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The number is an important mathematical constant that is the base of the natural logarithm. It is approximately equal to 2.71828,〔Oxford English Dictionary, 2nd ed.: (natural logarithm )〕 and is the limit of as approaches infinity, an expression that arises in the study of compound interest. It can also be calculated as the sum of the infinite series〔Encyclopedic Dictionary of Mathematics 142.D〕 : The constant can be defined in many ways. For example, can be defined as the unique positive number such that the graph of the function has unit slope at . The function is called the exponential function, and its inverse is the natural logarithm, or logarithm to base . The natural logarithm of a positive number can also be defined directly as the area under the curve between and , in which case is the number whose natural logarithm is 1. There are alternative characterizations. Sometimes called Euler's number after the Swiss mathematician Leonhard Euler, is not to be confused with , the Euler–Mascheroni constant, sometimes called simply ''Euler's constant''. The number is also known as Napier's constant, but Euler's choice of the symbol is said to have been retained in his honor. The constant was discovered by the Swiss mathematician Jacob Bernoulli while studying compound interest. The number is of eminent importance in mathematics, alongside 0, 1, and . All five of these numbers play important and recurring roles across mathematics, and are the five constants appearing in one formulation of Euler's identity. Like the constant , is irrational: it is not a ratio of integers. Also like , is transcendental: it is not a root of ''any'' non-zero polynomial with rational coefficients. The numerical value of truncated to 50 decimal places is : . ==History== The first references to the constant were published in 1618 in the table of an appendix of a work on logarithms by John Napier.〔 However, this did not contain the constant itself, but simply a list of logarithms calculated from the constant. It is assumed that the table was written by William Oughtred. The discovery of the constant itself is credited to Jacob Bernoulli,〔Jacob Bernoulli considered the problem of continuous compounding of interest, which led to a series expression for ''e''. See: Jacob Bernoulli (1690) "Quæstiones nonnullæ de usuris, cum solutione problematis de sorte alearum, propositi in Ephem. Gall. A. 1685" (Some questions about interest, with a solution of a problem about games of chance, proposed in the ''Journal des Savants'' (''Ephemerides Eruditorum Gallicanæ''), in the year (anno) 1685. * *), ''Acta eruditorum'', pp. 219-223. (On page 222 ), Bernoulli poses the question: ''"Alterius naturæ hoc Problema est: Quæritur, si creditor aliquis pecuniæ summam fænori exponat, ea lege, ut singulis momentis pars proportionalis usuræ annuæ sorti annumeretur; quantum ipsi finito anno debeatur?"'' (This is a problem of another kind: The question is, if some lender were to invest () sum of money () interest, let it accumulate, so that () every moment () were to receive () proportional part of () annual interest; how much would he be owed (the ) end of () year?) Bernoulli constructs a power series to calculate the answer, and then writes: ''" … quæ nostra serie (expression for a geometric series ) &c. major est. … si ''a''=''b'', debebitur plu quam 2½''a'' & minus quam 3''a''."'' ( … which our series (geometric series ) is larger (). … if ''a''=''b'', (lender ) will be owed more than 2½''a'' and less than 3''a''.) If ''a''=''b'', the geometric series reduces to the series for ''a'' × ''e'', so 2.5 < ''e'' < 3. ( * * The reference is to a problem which Jacob Bernoulli posed and which appears in the ''Journal des Sçavans'' of 1685 at the bottom of (page 314. ))〕 who attempted to find the value of the following expression (which is in fact ): : The first known use of the constant, represented by the letter , was in correspondence from Gottfried Leibniz to Christiaan Huygens in 1690 and 1691. Leonhard Euler introduced the letter as the base for natural logarithms, writing in a letter to Christian Goldbach of 25 November 1731.〔Lettre XV. Euler à Goldbach, dated November 25, 1731 in: P. H. Fuss, ed., ''Correspondance Mathématique et Physique de Quelques Célèbres Géomètres du XVIIIeme Siècle'' … (Mathematical and physical correspondence of some famous geometers of the 18th century), vol. 1, (St. Petersburg, Russia: 1843), pp. 56-60 ; see especially (page 58. ) From page 58: ''" … ( e denotat hic numerum, cujus logarithmus hyperbolicus est = 1), … "'' ( … (e denotes that number whose hyperbolic (natural ) logarithm is equal to 1) … )〕 Euler started to use the letter for the constant in 1727 or 1728, in an unpublished paper on explosive forces in cannons,〔Euler, ''(Meditatio in experimenta explosione tormentorum nuper instituta )''.〕 and the first appearance of in a publication was Euler's ''Mechanica'' (1736).〔Leonhard Euler, ''Mechanica, sive Motus scientia analytice exposita'' (St. Petersburg (Petropoli), Russia: Academy of Sciences, 1736), vol. 1, Chapter 2, Corollary 11, paragraph 171, p. 68. (From page 68: ) ''Erit enim seu ubi ''e'' denotat numerum, cuius logarithmus hyperbolicus est 1.'' (So it (''c'', the speed ) will be or , where ''e'' denotes the number whose hyperbolic (natural ) logarithm is 1.)〕 While in the subsequent years some researchers used the letter , was more common and eventually became the standard. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「E (mathematical constant)」の詳細全文を読む スポンサード リンク
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