E (mathematical constant)

e is the unique number a, such that the value of the derivative (the slope of the tangent line) of the exponential function f (x) = a^x (blue curve) at the point x = 0 is exactly 1. For comparison, functions 2^x (dotted curve) and 4^x (dashed curve) are shown; they are not tangent to the line of slope 1 (red).

The mathematical constant e is the unique real number such that the function e^x has the same value as the slope of the tangent line, for all values of x. A mathematical constant is a number usually a Real number, that arises naturally in Mathematics. In Mathematics, the real numbers may be described informally in several different ways In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change ^[1] More generally, the only functions equal to their own derivatives are of the form Ce^x, where C is a constant. In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change ^[2] The function e^x so defined is called the exponential function, and its inverse is the natural logarithm, or logarithm to base e. The exponential function is a function in Mathematics. The application of this function to a value x is written as exp( x) In Mathematics, if &fnof is a function from A to B then an inverse function for &fnof is a function in the opposite direction from B The natural logarithm, formerly known as the Hyperbolic logarithm is the Logarithm to the base e, where e is an irrational radix|basis (topologyIn Arithmetic, the base refers to the number b in an expression of the form b n. The number e is also commonly defined as the base of the natural logarithm (using an integral to define the latter), as the limit of a certain sequence, or as the sum of a certain series (see representations of e, below). The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space The limit of a sequence is one of the oldest concepts in Mathematical analysis. In Mathematics, a sequence is an ordered list of objects (or events In Mathematics, a series is often represented as the sum of a Sequence of terms That is a series is represented as a list of numbers with

The number e is one of the most important numbers in mathematics,^[3] alongside the additive and multiplicative identities 0 and 1, the constant π, and the imaginary unit i. Mathematics For any number x: x ·1 = 1· x = x (1 is the multiplicative identity IMPORTANT NOTICE Please note that Wikipedia is not a database to store the millions of digits of π please refrain from adding those to Wikipedia as it could cause technical problems Definition By definition the imaginary unit i is one solution (of two of the Quadratic equation

The number e is sometimes called Euler's number after the Swiss mathematician Leonhard Euler. Switzerland (English pronunciation; Schweiz Swiss German: Schwyz or Schwiiz Suisse Svizzera Svizra officially the Swiss Confederation A mathematician is a person whose primary area of study and research is the field of Mathematics. (e is not to be confused with γ – the Euler–Mascheroni constant, sometimes called simply Euler's constant. The Euler–Mascheroni constant (also called the Euler constant) is a Mathematical constant recurring in analysis and Number theory, usually )

Since e is transcendental, and therefore irrational, its value cannot be given exactly as a finite or eventually repeating decimal. In Mathematics, a transcendental number is a Complex number that is not algebraic, that is not a solution of a non-zero Polynomial equation In Mathematics, an irrational number is any Real number that is not a Rational number — that is it is a number which cannot be expressed as a fraction The numerical value of e truncated to 20 decimal places is:

2. The decimal ( base ten or occasionally denary) Numeral system has ten as its base. 71828 18284 59045 23536. . .

Part of a series of articles on
The mathematical constant, e

Natural logarithm

Applications in Compound interest · Euler's identity & Euler's formula · Half-lives & Exponential growth/decay

Defining e Proof that e is irrational · Representations of e · Lindemann–Weierstrass theorem

People John Napier · Leonhard Euler

Schanuel's conjecture

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. The natural logarithm, formerly known as the Hyperbolic logarithm is the Logarithm to the base e, where e is an irrational Compound interest is the concept of adding accumulated Interest back to the principal so that interest is earned on interest from that moment on In Mathematical analysis, Euler's identity, named after Leonhard Euler, is the equation e^{i \pi} + 1 = 0 \\! where This article is about Euler's formula in Complex analysis. For Euler's formula in algebraic topology and polyhedral combinatorics see Euler characteristic Half-Life (computer-game page here It's already listed in the disambiguation page Exponential growth (including Exponential decay) occurs when the growth rate of a mathematical function is proportional to the function's current value A quantity is said to be subject to exponential decay if it decreases at a rate proportional to its value In Mathematics, the series representation of Euler's number e e = \sum_{n = 0}^{\infty} \frac{1}{n!}\! can be used to prove The Mathematical constant ''e'' can be represented in a variety of ways as a Real number. In Mathematics, the Lindemann–Weierstrass theorem is a result that is very useful in establishing the transcendence of numbers For other people with the same name see John Napier (disambiguation. In Mathematics, specifically Transcendence theory, Schanuel's conjecture is the following statement Given any n Complex numbers For other people with the same name see John Napier (disambiguation. ^[4] 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 ( March 5, 1575 – June 30, 1660) was an English Mathematician. The "discovery" of the constant itself is credited to Jacob Bernoulli, who attempted to find the value of the following expression (which is in fact 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. For other family members named Jacob see Bernoulli family. Jacob Bernoulli (also known as James or Jacques) ( Basel Christiaan Huygens (ˈhaɪgənz in English ˈhœyɣəns in Dutch) ( April 14, 1629 &ndash July 8, 1695) was a Dutch Leonhard Euler started to use the letter e for the constant in 1727, and the first use of e in a publication was Euler's Mechanica (1736). 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 after a, which he was already using for another number, but his reason for using vowels is unknown. In Phonetics, a vowel is a Sound in spoken Language, such as English ah! or oh!, pronounced with an open Vocal tract It is unlikely that Euler chose the letter because it is the first letter of his surname, since he was a very modest man and tried to give proper credit to the work of others.

Applications

The compound-interest problem

Jacob Bernoulli discovered this constant by studying a question about compound interest. For other family members named Jacob see Bernoulli family. Jacob Bernoulli (also known as James or Jacques) ( Basel Compound interest is the concept of adding accumulated Interest back to the principal so that interest is earned on interest from that moment on

One simple example is an account that starts with $1. 00 and pays 100% interest per year. If the interest is credited once, at the end of the year, the value is $2. 00; but if the interest is computed and added twice in the year, the $1 is multiplied by 1. 5 twice, yielding $1. 00×1. 5² = $2. 25. Compounding quarterly yields $1. 00×1. 25⁴ = $2. 4414…, and compounding monthly yields $1. 00×(1. 0833…)¹² = $2. 613035….

Bernoulli noticed that this sequence approaches a limit (the force of interest) for more and smaller compounding intervals. Compound interest is the concept of adding accumulated Interest back to the principal so that interest is earned on interest from that moment on Compounding weekly yields $2. 692597…, while compounding daily yields $2. 714567…, just two cents more. Using n as the number of compounding intervals, with interest of 1⁄n in each interval, the limit for large n is the number that came to be known as e; with continuous compounding, the account value will reach $2. 7182818…. More generally, an account that starts at $1, and yields (1+R) dollars at simple interest, will yield e^R dollars with continuous compounding.

Bernoulli trials

The number e itself also has applications to probability theory, where it arises in a way not obviously related to exponential growth. Probability theory is the branch of Mathematics concerned with analysis of random phenomena Suppose that a gambler plays a slot machine that pays out with a probability of one in n and plays it n times. Then, for large n (such as a million) the probability that the gambler will win nothing at all is (approximately) 1⁄e. Probability is the likelihood or chance that something is the case or will happen

This is an example of a Bernoulli trials process. In the theory of Probability and Statistics, a Bernoulli trial is an experiment whose outcome is random and can be either of two possible outcomes "success" Each time the gambler plays the slots, there is a one in one million chance of winning. Playing one million times is modelled by the binomial distribution, which is closely related to the binomial theorem. WikipediaWikiProject Probability#Standards for a discussion of standards used for probability distribution articles such as this one In Mathematics, the binomial theorem is an important Formula giving the expansion of powers of Sums Its simplest version says The probability of winning k times out of a million trials is;

$\binom{10^6}{k} \left(10^{-6}\right)^k(1-10^{-6})^{10^6-k}.$

In particular, the probability of winning zero times (k=0) is

$\left(1-\frac{1}{10^6}\right)^{10^6}.$

This is very close to the following limit for 1⁄e:

$\frac{1}{e} = \lim_{n\to\infty} \left(1-\frac{1}{n}\right)^n.$

Derangements

Another application of e, also discovered in part by Jacob Bernoulli along with Pierre Raymond de Montmort is in the problem of derangements, also known as the hat check problem. Pierre Raymond de Montmort, a French Mathematician, was born in Paris on 27 October 1678, and died there on 7 October In combinatorial Mathematics, a derangement is a Permutation in which none of the elements of the set appear in their original positions ^[5] Here n guests are invited to a party, and at the door each guest checks his hat with the butler who then places them into labeled boxes. But the butler does not know the name of the guests, and so must put them into boxes selected at random. The problem of de Montmort is: what is the probability that none of the hats gets put into the right box. The answer is:

$p_n = 1-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+\cdots+(-1)^n\frac{1}{n!}.$

As the number n of guests tends to infinity, p_n approaches 1⁄e. Furthermore, the number of ways the hats can be placed into the boxes so that none of the hats is in the right box is exactly n!⁄e, rounded to the nearest integer. ^[6]

Asymptotics

The number e occurs naturally in connection with many problems involving asymptotics. In pure and Applied mathematics, particularly the Analysis of algorithms, real analysis and engineering asymptotic analysis is a method of describing A prominent example is Stirling's formula for the asymptotics of the factorial function, in which both the numbers e and π enter:

$n! \sim \sqrt{2\pi n}\, \frac{n^n}{e^n}.$

A particular consequence of this is

$e = \lim_{n\to\infty} \frac{n}{\sqrt[n]{n!}}.$

e in calculus

The natural log at e, ln(e), is equal to 1

The principal motivation for introducing the number e, particularly in calculus, is to perform differential and integral calculus with exponential functions and logarithms. In Mathematics, Stirling's approximation (or Stirling's formula) is an approximation for large Factorials It is named in honour of James Stirling Definition The factorial function is formally defined by n!=\prod_{k=1}^n k IMPORTANT NOTICE Please note that Wikipedia is not a database to store the millions of digits of π please refrain from adding those to Wikipedia as it could cause technical problems Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space The exponential function is a function in Mathematics. The application of this function to a value x is written as exp( x) In Mathematics, the logarithm of a number to a given base is the power or Exponent to which the base must be raised in order to produce ^[7] A general exponential function y=a^x has derivative given as the limit:

$\frac{d}{dx}a^x=\lim_{h\to 0}\frac{a^{x+h}-a^x}{h}=\lim_{h\to 0}\frac{a^{x}a^{h}-a^x}{h}=a^x\left(\lim_{h\to 0}\frac{a^h-1}{h}\right).$

The limit on the right-hand side is independent of the variable x: it depends only on the base a. In Mathematics, the limit of a function is a fundamental concept in Calculus and analysis concerning the behavior of that function near a particular When the base is e, this limit is equal to one, and so e is symbolically defined by the equation:

$\frac{d}{dx}e^x = e^x.$

Consequently, the exponential function with base e is particularly suited to doing calculus. Choosing e, as opposed to some other number, as the base of the exponential function makes calculations involving the derivative much simpler.

Another motivation comes from considering the base-a logarithm. In Mathematics, the logarithm of a number to a given base is the power or Exponent to which the base must be raised in order to produce ^[8] Considering the definition of the derivative of log_ax as the limit:

$\frac{d}{dx}\log_a x = \lim_{h\to 0}\frac{\log_a(x+h)-\log_a(x)}{h}=\frac{1}{x}\left(\lim_{u\to 0}\frac{1}{u}\log_a(1+u)\right).$

Once again, there is an undetermined limit which depends only on the base a, and if that base is e, the limit is one. So symbolically,

$\frac{d}{dx}\log_e x=\frac{1}{x}.$

The logarithm in this special base is called the natural logarithm (often represented as "ln"), and it also behaves well under differentiation since there is no undetermined limit to carry through the calculations. The natural logarithm, formerly known as the Hyperbolic logarithm is the Logarithm to the base e, where e is an irrational

There are thus two ways in which to select a special number a=e. One way is to set the derivative of the exponential function a^x to a^x. The other way is to set the derivative of the base a logarithm to 1/x. In each case, one arrives at a convenient choice of base for doing calculus. In fact, these two bases are actually the same, the number e.

Alternative characterizations

See also: Representations of e

Other characterizations of e are also possible: one is as the limit of a sequence, another is as the sum of an infinite series, and still others rely on integral calculus. The Mathematical constant ''e'' can be represented in a variety of ways as a Real number. The limit of a sequence is one of the oldest concepts in Mathematical analysis. In Mathematics, a series is often represented as the sum of a Sequence of terms That is a series is represented as a list of numbers with The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space So far, the following two (equivalent) properties have been introduced:

1. The number e is the unique positive real number such that

$\frac{d}{dt}e^t = e^t.$

2. In Mathematics, the real numbers may be described informally in several different ways The number e is the unique positive real number such that

$\frac{d}{dt} \log_e t = \frac{1}{t}.$

The following three characterizations can be proven equivalent:

3. In Mathematics, the Exponential function can be characterized in many ways The number e is the limit

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

The area under the graph y = 1/x is equal to 1 over the interval 1 ≤ x ≤ e. In Mathematics, the concept of a " limit " is used to describe the Behavior of a function as its argument either "gets close"

4. The number e is the sum of the infinite series

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

where n! is the factorial of n. In Mathematics, a series is often represented as the sum of a Sequence of terms That is a series is represented as a list of numbers with Definition The factorial function is formally defined by n!=\prod_{k=1}^n k

5. The number e is the unique positive real number such that

$\int_{1}^{e} \frac{1}{t} \, dt = {1}$ .

Properties

Calculus

As in the motivation, the exponential function f(x) = e^x is important in part because it is the unique nontrivial function (up to multiplication by a constant) which is its own derivative

$\frac{d}{dx}e^x=e^x$

and therefore its own antiderivative as well:

$e^x= \int_{-\infty}^x e^t\,dt$

$= \int_{-\infty}^0 e^t\,dt + \int_{0}^x e^t\,dt$

$\qquad= 1 + \int_{0}^x e^t\,dt.$

Exponential-like functions

The number x=e is where the global maximum occurs for the function:

$f(x) = x^{1 \over x}.$

More generally, $x=\!\ \sqrt[n]{e}$ is where the global maximum occurs for the function

$\!\ f(x) = x^{1 \over {x^n}}$

The infinite tetration

$x^{x^{x^{\cdot^{\cdot^{\cdot}}}}}$

converges only if $e^{-e} \le x \le e^{1/e},$ due to a theorem of Leonhard Euler. The exponential function is a function in Mathematics. The application of this function to a value x is written as exp( x) In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change In Calculus, an antiderivative, primitive or indefinite integral of a function f is a function F whose Derivative In Mathematics, maxima and minima, known collectively as extrema, are the largest value (maximum or smallest value (minimum that In Mathematics, tetration (also known as hyper -4

Finally, the exponential function e^x is usually defined as

$e^{x} = 1 + {x \over 1!} + {x^{2} \over 2!} + {x^{3} \over 3!} + \cdots$

Number theory

The real number e is irrational (see proof that e is irrational), and furthermore is transcendental (Lindemann–Weierstrass theorem). The exponential function is a function in Mathematics. The application of this function to a value x is written as exp( x) In Mathematics, an irrational number is any Real number that is not a Rational number — that is it is a number which cannot be expressed as a fraction In Mathematics, the series representation of Euler's number e e = \sum_{n = 0}^{\infty} \frac{1}{n!}\! can be used to prove In Mathematics, a transcendental number is a Complex number that is not algebraic, that is not a solution of a non-zero Polynomial equation In Mathematics, the Lindemann–Weierstrass theorem is a result that is very useful in establishing the transcendence of numbers It was the first number to be proved transcendental without having been specifically constructed for this purpose (compare with Liouville number). In Number theory, a Liouville number is a Real number x with the property that for any positive Integer n, there exist integers The proof was given by Charles Hermite in 1873. Charles Hermite (ʃaʁl ɛʁˈmit ( December 24, 1822 &ndash January 14, 1901) was a French Mathematician who did It is conjectured to be normal. A different topic is treated in the article titled Normal number (computing.

Complex numbers

It features in Euler's formula, an important formula related to complex numbers:

$e^{ix} = \cos x + i\sin x,\,\!$

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

$e^{i\pi}+1 =0 .\,\!$

Consequently,

$e^{i\pi}=-1 \,\!$

from which it follows that, in the principal branch of the logarithm,

$\log_e (-1) = i\pi .\,\!$

Furthermore, using the laws for exponentiation,

$(\cos x + i\sin x)^n = \left(e^{ix}\right)^n = e^{inx} = \cos (nx) + i \sin (nx)$

which is de Moivre's formula. This article is about Euler's formula in Complex analysis. For Euler's formula in algebraic topology and polyhedral combinatorics see Euler characteristic Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In Mathematical analysis, Euler's identity, named after Leonhard Euler, is the equation e^{i \pi} + 1 = 0 \\! where In Mathematics, a principal branch is a function which selects one branch or "slice" of a Multi-valued function. De Moivre's formula, named after Abraham de Moivre, states that for any Complex number (and in particular for any Real number) x and any

The case,

$\cos (x) + i \sin (x).\,\!$

is commonly referred to as Cis(x).

Representations of e

Main article: Representations of e

The number e can be represented as a real number in a variety of ways: as an infinite series, an infinite product, a continued fraction, or a limit of a sequence. The Mathematical constant ''e'' can be represented in a variety of ways as a Real number. In Mathematics, the real numbers may be described informally in several different ways In Mathematics, a series is often represented as the sum of a Sequence of terms That is a series is represented as a list of numbers with In Mathematics, for a Sequence of numbers a 1 a 2 a 3. the infinite product In Mathematics, a continued fraction is an expression such as x = a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3 + \cfrac{1}{\ddots\}}}} The limit of a sequence is one of the oldest concepts in Mathematical analysis. The chief among these representations, particularly in introductory calculus courses is the limit

$\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n,$

given above, as well as the series

$e=\sum_{n=0}^\infty \frac{1}{n!}$

given by evaluating the above power series for e^x at x=1. Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives

Still other less common representations are also available. For instance, e can be represented as an infinite simple continued fraction:

$e=2+\cfrac{1}{ 1+\cfrac{1}{ {\mathbf 2}+\cfrac{1}{ 1+\cfrac{1}{ 1+\cfrac{1}{ {\mathbf 4}+\cfrac{1}{ \ddots } } } } }}$

Or, in a more compact form (sequence A003417 in OEIS):

$e = [[2; 1, \textbf{2}, 1, 1, \textbf{4}, 1, 1, \textbf{6}, 1, 1, \textbf{8}, 1, \ldots,1, \textbf{2n}, 1,\ldots]] \,$

Which can be written more harmoniously by allowing zero:^[9]

$e = [[ 1 , \textbf{0} , 1 , 1, \textbf{2}, 1, 1, \textbf{4}, 1 , 1 , \textbf{6}, 1, \ldots]] \,$

Many other series, sequence, continued fraction, and infinite product representations of e have also been developed. In Mathematics, a continued fraction is an expression such as x = a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3 + \cfrac{1}{\ddots\}}}} The On-Line Encyclopedia of Integer Sequences ( OEIS) also cited simply as Sloane's, is an extensive searchable Database of Integer sequences

Stochastic representations of e

In addition to the deterministic analytical expressions for representation of e, as described above, there are some stochastic protocols for estimation of e. In one such protocol, random samples $X 1, X 2,. . . , X n$ of size n from the uniform distribution on (0, 1) are used to approximate e. If

$U= \min { \left \{ n \mid X_1+X_2+...+X_n > 1 \right \} },$

then the expectation of U is e: $E (U) = e$ . ^[10]^[11] Thus sample averages of U variables will approximate e.

Known digits

The number of known digits of e has increased dramatically during the last decades. This is due both to the increase of performance of computers as well as to algorithmic improvements. ^[12]^[13]

**Number of known decimal digits of e**
Date	Decimal digits	Computation performed by
1748	18^[14]	Leonhard Euler
1853	137	William Shanks
1871	205	William Shanks
1884	346	J. William Shanks ( January 25, 1812 &ndash summer 1882 Houghton-le-Spring, Durham, England) was a British Amateur William Shanks ( January 25, 1812 &ndash summer 1882 Houghton-le-Spring, Durham, England) was a British Amateur M. Boorman
1946	808	?
1949	2,010	John von Neumann (on the ENIAC)
1961	100,265	Daniel Shanks & John W. ENIAC, short for Electronic Numerical Integrator And Computer, was the first general-purpose electronic Computer. Wrench
1994	10,000,000	Robert Nemiroff & Jerry Bonnell
May 1997	18,199,978	Patrick Demichel
August 1997	20,000,000	Birger Seifert
September 1997	50,000,817	Patrick Demichel
February 1999	200,000,579	Sebastian Wedeniwski
October 1999	869,894,101	Sebastian Wedeniwski
November 21, 1999	1,250,000,000	Xavier Gourdon
July 10, 2000	2,147,483,648	Shigeru Kondo & Xavier Gourdon
July 16, 2000	3,221,225,472	Colin Martin & Xavier Gourdon
August 2, 2000	6,442,450,944	Shigeru Kondo & Xavier Gourdon
August 16, 2000	12,884,901,000	Shigeru Kondo & Xavier Gourdon
August 21, 2003	25,100,000,000	Shigeru Kondo & Xavier Gourdon
September 18, 2003	50,100,000,000	Shigeru Kondo & Xavier Gourdon
April 27, 2007	100,000,000,000	Shigeru Kondo & Steve Pagliarulo

e in computer culture

In contemporary internet culture, individuals and organizations frequently pay homage to the number e. Events 164 BC - Judas Maccabaeus, son of Mattathias of the Hasmonean family restores the Temple in Jerusalem. Year 1999 ( MCMXCIX) was a Common year starting on Friday (link will display full 1999 Gregorian calendar) Events 48 BC - Battle of Dyrrhachium, Julius Caesar barely avoids a catastrophic defeat to Pompey in Macedonia. 2000 ( MM) was a Leap year that started on Saturday of the Common Era, in accordance with the Gregorian calendar. Events 622 - The beginning of the Islamic calendar. 1054 - Three Roman legates fractured relations between the Western and 2000 ( MM) was a Leap year that started on Saturday of the Common Era, in accordance with the Gregorian calendar. Events 338 BC - A Macedonian army led by Philip II defeated the combined forces of Athens and Thebes in the 2000 ( MM) was a Leap year that started on Saturday of the Common Era, in accordance with the Gregorian calendar. Events 1384 - The Hongwu Emperor of Ming China, Emperor Dong hears a case of a couple who tore paper money bills while fighting 2000 ( MM) was a Leap year that started on Saturday of the Common Era, in accordance with the Gregorian calendar. Events 1192 - Minamoto Yoritomo becomes Seii Tai Shōgun and the De facto ruler of Japan. Year 2003 ( MMIII) was a Common year starting on Wednesday of the Gregorian calendar. Events 96 - Nerva is proclaimed Roman Emperor after Domitian is assassinated Year 2003 ( MMIII) was a Common year starting on Wednesday of the Gregorian calendar. Events 1124 - David I becomes King of Scotland. 1296 - Battle of Dunbar: The Scots are defeated Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century. Cyberculture is the Culture that has emerged or is emerging from the use of Computer networks for communication, entertainment and business

For example, in the IPO filing for Google, 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 to the nearest dollar. Initial public offering (IPO, also referred to simply as a "public offering" is when a company issues Common stock or shares to the public for the first Google Inc is an American public corporation, earning revenue from advertising related to its Internet search, e-mail, online The United States dollar ( sign: $; code: USD) is the unit of Currency of the United States; it has also been Google was also responsible for a mysterious billboard^[15] that appeared in the heart of Silicon Valley, and later in Cambridge, Massachusetts; Seattle, Washington; and Austin, Texas. For the valley nicknamed "Silicone Valley" see San Fernando Valley. Cambridge Massachusetts is a City in the Greater Boston area of Massachusetts, United States. It read {first 10-digit prime found in consecutive digits of e}. com (now defunct). Solving this problem and visiting the advertised web site led to an even more difficult problem to solve, which in turn leads to Google Labs where the visitor is invited to submit a resume. Google Labs is a website demonstrating new Google projects "that aren't quite ready for prime time" ^[16] The first 10-digit prime in e is 7427466391, which starts at the 99th digit. ^[17] (A random stream of digits has a 98. 4% chance of starting a 10-digit prime sooner. )

In another instance, the eminent computer scientist Donald Knuth let the version numbers of his program METAFONT approach e. A computer scientist is a person that has acquired knowledge of Computer science, the study of the theoretical foundations of information and computation and their application Donald Ervin Knuth (kəˈnuːθ (born 10 January 1938) is a renowned computer scientist and Professor Emeritus of the Art of Computer Metafont is a Programming language used to define vector fonts. The versions are 2, 2. 7, 2. 71, 2. 718, and so forth.

Notes

^ Keisler, H. J. Derivatives of Exponential Functions and the Number e
^ Keisler, H. J. General Solution of First Order Differential Equation
^ Howard Whitley Eves (1969). An Introduction to the History of Mathematics. Holt, Rinehart & Winston.
^ O'Connor, J. J. , and Roberson, E. F. ; The MacTutor History of Mathematics archive: "The number e"; University of St Andrews Scotland (2001)
^ Grinstead, C. M. and Snell, J. L. Introduction to probability theory (published online under the GFDL), p. The GNU Free Documentation License ( GNU FDL or simply GFDL) is a Copyleft License for free documentation designed by the Free Software 85.
^ Knuth (1997) The Art of Computer Programming Volume I, Addison-Wesley, p. The Art of Computer Programming is a comprehensive Monograph written by Donald Knuth that covers many kinds of Programming Algorithms 183.
^ See, for instance, Kline, M. (1998) Calculus: An intuitive and physical approach, Dover, section 12. 3 "The Derived Functions of Logarithmic Functions. "
^ This is the approach taken by Klein (1998).
^ Hofstadter, D. R. , "Fluid Concepts and Creative Analogies: Computer Models of the Fundamental Mechanisms of Thought" Basic Books (1995)
^ Russell, K. G. (1991) Estimating the Value of e by Simulation The American Statistician, Vol. 45, No. 1. (Feb. , 1991), pp. 66-68.
^ Dinov, ID (2007) Estimating e using SOCR simulation, SOCR Hands-on Activities (retrieved December 26, 2007). Events 1481 - Battle of Westbrook - Holland defeats troops of Utrecht. Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century.
^ Sebah, P. and Gourdon, X. ; The constant e and its computation
^ Gourdon, X. ; Reported large computations with PiFast
^ New Scientist 21st July 2007 p. 40
^ First 10-digit prime found in consecutive digits of e - Brain Tags
^ Shea, Andrea. "Google Entices Job-Searchers with Math Puzzle", NPR. Retrieved on 2007-06-09. Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century. Events 53 - Roman Emperor Nero marries Claudia Octavia 62 - Claudia Octavia commits
^ Kazmierczak, Marcus (2004-07-29). "MMIV" redirects here For the Modest Mouse album see " Baron von Bullshit Rides Again " Events 1014 - Byzantine-Bulgarian Wars: Battle of Kleidion: Byzantine emperor Basil II inflicts a decisive defeat Math : Google Labs Problems. mkaz. com. Retrieved on 2007-06-09. Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century. Events 53 - Roman Emperor Nero marries Claudia Octavia 62 - Claudia Octavia commits

References

Maor, Eli; e: The Story of a Number, ISBN 0-691-05854-7

External links

The number e to 1 million places and 2 and 5 million places
Earliest Uses of Symbols for Constants
e the EXPONENTIAL - the Magic Number of GROWTH - Keith Tognetti, University of Wollongong, NSW, Australia
An Intuitive Guide To Exponential Functions & e
"The story of e", by Robin Wilson at Gresham College, 28 February 2007 (available for audio and video download)
Class Library for Numbers (part of the GiNaC distribution) includes example code for computing e to arbitrary precision. Gresham College is an unusual institution of higher learning off Holborn in central London. Events 202 BC - coronation ceremony of Liu Bang as Emperor Gaozu of Han takes place initiating four centuries of the Han Dynasty 's rule Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century. GiNaC is a free Computer algebra system released under the GNU General Public License.
The SOCR resource provides a hands-on activity and an interactive Java applet (Uniform E-Estimate Experiment) for computing e using a simulation based on uniform distribution. The Statistics Online Computational Resource (SOCR is a suite of online tools and interactive aids for hands-on learning and teaching concepts in statistical analyses and In Probability theory and Statistics, the continuous uniform distribution is a family of Probability distributions such that for each member of the

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