre Leader: bbc.co.uk/news/magazine-15482101 – Andrew D May 29 '13 at 17:02

I think you are correct to lean toward analysis. It is the usual first step to pure math as you describe your inclination. Linear algebra can be done on several levels, but I think is best done in a rigorous context after real analysis.

In real analysis you will become familiar with proofs, begin to accumulate a working, ingrained math vocabulary as well as learn about topology, metric spaces, concepts such as convergence, continuity, etc. There is also usually a component presenting the material of calculus on a rigorous basis. This will give you a feel for the difference between operational mechanics and the pure math behind it.

I would suggest you take a look at this free set of notes of lectures given by Vaughan Jones (Fields Medal winner, equiv. Nobel Prize in math). They are really beautiful, are self-contained, and build nicely from a level that does not require prior experience.

I'm not exactly too sure on what you mean by "basic single variable calculus", but for those who are not from the UK, I figure it's worth briefly outlining the calculus involved in A-Level maths (for those who aren't in the know, A-levels are (usually) spread out over two years, so I'll list them as under years 1 and 2):

• Year $1$ - Basic differentiation and integration of polynomials and functions of the form $x^a$ for $a \neq -1$ (as in most pupils will not even learn the definition of differentiation, just the rule), calculation of stationary points for cubic points and sufficient conditions for the turning point to be a local maxima/minima/stationary point
• Year $2$ - Differentiation and integration of functions of the form $x^{-1}$, trigonometric functions, the exponential function, and the logarithmic function. In this, the chain, product and quotient rules are covered, along with the methods of integration by substitution and integration by parts (without proof).

While the above list is not completely exhaustive, I feel like it covers the majority of what I covered at A-level (which was only a year ago, but a year at university studying maths is a long time!)

Anyways, as to answering the question, I feel like a few good places to start are:

• Calculus by Spivak - This is a book which acts as a good introduction to real analysis, building the whole theory of single valued calculus from first principles (this including a treatment of the real number systems)

• The Pleasures of Counting by T. W. Körner - While this isn't a textbook as such, I feel like its worth a read by any aspiring mathematician (the reader will also find that the author has a very pleasant writing style, which makes the read very enjoyable)

• If you haven't done Further Maths, then looking at the textbooks for the Further Pure courses should also be of interest - if you aren't home schooled then you should be able to get copies from your school, else I would personally recommend to get the MEI FP1 through to FP3 books, along with possibly the differential equations book.

As for linear algebra, I can't think of a really good textbook to do with it, but maybe someone else will have a good idea.

• Thanks but I am doing this to avoid the textbooks used in a level. – tridianprime May 29 '13 at 16:19
• Personally I would say there's nothing wrong with the A-level textbooks for Further Maths (the ones for just the ordinary A-level I don't rate nearly as highly) as they do provide quite a good introduction to new ideas (such as complex numbers, matrices, and eventually some vector calculus and multi-variable calculus). Why are you wanting to avoid using the textbooks at A-level? – Andrew D May 29 '13 at 16:26
• I had bad experiences with the core and mechanics ones. – tridianprime May 29 '13 at 16:33
• I will happily admit that the core and mechanics textbooks (along with the statistics ones) are awful (as they were, especially for self study), at least for MEI, and I can understand why that would put you off. While I can ask you to maybe reconsider for the further maths books, if you are personally paying for them, I wouldn't bother, but otherwise, maybe just get the one and see how it goes. – Andrew D May 29 '13 at 16:39
• Ill see if I can get a PDF and check it out. Thanks. I would be paying unfortunately though and my school because its secondary has no pauses for a second anything. – tridianprime May 29 '13 at 16:44

The standard outline in many schools after single variable calculus is linear algebra, multivariable calculus, and differential equations, followed by several core classes, which often include analysis and abstract algebra but can include complex analysis (which does not have real analysis as a prerequisite necessarily), PDE's, numerical methods, etc. After that, people tend to take several electives, but in graduate school, you start over, often with topology, real analysis/measure theory, and abstract algebra again.

You can skip multivariable calculus and go into real analysis if you really want to, but multivariable calculus is beautiful, rigorous, fascinating, challenging, and studied and developed by Euler, Gauss, etc. Stewart Calculus has a great description of vector valued functions with the tangent, normal, and binormal, etc.

Analysis is not just more advanced calculus; it's a different emphasis, and tends to be less geometric than multivariable calculus.

This is all written from a limited perspective in the U.S.

• I think PDE's and abstract algebra are a little overly ambitious at this stage...Start with calculus I'd say :)! – dreamer May 29 '13 at 15:58
• Wouldn't I cover multivafiable calc in a rigorous study of real analysis? – tridianprime May 29 '13 at 16:24
• If you choose good books, yes. Haha, I think I was just trying to sneak some geometry into your schedule. I love differential geometry, which is touched on in some multivariable books. – Brian Rushton May 29 '13 at 20:50

Start with calculus and after that do linear algebra. For calculus I can recommend you Calculus Early Transcendentals by Stewart, and for Linear Algebra I recommend Linear Algebra: A modern Introduction by Poole. I'd start with calculus though. That'll be challenging enough at this stage.

You should study both, so you I would advise you to start studying the one that interests you the most.

I prefer analysis so I'll recommend you a couple of books about the subject, I took a course in real analysis last year and one recommended book was the one by Bartle and Sherbert (http://www.amazon.co.uk/Introduction-Real-Analysis-Robert-Bartle/dp/0471433314/ref=sr_1_1?ie=UTF8&qid=1369842686&sr=8-1&keywords=bartle+sherbert), it seemed to me a good book, it introduces some concepts of topology which are really interesting and help you to work with abstract spaces and develops some important result in convergence of series of functions (ie, exchanging the order of operation if you have a good enough kind of convergence) that have been quite useful in all the courses I took in many branches of mathematics and physics (ie, ordinary differential equations), you might find some lacks but I think it's a good book to start. I'd also recommend you Calculus by Spivak (which is useful especially for practising with more "practical" exercises, not just theorethical ones).

I can't recommend you good books of algebra because I'm spanish and I used books which were written in spanish (and I don't think there are english translations). EDIT: I just saw in one of the comments the book of algebra by Hoffmann and Kunze and it reminded me that I tried to use it while studying the subject, it's a hard book but at the same time it can give you a complete understanding of what are your goals while developing all the important results of linear algebra.

But my most important advice should be that you don't have to have any hurry, you're quite young yet and you don't have to feel dissappointed or discouraged if you can't get as much from those books as you had planned originally, you have to work hard, try again or look in a different book if you didn't understand something. There isn't a perfect book, to achieve a good understanding of the subject is helpful looking for information in a bunch of different places but I recommend you the book I stated previously as a good place to start into the subject of analysis of a real variable and to help you get deeper later.

I hope this has been helpful.

• That book looks like it could be good. Ill look for a PDF or see of Ivan get it. – tridianprime May 29 '13 at 16:21

Look at the university websites - Cambridge has a reading list here which has good general section.

There is also the Princeton Companion to Mathematics, if you can get hold of it from a library, which has a range of articles across a variety of mathematical topics at different levels of sophistication. The articles have a bibliography - so when you find a subject that interests you, you can do some further reading.

I would suggest pursuing what interests you rather than getting too focussed on the curriculum.

Also, though challenging, you might explore Hardy's classic Course of Pure Mathematics, which isn't exactly a modern treatment. But there are lots of things in there (particularly in some of the exercises) which are worth knowing.

• I have seen the cambridge reading list before but I have never been able to get a good inclination of how useful they actually are or whether they are just "time fillers". – tridianprime May 29 '13 at 16:52

I would agree with many of the answers above that real analysis will strongly propel you forward in the world of pure math. Where I went to school it was required to study multivariable calculus before moving onto analysis (it probably would have been manageable to study both concurrently). I did find that multivariate calculus and linear algebra helped me conceptually when thinking about topological spaces which I found beneficial upon entering analysis (the actual course title was "Advanced Calculus").

• I always got the impression that in a rigorous study of real analysis you cover single and multi var. calculus. – tridianprime May 29 '13 at 17:06