See Also

E (mathematical constant)

The mathematical constant e is the base of the natural logarithm Natural logarithm

The natural logarithm, formerly known as the hyperbolic logarithm, is the logarithm [i] to the base e [i] ... 

. It is occasionally called Euler's number after the Swiss Switzerland

Switzerland , officially the Swiss Confederation, is a landlocked [i] Alpine country [i] in Central Europe [i] ... 

 mathematician Mathematician

A mathematician is a person whose primary area of study and research is the field of mathematics [i]. ... 

 Leonhard Euler Leonhard Euler

Leonhard Euler was a Swiss [i] mathematician [i] and physicist [i]. ... 

, or Napier's constant in honor of the Scottish Scotland

Scotland is a nation [i] in northwest Europe [i] and one of the constituent [i] countries [i] ... 

 mathematician John Napier John Napier

John Napier or Neper, nicknamed Marvellous Merchiston was a Scottish [i] mathematician [i] ... 

 who introduced logarithm Logarithm

The logarithm is the mathematical [i] operation that is the inverse [i] of ... 

s. e is one of the most important numbers in mathematics, alongside the additive and multiplicative identities 0 and 1, the imaginary unit i, and π Pi

The mathematical constant [i] p is an irrational [i] real number [i], approximately eq ... 

, the circumference to diameter ratio for any circle. It has a number of equivalent definitions; some of them are given below.

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Encyclopedia



The mathematical constant e is the base of the natural logarithm Natural logarithm

The natural logarithm, formerly known as the hyperbolic logarithm, is the logarithm [i] to the base e [i]... 

. It is occasionally called Euler's number after the Swiss Switzerland

Switzerland , officially the Swiss Confederation, is a landlocked [i] Alpine country [i] in Central Europe [i] ... 

 mathematician Mathematician

A mathematician is a person whose primary area of study and research is the field of mathematics [i]. ... 

 Leonhard Euler Leonhard Euler

Leonhard Euler was a Swiss [i] mathematician [i] and physicist [i]. ... 

, or Napier's constant in honor of the Scottish Scotland

Scotland is a nation [i] in northwest Europe [i] and one of the constituent [i] countries [i] ... 

 mathematician John Napier John Napier

John Napier or Neper, nicknamed Marvellous Merchiston was a Scottish [i] mathematician [i] ... 

 who introduced logarithm Logarithm

The logarithm is the mathematical [i] operation that is the inverse [i] of ... 

s. e is one of the most important numbers in mathematics, alongside the additive and multiplicative identities 0 and 1, the imaginary unit i, and π Pi

The mathematical constant [i] p is an irrational [i] real number [i], approximately eq ... 

, the circumference to diameter ratio for any circle. It has a number of equivalent definitions; some of them are given below. To the 20th decimal place:

e ˜ 2.71828 18284 59045 23536

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 natural logarithms calculated from the constant. It is assumed that the table was written by William Oughtred William Oughtred

William Oughtred was an English [i] mathematician [i].
... 

. The first indication of e as a constant was discovered by Jacob Bernoulli Jakob Bernoulli

Jakob Bernoulli , also known as Jacob, Jacques or James Bernoulli was a Swiss [i] ... 

, trying to find the value of the following expression:




The first known use of the constant, represented by the letter b, was in correspondence from Gottfried Leibniz Gottfried Leibniz

Gottfried Wilhelm Leibniz was a German [i] polymath [i] who wrote mostly in French and Latin.
... 

 to Christiaan Huygens Christiaan Huygens

Christiaan Huygens , was a Dutch [i] mathematician [i] and physicist [i] ... 

 in 1690 and 1691. Leonhard Euler Leonhard Euler

Leonhard Euler was a Swiss [i] mathematician [i] and physicist [i]. ... 

 started to use the letter e for the constant in 1727, and the first use of e in a publication was Euler's Mechanica . While in the subsequent years some researchers used the letter c, e was more common and eventually became the standard.

The exact reasons for the use of the letter e are unknown, but it may be because it is the first letter of the word exponential. Another possibility is that Euler used it because it was the first vowel Vowel

In phonetics [i], a vowel is a sound [i] in spoken language [i] that is characterized by an open configu ... 

 after a A

The letter A is the first letter in the Latin alphabet [i]. Its name in English [i] is ... 

, which he was already using for another number, but his reason for using vowels is unknown. It is unlikely that Euler chose the letter because it is his last initial, since he was a very modest man, and tried to give proper credit to the work of others.

Definitions

The three most common definitions of e are listed below.

  1. The limit


  2. The sum of the infinite series
  3. where n! is the factorial Factorial

    In mathematics [i], the factorial of a natural number [i] n is the product [i] of all positive [i] ... 

     of n.

  4. The unique real number e > 0 such that
  5. .


These definitions can be proved to be equivalent.

Properties

The exponential function Exponential function

The exponential function is one of the most important function [i]s in mathematics [i]. ... 

 f = ex is important in part because it is the unique nontrivial function which is its own derivative Derivative

In mathematics [i], the derivative is defined as the instantaneous rate of change of a function [i] ... 

, and therefore, its own primitive Antiderivative

In calculus [i], an antiderivative, primitive or indefinite integral of a function [i] ... 

:

and

, where C is the arbitrary constant of integration.

It is known that e is irrational . In fact, if m/n is any rational number with n > 1, then

where S is the smallest positive integer such that n divides S!.

It is also known that e is transcendental . It was the first number to be proved transcendental without having been specifically constructed for this purpose ; the proof was given by Charles Hermite in 1873. It is conjectured to be normal. It features in Euler's formula Euler's formula

Euler's formula, named after Leonhard Euler [i], is a mathematical [i] formula in complex analysis [i]... 

, one of the most important formulas in mathematics:

described by Richard Feynman Richard Feynman

Richard Phillips Feynman was an influential American [i] physicist [i] known for expandi... 

  as "[...] the most remarkable formula in mathematics [...], our jewel".

The special case with x = π is known as Euler's identity Euler's identity

In mathematical analysis [i], Euler's identity, named after Leonhard Euler [i], is the equation
... 

:

The following is an infinite simple continued fraction expansion of e :

The following is an infinite generalized continued fraction expansion of e:

The number e is also equal to the sum of the following infinite series:The number e is also given by several infinite product forms including Pippenger's product

and Guillera's product

where the nth factor is the nth root of the product

as well as the infinite product

The number e is equal to the limit of several infinite sequences:

and

.

The symmetric limit,

may be obtained by manipulation of the basic limit definition of e. Another limit is



where is the nth prime and is the primorial Primorial

For n ≥ 2, the primorial is the product of all prime number [i]s less than or equal to n. ... 

 of the nth prime.

It was shown by Euler that the infinite tetration Tetration

Tetration is iterated exponentiation, the first hyper operator [i] after exponentiation. ... 



converges only if

The number e is the global maximum Maxima and minima

In mathematics [i], maxima and minima, also known as extrema, are points in the domain [i] ... 

 of the function

The value of this function at e is

Non-mathematical uses of e

One of the most famous mathematical constants, e is also frequently referenced outside of mathematics. Some examples are:

  • In the IPO IPO

    Sorry, no overview for this topic 

     filing for Google Google

    Google Inc. is an American [i] public corporation [i], first incorporated [i]... 

    , in 2004, rather than a typical round-number amount of money, the company announced its intention to raise $2,718,281,828, which is e billion dollars United States dollar

    For details of current paper money [i] and coins, see Federal Reserve Note [i] and United States coinage [i] ... 

     to the nearest dollar.


  • Google was also responsible for a mysterious billboard that appeared in the heart of Silicon Valley Silicon Valley

    Silicon Valley is the southern part [i] of the San Francisco Bay Area [i] in Northern [i] ... 

    , and later in Cambridge, Massachusetts Cambridge, Massachusetts

    Cambridge is a city [i] in the Greater Boston [i] area of Massachusetts [i], United States [i]. ... 

    , Seattle, Washington Seattle, Washington

    Seattle is the largest city [i] in the Pacific Northwest [i] region of the United States [i]. ... 

    , and Austin, Texas Austin, Texas

    Austin is the state capital of Texas [i] and the county seat [i] of Travis County [i] ... 

     which read .com. Solving this problem and visiting the web site advertised led to an even more difficult problem to solve, which in turn leads to Google Labs Google Labs

    Google Labs is a website demonstrating new Google [i] projects "that aren't quite ready for prime time".... 

     where the visitor is invited to submit a resume. The first 10-digit prime in e is 7427466391, which surprisingly starts as late as at the 101st digit.


  • The famous computer scientist Donald Knuth Donald Knuth

    Donald Ervin Knuth is a renowned computer scientist [i] and [i] ... 

     let the version numbers of his program METAFONT METAFONT

    METAFONT is a programming language [i] used to define vector fonts [i]. ... 

     approach e .

References

  • Maor, Eli; e: The Story of a Number, ISBN 0-691-05854-7
  • O'Connor, J.J., and Roberson, E.F.; The MacTutor History of Mathematics archive: ; University of St Andrews Scotland
  • , Amer. Math. Monthly 112 729-734.
  • Jonathan Sondow, "A Geometric Proof that e Is Irrational and a New Measure of Its Irrationality," Amer. Math. Monthly 113 637-641

Notes

O'Connor, "The number e"

External links

  • and or
  • - Keith Tognetti, University of Wollongong, NSW, Australia