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e (mathematical constant)

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From Wikipedia, the free encyclopedia
2.71828…, base of natural logarithms
"Euler's number" redirects here. For other Euler's numbers, seeList of topics named after Leonhard Euler § Numbers.
For information on the similar looking E notation, seeScientific notation.
Constant value used in mathematics
Euler's number
e
2.71828...[1]
General information
TypeTranscendental
History
Discovered1685
ByJacob Bernoulli
First mentionQuæstiones nonnullæ de usuris, cum solutione problematis de sorte alearum, propositi in Ephem. Gall. A. 1685
Named after
Graph of the equationy = 1/x. Here,e is the unique number larger than 1 that makes the shadedarea under the curve equal to 1.
Part ofa series of articles on the
mathematical constante
Properties
Applications
Defininge
People
Related topics

The numbere is amathematical constant, approximately equal to 2.71828, that is thebase of thenatural logarithm andexponential function. It is sometimes calledEuler's number, after the Swiss mathematicianLeonhard Euler, though this can invite confusion withEuler numbers, or withEuler's constant, a different constant typically denotedγ{\displaystyle \gamma }. Alternatively,e can be calledNapier's constant afterJohn Napier.[2][3] The Swiss mathematicianJacob Bernoulli discovered the constant while studyingcompound interest.[4][5]

The numbere is of great importance in mathematics,[6] alongside 0, 1,π, andi. All five appear in one formulation ofEuler's identityeiπ+1=0{\displaystyle e^{i\pi }+1=0} and play important and recurring roles across mathematics.[7][8]e isirrational, meaning that it cannot be represented as a ratio of integers. Moreover, like the constantπ, it istranscendental, meaning that it is not a root of any non-zeropolynomial with rational coefficients.[3] To 30 decimal places, the value ofe is:[1]

2.718281828459045235360287471352

Definitions

[edit]

The numbere is thelimitlimn→∞(1+1n)n,{\displaystyle \lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)^{n},}an expression that arises in the computation ofcompound interest.

It is the sum of the infiniteseriese=∑n=0∞1n!=1+11+11⋅2+11⋅2⋅3+⋯.{\displaystyle e=\sum \limits _{n=0}^{\infty }{\frac {1}{n!}}=1+{\frac {1}{1}}+{\frac {1}{1\cdot 2}}+{\frac {1}{1\cdot 2\cdot 3}}+\cdots .}

It is the unique positive numbera such that the graph of the functiony =ax has aslope of 1 atx = 0.

One hase=exp⁡(1),{\displaystyle e=\exp(1),} whereexp{\displaystyle \exp } is the (natural)exponential function, the unique function that equals its ownderivative and satisfies the equationexp⁡(0)=1.{\displaystyle \exp(0)=1.} Therefore,e is also the base of thenatural logarithm, the inverse of the natural exponential function.

The numbere can also be characterized in terms of anintegral:[9]∫1edxx=1.{\displaystyle \int _{1}^{e}{\frac {dx}{x}}=1.}

For other characterizations, see§ Representations.

History

[edit]

The first references to this constant were published in 1618 in the table of an appendix of a work on logarithms byJohn Napier. However, this did not contain the constant itself, but simply a list oflogarithms to the basee{\displaystyle e}. It is assumed that the table was written byWilliam 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 ofe, but he did not recognizee itself as a quantity of interest.[5][10]

The constant itself was introduced byJacob Bernoulli in 1683, for solving the problem ofcontinuous compounding of interest.[11][12]In his solution, the constante occurs as thelimitlimn→∞(1+1n)n,{\displaystyle \lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)^{n},}wheren represents the number of intervals in a year on which the compound interest is evaluated (for example,n=12{\displaystyle n=12} for monthly compounding).

The first symbol used for this constant was the letterb byGottfried Leibniz in letters to Christiaan Huygens in 1690 and 1691.[13]

Leonhard Euler started to use the lettere for the constant in 1727 or 1728, in an unpublished paper on explosive forces in cannons,[14] and in a letter toChristian Goldbach on 25 November 1731.[15][16] The first appearance ofe in a printed publication was in Euler'sMechanica (1736).[17] It is unknown why Euler chose the lettere.[18] Although some researchers used the letterc in the subsequent years, the lettere was more common and eventually became standard.[2]

Euler proved thate is the sum of theinfinite seriese=∑n=0∞1n!=10!+11!+12!+13!+14!+⋯,{\displaystyle e=\sum _{n=0}^{\infty }{\frac {1}{n!}}={\frac {1}{0!}}+{\frac {1}{1!}}+{\frac {1}{2!}}+{\frac {1}{3!}}+{\frac {1}{4!}}+\cdots ,}wheren! is thefactorial ofn.[5] The equivalence of the two characterizations using the limit and the infinite series can be proved via thebinomial theorem.[19]

Applications

[edit]

Compound interest

[edit]
The effect of earning 20% annual interest on aninitial $1,000 investment at various compounding frequencies. The limiting curve on top is the graphy=1000e0.2t{\displaystyle y=1000e^{0.2t}}, wherey is in dollars,t in years, and 0.2 = 20%.

Jacob Bernoulli discovered this constant in 1683, while studying a question aboutcompound interest:[5]

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 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$1.00 × 1.52 = $2.25 at the end of the year. Compounding quarterly yields$1.00 × 1.254 = $2.44140625, and compounding monthly yields$1.00 × (1 + 1/12)12 = $2.613035.... If there aren compounding intervals, the interest for each interval will be100%/n and the value at the end of the year will be $1.00 × (1 + 1/n)n.[20][21]

Bernoulli noticed that this sequence approaches a limit (theforce of interest) with largern and, thus, smaller compounding intervals.[5] Compounding weekly (n = 52) yields $2.692596..., while compounding daily (n = 365) yields $2.714567... (approximately two cents more). The limit asn grows large is the number that came to be known ase. That is, withcontinuous compounding, the account value will reach $2.718281828... More generally, an account that starts at $1 and offers an annual interest rate ofR will, aftert years, yieldeRt dollars with continuous compounding. Here,R is the decimal equivalent of the rate of interest expressed as apercentage, so for 5% interest,R = 5/100 = 0.05.[20][21]

Bernoulli trials

[edit]
Graphs of probabilityP ofnot observing independent events each of probability1/n aftern Bernoulli trials, and1 −P  vsn ; it can be observed that asn increases, the probability of a1/n-chance event never appearing aftern tries rapidlyconverges to1/e.

The numbere itself also has applications inprobability 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 inn and plays itn times. Asn increases, the probability that gambler will lose alln bets approaches1/e, which is approximately 36.79%. Forn = 20, this is already 1/2.789509... (approximately 35.85%).

This is an example of aBernoulli trial process. Each time the gambler plays the slots, there is a one inn chance of winning. Playingn times is modeled by thebinomial distribution, which is closely related to thebinomial theorem andPascal's triangle. The probability of winningk times out ofn trials is:[22]

Pr[k wins of n]=(nk)(1n)k(1−1n)n−k.{\displaystyle \Pr[k~\mathrm {wins~of} ~n]={\binom {n}{k}}\left({\frac {1}{n}}\right)^{k}\left(1-{\frac {1}{n}}\right)^{n-k}.}

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

Pr[0 wins of n]=(1−1n)n.{\displaystyle \Pr[0~\mathrm {wins~of} ~n]=\left(1-{\frac {1}{n}}\right)^{n}.}

The limit of the above expression, asn tends to infinity, is precisely1/e.

Exponential growth and decay

[edit]
Further information:Exponential growth andExponential decay

Exponential growth is a process that increases quantity over time at an ever-increasing rate. It occurs when the instantaneousrate of change (that is, thederivative) of a quantity with respect to time isproportional to the quantity itself.[21] Described as a function, a quantity undergoing exponential growth is anexponential function of time, that is, the variable representing time is the exponent (in contrast to other types of growth, such asquadratic growth). If the constant of proportionality is negative, then the quantity decreases over time, and is said to be undergoingexponential decay instead. The law of exponential growth can be written in different but mathematically equivalent forms, by using a differentbase, for which the numbere is a common and convenient choice:x(t)=x0⋅ekt=x0⋅et/τ.{\displaystyle x(t)=x_{0}\cdot e^{kt}=x_{0}\cdot e^{t/\tau }.}Here,x0{\displaystyle x_{0}} denotes the initial value of the quantityx,k is the growth constant, andτ{\displaystyle \tau } is the time it takes the quantity to grow by a factor ofe.

Standard normal distribution

[edit]
Main article:Normal distribution

The normal distribution with zero mean and unit standard deviation is known as thestandard normal distribution,[23] given by theprobability density functionϕ(x)=12πe−12x2.{\displaystyle \phi (x)={\frac {1}{\sqrt {2\pi }}}e^{-{\frac {1}{2}}x^{2}}.}

The constraint of unit standard deviation (and thus also unit variance) results in the⁠1/2⁠ in the exponent, and the constraint of unit total area under the curveϕ(x){\displaystyle \phi (x)} results in the factor1/2π{\displaystyle \textstyle 1/{\sqrt {2\pi }}}. This function is symmetric aroundx = 0, where it attains its maximum value1/2π{\displaystyle \textstyle 1/{\sqrt {2\pi }}}, and hasinflection points atx = ±1.

Derangements

[edit]
Main article:Derangement

Another application ofe, also discovered in part by Jacob Bernoulli along withPierre Remond de Montmort, is in the problem ofderangements, also known as thehat check problem:[24]n 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 inton 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 thatnone of the hats gets put into the right box. This probability, denoted bypn{\displaystyle p_{n}\!}, is:

pn=1−11!+12!−13!+⋯+(−1)nn!=∑k=0n(−1)kk!.{\displaystyle 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!}}.}

Asn tends to infinity,pn approaches1/e. Furthermore, the number of ways the hats can be placed into the boxes so that none of the hats are in the right box isn!/e,rounded to the nearest integer, for every positive n.[25]

Optimal planning problems

[edit]

The maximum value ofxx{\displaystyle {\sqrt[{x}]{x}}} occurs atx=e{\displaystyle x=e}. Equivalently, for any value of the baseb > 1, it is the case that the maximum value ofx−1logb⁡x{\displaystyle x^{-1}\log _{b}x} occurs atx=e{\displaystyle x=e} (Steiner's problem, discussedbelow).

This is useful in the problem of a stick of lengthL that is broken inton equal parts. The value ofn that maximizes the product of the lengths is then either[26]

n=⌊Le⌋{\displaystyle n=\left\lfloor {\frac {L}{e}}\right\rfloor } or⌈Le⌉.{\displaystyle \left\lceil {\frac {L}{e}}\right\rceil .}

The quantityx−1logb⁡x{\displaystyle x^{-1}\log _{b}x} is also a measure ofinformation gleaned from an event occurring with probability1/x{\displaystyle 1/x} (approximately36.8%{\displaystyle 36.8\%} whenx=e{\displaystyle x=e}), so that essentially the same optimal division appears in optimal planning problems like thesecretary problem.

Asymptotics

[edit]

The numbere occurs naturally in connection with many problems involvingasymptotics. An example isStirling's formula for theasymptotics of thefactorial function, in which both the numberse andπ appear:[27]n!∼2πn(ne)n.{\displaystyle n!\sim {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}.}

As a consequence,[27]e=limn→∞nn!n.{\displaystyle e=\lim _{n\to \infty }{\frac {n}{\sqrt[{n}]{n!}}}.}

Properties

[edit]

Calculus

[edit]
See also:Characterizations of the exponential function
The graphs of the functionsx ↦ax are shown fora = 2 (dotted),a =e (blue), anda = 4 (dashed). They all pass through the point(0,1), but the red line (which has slope1) is tangent to onlyex there.
The value of the natural log function for argumente, i.e.lne, equals1.

The principal motivation for introducing the numbere, particularly incalculus, is to performdifferential andintegral calculus withexponential functions andlogarithms.[28] A general exponentialfunctiony =ax has a derivative, given by alimit:

ddxax=limh→0ax+h−axh=limh→0axah−axh=ax⋅(limh→0ah−1h).{\displaystyle {\begin{aligned}{\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{aligned}}}

The parenthesized limit on the right is independent of thevariablex. Its value turns out to be the logarithm ofa to basee. Thus, when the value ofa is settoe, this limit is equalto1, and so one arrives at the following simple identity:

ddxex=ex.{\displaystyle {\frac {d}{dx}}e^{x}=e^{x}.}

Consequently, the exponential function with basee is particularly suited to doing calculus.Choosinge (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-a logarithm (i.e.,logax),[28] for x > 0:

ddxloga⁡x=limh→0loga⁡(x+h)−loga⁡(x)h=limh→0loga⁡(1+h/x)x⋅h/x=1xloga⁡(limu→0(1+u)1u)=1xloga⁡e,{\displaystyle {\begin{aligned}{\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{aligned}}}

where the substitutionu =h/x was made. The base-a logarithm ofe is 1, ifa equalse. So symbolically,

ddxloge⁡x=1x.{\displaystyle {\frac {d}{dx}}\log _{e}x={\frac {1}{x}}.}

The logarithm with this special base is called thenatural logarithm, and is usually denoted asln; 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 numbersa. One way is to set the derivative of the exponential functionax equal toax, and solve fora. The other way is to set the derivative of the basea logarithm to1/x and solve fora. In each case, one arrives at a convenient choice of base for doing calculus. It turns out that these two solutions fora are actuallythe same: the numbere.

The five colored regions are of equal area, and define units ofhyperbolic angle along thehyperbolaxy=1.{\displaystyle xy=1.}

TheTaylor 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:[29]ex=∑n=0∞xnn!.{\displaystyle e^{x}=\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}.}Settingx=1{\displaystyle x=1} recovers the definition ofe as the sum of an infinite series.

The natural logarithm function can be defined as the integral from 1 tox{\displaystyle x} of1/t{\displaystyle 1/t}, and the exponential function can then be defined as the inverse function of the natural logarithm. The numbere is the value of the exponential function evaluated atx=1{\displaystyle x=1}, or equivalently, the number whose natural logarithm is 1. It follows thate is the unique positivereal number such that∫1e1tdt=1.{\displaystyle \int _{1}^{e}{\frac {1}{t}}\,dt=1.}

Becauseex is the unique function (up to multiplication by a constantK) that is equal to its ownderivative,

ddxKex=Kex,{\displaystyle {\frac {d}{dx}}Ke^{x}=Ke^{x},}

it is therefore its ownantiderivative as well:[30]

∫Kexdx=Kex+C.{\displaystyle \int Ke^{x}\,dx=Ke^{x}+C.}

Equivalently, the family of functions

y(x)=Kex{\displaystyle y(x)=Ke^{x}}

whereK is any real or complex number, is the full solution to thedifferential equation

y′=y.{\displaystyle y'=y.}

Inequalities

[edit]
Exponential functionsy = 2x andy = 4x intersect the graph ofy =x + 1, respectively, atx = 1 andx = −1/2. The numbere is the unique base such thaty =ex intersects only atx = 0. We may infer thate lies between 2 and 4.

The numbere is the unique real number such that(1+1x)x<e<(1+1x)x+1{\displaystyle \left(1+{\frac {1}{x}}\right)^{x}<e<\left(1+{\frac {1}{x}}\right)^{x+1}}for all positivex.[31]

Also, we have the inequalityex≥x+1{\displaystyle e^{x}\geq x+1}for all realx, with equality if and only ifx = 0. Furthermore,e is the unique base of the exponential for which the inequalityax ≥x + 1 holds for allx.[32] This is a limiting case ofBernoulli's inequality.

Exponential-like functions

[edit]
Theglobal maximum ofx√xoccurs atx =e.

Steiner's problem asks to find theglobal maximum for the function

f(x)=x1x.{\displaystyle f(x)=x^{\frac {1}{x}}.}

This maximum occurs precisely atx =e. (One can check that the derivative oflnf(x) is zero only for this value of x.)

Similarly,x = 1/e is where theglobal minimum occurs for the function

f(x)=xx.{\displaystyle f(x)=x^{x}.}

The infinitetetration

xxx⋅⋅⋅{\displaystyle x^{x^{x^{\cdot ^{\cdot ^{\cdot }}}}}} or∞x{\displaystyle {^{\infty }}x}

converges if and only ifx ∈ [(1/e)e,e1/e] ≈ [0.06599, 1.4447],[33][34] shown by a theorem ofLeonhard Euler.[35][36][37]

Number theory

[edit]

The real numbere isirrational.Euler proved this by showing that itssimple continued fraction expansion does not terminate.[38] (See alsoFourier'sproof thate is irrational.)

Furthermore, by theLindemann–Weierstrass theorem,e istranscendental, 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 withLiouville number); the proof was given byCharles Hermite in 1873.[39] The numbere is one of only a few transcendental numbers for which the exactirrationality exponent is known (given byμ(e)=2{\displaystyle \mu (e)=2}).[40]

Anunsolved problem thus far is the question of whether or not the numberse andπ arealgebraically independent. This would be resolved bySchanuel's conjecture – a currently unproven generalization of the Lindemann–Weierstrass theorem.[41][42]

It is conjectured thate isnormal, meaning that whene is expressed in anybase the possible digits in that base are uniformly distributed (occur with equal probability in any sequence of given length).[43]

Inalgebraic geometry, aperiod is a number that can be expressed as an integral of analgebraic function over an algebraicdomain. The constantπ is a period, but it is conjectured thate is not.[44]

Complex numbers

[edit]

Theexponential functionex may be written as aTaylor series[45][29]

ex=1+x1!+x22!+x33!+⋯=∑n=0∞xnn!.{\displaystyle e^{x}=1+{x \over 1!}+{x^{2} \over 2!}+{x^{3} \over 3!}+\cdots =\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}.}

Because this series isconvergent for everycomplex value ofx, it is commonly used to extend the definition ofex to the complex numbers.[46] This, with the Taylor series forsin andcosx, allows one to deriveEuler's formula:

eix=cos⁡x+isin⁡x,{\displaystyle e^{ix}=\cos x+i\sin x,}

which holds for every complexx.[46] The special case withx =π isEuler's identity:

eiπ+1=0,{\displaystyle e^{i\pi }+1=0,}which is considered to be an exemplar ofmathematical beauty as it shows a profound connection between the most fundamental numbers in mathematics. In addition, it is directly used ina proof thatπ istranscendental, which implies the impossibility ofsquaring the circle.[47][48] Moreover, the identity implies that, in theprincipal branch of the logarithm,[46]

ln⁡(−1)=iπ.{\displaystyle \ln(-1)=i\pi .}

Furthermore, using the laws for exponentiation,

(cos⁡x+isin⁡x)n=(eix)n=einx=cos⁡nx+isin⁡nx{\displaystyle (\cos x+i\sin x)^{n}=\left(e^{ix}\right)^{n}=e^{inx}=\cos nx+i\sin nx}

for any integern, which isde Moivre's formula.[49]

The expressions ofsin⁡(x){\displaystyle \sin(x)} andcos⁡(x){\displaystyle \cos(x)} in terms of theexponential function can be deduced from the Taylor series:[46]cos⁡x=eix+e−ix2,sin⁡x=eix−e−ix2i.{\displaystyle \cos x={\frac {e^{ix}+e^{-ix}}{2}},\qquad \sin x={\frac {e^{ix}-e^{-ix}}{2i}}.}

The expressioncos⁡x+isin⁡x{\textstyle \cos x+i\sin x}is sometimes abbreviated ascis(x){\displaystyle \mathrm {cis} (x)}.[49]

Entropy

[edit]

The constante{\displaystyle e} plays a distinguished role in the theory ofentropy inprobability theory andergodic theory.[50] The basic idea is to consider a partition of aprobability space into a finite number ofmeasurable sets,ξ=(A1,⋯,Ak){\displaystyle \xi =(A_{1},\cdots ,A_{k})}, the entropy of which is the expected information gained regarding the probability distribution by performing a random sample (or "experiment"). The entropy of the partition isH(ξ)=−∑i=1kp(Ai)ln⁡p(Ai).{\displaystyle H(\xi )=-\sum _{i=1}^{k}p(A_{i})\ln p(A_{i}).}The functionf(x)=−xln⁡x{\displaystyle f(x)=-x\ln x} is thus of fundamental importance, representing the amount of entropy contributed by a particular element of the partition,x=p(Ai){\displaystyle x=p(A_{i})}. This function is maximized whenx=1/e{\displaystyle x=1/e}. What this means, concretely, is that the entropy contribution of the particular eventAi{\displaystyle A_{i}} is maximized whenp(Ai)=1/e{\displaystyle p(A_{i})=1/e}; outcomes that are either too likely or too rare contribute less to the total entropy.

Representations

[edit]
Main article:List of representations ofe

The numbere can be represented in a variety of ways: as aninfinite series, aninfinite product, acontinued fraction, or alimit of a sequence. In addition to the limit and the series given above, there is also thesimple continued fraction

e=[2;1,2,1,1,4,1,1,6,1,...,1,2n,1,...],{\displaystyle e=[2;1,2,1,1,4,1,1,6,1,...,1,2n,1,...],}[51][52]

which written out looks like

e=2+11+12+11+11+14+11+11+⋱.{\displaystyle e=2+{\cfrac {1}{1+{\cfrac {1}{2+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{4+{\cfrac {1}{1+{\cfrac {1}{1+\ddots }}}}}}}}}}}}}}.}

The following infinite product evaluates toe:[26]e=21(43)1/2(6⋅85⋅7)1/4(10⋅12⋅14⋅169⋅11⋅13⋅15)1/8⋯.{\displaystyle 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 .}

Many other series, sequence, continued fraction, and infinite product representations ofe have been proved.

Stochastic representations

[edit]

In addition to exact analytical expressions for representation ofe, there are stochastic techniques for estimatinge. One such approach begins with an infinite sequence of independent random variablesX1,X2..., drawn from theuniform distribution on [0, 1]. LetV be the least numbern such that the sum of the firstn observations exceeds 1:

V=min{n∣X1+X2+⋯+Xn>1}.{\displaystyle V=\min \left\{n\mid X_{1}+X_{2}+\cdots +X_{n}>1\right\}.}

Then theexpected value ofV ise:E(V) =e.[53][54]

Known digits

[edit]

The number of known digits ofe has increased substantially since the introduction of the computer, due both to increasing performance of computers and to algorithmic improvements.[55][56]

Number of known decimal digits ofe
DateDecimal digitsComputation performed by
16901Jacob Bernoulli[11]
171413Roger Cotes[57]
174823Leonhard Euler[58]
1853137William Shanks[59]
1871205William Shanks[60]
1884346J. Marcus Boorman[61]
19492,010John von Neumann (on theENIAC)
1961100,265Daniel Shanks andJohn Wrench[62]
1981116,000Steve Wozniak on theApple II[63]

Since around 2010, the proliferation of modern high-speeddesktop computers has made it feasible for amateurs to compute trillions of digits ofe within acceptable amounts of time. On December 24, 2023, a record-setting calculation was made by Jordan Ranous, givinge to 35,000,000,000,000 digits.[64]

Computing the digits

[edit]

One way to compute the digits ofe is with the series[65]e=∑k=0∞1k!.{\displaystyle e=\sum _{k=0}^{\infty }{\frac {1}{k!}}.}

A faster method involves two recursive functionsp(a,b){\displaystyle p(a,b)} andq(a,b){\displaystyle q(a,b)}. The functions are defined as(p(a,b)q(a,b))={(1b),if b=a+1,(p(a,m)q(m,b)+p(m,b)q(a,m)q(m,b)),otherwise, where m=⌊(a+b)/2⌋.{\displaystyle {\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}}}

The expression1+p(0,n)q(0,n){\displaystyle 1+{\frac {p(0,n)}{q(0,n)}}} produces thenth partial sum of the series above. This method usesbinary splitting to computee with fewer single-digit arithmetic operations and thus reducedbit complexity. Combining this withfast Fourier transform-based methods of multiplying integers makes computing the digits very fast.[65]

In computer culture

[edit]

During the emergence ofinternet culture, individuals and organizations sometimes paid homage to the numbere.

In an early example, thecomputer scientistDonald Knuth let the version numbers of his programMetafont approache. The versions are 2, 2.7, 2.71, 2.718, and so forth.[66]

In another instance, theIPO filing forGoogle in 2004, rather than a typical round-number amount of money, the company announced its intention to raise 2,718,281,828USD, which ise billion dollarsrounded to the nearest dollar.[67]

Google was also responsible for a billboard[68]that appeared in the heart ofSilicon Valley, and later inCambridge, Massachusetts;Seattle, Washington; andAustin, Texas. It read "{first 10-digit prime found in consecutive digits ofe}.com". The first 10-digit prime ine is 7427466391, which starts at the 99th digit.[69] 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 ofe whose digits summed to 49. The fifth term in the sequence is 5966290435, which starts at the 127th digit.[70]Solving this second problem finally led to aGoogle Labs webpage where the visitor was invited to submit a résumé.[71]

The last release of the officialPython 2 interpreter has version number 2.7.18, a reference toe.[72]

In computing

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Inscientific computing, the constante{\displaystyle e} is often hard-coded. For example, thePython standard library includesmath.e = 2.718281828459045, a floating-point approximation ofe{\displaystyle e}. Despite this, it is generally morenumerically stable and efficient to use the built-in exponential function—such asmath.exp(x) in Python—rather than computingex{\displaystyle e^{x}} viapow(e, x), even whenx{\displaystyle x} is an integer.[73]

Most implementations of the exponential function use range reduction, lookup tables, and polynomial or rational approximations (such asPadé approximants or Taylor expansions) to achieve accurate results across a wide range of inputs.[74] In contrast, general-purpose exponentiation functions—likepow—may involve additional intermediate computations, such as logarithms and multiplications, and may accumulate more rounding error, particularly whene{\displaystyle e} is used in floating-point form.[75]

At very high precision, methods based onelliptic functions and fast convergence of theAGM andNewton's method can be used to compute the exponential function.[76] The digit expansion ofe{\displaystyle e} can then be obtained asexp⁡(1).{\displaystyle \exp(1).} Although this is asymptotically faster than other known methods for computing the exponential function, it is impractical because of high overhead cost.[74]

Tools such asy-cruncher are optimized for computing many digits of individual constants likee{\displaystyle e}, and use the Taylor series fore{\displaystyle e} because it converges very rapidly, especially when combined with various optimizations. In particular, the method ofbinary splitting applies to computing the series fore{\displaystyle e}, as opposed to the series forexp⁡(x){\displaystyle \exp(x)}, because the summands in the former series are simple rational numbers. This allows the complexity of computingn{\displaystyle n} digits ofe{\displaystyle e} to be reduced toO(nlog2⁡n){\displaystyle O(n\log ^{2}n)}, asymptotically the same as AGM methods, but much cheaper in practice.[77][78]

References

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  11. ^abJacob Bernoulli considered the problem of continuous compounding of interest, which led to a series expression fore. 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 theJournal 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. … sia =b, debebitur plu quam2½a & minus quam3a." ( … which our series [a geometric series] is larger [than]. … ifa=b, [the lender] will be owed more than2½a and less than3a.) Ifa =b, the geometric series reduces to the series fora ×e, so2.5 <e < 3. (** The reference is to a problem which Jacob Bernoulli posed and which appears in theJournal des Sçavans of 1685 at the bottom ofpage 314.)
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