## Most helpful math resources on the web

What are really helpful math resources out there on the web?

Please don't only post a link but a short description of what it does and why it is helpful.

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I edited your question a bit, my main goal was to remove the reference to WolframAlpha; and make the text easier to read. But feel free to revert if you like! (You can see the edit history and revert by clicking on "edited ... ago") – Ilya Nikokoshev Nov 4 2009 at 21:23
The answers below are great. Here's an idea I don't see that may be interesting to think about starting: An online example repository. This would be a place where one could upload and search various "first nontrivial example" notes. I think such a thing would really move mathematics along. – Jon Bannon Apr 20 2012 at 13:38

http://www.wikipedia.org

I have learned a lot of mathematics while reading Wikipedia. Allowing a wide audience to contribute to articles seems to work out well in the case of mathematics.

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See especially the mathematics portal en.wikipedia.org/wiki/Portal:Mathematics on which page there are links to more-specific mathematical portals, and to mathematics categories. – Rhubbarb Nov 11 2009 at 9:16
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For enumerative combinatorics, it's hard to beat Sloane's Online Encyclopedia of Integer Sequences.

It is what it says on the tin. A huge list of integer sequences, with references, links, formulas, and comments.

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Of course there's much more than enumerative combinatorics -- it's very strong in number theory, for example. – Charles Oct 7 2010 at 1:06

The Tricki

Quoting the site:

"Welcome to a brand new Wiki-style site that is intended to develop into a large store of useful mathematical problem-solving techniques. Some of these techniques will be very general, while others will concern particular subareas of mathematics. All of them will be techniques that are used regularly by mathematical problem-solvers, at every level of experience."

http://www.tricki.org/

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http://www.wolframalpha.com/

I'm just adding Wolfram Alpha to the fray so it can be voted on like other suggestions. For people who haven't heard of it, it's an online computational engine.

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Paul Stuart says: I think Wolfram Alpha is just a waste of computational effort. Whatever comes from the Wolfram corner I'd be wary. It ranges from acclaimed proofs to acclaimed "correct computation". Note they are infamous for making claims or assertions that aren't true. So, think twice. – Anton Geraschenko Nov 6 2009 at 18:21
Qiaochu Yuan says: In all honesty, I use Wolfram Alpha whenever I can't be bothered to remember what the real syntax for a command in Mathematica is. – Anton Geraschenko Nov 6 2009 at 18:22
dimitry (user 1087) says: @Qiaochu: That's just simply sad news altogether. There is a good point raised by Paul Stuart and that is the point raised about reliability. Wolfram Alpha, is a black box. God knows how reliable its answers are. If I aint completely sure, Wolfram had been (not unfairly) accused of being sloppy in many of his claims in his previous work on ANKS. If this is based on work in ANKS I'd be wary too. – Anton Geraschenko Nov 6 2009 at 18:25
Qiaochu Yuan says: Math Overflow is not a forum. It has no collective opinion on Wolfram, and this is not the place to discuss your opinion of his work. – Anton Geraschenko Nov 6 2009 at 18:28

Hosts high-level maths discussions, forums have inline LaTeX rendering.

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I use http://arxiv.org/ all the time.

Researchers post their articles here, so it is a great way to see if anyone have already a proof or an idea on something. Some people regularly access it through

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and they have rss feeds! – Andreas Holmstrom Oct 25 2009 at 4:56
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The book for this website is the "Concise Encyclopedia of Mathematics" by Eric W. Weisstein, CRC Press. The website is "Eric Weissteins's MathWorld", from which there are links to areas of mathematics and to other on-line mathematical tools. Both book and website take the form of a mathematics dictionary. This site is excellent, but beware that as it is associated with Wolfram Research, there is a bias towards the commercial mathematics package "Mathematica". – Rhubbarb Nov 11 2009 at 9:26

Everything by John Baez. In particular This Week's Finds in Mathematical Physics, the n-Category Cafe, and the n-Lab. He has an amazing ability to make even the most esoteric topics seem obvious and inevitable.

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I'm partial to both John Baez and the nLab, but the two don't have much intersection any more. The nLab is sometimes short on making really esoteric topics seem obvious. – Todd Trimble Aug 28 at 11:06

http://wiki.henryfarrell.net/wiki/index.php/Mathematics/Statistics

Large list of math blogs. Highly recommended in particular are Terence Tao's and Tim Gowers'.

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The link is broken and now we have mathblogging.org. – Junyan Xu Dec 20 2011 at 2:28

http://jmilne.org has lots of systematic, well-written courses.

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If you haven't figured this out already, you can read large portions of textbooks before you buy them to decide if they're what you need. (If you actually want free books, there's a separate question that addresses that.)

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One extremely useful way to use Google Books is as an "instant index." For example, I have the book ZZZ open in front of me. There is a term yyy I don't know the definition of. I look up yyy in the index of ZZZ. Ooops, no index! So I search for yyy at the Google Books page for ZZZ and order by pages. Done. – Sam Nead Mar 8 2010 at 14:24
An annoying feature of this site is that a lot of the book portions do not include the preface or introduction which is usually the first thing I want to read about a book. – Anonymous Mar 8 2010 at 16:41
The preface/introduction parts can be often found in the book preview on amazon.com. For the books published by the AMS these are often availble right on the AMS bookstore website (ams.org/bookstore). – mathphysicist Jun 4 2010 at 17:43

http://eom.springer.de

Very good articles with lots of references! (never mind the .de, it's in English!)

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Mathscinet, which contains summaries and reviews of published research papers. It's very useful when you want to get an idea of a paper without having to read it, and contains almost every paper ever published.

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I agree. But it's unfortunate that often the "summary and review" often consists of nothing more than the abstract of the article and the bibliography. – Grétar Amazeen Oct 27 2009 at 17:23
Of course every mathematician should be familiar with this. Be glad you are doing it today. Back in the day I would have to walk to the library, search through multiple index volumes of Math Reviews, and maybe find what I wanted. But I still did it once a week or so... – Gerald Edgar Oct 28 2009 at 13:51
Beyond the reviews at MathSciNet one of the most beautiful features of the website is that it allows you to page back and forward through the history of the topic at hand. You're reading the review and you can click on the link that offers you the papers that cite this paper. So if you happen across an old paper that's concerned with a topic you're interested in, and want to know if anyone has done anything more recently on that topic, you're just a click away. I don't know when that feature was added but when I was a young mathematician I would have killed for that feature. – Ryan Budney Nov 5 2009 at 4:15
I'm not in academia any more and so can't access Mathscinet (and JStor). It's very frustrating. – Anonymous Mar 8 2010 at 16:39
Finding forward and back references is indeed one of the most amazing and useful things about MathSciNet, but be aware that not everything will show up. Articles collected in books and older articles don't have their references in the database. The cutoff seems to be around 2000, but it may be different for different journals. – Tom Braden Jun 18 2010 at 12:43

Not exactly a resource, but a great way to listen to talks given by experts on the latest results in Computer Science and Math.

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http://maths.dept.shef.ac.uk/magic/index.php

Apparently UK has been building a depository/interactive system for graduate math courses. Click on "courses" to access archives. Many have lecture notes and other materials.

I found this recently. Have not actually personally used it, but potentially very useful.

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A similar "access grid" network comprises the Universities of Bath, Bristol, Oxford, Warwick and Imperial College London. The main site (which includes links to archived course material) can be found at tcc.maths.ox.ac.uk. – Nicholas Jackson Oct 24 2009 at 11:04

contains a lot of useful advice for people at various stages in their careers. In addition it contains a lot of discussions and explanations of the math that I find interesting.

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Allows you to search for research articles. Gives you direct links to all online versions of the article it can find. Strenghts include that it can often give you direct links to files hidden on obscure (non-arXiv) preprint servers or personal webpages, and if you sit on a computer with access to ScienceDirect, Springerlink etc, you get direct links to these artices, via your university library. Weaknesses include lots of errors due to the reliance on their "intelligent" search engine rather than correct metadata from publishers, but this is likely to improve over time.

To find what you really are looking for use the author tag, for example "infinite loop author:May" etc.

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Don't forget http://gigapedia.com/ for tons of e-books.

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Several people have flagged this answer as requiring moderator attention. Please use your own judgement as to the legality of any site on the internet, and remember linking is always okay. – Scott Morrison Mar 8 2010 at 2:24
We have enough police in this world already. – I. J. Kennedy Oct 27 2010 at 1:33
To quote a lower post by Henning Arnór Úlfarsson, "Alexandre Stefanov keeps an extensive list of free math books / lecture notes. The list is divided according to subject and updated frequently. I have found some very nice books there." (Link is mathbooks.110mb.com/mylist.php) This can work as a substitute for most of the titles one would want to find from gigapedia. – drvitek Oct 27 2010 at 5:10
Simon: try trillia.com/online-math/index.html for a modified version that somebody got. – drvitek May 3 2011 at 15:49

David Ben-Zvi takes electronic notes on the talks he attends and posts them publicly. This can often be the best source of information for a subject which has not yet been written down.

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The Art of Problem Solving

Mostly for the student, including high school. But has more advanced forums, too. Latex easily used.

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I learned more in high school from reading posts on Art of Problem Solving than from any other source. Of course, this is before I discovered the blogosphere. – Qiaochu Yuan Nov 5 2009 at 3:39

The nLab is an excellent resource, often containing more detail, explanation, and discussion than wikipedia, along with much more specialized and contemporary topics.

(nLab was mentioned in the answer by Justin Hilburn, but it was listed there after other resources, and I think people scanning under the one-resource-per-answer dictum will miss it.)

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Sloane's OEIS has already been mentioned.

A similarly useful site is ISC, Simon Plouffe's Inverse Symbolic Calculator.

Here you enter the decimal expansion of a number to as many places as you know, and the search engine makes suggestions of symbolic expressions that the expansion might be derived from. The answer might involve pi, e, sin, cosh, sqrt, ln, and so on.

Sometimes, it becomes difficult to calculate symbolically. Therefore, you can proceed numerically instead, and hope to recover the exact symbolic solution at the end, using ISC: sometimes proving that an answer is correct can be easier than calculating, or discovering, it in the first place.

It can also be useful for discovering simplifications of nested radicals, for example.

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I don't know if this reference is of sufficient generality:

Finite Calculus: A Tutorial for Solving Nasty Sums
http://www.stanford.edu/~dgleich/publications/finite-calculus.pdf

It is only a paper, but it describes the methods of the so-called "umbral calculus": a really useful technique to know.

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It's nice, but I don't think it ever comes to actual umbral calculus. Just working with finite difference operators and falling factorials is not umbral calculus yet. – darij grinberg Mar 7 2010 at 23:20
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edit by jc: As of May 11, 2010, the work has been completed!

This is a reference that is not yet complete, but it should be very useful when it finally does arrive:

Digital Library of Mathematical Functions (DLMF)
(book and associated website;
will replace Abramowitz & Stegun's Handbook of Mathematical Functions)
NIST / Cambridge University Press
expected 2009/2010
http://dlmf.nist.gov/

This will contain diagrams, tables, properties of, principal values of, and relationships between many important mathematical functions. For example, the trigonometric and other elementary functions are described, with very many formulae relating them.

The Handbook is very good; the Digital Library will be even better.

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Alexandre Stefanov keeps an extensive list of free math books / lecture notes. The list is divided according to subject and updated frequently. I have found some very nice books there.

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CiteULike (by Springer), to organize in a library the titles and abstracts of one's preferred papers and books.

http://www.citeulike.org/

(From the FAQ:) CiteULike is a free service to help you to store, organise and share the scholarly papers you are reading. When you see a paper on the web that interests you, you can click one button and have it added to your personal library. CiteULike automatically extracts the citation details, so there's no need to type them in yourself. It all works from within your web browser so there's no need to install any software. Because your library is stored on the server, you can access it from any computer with an Internet connection.

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www.optimization-online.org

The optimization community seems to prefer this specific online repository instead of the more broad one arxiv.

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It seems this link hasn't appeared above http://www.ams.org/mathweb/index.html The resources there are too rich to describe.

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