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00: General (and Elementary) Mathematics


Introduction

The Mathematics Subject Classification uses the classification 00 principally for non-subject-specific materials, such as conference proceedings, dictionaries, handbooks, and problem books. It also includes subject-specific items not typically noted in the Mathematical Reviews database by the AMS, such as elementary mathematics, recreational mathematics, and elementary applications of mathematics.

In general, such material is not included in this collection either. In many cases an occasion arises to treat an elementary question in a non-elementary way, and to so illustrate some branch of more advanced mathematics. When such an illustration is saved in this collection, it is classified according to the tool used (contrary to AMS guidelines for the use of the MSC!).

History

It seems difficult to contemplate a "history of general mathematics". One could of course consider a general history of mathematics, but such material is in 01: History and Biography.

See also this history of recreational mathematics.

Applications and related fields

Material from the pre-college curriculum is mostly absent from this collection. Elementary geometry is usually in 51: Geometry or perhaps 52: Convex Geometry. Ideas from secondary-school algebra are scattered among the Algebra pages. Some topics from trigonometry are included with 33: Special Functions. (You might also find this Short Course in Trigonometry useful.)

For each elementary college-level course there are a few files of some relevance. This includes:

Many other topics have, of course, an elementary component.

Material at the upper-division undergraduate or graduate levels can usually be identified with one or more subject areas in the MSC.

A few miscellaneous topics are approached mathematically in some applications pages at this site: astronomy and the calendar, music, etc.

Other pages in this collection likely to have the most interest to those first exploring mathematics might include the pages for 01: History and Biography, 11: Number Theory, and 51: Geometry.

Subfields

The only subfield other than conference proceedings (00B) is 00A, General and miscellaneous specific topics. This is further broken down into

Section 00A of the MR database is fairly small but section 00B is at least as large, primarily including collections of papers comparable to those classified above.

Prior to 1958 there was a related major heading of the MSC, "99: Miscellaneous".

Browse all (old) classifications for this area at the AMS.


Textbooks, reference works, and tutorials

The Mathematical Association of America has assembled a "Basic Library List", which covers all branches of mathematics (mostly at the undergraduate level) includes both general works and books for specific topics; the latter are occasionally duplicated on the appropriate index pages in this collection.

A comprehensive reference to some 500 mathematics topics is the three-volume "Encyclopedic dictionary of mathematics", (2nd edition 1993), translated from the Japanese; Shôkichi Iyanaga and Yukiyoshi Kawada, eds. (MIT press, 1977, 1750pp, ISBN 0-262-09016-3). Even more impressive -- and much more costly -- is the 5500-page Soviet "Encyclopaedia of mathematics" (1995 reprint: ISBN 1-55608-010-7, Kluwer Academic Publishers, 6 vol., MR96k:00002). A similar but more compact (and older) resource is the "Handbook of mathematics", edited by L. Kuipers and R. Timman. Pergamon Press, Oxford-New York-Toronto, Ont. 1969 782 pp. MR40#1223 (The original is Dutch; a German translation is available too.) Older yet but still of interest is the "Encyklopadie der mathematischen Wissenschaften" (Teubner, Leipzig, 1898 - 1908; second edition, 1939). A series with encyclopedic scope but with the mathematics user in mind (rather than the mathematician) is the multi-volume "Handbook of applicable mathematics", edited by Walter Ledermann, Steven Vajda, et al, John Wiley & Sons, Ltd., Chichester, 1980-1985 (Supplement 1990). ISBN 0-471-27704-5 et seq.

Well-known formularies, with long tables of integrals, mensuration formulas, and so on, include:

Mathematics dictionaries suffer from the tendency of mathematicians to use local definitions and symbols. Still, there are some excellent, compact sources. For example,

Among the journals appropriate for mention here are the nearest things to "coffee-table" magazines in mathematics. The Mathematical Intelligencer publishes history, culture, and opinion articles as well as short summaries of topics of immediate mathematical interest. Mathematics Magazine carries short articles of mathematics content sometimes appropriate for undergraduates. Quantum carries articles at a similar level but the emphasis includes physics and intermediate areas as well as mathematics proper. The Notices of the American Mathematical Society contains news items regarding professional issues (funding, employment, conferences, etc.) as well as biographical material and some expository work.

There are of course many research journals, some specific to individual disciplines in mathematics and some covering all fields. One may peruse the Tables of Contents of many current math journals. The profession is in the midst of a slow but dramatic change in the methods of research publication (as are all research disciplines). A list of electronic journals is available, and a list of other bibliographic resources. There is a preprint server at the AMS and a collection of subject-oriented preprint archives at Los Alamos National Laboratories; other preprint servers for specific disciplines are listed on the corresponding index pages.

For general bibliographic work, the preeminent tools are Mathematical Reviews and Zentralblatt für Mathematik. At least one of these is available in most university libraries. For work from the 1930s or before one must turn to the Jahrbuch über die Fortschritte der Mathematik. A comprehensive review of 19th century journal articles, the Royal Society Catalogue, has been reprinted by Kraus. Other reviewing journals have included the Dutch Revue Semestrielle des Publications Mathematiques (1893-1934) and the Soviet Referativnyi Zhurnal: Matematika (1953+); see this history of the mathematical literature. Some applied mathematics is included in the INSPEC database, Computing Reviews, Physical Reviews, Chemical Abstracts and so on.

The CompuMath Citation Index (part of the larger Science Citation Index) includes a database which tracks articles in over 300 leading mathematics journals; its unique feature is the inclusion of bibliographic data from each article -- which articles are cited by which others? (The SCI is also available over the World-Wide Web to subscribers. This includes all those affiliated with academic institutions in Great Britain. See also the Web of Science.)

For further information on mathematics literature see e.g.

Of course "textbooks" in recreational mathematics would be illogical, but the books by Martin Gardner would, arguably, be a rough equivalent. See also the article, "What is recreational mathematics?" by Charles W. Trigg, Math. Mag. 51 (1978), no. 1, 18--21. MR58#15766

Software and tables

For tools specific to particular disciplines (number theory, statistics, etc.) see the page for that discipline.

General mathematical tools include Mathematica, Maple, Macsyma, and so on. For information on such packages see 68W30: Symbolic computation.

Algorithms for general applications are also discussed in Numerical Analysis, but here are implementations in C, C++, Pascal, and Fortran, for several dozen common problems.

A unique piece of "software" is Plouffe's Inverter, the Inverse Symbolic Calculator: enter a real number, get some candidate symbolic expressions which evaluate to it.

At the risk of violating the guideline that these page refer only to mathematical content, let us mention the pivotal TeX software for mathematical typesetting, since not only is it the standard for formal math communication, but informal variants of it are also used in simple documents such as the files in this Collection. It has its own FAQ and newsgroup.

Tables for general mathematics (e.g. logarithm tables) have given way to calculators. Tables for specific areas of mathematics are discussed on the appropriate web pages. For tables of a multidisciplinary nature, consult the "Mathematical Tables and Other Aids to Computation", which eventually became the journal "Mathematics of Computation".

Other web sites with this focus

A highly recommended "front-end" to general mathematics sites on the Web is provided by MathSearch. One can search through collections of mathematical Web material (many at approximately the level of this Collection) by keyword. For this and other searching tools, see the search facility at this site.

A select set of pointers to general mathematics internet tools has been sorted by type of resource by the Exeter library.

Here are the AMS and Goettingen resource pages for area 00.

Other sites "focusing on general mathematics" might mean sites carrying information on some or all mathematical disciplines (without claiming to be comprehensive). These are usually inconsistent with the subject-specific focus of this collection, but many are interesting places to visit. Perhaps the Math Archives would be a reasonable place to begin surfing the web for such materials. Of course general information on mathematics and other subjects can be obtained from an internet index such as the one at Yahoo, or with a web search tool, but of course these are not limited to sites of interest to practicing mathematicians and their students. The quality of internet resources is known to be variable (and generally low); caveat lector!

It's not quite clear what constitutes "recreational mathematics", nor what should be listed as an appropriate website -- after all, some mathematicians think all of mathematics is a form of recreation! Usually the term refers to mathematics problems studied without particular regard to application or integration into the mathematical canon. There is a compilation of such problems from various sources organized by MathPro Press. Of course, one might study the mathematics behind recreation, that is, one might begin a mathematical analysis of games. While this could conceivably overlap, say, kinesthetics (see e.g. 70: Mechanics), most mathematical analysis of games concerns what are called "combinatorial games" such as chess or tic-tac-toe. This is one topic in what is (deceptively) called 91: Game Theory. There is a newsgroup alt.math.recreational but of wider circulation is the newsgroup rec.puzzles, which includes some recreational mathematical puzzles. The group has its own FAQ file, which discusses classic puzzles such as the Köningsberg bridge problem. See also the newsgroup rec.games.abstract.

Among resources for general mathematics, perhaps mention could be made of the USENET newsgroups sci.math, sci.math.research, k12.ed.math, alt.algebra.help, alt.math.undergrad; there are several regional (e.g. national) mathematics newsgroups, and a few subject-specific math groups (indicated on the appropriate pages in this Index). There is an archive of old sci.math.research posts (temporarily unavailable). While that newsgroup is moderated, the others are not, and signal-to-noise ratios vary :-) but posts back to March 1995 can be found through DejaNews. There is a Frequently Asked Questions list (FAQ) for sci.math. There are also many mailing lists, not all (yet) reviewed and placed to their proper Index pages; a partial listing is available, and you may wish to search for mailing lists at Tile.net, Liszt, Mailbase, or L-soft.

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Last modified 2000/07/14 by Dave Rusin. Mail: