Fiche : [LIVRE195]
Titre : E.W. WEISSTEIN, The CRC Concise Encyclopedia of Mathematics: A Treasure Trove of Mathematical Formulas, Facts, Figures, and Fun, CHAPMAN & HALL/CRC, 1998, 1825 pages.
Cité dans : [DIV354] Recherche sur les mots clés Partitions d'entiers ou Theory of Partitions, septembre 2002. Cité dans :[ART302]Auteur : Eric W. Weisstein
Année : 1998
Stockage : Bibliothèque LMP
Info. : Fiches LMP00-19
Mots_clés : mathématique
Référence : livres 2000 xx / LMP
Date_d'achat : 19 mai 2000
Prix : 74.09 euros (486 F)
Vers : Informations
Vers : Introduction
Lien : http://www.astro.virginia.edu/~ewww6n/math
Lien : http://mathworld.wolfram.com/
Lien : http://www.treasure-troves.com/
Editeur : http://www.crcpress.com/ - Hardcover Edition.
Lien : CRCINTRO.htm - 21 Ko, Introduction of "The CRC Concise Encyclopedia of Mathematics.."
Informations |
The hardcover edition is a compendium of mathematical definitions, formulas, figures,
tabulations, and references is written in an informal style intended to make it accessible to a
broad spectrum of readers with a wide range of mathematical backgrounds and interests. It
draws connections between many areas of mathematics and science and offers concise,
definitions, a multidisciplinary approach regarding the use of mathematics, 110,000
cross-references to other topics and sources (including many internet sites), and thousands of
illustrations, formulas and derivations.
Introduction |
The CRC Concise Encyclopedia of Mathematics is a compendium of mathematical definitions, formulas, figures, tabulations, and references. It is
written in an informal style intended to make it accessible to a broad spectrum of readers with a wide range of mathematical backgrounds and
interests. Although mathematics is a fascinating subject, it all too frequently is clothed in specialized jargon and dry formal exposition that make many
interesting and useful mathematical results inaccessible to laypeople. This problem is often further compounded by the difficulty in locating concrete
and easily understood examples. To give perspective to a subject, I find it helpful to learn why it is useful, how it is connected to other areas of
mathematics and science, and how it is actually implemented. While a picture may be worth a thousand words, explicit examples are worth at least a
few hundred! This work attempts to provide enough details to give the reader a flavor for a subject without getting lost in minutiae. While absolute
rigor may suffer somewhat, I hope the improvement in usefulness and readability will more than make up for the deficiencies of this approach.
The format of this work is somewhere between a handbook, a dictionary, and an encyclopedia. It differs from existing dictionaries of mathematics in
a number of important ways. First, the entire text and all the equations and figures are available in searchable electronic form on CD-ROM. Second,
the entries are extensively cross-linked and cross-referenced, not only to related entries but also to many external sites on the Internet. This makes
locating information very convenient. It also provides a highly efficient way to "navigate" from one related concept to another, a feature that is
especially powerful in the electronic version. Standard mathematical references, combined with a few popular ones, are also given at the end of most
entries to facilitate additional reading and exploration. In the interests of offering abundant examples, this work also contains a large number of
explicit formulas and derivations, providing a ready place to locate a particular formula, as well as including the framework for understanding where
it comes from.
The selection of topics in this work is more extensive than in most mathematical dictionaries (e.g., Borowski and Borwein's HarperCollins
Dictionary of Mathematics and Jeans and Jeans' Mathematics Dictionary). At the same time, the descriptions are more accessible than in
"technical" mathematical encyclopedias (e.g., Hazewinkel's Encyclopaedia of Mathematics and Iyanaga's Encyclopedic Dictionary of
Mathematics). While the latter remain models of accuracy and rigor, they are not terribly useful to the undergraduate, research scientist, or
recreational mathematician. In this work, the most useful, interesting, and entertaining (at least to my mind) aspects of topics are discussed in addition
to their technical definitions. For example, in my entry for pi (), the definition in terms of the diameter and circumference of a circle is
supplemented by a great many formulas and series for pi, including some of the amazing discoveries of Ramanujan. These formulas are
comprehensible to readers with only minimal mathematical background, and are interesting to both those with and without formal mathematics
training. However, they have not previously been collected in a single convenient location. For this reason, I hope that, in addition to serving as a
reference source, this work has some of the same flavor and appeal of Martin Gardner's delightful Scientific American columns.
Everything in this work has been compiled by me alone. I am an astronomer by training, but have picked up a fair bit of mathematics along the way.
It never ceases to amaze me how mathematical connections weave their way through the physical sciences. It frequently transpires that some piece
of recently acquired knowledge turns out to be just what I need to solve some apparently unrelated problem. I have therefore developed the habit of
picking up and storing away odd bits of information for future use. This work has provided a mechanism for organizing what has turned out to be a
fairly large collection of mathematics. I have also found it very difficult to find clear yet accessible explanations of technical mathematics unless I
already have some familiarity with the subject. I hope this encyclopedia will provide jumping-off points for people who are interested in the subjects
listed here but who, like me, are not necessarily experts.
The encyclopedia has been compiled over the last 11 years or so, beginning in my college years and continuing during graduate school. The initial
document was written in Microsoft Word on a Mac Plus computer, and had reached about 200 pages by the time I started graduate school in
1990. When Andrew Treverrow made his OzTEX program available for the Mac, I began the task of converting all my documents to TEX, resulting
in a vast improvement in readability. While undertaking the Word to TEX conversion, I also began cross-referencing entries, anticipating that
eventually I would be able to convert the entire document to hypertext. This hope was realized beginning in 1995, when the Internet explosion was
in full swing and I learned of Nikos Drakos and Ross Moore's excellent TEX to HTML converter, LATEX2HTML. After some additional effort, I
was able to post an HTML version of my encyclopedia to the World Wide Web, currently located at http://www.treasure-troves.com/math/ .
The selection of topics included in this compendium is not based on any fixed set of criteria, but rather reflects my own random walk through
mathematics. In truth, there is no good way of selecting topics in such a work. The mathematician James Sylvester may have summed up the
situation most aptly. According to Sylvester (as quoted in the introduction to Ian Stewart's book From Here to Infinity), "Mathematics is not a
book confined within a cover and bound between brazen clasps, whose contents it needs only patience to ransack; it is not a mine, whose treasures
may take long to reduce into possession, but which fill only a limited number of veins and lodes; it is not a soil, whose fertility can be exhausted by
the yield of successive harvests; it is not a continent or an ocean, whose area can be mapped out and its contour defined; it is as limitless as that
space which it finds too narrow for its aspiration; its possibilities are as infinite as the worlds which are forever crowding in and multiplying upon the
astronomer's gaze; it is as incapable of being restricted within assigned boundaries or being reduced to definitions of permanent validity, as the
consciousness of life."
Several of Sylvester's points apply particularly to this undertaking. As he points out, mathematics itself cannot be confined to the pages of a book.
The results of mathematics, however, are shared and passed on primarily through the printed (and now electronic) medium. While there is no danger
of mathematical results being lost through lack of dissemination, many people miss out on fascinating and useful mathematical results simply because
they are not aware of them. Not only does collecting many results in one place provide a single starting point for mathematical exploration, but it
should also lessen the aggravation of encountering explanations for new concepts which themselves use unfamiliar terminology. In this work, the
reader is only a cross-reference (or a mouse click) away from the necessary background material. As to Sylvester's second point, the very fact that
the quantity of mathematics is so great means that any attempt to catalog it with any degree of completeness is doomed to failure. This certainly does
not mean that it's not worth trying. Strangely, except for relatively small works usually on particular subjects, there do not appear to have been any
substantial attempts to collect and display in a place of prominence the treasure trove of mathematical results that have been discovered (invented?)
over the years (one notable exception being Sloane and Plouffe's Encyclopedia of Integer Sequences). This work, the product of the "gazing" of a
single astronomer, attempts to fill that omission.
Finally, a few words about logistics. Because of the alphabetical listing of entries in the encyclopedia, neither table of contents nor index are
included. In many cases, a particular entry of interest can be located from a cross-reference (indicated in SMALL CAPS TYPEFACE in the text) in a
related article. In addition, most articles are followed by a "see also" list of related entries for quick navigation. This can be particularly useful if you
are looking for a specific entry (say, "Zeno's Paradoxes"), but have forgotten the exact name. By examining the "see also" list at bottom of the entry
for "Paradox," you will likely recognize Zeno's name and thus quickly locate the desired entry.
The alphabetization of entries contains a few peculiarities which need mentioning. All entries beginning with a numeral are ordered by increasing
value and appear before the first entry for "A." In multiple-word entries containing a space or dash, the space or dash is treated as a character which
precedes "a," so entries appear in the following order: "Sum," "Sum P...," "Sum-P...," and "Summary." One exception is that in a series of entries
where a trailing "s" appears in some and not others, the trailing "s" is ignored in the alphabetization. Therefore, entries involving Euclid would be
alphabetized as follows: "Euclid's Axioms," "Euclid Number," "Euclidean Algorithm." Because of the non-standard nomenclature that ensues from
naming mathematical results after their discoverers, an important result such as the "Pythagorean Theorem" is written variously as "Pythagoras's
Theorem," the "Pythagoras Theorem," etc. In this encyclopedia, I have endeavored to use the most widely accepted form. I have also tried to
consistently give entry titles in the singular (e.g., "Knot" instead of "Knots").
In cases where the same word is applied in different contexts, the context is indicated in parentheses or appended to the end. Examples of the first
type are "Crossing Number (Graph)" and "Crossing Number (Link)." Examples of the second type are "Convergent Sequence" and "Convergent
Series." In the case of an entry like "Euler Theorem," which may describe one of three or four different formulas, I have taken the liberty of adding
descriptive words ("Euler's Something Theorem") to all variations, or kept the standard name for the most commonly used variant and added
descriptive words for the others. In cases where specific examples are derived from a general concept, em dashes (--) are used (for example,
"Fourier Series," "Fourier Series--Power Series," "Fourier Series--Square Wave," "Fourier Series--Triangle"). The decision to put a possessive 's at
the end of a name or to use a lone trailing apostrophe is based on whether the final "s" is voiced. "Gauss's Theorem" is therefore written out, whereas
"Archimedes' Recurrence Formula" is not. Finally, given the absence of a definitive stylistic convention, plurals of numerals are written without an
apostrophe (e.g., 1990s instead of 1990's).
In an endeavor of this magnitude, errors and typographical mistakes are inevitable. The blame for these lies with me alone. Although the current
length makes extensive additions in a printed version problematic, I plan to continue updating, correcting, and improving the work.
Eric Weisstein
Charlottesville, Virginia
May 20, 1999
Acknowledgments |
Although I alone have compiled and typeset this work, many people have contributed indirectly and directly to its creation. I have not yet had the
good fortune to meet Donald Knuth of Stanford University, but he is unquestionably the person most directly responsible for making this work
possible. Before his mathematical typesetting program TEX, it would have been impossible for a single individual to compile such a work as this.
Had Prof. Bateman owned a personal computer equipped with TEX, perhaps his shoe box of notes would not have had to await the labors of
Erdelyi, Magnus, Oberhettinger, and Tricomi to become a three-volume work on mathematical functions. Andrew Trevorrow's shareware
implementation of TEX for the Macintosh, OzTEX ( http://www.kagi.com/authors/akt/oztex.html ), was also of fundamental importance. Nikos Drakos and
Ross Moore have provided another building block for this work by developing the LATEX2HTML program ( http://www-dsed.llnl.gov/files/programs/unix/latex2html/manual/manual.html ),
which has allowed me to easily maintain and update an on-line version of the encyclopedia long before it existed in book form.
I would like to thank Steven Finch of MathSoft, Inc., for his interesting on-line essays about mathematical constants ( http://www.mathsoft.com/asolve/constant/constant.html ),
and also for his kind permission to reproduce excerpts from some of these essays. I hope that
Steven w/ill someday publish his detailed essays in book form. Thanks also to Neil Sloane and Simon Plouffe for compiling and making available the
printed and on-line ( http://www.research.att.com/~njas/sequences/ ) versions of the Encyclopedia of Integer Sequences, an immensely valuable
compilation of useful information which represents a truly mind-boggling investment of labor.
Thanks to Robert Dickau, Simon Plouffe, and Richard Schroeppel for reading portions of the manuscript and providing a number of helpful
suggestions and additions. Thanks also to algebraic topologist Ryan Budney for sharing some of his expertise, to Charles Walkden for his helpful
comments about dynamical systems theory, and to Lambros Lambrou for his contributions. Thanks to David W. Wilson for a number of helpful
comments and corrections. Thanks to Dale Rolfsen, compiler James Bailey, and artist Ali Roth for permission to reproduce their beautiful knot and
link diagrams. Thanks to Gavin Theobald for providing diagrams of his masterful polygonal dissections. Thanks to Wolfram Research, not only for
creating an indispensable mathematical tool in Mathematica, but also for permission to include figures from the Mathematica book and
MathSource repository for the braid, conical spiral, Enneper's Minimal Surface, Hadamard matrix, helicoid, helix, Henneberg's minimal surface,
hyperbolic polyhedra, Klein bottle, Maeder's owl minimal surface, Penrose tiles, polyhedron, and Scherk's minimal surfaces entries.
Particular thanks to Tim Buchowski, Patrick De Geest, Robert Dickau, Dewi Jones, Lloyd Rowsey, and Paul Stanford for their help in correcting
typographical errors from the first hardcover printing of this work.
Many thanks to Joshua Kempner for design and layout of the book's cover illustration and for layout and graphic design on the CD-ROM.
Sincere thanks to Judy Schroeder for her skill and diligence in the monumental task of proofreading the entire document for syntax. Thanks also to
Bob Stern, my executive editor from CRC Press, for his encouragement, and to Mimi Williams of CRC Press for her careful reading of the
manuscript for typographical and formatting errors. As this encyclopedia's entry on Proofreading Mistakes shows, the number of mistakes that are
expected to remain after three independent proofreadings is much lower than the original number, but unfortunately still nonzero. Many thanks to the
library staff at the University of Virginia, who have provided invaluable assistance in tracking down many an obscure citation. Finally, I would like to
thank the hundreds of people who took the time to e-mail me comments and suggestions while this work was in its formative stages. Your continued
comments and feedback are very welcome.
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