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Encyclopedia > E (mathematical constant)

e is the unique number such that the value of the derivative of f (x) = ex (blue curve) at the point x = 0 is exactly 1. For comparison, functions 2x (dotted curve) and 4x (dashed curve) are shown; they are not tangent to the line of slope 1 (red).
e is the unique number such that the value of the derivative of f (x) = ex (blue curve) at the point x = 0 is exactly 1. For comparison, functions 2x (dotted curve) and 4x (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 value of the derivative (slope of the tangent line) of f(x) = ex at the point x = 0 is exactly 1. The function ex so defined is called the exponential function, and its inverse is the natural logarithm, or logarithm to base e. Image File history File links Exp_derivative_at_0. ... A mathematical constant is a quantity, usually a real number or a complex number, that arises naturally in mathematics and does not change. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... For a non-technical overview of the subject, see Calculus. ... In mathematics, the word tangent has two distinct but etymologically-related meanings: one in geometry and one in trigonometry. ... The exponential function is one of the most important functions in mathematics. ... In mathematics, an inverse function is in simple terms a function which does the reverse of a given function. ... The natural logarithm, formerly known as the hyperbolic logarithm, is the logarithm to the base e, where e is equal to 2. ... In mathematics, the base or radix is the number of various unique symbols (digits), including zero, that a positional numeral system uses to represent numbers in a given counting system. ...


The number e is one of the most important in mathematics,[1] alongside the additive and multiplicative identities 0 and 1, the imaginary unit i, and π, the circumference to diameter ratio for any circle in a plane. It has a number of equivalent definitions; some of them are given below. For other uses, see zero or 0. ... Look up one in Wiktionary, the free dictionary. ... In mathematics, the imaginary unit (or sometimes the Latin or the Greek iota, see below) allows the real number system to be extended to the complex number system . ... When a circles diameter is 1, its circumference is Ï€. The mathematical constant Ï€ is an irrational real number, approximately equal to 3. ...


The number e is occasionally called Euler's number after the Swiss mathematician Leonhard Euler, or Napier's constant in honor of the Scottish mathematician John Napier who introduced logarithms. (e is not to be confused with γ – the Euler–Mascheroni constant, sometimes called simply Euler's constant.) Leonhard Euler, one of the greatest mathematicians of all time A mathematician is a person whose primary area of study and research is the field of mathematics. ... Leonhard Paul Euler (pronounced Oiler; IPA ) (April 15, 1707 – September 18 [O.S. September 7] 1783) was a pioneering Swiss mathematician and physicist, who spent most of his life in Russia and Germany. ... Motto (Latin) No one provokes me with impunity Cha togar mfhearg gun dioladh (Scottish Gaelic) Wha daur meddle wi me?(Scots)1 Anthem (Multiple unofficial anthems) Scotlands location in Europe Capital Edinburgh Largest city Glasgow Official languages English (de facto)1; Gaelic[1]2 and Scots3 (recognised minority... For other people with the same name, see John Napier (disambiguation). ... Logarithms to various bases: is to base e, is to base 10, and is to base 1. ... The Euler-Mascheroni constant is a mathematical constant, used mainly in number theory. ...


Since e is transcendental, and therefore irrational, its value cannot be given exactly as a finite or eventually repeating decimal. The numerical value of e truncated to 20 decimal places is: In mathematics, a transcendental number is any complex number that is not algebraic, that is, not the solution of a non-zero polynomial equation with integer (or, equivalently, rational) coefficients. ... 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 m/n, where m and n are integers. ... The decimal (base ten or occasionally denary) numeral system has ten as its base. ...

2.71828 18284 59045 23536...

Contents

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. 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): For other people with the same name, see John Napier (disambiguation). ... William Oughtred William Oughtred (March 5, 1575 – June 30, 1660) was an English mathematician. ... Jakob Bernoulli. ...

lim_{ntoinfty} 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 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. It has been suggested that this article be split into multiple articles. ... Christiaan Huygens (pronounced in English (IPA): ; in Dutch: ) (April 14, 1629 – July 8, 1695), was a Dutch mathematician, astronomer and physicist; born in The Hague as the son of Constantijn Huygens. ... Leonhard Paul Euler (pronounced Oiler; IPA ) (April 15, 1707 – September 18 [O.S. September 7] 1783) was a pioneering Swiss mathematician and physicist, who spent most of his life in Russia and Germany. ... Events January 26 - Stanislaus I of Poland abdicates his throne. ...


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. 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.[2] The term exponential may refer to any of several topics in mathematics: Exponential distribution Exponential function Exponential growth, exponential decay Exponential time Matrix exponential Exponential map (in differential geometry) All relate in some fashion to exponents. ... Note: This page contains IPA phonetic symbols in Unicode. ...


Applications

The compound-interest problem

Jacob Bernoulli discovered this constant by studying a question about compound interest. Jakob Bernoulli. ... Compound interest refers to the fact that whenever interest is calculated, it is based not only on the original principal, but also on any unpaid interest that has been added to the principal. ...


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 for more and smaller compounding intervals. 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 eR 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. Suppose that a gambler plays a slot machine with a one in n probability 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 theory is the branch of mathematics concerned with analysis of random phenomena. ... Probability is the likelihood that something is the case or will happen. ...


This is an example of a Bernoulli trials process. 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. The probability of winning k times out of a million trials is; 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, called success and failure. ... In probability theory and statistics, the binomial distribution is the discrete probability distribution of the number of successes in a sequence of n independent yes/no experiments, each of which yields success with probability p. ... In mathematics, the binomial theorem is an important formula giving the expansion of powers of sums. ...

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_{ntoinfty} 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.[3] 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 labelled 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: Pierre Raymond de Montmort was born in Paris on Oct. ... In combinatorics, a derangement is a permutation φ of a set S (i. ...

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

As the number n of guests tends to infinity, pn 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.[4]


e in calculus

The natural log at e, Ln(e), is equal to 1
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.[5] A general exponential function y=ax has derivative given as the limit: Image File history File links This is a lossless scalable vector image. ... Image File history File links This is a lossless scalable vector image. ... Calculus (from Latin, pebble or little stone) is a branch of mathematics that includes the study of limits, derivatives, integrals, and infinite series, and constitutes a major part of modern university education. ... For a non-technical overview of the subject, see Calculus. ... This article deals with the concept of an integral in calculus. ... The exponential function is one of the most important functions in mathematics. ... Logarithms to various bases: is to base e, is to base 10, and is to base 1. ... In mathematics, the limit of a function is a fundamental concept in mathematical analysis. ...

frac{d}{dx}a^x=lim_{hto 0}frac{a^{x+h}-a^x}{h}=lim_{hto 0}frac{a^{x}a^{h}-a^x}{h}=a^xleft(lim_{hto 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. 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.

As a consequence, 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.[6] Considering the definition of the derivative of logax as the limit: Logarithms to various bases: is to base e, is to base 10, and is to base 1. ...

frac{d}{dx}log_a x = lim_{hto 0}frac{log_a(x+h)-log_a(x)}{h}=frac{1}{x}left(lim_{uto 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 equal to 2. ...


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 ax to ax. 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 ostensibly different bases are actually the same, the number e.


Alternative characterizations

The area under the graph y=1/x is equal to 1 over the interval 1 ≤ x ≤ e.
The area under the graph y=1/x is equal to 1 over the interval 1 ≤ x ≤ 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. So far, the following two (equivalent) properties have been introduced: Image File history File links This is a lossless scalable vector image. ... Image File history File links This is a lossless scalable vector image. ... The limit of a sequence is one of the oldest concepts in mathematical analysis. ... In mathematics, a series is a sum of a sequence of terms. ... This article deals with the concept of an integral in calculus. ...


1. The number e is the unique positive real number such that In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...

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

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

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

The following alternative characterizations can also be proven to be equivalent: In mathematics, the exponential function can be characterized in many ways. ...


3. The number e is the limit Wikibooks Calculus has a page on the topic of Limits In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements, as...

e = lim_{ntoinfty} left( 1 + frac{1}{n} right)^n

4. The number e is the sum of the infinite series In mathematics, a series is a sum of a sequence of terms. ...

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. For factorial rings in mathematics, see unique factorisation domain. ...


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

int_{1}^{e} frac{1}{t} , dt = {1}

(that is, the number e such that the area under the hyperbola f(t) = 1 / t from 1 to e is equal to 1). In mathematics, a hyperbola (Greek literally overshooting or excess) is a type of conic section defined as the intersection between a right circular conical surface and a plane which cuts through both halves of the cone. ...


Properties

Calculus

As in the motivation, the exponential function f(x) = ex is important in part because it is the unique nontrivial function (up to multiplication by a constant) which is its own derivative, and therefore its own antiderivative as well: The exponential function is one of the most important functions in mathematics. ... For a non-technical overview of the subject, see Calculus. ... In calculus, an antiderivative, primitive or indefinite integral of a function f is a function F whose derivative is equal to f, i. ...

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

and

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 A graph illustrating local min/max and global min/max points In mathematics, a point x* is a local maximum of a function f if there exists some &#949; > 0 such that f(x*) &#8805; f(x) for all x with |x-x*| < &#949;. Stated less formally, a local maximum...

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 Tetration (also exponential map, hyperpower, power tower, super-exponentiation, and hyper4) is iterated exponentiation, the first hyper operator after exponentiation. ...

x^{x^{x^{cdot^{cdot^{cdot}}}}}

converges only if e^{-e} le x le e^{1/e}, due to a theorem of Leonhard Euler. Leonhard Paul Euler (pronounced Oiler; IPA ) (April 15, 1707 – September 18 [O.S. September 7] 1783) was a pioneering Swiss mathematician and physicist, who spent most of his life in Russia and Germany. ...


ex 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). It was the first number to be proved transcendental without having been specifically constructed for this purpose (compare with Liouville number). The proof was given by Charles Hermite in 1873. It is conjectured to be normal. 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 m/n, where m and n are integers. ... In mathematics, the series expansion of the number e can be used to prove that e is irrational. ... In mathematics, a transcendental number is any complex number that is not algebraic, that is, not the solution of a non-zero polynomial equation with integer (or, equivalently, rational) coefficients. ... In mathematics, the Lindemann–Weierstrass theorem states that if α1,...,αn are algebraic numbers which are linearly independent over the rational numbers, then are algebraically independent over the algebraic numbers; in other words the set has transcendence degree n over . ... In number theory, a Liouville number is a real number x with the property that, for any positive integer n, there exist integers p and q with q > 1 and such that 0 < |x − p/q| < 1/qn. ... Charles Hermite (pronounced in IPA, , or phonetically air-meet) (December 24, 1822 - January 14, 1901) was a French mathematician who did research on number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra. ... 1873 (MDCCCLXXIII) was a common year starting on Wednesday (see link for calendar). ... In mathematics, a normal number is, roughly speaking, a real number whose digits show a random distribution with all digits being equally likely. ...


Complex numbers

It features in Euler's formula, an important formula related to complex numbers: Eulers formula, named after Leonhard Euler, is a mathematical formula in complex analysis that shows a deep relationship between the trigonometric functions and the complex exponential function. ... The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ...

e^{ix} = cos x + isin x,,!

The special case with x = π is known as Euler's identity: For other meanings, see List of topics named after Leonhard Euler In mathematical analysis, Eulers identity, named after Leonhard Euler, is the equation where is Eulers number, the base of the natural logarithm, is the imaginary unit, one of the two complex numbers whose square is negative one...

e^{ipi}+1 =0 .,!

Furthermore, using the laws for exponentiation,

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

which is de Moivre's formula. de Moivres formula, named after Abraham de Moivre, states that for any complex number (and, in particular, for any real number) x and any integer n it holds that The formula is important because it connects complex numbers (i stands for the imaginary unit) and trigonometry. ...


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 chief among these representations, particularly in introductory calculus courses is the limit 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 as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, a series is a sum of a sequence of terms. ... In mathematics, for a sequence of numbers a1, a2, a3, ... the infinite product is defined to be the limit of the partial products a1a2. ... In mathematics, a continued fraction is an expression such as where a0 is some integer and all the other numbers an are positive integers. ... The limit of a sequence is one of the oldest concepts in mathematical analysis. ... Calculus (from Latin, pebble or little stone) is a branch of mathematics that includes the study of limits, derivatives, integrals, and infinite series, and constitutes a major part of modern university education. ...

lim_{ntoinfty}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 ex at x=1.


Still other less common representations are also available. For instance, e can be represented as an infinite simple continued fraction: In mathematics, a continued fraction is an expression such as where a0 is some integer and all the other numbers an are positive integers. ...

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): The On-Line Encyclopedia of Integer Sequences (OEIS) is an extensive searchable database of integer sequences, freely available on the Web. ...

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

Many other series, sequence, continued fraction, and infinte product representations of e have also been developed.


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.[7][8]

Number of known decimal digits of e
Date Decimal digits Computation performed by
April 27 2007 100,000,000,000 Shigeru Kondo & Steve Pagliarulo
September 18 2003 50,100,000,000 Shigeru Kondo & Xavier Gourdon
August 21 2003 25,100,000,000 Shigeru Kondo & Xavier Gourdon
August 16 2000 12,884,901,000 Shigeru Kondo & Xavier Gourdon
August 2000 6,442,450,944 Shigeru Kondo & Xavier Gourdon
July 16 2000 3,221,225,472 Colin Martin & Xavier Gourdon
July 11 2000 2,147,483,648 Shigeru Kondo & Xavier Gourdon
November 21 1999 1,250,000,000 Xavier Gourdon
October 1999 869,894,101 Sebastian Wedeniwski
February 1999 200,000,579 Sebastian Wedeniwski
September 1997 50,000,817 Patrick Demichel
August 1997 20,000,000 Birger Seifert
May 1997 18,199,978 Patrick Demichel
1994 10,000,000 Robert Nemiroff & Jerry Bonnell
1961 100,265 Daniel Shanks & John W. Wrench
1949 2,010 John von Neumann (on the ENIAC)
1946 808  ?
1884 346 J. M. Boorman
1871 205 William Shanks
1853 137 William Shanks
1748 18[9] Leonhard Euler

John von Neumann (Hungarian Margittai Neumann János Lajos) (born December 28, 1903 in Budapest, Austria-Hungary; died February 8, 1957 in Washington D.C., United States) was a Hungarian-born American mathematician who made contributions to quantum physics, functional analysis, set theory, topology, economics, computer science, numerical analysis, hydrodynamics... ENIAC ENIAC, short for Electronic Numerical Integrator And Computer,[1] was the first large-scale, electronic, digital computer capable of being reprogrammed to solve a full range of computing problems,[2] although earlier computers had been built with some of these properties. ... William Shanks (January 25, 1812 -- 1882 in Houghton-le-Spring, Durham, England) was a British amateur mathematician. ... Leonhard Paul Euler (pronounced Oiler; IPA ) (April 15, 1707 – September 18 [O.S. September 7] 1783) was a pioneering Swiss mathematician and physicist, who spent most of his life in Russia and Germany. ...

e in computer culture

In contemporary internet culture, individuals and organizations frequently pay homage to the number e. This article is about the Internet An internet is a more general term for any set of interconnected computer networks that are connected by internetworking Graphic representation of the WWW information network structure around Wikipedia, as represented by hyperlinks The Internet, or simply the Net, is the publicly available worldwide...


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. Google was also responsible for a mysterious billboard[10] that appeared in the heart of Silicon Valley, and later in Cambridge, Massachusetts; Seattle, Washington; and Austin, Texas. It read {first 10-digit prime found in consecutive digits of e}.com. 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.[11] The first 10-digit prime in e is 7427466391, which starts at the 101st digit.[12] (A random stream of digits has a 98.4% chance of starting a 10-digit prime sooner.) Wikipedia does not yet have an article with this exact name. ... Google Inc. ... ISO 4217 Code USD User(s) the United States, the British Indian Ocean Territory,[1] the British Virgin Islands, East Timor, Ecuador, El Salvador, the Marshall Islands, Micronesia, Palau, Panama, Caicos Islands, and the insular areas of the United States Inflation 2. ... A view of downtown San Jose, the self-proclaimed Capital of Silicon Valley. ... Location in Massachusetts Coordinates: , Country United States State Massachusetts County Middlesex County Settled 1630 Incorporated 1636 Government  - Type Mayor-council city  - Mayor Kenneth Reeves (D) Area  - City  7. ... “Seattle” redirects here. ... Nickname: Location in the state of Texas Coordinates: , Country United States State Texas Counties Travis County Government  - Mayor Will Wynn Area  - City  296. ... Google Labs is a website demonstrating new Google projects that arent quite ready for prime time. It serves as a testing ground for new services being developed. ...


In another instance, the eminent computer scientist Donald Knuth let the version numbers of his program METAFONT approach e. The versions are 2, 2.7, 2.71, 2.718, and so forth. Computer science (informally: CS or compsci) is, in its most general sense, the study of computation and information processing, both in hardware and in software. ... Donald Ervin Knuth ( or Ka-NOOTH[1], Chinese: [2]) (b. ... METAFONT is a programming language used to define vector fonts. ...


Notes

  1. ^ Howard Whitley Eves (1969). An Introduction to the History of Mathematics. Holt, Rinehart & Winston. 
  2. ^ O'Connor, J.J., and Roberson, E.F.; The MacTutor History of Mathematics archive: "The number e"; University of St Andrews Scotland (2001)
  3. ^ Grinstead, C.M. and Snell, J.L. Introduction to probability theory (published online under the GFDL), p. 85.
  4. ^ Knuth (1997) The Art of Computer Programming Volume I, Addison-Wesley, p. 183.
  5. ^ See, for instance, Kline, M. (1998) Calculus: An intuitive and physical approach, Dover, section 12.3 "The Derived Functions of Logarithmic Functions."
  6. ^ This is the approach taken by Klein (1998).
  7. ^ Sebah, P. and Gourdon, X.; The constant e and its computation
  8. ^ Gourdon, X.; Reported large computations with PiFast
  9. ^ New Scientist 21st July 2007 p.40
  10. ^ Archive copy at the Internet Archive Wayback Machine
  11. ^ Shea, Andrea. "Google Entices Job-Searchers with Math Puzzle", NPR. Retrieved on 2007-06-09. 
  12. ^ Kazmierczak, Marcus (2004-07-29). Math : Google Labs Problems. mkaz.com. Retrieved on 2007-06-09.

GFDL redirects here. ... Cover of books The Art of Computer Programming[1] is a comprehensive monograph written by Donald Knuth which covers many kinds of programming algorithms and their analysis. ... The logo of Internet Archive The Internet Archive (IA) is a non-profit organization dedicated to maintaining an on-line library and archive of Web and multimedia resources. ... Year 2007 (MMVII) is now the current year, a common year starting on Monday of the Gregorian calendar and the AD/CE era. ... June 9 is the 160th day of the year in the Gregorian calendar (161st in leap years), with 205 days remaining. ... shelby was here 2004 (MMIV) was a leap year starting on Thursday of the Gregorian calendar. ... is the 210th day of the year (211th in leap years) in the Gregorian calendar. ... Year 2007 (MMVII) is now the current year, a common year starting on Monday of the Gregorian calendar and the AD/CE era. ... June 9 is the 160th day of the year in the Gregorian calendar (161st in leap years), with 205 days remaining. ...

References

External links


  Results from FactBites:
 
The Constant e (338 words)
"e" is a numerical constant that is equal to 2.71828.
The value of "e" is found in many mathematical formulas such as those describing a nonlinear increase or decrease such as growth or decay (including compound interest), the statistical "bell curve," the shape of a hanging cable or a standing arch.
The number e was first studied by the Swiss mathematician Leonhard Euler in the 1720s, although its existence was more or less implied in the work of John Napier, the inventor of logarithms, in 1614.
Article about "E" in the English Wikipedia on 24-Jul-2004 (502 words)
In biochemistry, E is the symbol for glutamic acid and also often an abbreviation for enzyme.
In computing, the letter e (with or without a subsequent hyphen) is often used as a prefix for other words to imply "electronic," such as e-mail or e-commerce.
E is often used as a digit meaning fourteen in hexadecimal and other positional numeral systems with a radix of 15 or greater.
  More results at FactBites »

 

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