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The number is a
mathematical constant A mathematical constant is a number whose value is fixed by an unambiguous definition, often referred to by a special symbol (e.g., an Letter (alphabet), alphabet letter), or by mathematicians' names to facilitate using it across multiple mathem ...
approximately equal to 2.71828 that is the base of the
natural logarithm The natural logarithm of a number is its logarithm to the base of a logarithm, base of the e (mathematical constant), mathematical constant , which is an Irrational number, irrational and Transcendental number, transcendental number approxima ...
and exponential function. It is sometimes called Euler's number, after the Swiss mathematician
Leonhard Euler Leonhard Euler ( ; ; ; 15 April 170718 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential ...
, though this can invite confusion with
Euler numbers Leonhard Euler ( ; ; ; 15 April 170718 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential ...
, or with Euler's constant, a different constant typically denoted \gamma. Alternatively, can be called Napier's constant after
John Napier John Napier of Merchiston ( ; Latinisation of names, Latinized as Ioannes Neper; 1 February 1550 – 4 April 1617), nicknamed Marvellous Merchiston, was a Scottish landowner known as a mathematician, physicist, and astronomer. He was the 8 ...
. The Swiss mathematician
Jacob Bernoulli Jacob Bernoulli (also known as James in English or Jacques in French; – 16 August 1705) was a Swiss mathematician. He sided with Gottfried Wilhelm Leibniz during the Leibniz–Newton calculus controversy and was an early proponent of Leibniz ...
discovered the constant while studying compound interest.Extract of page 166
/ref> The number is of great importance in mathematics, alongside 0, 1, , and . All five appear in one formulation of Euler's identity e^+1=0 and play important and recurring roles across mathematics. Like the constant , is
irrational Irrationality is cognition, thinking, talking, or acting without rationality. Irrationality often has a negative connotation, as thinking and actions that are less useful or more illogical than other more rational alternatives. The concept of ...
, 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 In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...
with rational coefficients. To 30 decimal places, the value of is:


Definitions

The number is the limit \lim_\left(1+\frac 1n\right)^n, an expression that arises in the computation of
compound interest Compound interest is interest accumulated from a principal sum and previously accumulated interest. It is the result of reinvesting or retaining interest that would otherwise be paid out, or of the accumulation of debts from a borrower. Compo ...
. It is the sum of the infinite series e = \sum\limits_^ \frac = 1 + \frac + \frac + \frac + \cdots. It is the unique positive number such that the graph of the function has a
slope In mathematics, the slope or gradient of a Line (mathematics), line is a number that describes the direction (geometry), direction of the line on a plane (geometry), plane. Often denoted by the letter ''m'', slope is calculated as the ratio of t ...
of 1 at . One has e=\exp(1), where \exp is the (natural) exponential function, the unique function that equals its own
derivative In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...
and satisfies the equation \exp(0)=1. Since the exponential function is commonly denoted as x\mapsto e^x, one has also e=e^1. The
logarithm In mathematics, the logarithm of a number is the exponent by which another fixed value, the base, must be raised to produce that number. For example, the logarithm of to base is , because is to the rd power: . More generally, if , the ...
of base can be defined as the
inverse function In mathematics, the inverse function of a function (also called the inverse of ) is a function that undoes the operation of . The inverse of exists if and only if is bijective, and if it exists, is denoted by f^ . For a function f\colon ...
of the function x\mapsto b^x. Since b=b^1, one has \log_b b= 1. The equation e=e^1 implies therefore that is the base of the natural logarithm. The number can also be characterized in terms of an
integral In mathematics, an integral is the continuous analog of a Summation, sum, which is used to calculate area, areas, volume, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental oper ...
: \int_1^e \frac x =1. For other characterizations, see .


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 John Napier of Merchiston ( ; Latinisation of names, Latinized as Ioannes Neper; 1 February 1550 – 4 April 1617), nicknamed Marvellous Merchiston, was a Scottish landowner known as a mathematician, physicist, and astronomer. He was the 8 ...
. However, this did not contain the constant itself, but simply a list of logarithms to the base e. It is assumed that the table was written by William Oughtred. In 1661,
Christiaan Huygens Christiaan Huygens, Halen, Lord of Zeelhem, ( , ; ; also spelled Huyghens; ; 14 April 1629 – 8 July 1695) was a Dutch mathematician, physicist, engineer, astronomer, and inventor who is regarded as a key figure in the Scientific Revolution ...
studied how to compute logarithms by geometrical methods and calculated a quantity that, in retrospect, is the base-10 logarithm of , but he did not recognize itself as a quantity of interest. The constant itself was introduced by
Jacob Bernoulli Jacob Bernoulli (also known as James in English or Jacques in French; – 16 August 1705) was a Swiss mathematician. He sided with Gottfried Wilhelm Leibniz during the Leibniz–Newton calculus controversy and was an early proponent of Leibniz ...
in 1683, for solving the problem of continuous compounding of interest.Jacob Bernoulli considered the problem of continuous compounding of interest, which led to a series expression for . 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 sum of money tinterest, let it accumulate, so that tevery moment twere to receive proportional part of tsannual interest; how much would be owing t theend of heyear?) Bernoulli constructs a power series to calculate the answer, and then writes: " … quæ nostra serie athematical expression for a geometric series&c. major est. … si , debebitur plu quam & minus quam ." ( … which our series geometric seriesis larger han … if , he lenderwill be owed more than and less than .) If , the geometric series reduces to the series for , so . (** The reference is to a problem which Jacob Bernoulli posed and which appears in the ''Journal des Sçavans'' of 1685 at the bottom o
page 314.
In his solution, the constant occurs as the limit \lim_ \left( 1 + \frac \right)^n, where represents the number of intervals in a year on which the compound interest is evaluated (for example, n=12 for monthly compounding). The first symbol used for this constant was the letter by
Gottfried Leibniz Gottfried Wilhelm Leibniz (or Leibnitz; – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat who is credited, alongside Isaac Newton, Sir Isaac Newton, with the creation of calculus in ad ...
in letters to
Christiaan Huygens Christiaan Huygens, Halen, Lord of Zeelhem, ( , ; ; also spelled Huyghens; ; 14 April 1629 – 8 July 1695) was a Dutch mathematician, physicist, engineer, astronomer, and inventor who is regarded as a key figure in the Scientific Revolution ...
in 1690 and 1691.
Leonhard Euler Leonhard Euler ( ; ; ; 15 April 170718 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential ...
started to use the letter for the constant in 1727 or 1728, in an unpublished paper on explosive forces in cannons,Euler,
Meditatio in experimenta explosione tormentorum nuper instituta
'. (English: Written for the number of which the logarithm has the unit, e, that is 2,7182817...")
and in a letter to Christian Goldbach on 25 November 1731. The first appearance of in a printed publication was in Euler's '' Mechanica'' (1736). It is unknown why Euler chose the letter . Although some researchers used the letter in the subsequent years, the letter was more common and eventually became standard. Euler proved that is the sum of the
infinite series In mathematics, a series is, roughly speaking, an addition of infinitely many terms, one after the other. The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathemati ...
e = \sum_^\infty \frac = \frac + \frac + \frac + \frac + \frac + \cdots , where is the
factorial In mathematics, the factorial of a non-negative denoted is the Product (mathematics), product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times ...
of . The equivalence of the two characterizations using the limit and the infinite series can be proved via the
binomial theorem In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, the power expands into a polynomial with terms of the form , where the exponents and a ...
.


Applications


Compound interest

Jacob Bernoulli discovered this constant in 1683, while studying a question about
compound interest Compound interest is interest accumulated from a principal sum and previously accumulated interest. It is the result of reinvesting or retaining interest that would otherwise be paid out, or of the accumulation of debts from a borrower. Compo ...
: 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 at the end of the year. Compounding quarterly yields , and compounding monthly yields . If there are compounding intervals, the interest for each interval will be and the value at the end of the year will be $1.00 × . Bernoulli noticed that this sequence approaches a limit (the force of interest) with larger and, thus, smaller compounding intervals. Compounding weekly () yields $2.692596..., while compounding daily () yields $2.714567... (approximately two cents more). The limit as grows large is the number that came to be known as . 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 will, after years, yield dollars with continuous compounding. Here, is the decimal equivalent of the rate of interest expressed as a ''percentage'', so for 5% interest, .


Bernoulli trials

The number itself also has applications in
probability theory Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...
, 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 and plays it times. As increases, the probability that gambler will lose all bets approaches , which is approximately 36.79%. For , this is already 1/2.789509... (approximately 35.85%). This is an example of a
Bernoulli trial In the theory of probability and statistics, a Bernoulli trial (or binomial trial) is a random experiment with exactly two possible outcomes, "success" and "failure", in which the probability of success is the same every time the experiment is ...
process. Each time the gambler plays the slots, there is a one in chance of winning. Playing times is modeled by the
binomial distribution In probability theory and statistics, the binomial distribution with parameters and is the discrete probability distribution of the number of successes in a sequence of statistical independence, independent experiment (probability theory) ...
, which is closely related to the
binomial theorem In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, the power expands into a polynomial with terms of the form , where the exponents and a ...
and
Pascal's triangle In mathematics, Pascal's triangle is an infinite triangular array of the binomial coefficients which play a crucial role in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Bla ...
. The probability of winning times out of trials is: :\Pr ~\mathrm~n= \binom \left(\frac\right)^k\left(1 - \frac\right)^. In particular, the probability of winning zero times () is :\Pr ~\mathrm~n= \left(1 - \frac\right)^. The limit of the above expression, as tends to infinity, is precisely .


Exponential growth and decay

Exponential growth Exponential growth occurs when a quantity grows as an exponential function of time. The quantity grows at a rate directly proportional to its present size. For example, when it is 3 times as big as it is now, it will be growing 3 times as fast ...
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 In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...
) of a quantity with respect to time is proportional to the quantity itself. 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 A quantity is subject to exponential decay if it decreases at a rate proportional to its current value. Symbolically, this process can be expressed by the following differential equation, where is the quantity and (lambda Lambda (; uppe ...
instead. The law of exponential growth can be written in different but mathematically equivalent forms, by using a different base, for which the number is a common and convenient choice: x(t) = x_0\cdot e^ = x_0\cdot e^. Here, x_0 denotes the initial value of the quantity , is the growth constant, and \tau is the time it takes the quantity to grow by a factor of .


Standard normal distribution

The normal distribution with zero mean and unit standard deviation is known as the ''standard normal distribution'', given by the
probability density function In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a Function (mathematics), function whose value at any given sample (or point) in the sample space (the s ...
\phi(x) = \frac e^. The constraint of unit standard deviation (and thus also unit variance) results in the in the exponent, and the constraint of unit total area under the curve \phi(x) results in the factor \textstyle 1/\sqrt. This function is symmetric around , where it attains its maximum value \textstyle 1/\sqrt, and has inflection points at .


Derangements

Another application of , 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'': 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 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 p_n\!, is: :p_n = 1 - \frac + \frac - \frac + \cdots + \frac = \sum_^n \frac. As tends to infinity, approaches . 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 rounded to the nearest integer, for every positive .


Optimal planning problems

The maximum value of \sqrt /math> occurs at x = e. Equivalently, for any value of the base , it is the case that the maximum value of x^\log_b x occurs at x = e ( Steiner's problem, discussed below). This is useful in the problem of a stick of length that is broken into equal parts. The value of that maximizes the product of the lengths is then either :n = \left\lfloor \frac \right\rfloor or \left\lceil \frac \right\rceil. The quantity x^\log_b x is also a measure of
information Information is an Abstraction, abstract concept that refers to something which has the power Communication, to inform. At the most fundamental level, it pertains to the Interpretation (philosophy), interpretation (perhaps Interpretation (log ...
gleaned from an event occurring with probability 1/x (approximately 36.8\% when x=e), so that essentially the same optimal division appears in optimal planning problems like the secretary problem.


Asymptotics

The number 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 and appear: The principal motivation for introducing the number , particularly in
calculus Calculus is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations. Originally called infinitesimal calculus or "the ...
, is to perform differential and integral calculus with exponential functions and
logarithm In mathematics, the logarithm of a number is the exponent by which another fixed value, the base, must be raised to produce that number. For example, the logarithm of to base is , because is to the rd power: . More generally, if , the ...
s. A general exponential has a derivative, given by a limit: :\begin \fraca^x &= \lim_\frac = \lim_\frac \\ &= a^x \cdot \left(\lim_\frac\right). \end The parenthesized limit on the right is independent of the Its value turns out to be the logarithm of to base . Thus, when the value of is set this limit is equal and so one arrives at the following simple identity: :\frace^x = e^x. Consequently, the exponential function with base is particularly suited to doing calculus. (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- logarithm (i.e., ), for : :\begin \frac\log_a x &= \lim_\frac \\ &= \lim_\frac \\ &= \frac\log_a\left(\lim_(1 + u)^\frac\right) \\ &= \frac\log_a e, \end where the substitution was made. The base- logarithm of is 1, if equals . So symbolically, :\frac\log_e x = \frac. The logarithm with this special base is called the
natural logarithm The natural logarithm of a number is its logarithm to the base of a logarithm, base of the e (mathematical constant), mathematical constant , which is an Irrational number, irrational and Transcendental number, transcendental number approxima ...
, and is usually denoted as ; 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 . One way is to set the derivative of the exponential function equal to , and solve for . The other way is to set the derivative of the base logarithm to and solve for . In each case, one arrives at a convenient choice of base for doing calculus. It turns out that these two solutions for are actually ''the same'': the number . The
Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
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: e^x = \sum_^\infty \frac. Setting x = 1 recovers the definition of as the sum of an infinite series. The natural logarithm function can be defined as the integral from 1 to x of 1/t, and the exponential function can then be defined as the inverse function of the natural logarithm. The number is the value of the exponential function evaluated at x = 1, or equivalently, the number whose natural logarithm is 1. It follows that is the unique positive real number such that \int_1^e \frac \, dt = 1. Because is the unique function (
up to Two Mathematical object, mathematical objects and are called "equal up to an equivalence relation " * if and are related by , that is, * if holds, that is, * if the equivalence classes of and with respect to are equal. This figure of speech ...
multiplication by a constant ) that is equal to its own
derivative In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...
, \fracKe^x = Ke^x, it is therefore its own
antiderivative In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a continuous function is a differentiable function whose derivative is equal to the original function . This can be stated ...
as well: \int Ke^x\,dx = Ke^x + C . Equivalently, the family of functions y(x) = Ke^x where is any real or complex number, is the full solution to the differential equation y' = y .


Inequalities

The number is the unique real number such that \left(1 + \frac\right)^x < e < \left(1 + \frac\right)^ for all positive . Also, we have the inequality e^x \ge x + 1 for all real , with equality if and only if . Furthermore, is the unique base of the exponential for which the inequality holds for all . This is a limiting case of Bernoulli's inequality.


Exponential-like functions

Steiner's problem asks to find the global maximum for the function f(x) = x^\frac . This maximum occurs precisely at . (One can check that the derivative of is zero only for this value of .) Similarly, is where the global minimum occurs for the function f(x) = x^x . The infinite
tetration In mathematics, tetration (or hyper-4) is an operation (mathematics), operation based on iterated, or repeated, exponentiation. There is no standard mathematical notation, notation for tetration, though Knuth's up arrow notation \uparrow \upa ...
: x^ or x converges if and only if , shown by a theorem of
Leonhard Euler Leonhard Euler ( ; ; ; 15 April 170718 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential ...
.


Number theory

The real number is
irrational Irrationality is cognition, thinking, talking, or acting without rationality. Irrationality often has a negative connotation, as thinking and actions that are less useful or more illogical than other more rational alternatives. The concept of ...
. Euler proved this by showing that its
simple continued fraction A simple or regular continued fraction is a continued fraction with numerators all equal one, and denominators built from a sequence \ of integer numbers. The sequence can be finite or infinite, resulting in a finite (or terminated) continued fr ...
expansion does not terminate. (See also Fourier's proof that is irrational.) Furthermore, by the Lindemann–Weierstrass theorem, 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. The number is one of only a few transcendental numbers for which the exact irrationality exponent is known (given by \mu(e)=2). An unsolved problem thus far is the question of whether or not the numbers and are algebraically independent. This would be resolved by Schanuel's conjecture – a currently unproven generalization of the Lindemann–Weierstrass theorem. It is conjectured that is normal, meaning that when is expressed in any base the possible digits in that base are uniformly distributed (occur with equal probability in any sequence of given length). In
algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
, a '' period'' is a number that can be expressed as an integral of an algebraic function over an algebraic domain. The constant is a period, but it is conjectured that is not.


Complex numbers

The exponential function may be written as a
Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
e^ = 1 + + + + \cdots = \sum_^ \frac. Because this series is convergent for every
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
value of , it is commonly used to extend the definition of to the complex numbers. This, with the Taylor series for and , allows one to derive Euler's formula: e^ = \cos x + i\sin x , which holds for every complex . The special case with is Euler's identity: e^ + 1 = 0 , 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 is transcendental, which implies the impossibility of
squaring the circle Squaring the circle is a problem in geometry first proposed in Greek mathematics. It is the challenge of constructing a square (geometry), square with the area of a circle, area of a given circle by using only a finite number of steps with a ...
. Moreover, the identity implies that, in the principal branch of the logarithm, \ln (-1) = i\pi . Furthermore, using the laws for exponentiation, (\cos x + i\sin x)^n = \left(e^\right)^n = e^ = \cos nx + i \sin nx for any integer , which is de Moivre's formula. The expressions of and in terms of the exponential function can be deduced from the Taylor series: \cos x = \frac , \qquad \sin x = \frac. The expression \cos x + i \sin x is sometimes abbreviated as .


Representations

The number can be represented in a variety of ways: as an
infinite series In mathematics, a series is, roughly speaking, an addition of infinitely many terms, one after the other. The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathemati ...
, an infinite product, a continued fraction, or a
limit of a sequence As the positive integer n becomes larger and larger, the value n\times \sin\left(\tfrac1\right) becomes arbitrarily close to 1. We say that "the limit of the sequence n \times \sin\left(\tfrac1\right) equals 1." In mathematics, the li ...
. In addition to the limit and the series given above, there is also the
simple continued fraction A simple or regular continued fraction is a continued fraction with numerators all equal one, and denominators built from a sequence \ of integer numbers. The sequence can be finite or infinite, resulting in a finite (or terminated) continued fr ...
: e = ; 1, 2, 1, 1, 4, 1, 1, 6, 1, ..., 1, 2n, 1, ... which written out looks like :e = 2 + \cfrac . The following infinite product evaluates to : e = \frac \left(\frac\right)^ \left(\frac\right)^ \left(\frac\right)^ \cdots. Many other series, sequence, continued fraction, and infinite product representations of have been proved.


Stochastic representations

In addition to exact analytical expressions for representation of , there are stochastic techniques for estimating . One such approach begins with an infinite sequence of independent random variables , ..., drawn from the uniform distribution on
, 1 The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
Let be the least number such that the sum of the first observations exceeds 1: :V = \min\left\. Then the
expected value In probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first Moment (mathematics), moment) is a generalization of the weighted average. Informa ...
of is : .


Known digits

The number of known digits of has increased substantially since the introduction of the computer, due both to increasing performance of computers and to algorithmic improvements. Since around 2010, the proliferation of modern high-speed
desktop computer A desktop computer, often abbreviated as desktop, is a personal computer designed for regular use at a stationary location on or near a desk (as opposed to a portable computer) due to its size and power requirements. The most common configuratio ...
s has made it feasible for amateurs to compute trillions of digits of within acceptable amounts of time. On Dec 24, 2023, a record-setting calculation was made by Jordan Ranous, giving to 35,000,000,000,000 digits.


Computing the digits

One way to compute the digits of is with the series e=\sum_^\infty \frac. A faster method involves two recursive functions p(a,b) and q(a,b). The functions are defined as \binom= \begin \binom, & \textb=a+1\text \\ \binom, & \textm=\lfloor(a+b)/2\rfloor .\end The expression 1+\frac produces the th partial sum of the series above. This method uses binary splitting to compute with fewer single-digit arithmetic operations and thus reduced bit complexity. Combining this with
fast Fourier transform A fast Fourier transform (FFT) is an algorithm that computes the discrete Fourier transform (DFT) of a sequence, or its inverse (IDFT). A Fourier transform converts a signal from its original domain (often time or space) to a representation in ...
-based methods of multiplying integers makes computing the digits very fast.


In computer culture

During the emergence of internet culture, individuals and organizations sometimes paid homage to the number . In an early example, the
computer scientist A computer scientist is a scientist who specializes in the academic study of computer science. Computer scientists typically work on the theoretical side of computation. Although computer scientists can also focus their work and research on ...
Donald Knuth Donald Ervin Knuth ( ; born January 10, 1938) is an American computer scientist and mathematician. He is a professor emeritus at Stanford University. He is the 1974 recipient of the ACM Turing Award, informally considered the Nobel Prize of comp ...
let the version numbers of his program Metafont approach . The versions are 2, 2.7, 2.71, 2.718, and so forth. In another instance, the IPO filing for
Google Google LLC (, ) is an American multinational corporation and technology company focusing on online advertising, search engine technology, cloud computing, computer software, quantum computing, e-commerce, consumer electronics, and artificial ...
in 2004, rather than a typical round-number amount of money, the company announced its intention to raise 2,718,281,828
USD The United States dollar (symbol: $; currency code: USD) is the official currency of the United States and several other countries. The Coinage Act of 1792 introduced the U.S. dollar at par with the Spanish silver dollar, divided it int ...
, which is billion dollars rounded to the nearest dollar. Google was also responsible for a billboard that appeared in the heart of
Silicon Valley Silicon Valley is a region in Northern California that is a global center for high technology and innovation. Located in the southern part of the San Francisco Bay Area, it corresponds roughly to the geographical area of the Santa Clara Valley ...
, and later in
Cambridge, Massachusetts Cambridge ( ) is a city in Middlesex County, Massachusetts, United States. It is a suburb in the Greater Boston metropolitan area, located directly across the Charles River from Boston. The city's population as of the 2020 United States census, ...
;
Seattle, Washington Seattle ( ) is the List of municipalities in Washington, most populous city in the U.S. state of Washington (state), Washington and in the Pacific Northwest region of North America. With a population of 780,995 in 2024, it is the List of Unit ...
; and
Austin, Texas Austin ( ) is the List of capitals in the United States, capital city of the U.S. state of Texas. It is the county seat and most populous city of Travis County, Texas, Travis County, with portions extending into Hays County, Texas, Hays and W ...
. It read ".com". The first 10-digit prime in is 7427466391, which starts at the 99th digit. 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 whose digits summed to 49. The fifth term in the sequence is 5966290435, which starts at the 127th digit. Solving this second problem finally led to a Google Labs webpage where the visitor was invited to submit a résumé. The last release of the official Python 2 interpreter has version number 2.7.18, a reference to ''e''.


References


Further reading

*
Commentary on Endnote 10
of the book '' Prime Obsession'' for another stochastic representation *


External links

*