E (mathematical constant)

The number is the limit \lim_{n\to \infty}\left(1+\frac 1n\right)^n, an expression that arises in the computation of compound interest. It is the sum of the infinite series 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 number such that the graph of the function has a slope of 1 at . One has e=\exp(1), where \exp is the (natural) exponential function, the unique function that equals its own derivative and satisfies the equation \exp(0)=1. Therefore, is also the base of the natural logarithm, the inverse of the natural exponential function. The number can also be characterized in terms of an integral: \int_1^e \frac {dx}x =1. For other characterizations, see . == History ==

The first references to this 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 e. 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 , but he did not recognize itself as a quantity of interest. The constant itself was introduced by Jacob Bernoulli in 1683, for solving the problem of continuous compounding of interest. In his solution, the constant occurs as the limit \lim_{n\to \infty} \left( 1 + \frac{1}{n} \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 in letters to Christiaan Huygens in 1690 and 1691. Leonhard Euler started to use the letter for the constant in 1727 or 1728, in an unpublished paper on explosive forces in cannons, and in a letter to Christian Goldbach on 25 November 1731.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 e = \sum_{n = 0}^\infty \frac{1}{n!} = \frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + \cdots , where is the factorial of . == Applications ==

Compound interest Jacob Bernoulli discovered this constant in 1683, while studying a question about compound interest: :\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 () is :\Pr[0~\mathrm{wins~of}~n] = \left(1 - \frac{1}{n}\right)^{n}. The limit of the above expression, as tends to infinity, is precisely . Exponential growth and decay 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. 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 is a common and convenient choice: x(t) = x_0\cdot e^{kt} = x_0\cdot e^{t/\tau}. 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 \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 the in the exponent, and the constraint of unit total area under the curve \phi(x) results in the factor \textstyle 1/\sqrt{2\pi}. This function is symmetric around , where it attains its maximum value \textstyle 1/\sqrt{2\pi}, 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{1}{1!} + \frac{1}{2!} - \frac{1}{3!} + \cdots + \frac{(-1)^n}{n!} = \sum_{k = 0}^n \frac{(-1)^k}{k!}. 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[x]{x} occurs at x = e. Equivalently, for any value of the base , it is the case that the maximum value of x^{-1}\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{L}{e} \right\rfloor or \left\lceil \frac{L}{e} \right\rceil. The quantity x^{-1}\log_b x is also a measure of information 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 pi| appear: n! \sim \sqrt{2\pi n} \left(\frac{n}{e}\right)^n. As a consequence, e = \lim_{n\to\infty} \frac{n}{\sqrt[n]{n!}} . == Properties ==

Calculus The principal motivation for introducing the number , particularly in calculus, is to perform differential and integral calculus with exponential functions and logarithms. A general exponential has a derivative, given by a limit: :\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} 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: :\frac{d}{dx}e^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{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} where the substitution was made. The base- logarithm of is 1, if equals . So symbolically, :\frac{d}{dx}\log_e x = \frac{1}{x}. The logarithm with this special base is called the natural logarithm, 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 . along the 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: e^x = \sum_{n=0}^\infty \frac{x^n}{n!}. 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{1}{t} \, dt = 1. Because is the unique function (up to multiplication by a constant ) that is equal to its own derivative, \frac{d}{dx}Ke^x = Ke^x, it is therefore its own antiderivative 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{1}{x}\right)^x for all positive . Also, there is 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 of Steiner's problem asks to find the global maximum for the function f(x) = x^\frac{1}{x} . 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 : x^{x^{x^{\cdot^{\cdot^{\cdot}}}}} or {^\infty}x converges if and only if , shown by a theorem of Leonhard Euler. Number theory The real number is irrational. Euler proved this by showing that its simple continued fraction 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, 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 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 is convergent for every complex 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^{ix} = \cos x + i\sin x , which holds for every complex . Moreover, the identity implies that, in the principal branch of the logarithm, Entropy The constant e plays a distinguished role in the theory of entropy in probability theory and ergodic theory. The basic idea is to consider a partition of a probability space into a finite number of measurable sets, \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 is H(\xi) = -\sum_{i=1}^k p(A_i)\ln p(A_i). The function 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(A_i). This function is maximized when x=1/e. What this means, concretely, is that the entropy contribution of the particular event A_i is maximized when p(A_i)=1/e; outcomes that are either too likely or too rare contribute less to the total entropy. == Representations ==

The number 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 : e = [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, ..., 1, 2n, 1, ...], which written out looks like :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 to : 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 computers has made it feasible for amateurs to compute trillions of digits of within acceptable amounts of time. On December 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_{k=0}^\infty \frac{1}{k!}. A faster method involves two recursive functions p(a,b) and q(a,b). The functions are defined as \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 expression 1+\frac{p(0,n)}{q(0,n)} 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-based methods of multiplying integers makes computing the digits very fast. == In computer culture ==

Both individuals and organizations have paid homage in Internet culture. In an early example, the computer scientist Donald Knuth 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 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 billion dollars rounded to the nearest dollar. Google was also responsible for a billboard 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 }.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. ==In computing==

In scientific computing, the constant e is often hard-coded. For example, the Python standard library includes math.e = 2.718281828459045, a floating-point approximation of e. Despite this, it is generally more numerically stable and efficient to use the built-in exponential function—such as math.exp(x) in Python—rather than computing e^x via pow(e, x), even when x is an integer. Most implementations of the exponential function use range reduction, lookup tables, and polynomial or rational approximations (such as Padé approximants or Taylor expansions) to achieve accurate results across a wide range of inputs. In contrast, general-purpose exponentiation functions—like pow—may involve additional intermediate computations, such as logarithms and multiplications, and may accumulate more rounding error, particularly when e is used in floating-point form. At very high precision, methods based on elliptic functions and fast convergence of the AGM and Newton's method can be used to compute the exponential function. The digit expansion of e can then be obtained as \exp(1). Although this is asymptotically faster than other known methods for computing the exponential function, it is impractical because of high overhead cost. == References ==