41. Mathematical Constants Subject Re How did we come by e? From Matthew P Wiener Date 1998/08/12 Newsgroups sci.math http://pauillac.inria.fr/algo/bsolve/constant/e/wiener.html

Mathematical Constants by Steven R. Finch Clay Mathematics Institute Book Fellow My website is smaller than it once was. Please visit again, however, since new materials will continue to appear occasionally. * My book Mathematical Constants is now available for online purchase from Cambridge University Press (in the United Kingdom and in North America ). It is far more encompassing and detailed than my website ever was. It is also lovingly edited and beautifully produced - many thanks to Cambridge! - please support us in our publishing venture. Thank you. (If you wish, see several very kind reviews . You can also search the book via Amazon and Google by keyword.) Here are errata and addenda to the book (last updated 8/11/2005), as well sample essays from the book about integer compositions optimal stopping and Reuleaux triangles . Here also are recent supplementary materials, organized by topic: Number Theory and Combinatorics Bipartite, k-colorable and k-colored graphs Transitive relations, topologies and partial orders Series-parallel networks Two asymptotic series ... Quadratic Dirichlet L-series Elliptic curves over Q Inequalities and Approximation Hardy-Littlewood maximal inequalities Bessel function zeroes Nash's inequality Uncertainty inequalities ... Moduli of continuity Real and Complex Analysis Radii in geometric function theory Numerical radii of linear operators Coefficient estimates for univalent functions Planar harmonic mappings ... Constant of interpolation Probability and Stochastic Processes

42. EXSLT - Math:constant - Implementer Page The mathconstant function returns the specified constant to a set precision. Builtin support for mathconstant is available in the following XSLT http://www.exslt.org/math/functions/constant/math.constant.html

EXSLT math constant - Implementer Page User Page How To Downloads Modules Dates and Times Dynamic Common Functions ... Contact Version: Status: implemented User Page: index.html XML Definition: math.constant.xml Function Package: math.constant.zip Function Syntax number math:constant string number Template Syntax math:constant string number The math:constant function returns the specified constant to a set precision. The possible constants are: PI E An implementation of this extension function in the EXSLT math namespace must conform to the behaviour described in this document. Implementations Built-in support for math:constant is available in the following XSLT processors: Processor Processor Version Implemented Version 4XSLT, from 4Suite. libxslt from Daniel Veillard et al. The following implementations of math:constant are available: Language Implemented Version Creator Date Download XSLT Template James Fuller math.constant.template.xsl Javascript ... math.constant.function.xsl Change History Submitted: Creator: James Fuller http://www.ruminate.co.uk

43. E -- From MathWorld e, and Euler s constant. math. Comput. 50, 275281, 1988. HJ and Knox, JA New Closed-Form Approximations to the Logarithmic constant e. math. http://mathworld.wolfram.com/e.html

INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Alphabetical Index DESTINATIONS About MathWorld About the Author Headline News ... Random Entry CONTACT Contribute an Entry Send a Message to the Team MATHWORLD - IN PRINT Order book from Amazon Number Theory Constants e ... Sondow e The constant is base of the natural logarithm , whose symbol honors Euler . This means that is the unique number with the property that the area of the region bounded by the hyperbola , the x -axis , and the vertical lines and is 1. In other words, With the possible exception of is the most important constant in mathematics since it appears in myriad mathematical contexts involving limits and derivatives . The numerical value of is (Sloane's can be defined by the limit (illustrated above), or by the infinite series as first published by Newton (1669; reprinted in Whitehead 1968, p. 225). also has a surprising connection with the product-of-primes primorial function, and is given by the unusual limit (Brothers and Knox 1998). Euler proved that is irrational by proving that has an infinite simple continued fraction ( ; Nagell 1951), and

44. Feigenbaum Constant -- From MathWorld Briggs, K. A Precise Calculation of the Feigenbaum Constants. math. Comput. 57, 435439, 1991. Briggs, KM Feigenbaum Scaling in Discrete Dynamical http://mathworld.wolfram.com/FeigenbaumConstant.html

INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Alphabetical Index DESTINATIONS About MathWorld About the Author Headline News ... Random Entry CONTACT Contribute an Entry Send a Message to the Team MATHWORLD - IN PRINT Order book from Amazon Applied Mathematics Complex Systems Chaos ... Trott Feigenbaum Constant The Feigenbaum constant is a universal constant for functions approaching chaos via period doubling . It was discovered by Feigenbaum in 1975 (Feigenbaum 1979) while studying the fixed points of the iterated function and characterizes the geometric approach of the bifurcation parameter to its limiting value as the parameter is increased for fixed several hundred times for a series of discrete but closely spaced values of , discarding the first hundred or so points before the iteration has settled down to its fixed points, and then plotting the points remaining. A similar plot that more directly shows the cycle may be constructed by plotting as a function of . The plot above (Trott) shows the resulting curves for , 2, and 4.

45. Curl 3.0: Package CURL.LANGUAGE.MATH-CONSTANT CURL.LANGUAGE.mathconstant. Variables e half-pi infinity inv-half-pi inv-pi inv-quarter-pi inv-sqrt-2 ln-of-10 ln-of-2 log10-of-e http://km0330.keymachine.de/friedger/doc/CURL.LANGUAGE.MATH-CONSTANT.html

CURL.LANGUAGE.MATH-CONSTANT Variables e half-pi infinity inv-half-pi ... sqrt-2

46. Curl 3.0: Package CURL.LANGUAGE.MATH-CONSTANT Package CURL.LANGUAGE.mathconstant. Variables. double, e = 2.71828. double, half-pi = 1.5708. double, infinity =. double, inv-half-pi = 0.63662 http://km0330.keymachine.de/friedger/doc/CURL.LANGUAGE.MATH-CONSTANT/

Overview Package Class Index Help Curl TM Package CURL.LANGUAGE.MATH-CONSTANT Variables double e double half-pi double infinity double inv-half-pi double inv-pi double inv-quarter-pi double inv-sqrt-2 double ln-of-10 double ln-of-2 double log10-of-e double log2-of-e double nan double pi double quarter-pi double sqrt-2

47. Mathematics Archives - Numbers All numbers are not created equal; that certain constants appear at all and then echo throughout Euler s constant gamma and Apery s constant. Mayan math http://archives.math.utk.edu/subjects/numbers.html

Numbers Facts about the number 17: 17 in history, computing, astronomy, etc. The 47 Society The 47 Society is an international interest-group that follows the occurence and recurrence of the quintessential random number: 47. Many suspect that the coinciential nature of 47 carries some mystical, metaphysical and/or scientific significance. What is special about the number 73939133? Aesthetics of the Prime Sequence Hear and see the prime numbers! A Common Book of p The number p has been the subject of a great deal of mathematical (and popular) folklore. It's been worshipped, maligned, and misunderstood. Overestimated, underestimated, and legislated. Of interest to scholars, crackpots, and everyday people. Continued Fractions A senior Honor's Project at Calvin College by Adam Van Tuyl which gives the history, theory, applications and bibliography on the thery of continued fractions. In the section on applications there are a number of interactive programs that convert rationals (or quadratic irrationals) into a simple continued fraction, as well as the converse. Data Powers of Ten A petabyte?

48. Mathematical Constants This is the most well known mathematical constant, and is the ratio of the circumference of a circle to its Mudd math Fun Facts Continuum Hypothesis http://dspace.dial.pipex.com/town/way/po28/maths/constant.htm

Mathematical Constants This is the most well known mathematical constant, and is the ratio of the circumference of a circle to its diameter - the fact that this is the same for all circles is amazing in itself. The value has been estimated since biblical times, Pi through the ages , and a (wrong) value was almost written into a law You will know that can be approximated by which only gives the value correct to 2 decimal places. is correct to 6 decimal places, and to 9 decimal places. has been calculated to over 50 billion digits . See Table of current records for the computation of constants for the current record for and other constants. is not only irrational (it cannot be written exactly as the ratio of two integers) it is also transcendental (it is not the solution of any polynomial equation with rational coefficients). You will find proofs in Transcendence of Pi and in T E X format Not many transcendental numbers are known ( e and Liouville's number are another two examples) but in fact in 1874 Cantor showed that almost all real numbers are transcendental. You can see some of the proofs if you follow the links at

49. Kaprekar's Constant Recreational math became his hobby as a child he enjoyed spending time solving math In 1946 he discovered Kaprekar s constant which was named after him. http://math.about.com/od/recreationalmath/a/kaprekar.htm

var zLb=8; var zIoa2 = new Array('Elsewhere on the Web','From Wolfram Research: Kaprekar','http://mathworld.wolfram.com/KaprekarNumber.html','University of Waterloo','http://www.cs.uwaterloo.ca/journals/JIS/VOL3/iann2a.html'); zJs=10 zJs=11 zJs=12 zJs=13 zc(5,'jsc',zJs,9999999,'') About Homework Help Mathematics Recreational Math Kaprekar's Constant Homework Help Mathematics Essentials Grade By Grade Goals ... Help zau(256,140,140,'el','http://z.about.com/0/ip/417/C.htm','');w(xb+xb+' ');zau(256,140,140,'von','http://z.about.com/0/ip/496/7.htm','');w(xb+xb); FREE Newsletter Sign Up Now for the Mathematics newsletter! See Online Courses Search Mathematics Stay up to date! Compare Prices Email to a friend ... Print this page Elsewhere on the Web From Wolfram Research: Kaprekar University of Waterloo Most Popular Area and Surface Formulas Worksheets in PDF and Word multiplication table tricks Multiplication Chart, Multiplication Table ... Facts to 20 - PDF What's Hot Exponents 11th grade math, Eleventh grade math Polygons Shape Book ... What's Needed? Related Topics Physics Chemistry Teen Advice Homework / Study Tips ... Homeschooling Kaprekar's Constant From Deb Russell Your Guide to Mathematics FREE Newsletter.

50. GSL-- 0.6 - GSL--: Constant.hh File Reference mathematical constants. Author Christian Holm. Date Wed Mar 12 152419 2003. include gsl/gsl_math.h . Include dependency graph for math/constant.hh http://cholm.home.cern.ch/cholm/misc/gslmm/math_2constant_8hh.html

Examples Modules Namespaces Hierarchy ... math math/constant.hh File Reference Detailed Description Mathematical constants. Author: Christian Holm Date: Wed Mar 12 15:24:19 2003 Include dependency graph for math/constant.hh: Namespaces namespace gslmm Top of page Last update Wed Jun 1 11:39:14 2005 Christian Holm Created by DoxyGen 1.4.2

51. GSL-- 0.6 - GSL--: Gslmm::constant Struct Reference Physical constants. include gslmm/math/constant.hh . Collaboration diagram for gslmmconstant. Collaboration graph. legend. List of all members. http://cholm.home.cern.ch/cholm/misc/gslmm/structgslmm_1_1constant.html

Examples Modules Namespaces Hierarchy ... constant gslmm::constant Struct Reference Physical Constants Collaboration diagram for gslmm::constant: legend List of all members. Detailed Description Mathematical contants. Static Public Attributes static const double e static const double static const double static const double static const double static const double static const double pi static const double static const double static const double static const double static const double static const double static const double static const double static const double static const double euler static const double static const double static const double Member Data Documentation const double gslmm::constant::e [static] The base of exponentials, const double gslmm::constant::euler [static] Euler's static constant, const double gslmm::constant::half_of_pi [static] divided by two, const double gslmm::constant::log10_of_e [static] The base-10 logarithm of e, const double gslmm::constant::log2_of_e [static] The base-2 logarithm of e, const double gslmm::constant::log_of_10 [static] The natural logarithm of ten

52. Math Forum - Ask Dr. Math Ask Dr. math Questions and Answers from our Archives The surface of a sphere is one good example; it has a constant curvature. http://mathforum.org/library/drmath/view/55198.html

Associated Topics Dr. Math Home Search Dr. Math Einstein, Curved Space, and Pi Date: 10/09/1999 at 13:35:35 From: Steve Subject: Pi and curved space... I am in 8th grade and recently became interested in Einstein's General and Special Relativity theories. (Just so you know my question is NOT about Einstein's Relativity but how it relates to pi). In Einstein's general relativity, space is described as being curved. If space is curved, then Euclidean geometry doesn't apply. And since one of the parts to Euclidean Geometry is that the relation between a circle's circumference and its diameter (pi) is always 3.1415926535897923... doesn't that mean that the value of pi changes when space is curved? Doesn't that also mean that sometimes the value of pi can actually be rational? Do you know of any methods to calculate pi in curved space? Thanks for your help. http://mathforum.org/dr.math/ Associated Topics High School Geometry High School Higher-Dimensional Geometry Middle School Geometry Middle School Higher-Dimensional Geometry ... Middle School Pi Search the Dr. Math Library: Find items containing (put spaces between keywords): Click only once for faster results: [ Choose "whole words" when searching for a word like age.

53. JavaScript: The Math Object var my_pie=math.PI;. This returns the pi constant, a number with lots of decimals, PI, Returns the constant pi. math.SQRT2, Returns the square root of 2 http://www.pageresource.com/jscript/jmath.htm

The Math Object Helpful methods for calculations JS Main Basics Advanced Complete List ... Advanced /The Math Object Browser Compatibility: NS 4+, IE 4+ The Math object of JavaScript allows you to perform certain calculations by using method functions of the Math object. Also, it provides a few constants such as pi. The easiest way to get a value is to define a variable and set its value to a property or function of the Math Object. If you want to use a property, which returns a constant value (such as pi), you would write something like this: var my_car=Math.property; You would replace the word "property" with the property you want to use. So, if you want to use the value of pi in a variable, you would use the PI property of the Math object. You would write: var my_pie=Math.PI; This returns the pi constant, a number with lots of decimals, or 3.14..... If you want to use a member function, which performs a calculation, you would write something like this: var my_house=Math.function(x); Most of the member functions have one or more parameters, which is what the "x" is for. You can replace x with a number or variable. You would replace the word "function" with the function you want to use. For instance, if you want the square root of a number, you can call the square root member function of the Math object with the number as the parameter: var my_money=Math.sqrt(2);

54. Is The Speed Of Light Constant? If it wasn t zero, the speed of light would not be constant; but from a theoretical point Can we tell if the speed of light is constant in those units? http://math.ucr.edu/home/baez/physics/Relativity/SpeedOfLight/speed_of_light.htm

[Physics FAQ] Updated 1997 by Steve Carlip. Original by Philip Gibbs 1996. Is The Speed of Light Constant? There are a number of senses to the meaning of this question and so there are a number of different answers. Firstly . . . Does the speed of light change in air or water? Yes. Light is slowed down in transparent media such as air, water and glass. The ratio by which it is slowed is called the refractive index of the medium and is always greater than one. This was discovered by Jean Foucault in 1850. When people talk about "the speed of light" in a general context, they usually mean the speed of light in a vacuum. This quantity is also referred to as c Is c , the speed of light in vacuum, constant? At the 1983 Conference Generale des Poids et Mesures , the following SI (Systeme International) definition of the metre was adopted: The metre is the length of the path travelled by light in vacuum during a time interval of 1/299 792 458 of a second. This defines the speed of light in vacuum to be exactly 299,792,458 m/s. This provides a very short answer to the question "Is

55. Calculus I (Math 2413) - Extras - Constant Of Integration Online Notes / Calculus I (math 2413) / Extras / constant of Integration So, without a proper understanding of the constant of integration, http://tutorial.math.lamar.edu/AllBrowsers/2413/ConstantOfIntegration.asp

MPBodyInit('ConstantOfIntegration_files') Paul's Online Math Notes Online Notes / Calculus I (Math 2413) / Extras / Constant of Integration Constants of Integration In this section we need to address a couple of topics about the constant of integration.  Throughout most calculus classes we play pretty fast and loose with it and because of that many students don’t really understand it or how it can be important. First, let’s address how we play fast and loose with it.  Recall that technically when we integrate a sum or difference we are actually doing multiple integrals.  For instance, MPSetEqnAttrs('eq0001','',3,[[178,18,6,-1,-1],[237,24,8,-1,-1],[297,29,10,-1,-1],[],[],[],[743,75,25,-3,-3]]) MPSetEqnAttrs('eq0001','',3,[[178,18,6,-1,-1],[237,24,8,-1,-1],[297,29,10,-1,-1],[],[],[],[743,75,25,-3,-3]]); MPNNCalcTopLeft(document.mpeq0001ph,'1') MPDeleteCode('eq0001') Upon evaluating each of these integrals we should get a constant of integration for each integral since we really are doing two integrals. MPSetEqnAttrs('eq0002','',3,[[178,54,20,-1,-1],[237,73,26,-1,-1],[297,90,33,-1,-1],[],[],[],[743,226,82,-3,-3]])

56. Constant Field Values (Math 1.0 API) constant Field Values org.apache.commons.math.analysis.PolynomialSplineFunction. static final long, serialVersionUID, 7011031166416885789l http://jakarta.apache.org/commons/math/api/constant-values.html

Overview Package Class Use Tree Deprecated Index Help PREV NEXT FRAMES NO FRAMES All Classes Constant Field Values Contents org.apache.* org.apache.* org.apache.commons.math. ConvergenceException static final long serialVersionUID org.apache.commons.math. FunctionEvaluationException static final long serialVersionUID org.apache.commons.math. MathConfigurationException static final long serialVersionUID org.apache.commons.math. MathException static final long serialVersionUID org.apache.commons.math.analysis. BisectionSolver static final long serialVersionUID org.apache.commons.math.analysis. BrentSolver static final long serialVersionUID org.apache.commons.math.analysis. NewtonSolver static final long serialVersionUID org.apache.commons.math.analysis. PolynomialFunction static final long serialVersionUID org.apache.commons.math.analysis. PolynomialSplineFunction static final long serialVersionUID org.apache.commons.math.analysis. SecantSolver static final long serialVersionUID org.apache.commons.math.analysis. UnivariateRealSolverImpl static final long serialVersionUID org.apache.commons.math.complex.

57. Constant Field Values (Math 1.2-dev API) constant Field Values org.apache.commons.math.stat.descriptive. DescriptiveStatistics. public static final int, INFINITE_WINDOW, 1 http://jakarta.apache.org/commons/math/apidocs/constant-values.html

Overview Package Class Use Tree Deprecated Index Help PREV NEXT FRAMES NO FRAMES All Classes Constant Field Values Contents org.apache.* org.apache.* org.apache.commons.math.random. ValueServer public static final int public static final int public static final int public static final int public static final int public static final int org.apache.commons.math.stat.descriptive. DescriptiveStatistics public static final int org.apache.commons.math.util. ResizableDoubleArray public static final int public static final int Overview Package Class Use Tree Deprecated Index Help PREV NEXT FRAMES NO FRAMES All Classes

58. Fine Structure Constant, Alpha The fine structure constant itself is usually denoted by alpha with no subscript. Tour through mathNet Links to the mathematical world General http://www.fine-structure-constant.org/

James G. Gilson, Fine Structure Constant The fine structure constant, a 20th century mystery The question of how to calculate the numerical value of the fine structure constant from theory was one of the most outstandingly difficult problems in mathematical physics for the greater part of the 20th century. There were many unsuccessful attempts by researchers including famous ones such as P. A. M. Dirac to find a formula for the fine structure constant. See John Baez's website page Open Questions in Physics and also the PhysLINK website. Substantial progress with this fundamental problem is outlined on this web page. R.P. Feynman in a remarkable intuitional leap some 40 years ago as can be seen from the following much quoted extract from one of Feynman's books. fundamental quantum coupling constants are given by the very simple formula depending on two integer parameters n and n ,n ) = n The values of the quantum coupling constants are often denoted by the lower case Greek letter alpha alpha with no subscript.

59. Earliest Uses Of Symbols For Constants In a post in sci.math in 1995, Weihwa Huang wrote I believe that e was not named The name Euler s constant may be inappropriate for this number, http://members.aol.com/jeff570/constants.html

Earliest Uses of Symbols for Constants Last revision: Dec. 29, 2001 for 3.14159... Early writers indicated this constant as a ratio of two values. William Oughtred (1574-1660) designated the ratio by the fraction lower case pi over lower case delta in Clavis mathematicae. The symbolism appears in the editions of this book of 1647, 1648, 1652, 1667, 1693, and 1694 (Cajori vol. 2, page 9). Cajori writes that "perhaps the earliest use of a single letter to represent the ratio of the length of a circle to its diameter" occurs in 1689 in Mathesis enucleata by J. Christoph Sturm, who used e for 3.14159.... si diameter alicuius circuli ponatur a, circumferentiam appellari posse ea (quaecumque enim inter eas fuerit ratio, illius nomen potest designari littera e Cajori cites a note by A. Krazer in Euleri opera omnia as a reference for the above. The first person to use to represent the ratio of the circumference to the diameter (3.14159...) was William Jones (1675-1749) in 1706 in Synopsis palmariorum mathesios. It is believed he used the Greek letter pi because it is the first letter in perimetron (= perimeter). From Cajori (vol. 2, page 9):

60. Proportional: Definition, Synonyms And Much More From Answers.com In this case, k is called the proportionality constant or constant of proportionality of the relation. (math.) Proportionale. (Greek) http://www.answers.com/topic/proportionality

showHide_TellMeAbout2('false'); Business Entertainment Games Health ... More... On this page: Dictionary Thesaurus WordNet Wikipedia Translations Best of Web Mentioned In Or search: - The Web - Images - News - Blogs - Shopping proportional Dictionary pro·por·tion·al prə-p´r shə-nəl, -pōr adj. Forming a relationship with other parts or quantities; being in proportion. Properly related in size, degree, or other measurable characteristics; corresponding: Punishment ought to be proportional to the crime. Mathematics. Having the same or a constant ratio. n. One of the quantities in a mathematical proportion. pro·por tion·al i·ty -shə-năl Ä­-tē n. pro·por tion·al·ly adv. Thesaurus proportional adjective Properly or correspondingly related in size, amount, or scale: commensurable commensurate proportionate Idioms: in proportion. See big/small/amount Characterized by or displaying symmetry, especially correspondence in scale or measure: balanced proportionate regular symmetric ... symmetrical See same/different/compare WordNet Note: click on a word meaning below to see its connections and related words. The noun proportional has one meaning: Meaning #1 one of the quantities in a mathematical proportion The adjective proportional has 2 meanings: Meaning #1 properly related in size or degree or other measurable characteristics; usually followed by `to'

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