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The following data may be collected and linked to your identity:. The following data may be collected but it is not linked to your identity:. Privacy practices may vary, for example, based on the features you use or your age. Learn More. Playing n times is modeled by the binomial distribution , which is closely related to the binomial theorem and Pascal's triangle. The probability of winning k times out of n trials is:.
The normal distribution with zero mean and unit standard deviation is known as the standard normal distribution , 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 :  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 he 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. 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 n!
This is useful in the problem of a stick of length L that is broken into n equal parts. The value of n that maximizes the product of the lengths is then either . The number e occurs naturally in connection with many problems involving asymptotics.
The principal motivation for introducing the number e , particularly in calculus , is to perform differential and integral calculus with exponential functions and logarithms. The parenthesized limit on the right is independent of the variable x. Its value turns out to be the logarithm of a to base e.
Thus, when the value of a is set to e , this limit is equal to 1 , and so one arrives at the following simple identity:. Consequently, the exponential function with base e is particularly suited to doing calculus. 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. The base- a logarithm of e is 1, if a equals e. So symbolically,. The logarithm with this special base is called the natural logarithm , and is denoted as ln ; 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 a. One way is to set the derivative of the exponential function a x equal to a x , and solve for a. In each case, one arrives at a convenient choice of base for doing calculus. It turns out that these two solutions for a are actually the same : the number e. Other characterizations of e are also possible: one is as the limit of a sequence , another is as the sum of an infinite series, and still others rely on integral calculus.
So far, the following two equivalent properties have been introduced:. The following four characterizations can be proved to be equivalent :. As in the motivation, the exponential function e x is important in part because it is the unique nontrivial function that is its own derivative up to multiplication by a constant :. Steiner's problem asks to find the global maximum for the function. The value of this maximum is  1. The real number e is irrational.
Euler proved this by showing that its simple continued fraction expansion is infinite. Furthermore, by the Lindemann—Weierstrass theorem , e is transcendental , meaning that it is not a solution of any non-constant 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 It is conjectured that e is normal , meaning that when e is expressed in any base the possible digits in that base are uniformly distributed occur with equal probability in any sequence of given length.
It is conjectured that e is not a Kontsevich-Zagier period. The exponential function e x may be written as a Taylor series. Because this series is convergent for every complex value of x , it is commonly used to extend the definition of e x to the complex numbers. This, with the Taylor series for sin and cos x , allows one to derive Euler's formula :.
The expressions of sin x and cos x in terms of the exponential function can be deduced:. The number e can be represented in a variety of ways: as an infinite series , an infinite product , a continued fraction , or a limit of a sequence.
Two of these representations, often used in introductory calculus courses, are the limit. Less common is the continued fraction. This continued fraction for e converges three times as quickly: [ citation needed ]. 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 X 1 , X Let V be the least number n such that the sum of the first n observations exceeds The number of known digits of e has increased substantially during the last decades.
This is due both to the increased performance of computers and to algorithmic improvements. Since around , the proliferation of modern high-speed desktop computers has made it feasible for most amateurs to compute trillions of digits of e within acceptable amounts of time. Combining this with Fast Fourier Transform -based methods of multiplying integers makes computing the digits very fast.
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. In another instance, the IPO filing for Google in , rather than a typical round-number amount of money, the company announced its intention to raise 2,,, USD , which is e 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.
The first digit prime in e is , which starts at the 99th digit. It turned out that the sequence consisted of digit numbers found in consecutive digits of e whose digits summed to The fifth term in the sequence is , which starts at the th digit.
From Wikipedia, the free encyclopedia. For other uses, see E disambiguation. Main article: Normal distribution. Main article: Derangement. Main article: List of representations of e. Retrieved Sterling Publishing Company. ISBN MacTutor History of Mathematics. Mathematician's Delight.
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