E (константа)

График једначине $y=1/x.$ Овдје је e јединствени број већи од 1, што чини осенчену површину једнаком 1.

Број e, познат као Ојлеров број или Неперова константа, основа је природног логаритма и један од најзначајнијих бројева у савременој математици, поред неутрала сабирања и множења 0 и 1, имагинарне јединице број i и броја пи. Осим што је ирационалан и реалан, овај број је још и трансцедентан. До тридесетог децималног места, овај број износи:

e ≈ 2,71828 18284 59045 23536 02874 71352...

It is the base of the natural logarithms. It is the limit of (1 + 1/n)ⁿ as n approaches infinity, an expression that arises in the study of compound interest. It can also be calculated as the sum of the infinite series

$$e=\sum \limits _{n=0}^{\infty }{\frac {1}{n!}}=1+{\frac {1}{1}}+{\frac {1}{1\cdot 2}}+{\frac {1}{1\cdot 2\cdot 3}}+\cdots .$$

It is also the unique positive number a such that the graph of the function y = a^x has a slope of 1 at x = 0.

The (natural) exponential function f(x) = e^x is the unique function f that equals its own derivative and satisfies the equation f(0) = 1; hence one can also define e as f(1). The natural logarithm, or logarithm to base e, is the inverse function to the natural exponential function. The natural logarithm of a number k > 1 can be defined directly as the area under the curve y = 1/x between x = 1 and x = k, in which case e is the value of k for which this area equals one (see image). There are various other characterizations.

e is sometimes called Euler's number (not to be confused with Euler's constant $\gamma$), after the Swiss mathematician Leonhard Euler, or Napier's constant, after John Napier. The constant was discovered by the Swiss mathematician Jacob Bernoulli while studying compound interest.

The number e is of great importance in mathematics, alongside 0, 1, π, and i. All five appear in one formulation of Euler's identity, and play important and recurring roles across mathematics. Like the constant π, e is irrational (that is, it cannot be represented as a ratio of integers) and transcendental (that is, it is not a root of any non-zero polynomial with rational coefficients).

Дефиниције

Број e се може представити као:

1. Гранична вредност бесконачног низа:

   $e=\lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)^{n}$

2. Сума бесконачног низа:

   $e=\sum _{n=0}^{\infty }{\frac {1}{n!}}={\frac {1}{0!}}+{\frac {1}{1!}}+{\frac {1}{2!}}+{\frac {1}{3!}}+{\frac {1}{4!}}+\cdots$

   Где је n! факторијел n.

3. Позитивна вредност која задовољава следећу једначину:

   $\int _{1}^{e}{\frac {1}{t}}\,dt={1}$

   Може се доказати да су наведена три исказа еквивалентна.

4. Овај број се среће и као део Ојлеровог идентитета:

   $e^{\mathrm {i} \cdot \pi }=-1$

Историја

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 to the base $e$. It is assumed that the table was written by William Oughtred.

The discovery of the constant itself is credited to Jacob Bernoulli in 1683, who attempted to find the value of the following expression (which is equal to e):

$\lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)^{n}.$

The first known use of the constant, represented by the letter b, was in correspondence from Gottfried Leibniz to Christiaan Huygens in 1690 and 1691. Leonhard Euler introduced the letter e as the base for natural logarithms, writing in a letter to Christian Goldbach on 25 November 1731. Euler started to use the letter e for the constant in 1727 or 1728, in an unpublished paper on explosive forces in cannons, while the first appearance of e in a publication was in Euler's Mechanica (1736). Although some researchers used the letter c in the subsequent years, the letter e was more common and eventually became standard.

Литература

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• Commentary on Endnote 10 of the book Prime Obsession for another stochastic representation
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• Karney, C. F. F. (2016). „Sampling exactly from the normal distribution”. ACM Transactions on Mathematical Software. 42 (1): 3:1—14. S2CID 14252035. arXiv:1303.6257 . doi:10.1145/2710016. 
• Kinderman, Albert J.; Monahan, John F. (1977). „Computer Generation of Random Variables Using the Ratio of Uniform Deviates”. ACM Transactions on Mathematical Software. 3 (3): 257—260. S2CID 12884505. doi:10.1145/355744.355750. 
• Krishnamoorthy, Kalimuthu (2006). Handbook of Statistical Distributions with Applications. Chapman & Hall/CRC. ISBN 978-1-58488-635-8. 
• Kruskal, William H.; Stigler, Stephen M. (1997). Spencer, Bruce D., ур. Normative Terminology: 'Normal' in Statistics and Elsewhere. Statistics and Public Policy. Oxford University Press. ISBN 978-0-19-852341-3. 
• Laplace, Pierre-Simon de (1774). „Mémoire sur la probabilité des causes par les événements”. Mémoires de l'Académie Royale des Sciences de Paris (Savants étrangers), Tome 6: 621—656.  Translated by Stephen M. Stigler in Statistical Science 1 (3), 1986: JSTOR 2245476.

Спољашње везе

• The number e to 1 million places and NASA.gov 2 and 5 million places
• e Approximations – Wolfram MathWorld
• Earliest Uses of Symbols for Constants Jan. 13, 2008
• "The story of e", by Robin Wilson at Gresham College, 28 February 2007 (available for audio and video download)
• e Search Engine 2 billion searchable digits of e, π and √2