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Template:Pp-move-indef Template:Infobox mathematical constant
Template:E (mathematical constant)
The number Template:Mvar is a mathematical constant approximately equal to 2.71828 that is the base of the natural logarithm and exponential function. It is sometimes called Euler's number, after the Swiss mathematician Leonhard Euler, though this can invite confusion with Euler numbers, or with Euler's constant, a different constant typically denoted <math>\gamma</math>. Alternatively, Template:Mvar can be called Napier's constant after John Napier.<ref name="Miller"/><ref name="Weisstein">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> The Swiss mathematician Jacob Bernoulli discovered the constant while studying compound interest.<ref name="Pickover">Template:Cite book Extract of page 166</ref><ref name="OConnor">Template:MacTutor</ref>
The number Template:Mvar is of great importance in mathematics,<ref>Template:Cite book</ref> alongside 0, 1, [[Pi|Template:Pi]], and Template:Mvar. All five appear in one formulation of Euler's identity <math>e^{i\pi}+1=0</math> and play important and recurring roles across mathematics.<ref>Template:Cite book</ref><ref>Template:Cite book</ref> Like the constant Template:Pi, Template:Mvar is irrational, meaning that it cannot be represented as a ratio of integers, and moreover it is transcendental, meaning that it is not a root of any non-zero polynomial with rational coefficients.<ref name="Weisstein" /> To 30 decimal places, the value of Template:Mvar is:<ref name="OEIS decimal expansion">Template:Cite OEIS</ref> Template:Block indent
DefinitionsEdit
The number Template:Mvar is the limit <math display= block>\lim_{n\to \infty}\left(1+\frac 1n\right)^n,</math> an expression that arises in the computation of compound interest.
It is the sum of the infinite series <math display ="block">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.</math>
It is the unique positive number Template:Mvar such that the graph of the function Template:Math has a slope of 1 at Template:Math.
One has <math display=block>e=\exp(1),</math> where <math>\exp</math> is the (natural) exponential function, the unique function that equals its own derivative and satisfies the equation <math>\exp(0)=1.</math> Since the exponential function is commonly denoted as <math>x\mapsto e^x,</math> one has also <math display=block>e=e^1.</math>
The logarithm of base Template:Mvar can be defined as the inverse function of the function <math>x\mapsto b^x.</math> Since <math>b=b^1,</math> one has <math>\log_b b= 1.</math> The equation <math>e=e^1</math> implies therefore that Template:Mvar is the base of the natural logarithm.
The number Template:Mvar can also be characterized in terms of an integral:<ref>Template:Dlmf</ref> <math display = block>\int_1^e \frac {dx}x =1.</math>
For other characterizations, see Template:Slink.
HistoryEdit
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 <math>e</math>. It is assumed that the table was written by William Oughtred. In 1661, Christiaan Huygens studied how to compute logarithms by geometrical methods and calculated a quantity that, in retrospect, is the base-10 logarithm of Template:Mvar, but he did not recognize Template:Mvar itself as a quantity of interest.<ref name="OConnor" /><ref>Template:Cite journal</ref>
The constant itself was introduced by Jacob Bernoulli in 1683, for solving the problem of continuous compounding of interest.<ref name="Bernoulli, 1690">Jacob Bernoulli considered the problem of continuous compounding of interest, which led to a series expression for Template:Mvar. 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. 219–23. 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 [a] sum of money [at] interest, let it accumulate, so that [at] every moment [it] were to receive [a] proportional part of [its] annual interest; how much would be owing [at the] end of [the] year?) Bernoulli constructs a power series to calculate the answer, and then writes: " … quæ nostra serie [mathematical expression for a geometric series] &c. major est. … si Template:Math, debebitur plu quam Template:Math & minus quam Template:Math." ( … which our series [a geometric series] is larger [than]. … if Template:Math, [the lender] will be owed more than Template:Math and less than Template:Math.) If Template:Math, the geometric series reduces to the series for Template:Math, so Template:Math. (** 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.)</ref><ref>Template:Cite book</ref> In his solution, the constant Template:Mvar occurs as the limit <math display="block">\lim_{n\to \infty} \left( 1 + \frac{1}{n} \right)^n,</math> where Template:Mvar represents the number of intervals in a year on which the compound interest is evaluated (for example, <math>n=12</math> for monthly compounding).
The first symbol used for this constant was the letter Template:Mvar by Gottfried Leibniz in letters to Christiaan Huygens in 1690 and 1691.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>
Leonhard Euler started to use the letter Template:Mvar for the constant in 1727 or 1728, in an unpublished paper on explosive forces in cannons,<ref name="Meditatio">Euler, Meditatio in experimenta explosione tormentorum nuper instituta. {{#invoke:Lang|lang}} (English: Written for the number of which the logarithm has the unit, e, that is 2,7182817...")</ref> and in a letter to Christian Goldbach on 25 November 1731.<ref>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 p. 58. From p. 58: " … ( e denotat hic numerum, cujus logarithmus hyperbolicus est = 1), … " ( … (e denotes that number whose hyperbolic [i.e., natural] logarithm is equal to 1) … )</ref><ref>Template:Cite book</ref> The first appearance of Template:Mvar in a printed publication was in Euler's Mechanica (1736).<ref>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 <math>\frac{dc}{c} = \frac{dy ds}{rdx}</math> seu <math>c = e^{\int\frac{dy ds}{rdx}}</math> ubi Template:Mvar denotat numerum, cuius logarithmus hyperbolicus est 1. (So it [i.e., Template:Mvar, the speed] will be <math>\frac{dc}{c} = \frac{dy ds}{rdx}</math> or <math>c = e^{\int\frac{dy ds}{rdx}}</math>, where Template:Mvar denotes the number whose hyperbolic [i.e., natural] logarithm is 1.)</ref> It is unknown why Euler chose the letter Template:Mvar.<ref>Template:Cite book p. 124.</ref> Although some researchers used the letter Template:Mvar in the subsequent years, the letter Template:Mvar was more common and eventually became standard.<ref name="Miller">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>
Euler proved that Template:Mvar is the sum of the infinite series <math display="block">e = \sum_{n = 0}^\infty \frac{1}{n!} = \frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + \cdots ,</math> where Template:Math is the factorial of Template:Mvar.<ref name="OConnor" /> The equivalence of the two characterizations using the limit and the infinite series can be proved via the binomial theorem.<ref>Template:Cite book</ref>
ApplicationsEdit
Compound interestEdit
Jacob Bernoulli discovered this constant in 1683, while studying a question about compound interest:<ref name="OConnor" />
<templatestyles src="Template:Blockquote/styles.css" />
An account starts with $1.00 and pays 100 percent interest per year. If the interest is credited once, at the end of the year, the value of the account at year-end will be $2.00. What happens if the interest is computed and credited more frequently during the year?{{#if:|{{#if:|}}
— {{#if:|, in }}Template:Comma separated entries}}
{{#invoke:Check for unknown parameters|check|unknown=Template:Main other|preview=Page using Template:Blockquote with unknown parameter "_VALUE_"|ignoreblank=y| 1 | 2 | 3 | 4 | 5 | author | by | char | character | cite | class | content | multiline | personquoted | publication | quote | quotesource | quotetext | sign | source | style | text | title | ts }}
If the interest is credited twice in the year, the interest rate for each 6 months will be 50%, so the initial $1 is multiplied by 1.5 twice, yielding Template:Nowrap at the end of the year. Compounding quarterly yields Template:Nowrap, and compounding monthly yields Template:Nowrap. If there are Template:Mvar compounding intervals, the interest for each interval will be Template:Math and the value at the end of the year will be $1.00 × Template:Math.<ref name="Gonick"/><ref name=":0" />
Bernoulli noticed that this sequence approaches a limit (the force of interest) with larger Template:Mvar and, thus, smaller compounding intervals.<ref name="OConnor" /> Compounding weekly (Template:Math) yields $2.692596..., while compounding daily (Template:Math) yields $2.714567... (approximately two cents more). The limit as Template:Mvar grows large is the number that came to be known as Template:Mvar. That is, with continuous compounding, the account value will reach $2.718281828... More generally, an account that starts at $1 and offers an annual interest rate of Template:Mvar will, after Template:Mvar years, yield Template:Math dollars with continuous compounding. Here, Template:Mvar is the decimal equivalent of the rate of interest expressed as a percentage, so for 5% interest, Template:Math.<ref name="Gonick">Template:Cite book</ref><ref name=":0" />
Bernoulli trialsEdit
The number Template:Mvar itself also has applications in probability theory, in a way that is not obviously related to exponential growth. Suppose that a gambler plays a slot machine that pays out with a probability of one in Template:Mvar and plays it Template:Mvar times. As Template:Mvar increases, the probability that gambler will lose all Template:Mvar bets approaches Template:Math, which is approximately 36.79%. For Template:Math, this is already 1/2.789509... (approximately 35.85%).
This is an example of a Bernoulli trial process. Each time the gambler plays the slots, there is a one in Template:Mvar chance of winning. Playing Template:Mvar times is modeled by the binomial distribution, which is closely related to the binomial theorem and Pascal's triangle. The probability of winning Template:Mvar times out of Template:Mvar trials is:<ref>Template:Cite book</ref>
- <math>\Pr[k~\mathrm{wins~of}~n] = \binom{n}{k} \left(\frac{1}{n}\right)^k\left(1 - \frac{1}{n}\right)^{n-k}.</math>
In particular, the probability of winning zero times (Template:Math) is
- <math>\Pr[0~\mathrm{wins~of}~n] = \left(1 - \frac{1}{n}\right)^{n}.</math>
The limit of the above expression, as Template:Mvar tends to infinity, is precisely Template:Math.
Exponential growth and decayEdit
Template:Further Exponential growth is a process that increases quantity over time at an ever-increasing rate. It occurs when the instantaneous rate of change (that is, the derivative) of a quantity with respect to time is proportional to the quantity itself.<ref name=":0">Template:Cite book</ref> Described as a function, a quantity undergoing exponential growth is an exponential function of time, that is, the variable representing time is the exponent (in contrast to other types of growth, such as quadratic growth). If the constant of proportionality is negative, then the quantity decreases over time, and is said to be undergoing exponential decay instead. The law of exponential growth can be written in different but mathematically equivalent forms, by using a different base, for which the number Template:Mvar is a common and convenient choice: <math display="block">x(t) = x_0\cdot e^{kt} = x_0\cdot e^{t/\tau}.</math> Here, <math>x_0</math> denotes the initial value of the quantity Template:Mvar, Template:Mvar is the growth constant, and <math>\tau</math> is the time it takes the quantity to grow by a factor of Template:Mvar.
Standard normal distributionEdit
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}}
The normal distribution with zero mean and unit standard deviation is known as the standard normal distribution,Template:R given by the probability density function <math display="block"> \phi(x) = \frac{1}{\sqrt{2\pi}} e^{-\frac{1}{2} x^2}. </math>
The constraint of unit standard deviation (and thus also unit variance) results in the Template:Frac2 in the exponent, and the constraint of unit total area under the curve <math>\phi(x)</math> results in the factor <math>\textstyle 1/\sqrt{2\pi}</math>. This function is symmetric around Template:Math, where it attains its maximum value <math>\textstyle 1/\sqrt{2\pi}</math>, and has inflection points at Template:Math.
DerangementsEdit
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Another application of Template:Mvar, also discovered in part by Jacob Bernoulli along with Pierre Remond de Montmort, is in the problem of derangements, also known as the hat check problem:<ref>Template:Cite book</ref> Template:Mvar guests are invited to a party and, at the door, the guests all check their hats with the butler, who in turn places the hats into Template:Mvar boxes, each labelled with the name of one guest. But the butler has not asked the identities of the guests, and so puts the hats into boxes selected at random. The problem of de Montmort is to find the probability that none of the hats gets put into the right box. This probability, denoted by <math>p_n\!</math>, is:
- <math>p_n = 1 - \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!} + \cdots + \frac{(-1)^n}{n!} = \sum_{k = 0}^n \frac{(-1)^k}{k!}.</math>
As Template:Mvar tends to infinity, Template:Math approaches Template:Math. Furthermore, the number of ways the hats can be placed into the boxes so that none of the hats are in the right box is Template:Math rounded to the nearest integer, for every positive Template:Mvar.<ref>Template:Cite book</ref>
Optimal planning problemsEdit
The maximum value of <math>\sqrt[x]{x}</math> occurs at <math>x = e</math>. Equivalently, for any value of the base Template:Math, it is the case that the maximum value of <math>x^{-1}\log_b x</math> occurs at <math>x = e</math> (Steiner's problem, discussed below).
This is useful in the problem of a stick of length Template:Mvar that is broken into Template:Mvar equal parts. The value of Template:Mvar that maximizes the product of the lengths is then either<ref name="Finch-2003-p14">Template:Cite book</ref>
- <math>n = \left\lfloor \frac{L}{e} \right\rfloor</math> or <math>\left\lceil \frac{L}{e} \right\rceil.</math>
The quantity <math>x^{-1}\log_b x</math> is also a measure of information gleaned from an event occurring with probability <math>1/x</math> (approximately <math>36.8\%</math> when <math>x=e</math>), so that essentially the same optimal division appears in optimal planning problems like the secretary problem.
AsymptoticsEdit
The number Template:Mvar occurs naturally in connection with many problems involving asymptotics. An example is Stirling's formula for the asymptotics of the factorial function, in which both the numbers Template:Mvar and [[pi|Template:Pi]] appear:<ref name="greg">Template:Cite book</ref> <math display="block>n! \sim \sqrt{2\pi n} \left(\frac{n}{e}\right)^n.</math>
As a consequence,<ref name="greg"/> <math display="block>e = \lim_{n\to\infty} \frac{n}{\sqrt[n]{n!}} .</math>
PropertiesEdit
CalculusEdit
The principal motivation for introducing the number Template:Mvar, particularly in calculus, is to perform differential and integral calculus with exponential functions and logarithms.<ref name="kline">Template:Cite book</ref> A general exponential Template:Nowrap has a derivative, given by a limit:
- <math>\begin{align}
\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 \cdot \left(\lim_{h\to 0}\frac{a^h - 1}{h}\right).
\end{align}</math>
The parenthesized limit on the right is independent of the Template:Nowrap Its value turns out to be the logarithm of Template:Mvar to base Template:Mvar. Thus, when the value of Template:Mvar is set Template:Nowrap this limit is equal Template:Nowrap and so one arrives at the following simple identity:
- <math>\frac{d}{dx}e^x = e^x.</math>
Consequently, the exponential function with base Template:Mvar is particularly suited to doing calculus. Template:Nowrap (as opposed to some other number) as the base of the exponential function makes calculations involving the derivatives much simpler.
Another motivation comes from considering the derivative of the base-Template:Mvar logarithm (i.e., Template:Math),Template:R for Template:Math:
- <math>\begin{align}
\frac{d}{dx}\log_a x
&= \lim_{h\to 0}\frac{\log_a(x + h) - \log_a(x)}{h} \\
&= \lim_{h\to 0}\frac{\log_a(1 + h/x)}{x\cdot h/x} \\
&= \frac{1}{x}\log_a\left(\lim_{u\to 0}(1 + u)^\frac{1}{u}\right) \\
&= \frac{1}{x}\log_a e,
\end{align}</math>
where the substitution Template:Math was made. The base-Template:Mvar logarithm of Template:Mvar is 1, if Template:Mvar equals Template:Mvar. So symbolically,
- <math>\frac{d}{dx}\log_e x = \frac{1}{x}.</math>
The logarithm with this special base is called the natural logarithm, and is usually denoted as Template:Math; it behaves well under differentiation since there is no undetermined limit to carry through the calculations.
Thus, there are two ways of selecting such special numbers Template:Mvar. One way is to set the derivative of the exponential function Template:Math equal to Template:Math, and solve for Template:Mvar. The other way is to set the derivative of the base Template:Mvar logarithm to Template:Math and solve for Template:Mvar. In each case, one arrives at a convenient choice of base for doing calculus. It turns out that these two solutions for Template:Mvar are actually the same: the number Template:Mvar.
The Taylor series for the exponential function can be deduced from the facts that the exponential function is its own derivative and that it equals 1 when evaluated at 0:<ref name="strangherman">Template:Cite book</ref> <math display="block">e^x = \sum_{n=0}^\infty \frac{x^n}{n!}.</math> Setting <math>x = 1</math> recovers the definition of Template:Mvar as the sum of an infinite series.
The natural logarithm function can be defined as the integral from 1 to <math>x</math> of <math>1/t</math>, and the exponential function can then be defined as the inverse function of the natural logarithm. The number Template:Mvar is the value of the exponential function evaluated at <math>x = 1</math>, or equivalently, the number whose natural logarithm is 1. It follows that Template:Mvar is the unique positive real number such that <math display="block">\int_1^e \frac{1}{t} \, dt = 1.</math>
Because Template:Math is the unique function (up to multiplication by a constant Template:Mvar) that is equal to its own derivative,
<math display="block">\frac{d}{dx}Ke^x = Ke^x,</math>
it is therefore its own antiderivative as well:<ref>Template:Cite book</ref>
<math display="block">\int Ke^x\,dx = Ke^x + C .</math>
Equivalently, the family of functions
<math display="block">y(x) = Ke^x</math>
where Template:Mvar is any real or complex number, is the full solution to the differential equation
<math display="block">y' = y .</math>
InequalitiesEdit
The number Template:Mvar is the unique real number such that <math display="block">\left(1 + \frac{1}{x}\right)^x < e < \left(1 + \frac{1}{x}\right)^{x+1}</math> for all positive Template:Mvar.<ref>Template:Cite book</ref>
Also, we have the inequality <math display="block">e^x \ge x + 1</math> for all real Template:Mvar, with equality if and only if Template:Math. Furthermore, Template:Mvar is the unique base of the exponential for which the inequality Template:Math holds for all Template:Mvar.<ref>A standard calculus exercise using the mean value theorem; see for example Apostol (1967) Calculus, § 6.17.41.</ref> This is a limiting case of Bernoulli's inequality.
Exponential-like functionsEdit
Steiner's problem asks to find the global maximum for the function
<math display="block"> f(x) = x^\frac{1}{x} .</math>
This maximum occurs precisely at Template:Math. (One can check that the derivative of Template:Math is zero only for this value of Template:Mvar.)
Similarly, Template:Math is where the global minimum occurs for the function
<math display="block"> f(x) = x^x .</math>
The infinite tetration
- <math> x^{x^{x^{\cdot^{\cdot^{\cdot}}}}} </math> or <math>{^\infty}x</math>
converges if and only if Template:Math,<ref>Template:Cite OEIS</ref><ref>Template:Cite OEIS</ref> shown by a theorem of Leonhard Euler.<ref>Euler, L. "De serie Lambertina Plurimisque eius insignibus proprietatibus." Acta Acad. Scient. Petropol. 2, 29–51, 1783. Reprinted in Euler, L. Opera Omnia, Series Prima, Vol. 6: Commentationes Algebraicae. Leipzig, Germany: Teubner, pp. 350–369, 1921. (facsimile)</ref><ref>Template:Cite journal</ref><ref>Template:Cite journal</ref>
Number theoryEdit
The real number Template:Mvar is irrational. Euler proved this by showing that its simple continued fraction expansion does not terminate.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> (See also Fourier's [[proof that e is irrational|proof that Template:Mvar is irrational]].)
Furthermore, by the Lindemann–Weierstrass theorem, Template:Mvar is transcendental, meaning that it is not a solution of any non-zero polynomial equation with rational coefficients. 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.<ref>Template:Cite book</ref> The number Template:Mvar is one of only a few transcendental numbers for which the exact irrationality exponent is known (given by <math>\mu(e)=2</math>).<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>
An unsolved problem thus far is the question of whether or not the numbers Template:Mvar and Template:Mvar are algebraically independent. This would be resolved by Schanuel's conjecture – a currently unproven generalization of the Lindemann–Weierstrass theorem.<ref>Template:Cite book</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>
It is conjectured that Template:Mvar is normal, meaning that when Template:Mvar is expressed in any base the possible digits in that base are uniformly distributed (occur with equal probability in any sequence of given length).<ref>Template:Cite book</ref>
In algebraic geometry, a period is a number that can be expressed as an integral of an algebraic function over an algebraic domain. The constant Template:Pi is a period, but it is conjectured that Template:Mvar is not.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>
Complex numbersEdit
The exponential function Template:Math may be written as a Taylor series<ref>Template:Cite book</ref><ref name="strangherman"></ref>
<math display="block"> e^{x} = 1 + {x \over 1!} + {x^{2} \over 2!} + {x^{3} \over 3!} + \cdots = \sum_{n=0}^{\infty} \frac{x^n}{n!}.</math>
Because this series is convergent for every complex value of Template:Mvar, it is commonly used to extend the definition of Template:Math to the complex numbers.<ref name="Dennery">Template:Cite book</ref> This, with the Taylor series for [[trigonometric functions|Template:Math and Template:Math]], allows one to derive Euler's formula:
<math display="block">e^{ix} = \cos x + i\sin x ,</math>
which holds for every complex Template:Mvar.<ref name="Dennery"/> The special case with Template:Math is Euler's identity:
<math display="block">e^{i\pi} + 1 = 0 ,</math> which is considered to be an exemplar of mathematical beauty as it shows a profound connection between the most fundamental numbers in mathematics. In addition, it is directly used in a proof that Template:Pi is transcendental, which implies the impossibility of squaring the circle.<ref>Template:Cite arXiv</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> Moreover, the identity implies that, in the principal branch of the logarithm,<ref name="Dennery"/>
<math display="block">\ln (-1) = i\pi .</math>
Furthermore, using the laws for exponentiation,
<math display="block">(\cos x + i\sin x)^n = \left(e^{ix}\right)^n = e^{inx} = \cos nx + i \sin nx</math>
for any integer Template:Mvar, which is de Moivre's formula.<ref name="Sultan"/>
The expressions of Template:Math and Template:Math in terms of the exponential function can be deduced from the Taylor series:<ref name="Dennery"/> <math display="block">
\cos x = \frac{e^{ix} + e^{-ix}}{2} , \qquad
\sin x = \frac{e^{ix} - e^{-ix}}{2i}.
</math>
The expression <math display=inline>\cos x + i \sin x</math> is sometimes abbreviated as Template:Math.<ref name="Sultan">Template:Cite book</ref>
RepresentationsEdit
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}}
The number Template:Mvar can be represented in a variety of ways: as an infinite series, an infinite product, a continued fraction, or a limit of a sequence. In addition to the limit and the series given above, there is also the simple continued fraction
- <math>
e = [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, ..., 1, 2n, 1, ...],
</math><ref>Template:Cite book</ref><ref name="OEIS continued fraction">Template:Cite OEIS</ref>
which written out looks like
- <math>e = 2 +
\cfrac{1}
{1 + \cfrac{1}
{2 + \cfrac{1}
{1 + \cfrac{1}
{1 + \cfrac{1}
{4 + \cfrac{1}
{1 + \cfrac{1}
{1 + \ddots}
}
}
}
}
}
}
. </math>
The following infinite product evaluates to Template:Mvar:<ref name="Finch-2003-p14"/> <math display="block">e = \frac{2}{1} \left(\frac{4}{3}\right)^{1/2} \left(\frac{6 \cdot 8}{5 \cdot 7}\right)^{1/4} \left(\frac{10 \cdot 12 \cdot 14 \cdot 16}{9 \cdot 11 \cdot 13 \cdot 15}\right)^{1/8} \cdots.</math>
Many other series, sequence, continued fraction, and infinite product representations of Template:Mvar have been proved.
Stochastic representationsEdit
In addition to exact analytical expressions for representation of Template:Mvar, there are stochastic techniques for estimating Template:Mvar. One such approach begins with an infinite sequence of independent random variables Template:Math, Template:Math..., drawn from the uniform distribution on [0, 1]. Let Template:Mvar be the least number Template:Mvar such that the sum of the first Template:Mvar observations exceeds 1:
- <math>V = \min\left\{ n \mid X_1 + X_2 + \cdots + X_n > 1 \right\}.</math>
Then the expected value of Template:Mvar is Template:Mvar: Template:Math.<ref>Template:Cite journal</ref><ref>Dinov, ID (2007) Estimating e using SOCR simulation, SOCR Hands-on Activities (retrieved December 26, 2007).</ref>
Known digitsEdit
The number of known digits of Template:Mvar has increased substantially since the introduction of the computer, due both to increasing performance of computers and to algorithmic improvements.<ref>Sebah, P. and Gourdon, X.; The constant Template:Mvar and its computation</ref><ref>Gourdon, X.; Reported large computations with PiFast</ref>
| Date | Decimal digits | Computation performed by |
|---|---|---|
| 1690 | 1 | Jacob Bernoulli<ref name="Bernoulli, 1690" /> |
| 1714 | 13 | Roger Cotes<ref>Roger Cotes (1714) "Logometria," Philosophical Transactions of the Royal Society of London, 29 (338) : 5–45; see especially the bottom of page 10. From page 10: "Porro eadem ratio est inter 2,718281828459 &c et 1, … " (Furthermore, by the same means, the ratio is between 2.718281828459… and 1, … )</ref> |
| 1748 | 23 | Leonhard Euler<ref>Leonhard Euler, Introductio in Analysin Infinitorum (Lausanne, Switzerland: Marc Michel Bousquet & Co., 1748), volume 1, page 90.</ref> |
| 1853 | 137 | William Shanks<ref>William Shanks, Contributions to Mathematics, ... (London, England: G. Bell, 1853), page 89.</ref> |
| 1871 | 205 | William Shanks<ref>William Shanks (1871) "On the numerical values of Template:Mvar, Template:Math, Template:Math, Template:Math, and Template:Math, also on the numerical value of Template:Mvar the modulus of the common system of logarithms, all to 205 decimals," Proceedings of the Royal Society of London, 20 : 27–29.</ref> |
| 1884 | 346 | J. Marcus Boorman<ref>J. Marcus Boorman (October 1884) "Computation of the Naperian base," Mathematical Magazine, 1 (12) : 204–205.</ref> |
| 1949 | 2,010 | John von Neumann (on the ENIAC) |
| 1961 | 100,265 | Daniel Shanks and John Wrench<ref name="We have computed e on a 7090 to 100,265D by the obvious program.">Template:Cite journal</ref> |
| 1978 | 116,000 | Steve Wozniak on the Apple II<ref name="wozniak198106">Template:Cite magazine</ref> |
Since around 2010, the proliferation of modern high-speed desktop computers has made it feasible for amateurs to compute trillions of digits of Template:Mvar within acceptable amounts of time. On Dec 24, 2023, a record-setting calculation was made by Jordan Ranous, giving Template:Mvar to 35,000,000,000,000 digits.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>
Computing the digitsEdit
One way to compute the digits of Template:Mvar is with the series<ref name="Finch-2005">Template:Cite book</ref> <math display=block>e=\sum_{k=0}^\infty \frac{1}{k!}.</math>
A faster method involves two recursive functions <math>p(a,b)</math> and <math>q(a,b)</math>. The functions are defined as <math display=block>\binom{p(a,b)}{q(a,b)}= \begin{cases} \binom{1}{b}, & \text{if }b=a+1\text{,} \\ \binom{p(a,m)q(m,b)+p(m,b)}{q(a,m)q(m,b)}, & \text{otherwise, where }m=\lfloor(a+b)/2\rfloor .\end{cases}</math>
The expression <math display=block>1+\frac{p(0,n)}{q(0,n)}</math> produces the Template:Mvarth partial sum of the series above. This method uses binary splitting to compute Template:Mvar with fewer single-digit arithmetic operations and thus reduced bit complexity. Combining this with fast Fourier transform-based methods of multiplying integers makes computing the digits very fast.<ref name="Finch-2005" />
In computer cultureEdit
During the emergence of internet culture, individuals and organizations sometimes paid homage to the number Template:Mvar.
In an early example, the computer scientist Donald Knuth let the version numbers of his program Metafont approach Template:Mvar. The versions are 2, 2.7, 2.71, 2.718, and so forth.<ref>Template:Cite journal</ref>
In another instance, 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 USD, which is Template:Mvar billion dollars rounded to the nearest dollar.<ref>Template:Cite journal</ref>
Google was also responsible for a billboard<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> 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 Template:Mvar}.com". The first 10-digit prime in Template:Mvar is 7427466391, which starts at the 99th digit.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> Solving this problem and visiting the advertised (now defunct) website led to an even more difficult problem to solve, which consisted of finding the fifth term in the sequence 7182818284, 8182845904, 8747135266, 7427466391. It turned out that the sequence consisted of 10-digit numbers found in consecutive digits of Template:Mvar whose digits summed to 49. The fifth term in the sequence is 5966290435, which starts at the 127th digit.<ref>The first 10-digit prime in Template:Math Template:Webarchive. Explore Portland Community. Retrieved on 2020-12-09.</ref> Solving this second problem finally led to a Google Labs webpage where the visitor was invited to submit a résumé.<ref>Template:Cite news</ref>
The last release of the official Python 2 interpreter has version number 2.7.18, a reference to e.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>
ReferencesEdit
Further readingEdit
- Template:Cite book
- Commentary on Endnote 10 of the book Prime Obsession for another stochastic representation
- Template:Cite journal
External linksEdit
Template:Sister project Template:Sister project
- The number Template:Mvar to 1 million places and NASA.gov 2 and 5 million places
- Template:Mvar Approximations – Wolfram MathWorld
- Earliest Uses of Symbols for Constants Jan. 13, 2008
- "The story of Template:Mvar", by Robin Wilson at Gresham College, 28 February 2007 (available for audio and video download)
- Template:Mvar Search Engine 2 billion searchable digits of Template:Mvar, Template:Pi and Template:Radic
Template:Irrational number Template:Authority control Template:Good article