e (mathematical constant)

The number e is a mathematical constant approximately equal to 2.71828 that is the base of the natural logarithm and exponential function.

[2][3] The Swiss mathematician Jacob Bernoulli discovered the constant while studying compound interest.

The first references to the constant were published in 1618 in the table of an appendix of a work on logarithms by John Napier.

[5][10] The constant itself was introduced by Jacob Bernoulli in 1683, for solving the problem of continuous compounding of interest.

is the factorial of n.[5] The equivalence of the two characterizations using the limit and the infinite series can be proved via the binomial theorem.

The limit as n grows large is the number that came to be known as e. 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 R will, after t years, yield eRt dollars with continuous compounding.

[20][21] The number e itself also has applications in probability theory, in a way that is not obviously related to exponential growth.

Exponential growth is a process that increases quantity over time at an ever-increasing rate.

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 e is a common and convenient choice:

is the time it takes the quantity to grow by a factor of e. The normal distribution with zero mean and unit standard deviation is known as the standard normal distribution,[23] given by the probability density function

Another application of e, 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:[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 into n 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.

An example is Stirling's formula for the asymptotics of the factorial function, in which both the numbers e and π appear:[27]

Choosing e (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., loga x),[28] for x > 0: where the substitution u = h/x was made.

Euler proved this by showing that its simple continued fraction expansion does not terminate.

Furthermore, by the Lindemann–Weierstrass theorem, e is transcendental, meaning that it is not a solution of any non-zero polynomial equation with rational coefficients.

In addition, it is directly used in a proof that π is transcendental, which implies the impossibility of squaring the circle.

[49] The expressions of cos x and sin x in terms of the exponential function can be deduced from the Taylor series:[46]

In addition to the limit and the series given above, there is also the simple continued fraction which written out looks like The following infinite product evaluates to e:[26]

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

In addition to exact analytical expressions for representation of e, there are stochastic techniques for estimating e. One such approach begins with an infinite sequence of independent random variables X1, X2..., drawn from the uniform distribution on [0, 1].

Let V be the least number n such that the sum of the first n observations exceeds 1: Then the expected value of V is e: E(V) = e.[52][53] The number of known digits of e has increased substantially since the introduction of the computer, due both to increasing performance of computers and to algorithmic improvements.

This method uses binary splitting to compute e with fewer single-digit arithmetic operations and thus reduced bit complexity.

[64] During the emergence of internet culture, individuals and organizations sometimes paid homage to the number e. In an early example, the computer scientist Donald Knuth let the version numbers of his program Metafont approach e. The versions are 2, 2.7, 2.71, 2.718, and so forth.

[65] 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 e billion dollars rounded to the nearest dollar.

[66] Google was also responsible for a billboard[67] that appeared in the heart of Silicon Valley, and later in Cambridge, Massachusetts; Seattle, Washington; and Austin, Texas.

[69] Solving this second problem finally led to a Google Labs webpage where the visitor was invited to submit a résumé.