Английская Википедия:E (mathematical constant)

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Файл:Hyperbola E.svg

Graph of the equation Шаблон:Math. Here, Шаблон:Mvar is the unique number larger than 1 that makes the shaded area under the curve equal to 1.

The number Шаблон:Mvar is a mathematical constant approximately equal to 2.71828 that can be characterized in many ways. It is the base of natural logarithms. It is the limit of Шаблон:Math as Шаблон:Mvar approaches infinity, an expression that arises in the computation of compound interest. It can also be calculated as the sum of the infinite series <math display ="block">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.</math>

It is also the unique positive number Шаблон:Mvar such that the graph of the function Шаблон:Math has a slope of 1 at Шаблон:Math.

The (natural) exponential function Шаблон:Math is the unique function Шаблон:Mvar that equals its own derivative and satisfies the equation Шаблон:Math; hence one can also define Шаблон:Mvar as Шаблон:Math. The natural logarithm, or logarithm to base Шаблон:Mvar, is the inverse function to the natural exponential function. The natural logarithm of a number Шаблон:Math can be defined directly as the area under the curve Шаблон:Math between Шаблон:Math and Шаблон:Math, in which case Шаблон:Mvar is the value of Шаблон:Mvar for which this area equals Шаблон:Math (see image). There are various other characterizations.

The number Шаблон:Mvar is sometimes called Euler's number, after the Swiss mathematician Leonhard Euler, though this can invite confusion with Euler's constant, a different constant typically denoted <math>\gamma</math>. Alternatively, Шаблон:Mvar can be called Napier's constant after John Napier. The constant was discovered by the Swiss mathematician Jacob Bernoulli while studying compound interest.

The number Шаблон:Mvar is of great importance in mathematics, alongside 0, 1, [[Pi|Шаблон:Pi]], and Шаблон:Mvar. All five appear in one formulation of Euler's identity <math>e^{i\pi}+1=0</math> and play important and recurring roles across mathematics. Like the constant Шаблон:Pi, Шаблон:Mvar is irrational, 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 with rational coefficients. To 40 decimal places, the value of Шаблон:Mvar is: Шаблон:Block indent

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. However, this did not contain the constant itself, but simply a list of logarithms to the base <math>e</math>. 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 Шаблон:Mvar, but he did not recognize Шаблон:Mvar 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 Шаблон:Mvar occurs as the limit <math display="block">\lim_{n\to \infty} \left( 1 + \frac{1}{n} \right)^n,</math> where Шаблон:Mvar represents the number of intervals in a year on which the compound interest is evaluated (for example, <math>n=12</math> for monthly compounding).

The first symbol used for this constant was the letter Шаблон:Mvar by Gottfried Leibniz in letters to Christiaan Huygens in 1690 and 1691.

Leonhard Euler started to use the letter Шаблон:Mvar 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 Шаблон:Mvar in a printed publication was in Euler's Mechanica (1736). It is unknown why Euler chose the letter Шаблон:Mvar. Although some researchers used the letter Шаблон:Mvar in the subsequent years, the letter Шаблон:Mvar was more common and eventually became standard.

Euler proved that Шаблон:Mvar is the sum of the infinite series <math display="block">e = \sum_{n = 0}^\infty \frac{1}{n!} = \frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + \cdots ,</math> where Шаблон:Math is the factorial of Шаблон:Mvar. The equivalence of the two characterizations using the limit and the infinite series can be proved via the binomial theorem.

Applications

Compound interest

Файл:Compound Interest with Varying Frequencies.svg

The effect of earning 20% annual interest on an Шаблон:Nowrap investment at various compounding frequencies. The limiting curve on top is the graph <math>y=1000e^{0.2t}</math>, where Шаблон:Mvar is in dollars, Шаблон:Mvar in years, and 0.2 = 20%.

Jacob Bernoulli discovered this constant in 1683, while studying a question about compound interest: Шаблон:Blockquote

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 Шаблон:Nowrap at the end of the year. Compounding quarterly yields Шаблон:Nowrap, and compounding monthly yields Шаблон:Nowrap. If there are Шаблон:Mvar compounding intervals, the interest for each interval will be Шаблон:Math and the value at the end of the year will be $1.00 × Шаблон:Math.

Bernoulli noticed that this sequence approaches a limit (the force of interest) with larger Шаблон:Mvar and, thus, smaller compounding intervals. Compounding weekly (Шаблон:Math) yields $2.692596..., while compounding daily (Шаблон:Math) yields $2.714567... (approximately two cents more). The limit as Шаблон:Mvar grows large is the number that came to be known as Шаблон:Mvar. 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 Шаблон:Mvar will, after Шаблон:Mvar years, yield Шаблон:Math dollars with continuous compounding. Here, Шаблон:Mvar is the decimal equivalent of the rate of interest expressed as a percentage, so for 5% interest, Шаблон:Math.

Bernoulli trials

Файл:Bernoulli trial sequence.svg

Graphs of probability Шаблон:Mvar of Шаблон:Em observing independent events each of probability Шаблон:Math after Шаблон:Mvar Bernoulli trials, and Шаблон:Math vs Шаблон:Mvar ; it can be observed that as Шаблон:Mvar increases, the probability of a Шаблон:Math-chance event never appearing after n tries rapidly Шаблон:Nowrap

The number Шаблон:Mvar itself also has applications in probability 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 in Шаблон:Mvar and plays it Шаблон:Mvar times. As Шаблон:Mvar increases, the probability that gambler will lose all Шаблон:Mvar bets approaches Шаблон:Math. For Шаблон:Math, this is already approximately 1/2.789509....

This is an example of a Bernoulli trial process. Each time the gambler plays the slots, there is a one in Шаблон:Mvar chance of winning. Playing Шаблон:Mvar times is modeled by the binomial distribution, which is closely related to the binomial theorem and Pascal's triangle. The probability of winning Шаблон:Mvar times out of Шаблон:Mvar trials is:

<math>\Pr[k~\mathrm{wins~of}~n] = \binom{n}{k} \left(\frac{1}{n}\right)^k\left(1 - \frac{1}{n}\right)^{n-k}.</math>

In particular, the probability of winning zero times (Шаблон:Math) is

<math>\Pr[0~\mathrm{wins~of}~n] = \left(1 - \frac{1}{n}\right)^{n}.</math>

The limit of the above expression, as Шаблон:Mvar tends to infinity, is precisely Шаблон:Math.

Exponential growth and decay

Шаблон:Further 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 Шаблон:Mvar is a common and convenient choice: <math display="block">x(t) = x_0\cdot e^{kt} = x_0\cdot e^{t/\tau}.</math> Here, <math>x_0</math> denotes the initial value of the quantity Шаблон:Mvar, Шаблон:Mvar is the growth constant, and <math>\tau</math> is the time it takes the quantity to grow by a factor of Шаблон:Mvar.

Standard normal distribution

The normal distribution with zero mean and unit standard deviation is known as the standard normal distribution,Шаблон:R given by the probability density function <math display="block"> \phi(x) = \frac{1}{\sqrt{2\pi}} e^{-\frac{1}{2} x^2}. </math>

The constraint of unit standard deviation (and thus also unit variance) results in the Шаблон:Frac2 in the exponent, and the constraint of unit total area under the curve <math>\phi(x)</math> results in the factor <math>\textstyle 1/\sqrt{2\pi}</math>. This function is symmetric around Шаблон:Math, where it attains its maximum value <math>\textstyle 1/\sqrt{2\pi}</math>, and has inflection points at Шаблон:Math.

Derangements

Шаблон:Main Another application of Шаблон:Mvar, 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: Шаблон:Mvar 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 Шаблон:Mvar 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 <math>p_n\!</math>, is:

<math>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!}.</math>

As Шаблон:Mvar tends to infinity, Шаблон:Math approaches Шаблон:Math. 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 Шаблон:Math rounded to the nearest integer, for every positive Шаблон:Mvar.

Optimal planning problems

The maximum value of <math>\sqrt[x]{x}</math> occurs at <math>x = e</math>. Equivalently, for any value of the base Шаблон:Math, it is the case that the maximum value of <math>x^{-1}\log_b x</math> occurs at <math>x = e</math> (Steiner's problem, discussed below).

This is useful in the problem of a stick of length Шаблон:Mvar that is broken into Шаблон:Mvar equal parts. The value of Шаблон:Mvar that maximizes the product of the lengths is then either

<math>n = \left\lfloor \frac{L}{e} \right\rfloor</math> or <math>\left\lceil \frac{L}{e} \right\rceil.</math>

The quantity <math>x^{-1}\log_b x</math> is also a measure of information gleaned from an event occurring with probability <math>1/x</math>, so that essentially the same optimal division appears in optimal planning problems like the secretary problem.

Asymptotics

The number Шаблон:Mvar 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 Шаблон:Mvar and [[pi|Шаблон:Pi]] appear: <math display="block>n! \sim \sqrt{2\pi n} \left(\frac{n}{e}\right)^n.</math>

As a consequence, <math display="block>e = \lim_{n\to\infty} \frac{n}{\sqrt[n]{n!}} .</math>

Properties

Calculus

Файл:Exp derivative at 0.svg

The graphs of the functions Шаблон:Math are shown for Шаблон:Math (dotted), Шаблон:Math (blue), and Шаблон:Math (dashed). They all pass through the point Шаблон:Math, but the red line (which has slope Шаблон:Math) is tangent to only Шаблон:Math there.

Файл:Ln+e.svg

The value of the natural log function for argument Шаблон:Mvar, i.e. Шаблон:Math, equals Шаблон:Math

The principal motivation for introducing the number Шаблон:Mvar, particularly in calculus, is to perform differential and integral calculus with exponential functions and logarithms. A general exponential Шаблон:Nowrap has a derivative, given by a limit:

<math>\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}</math>

The parenthesized limit on the right is independent of the Шаблон:Nowrap Its value turns out to be the logarithm of Шаблон:Mvar to base Шаблон:Mvar. Thus, when the value of Шаблон:Mvar is set Шаблон:Nowrap this limit is equal Шаблон:Nowrap and so one arrives at the following simple identity:

<math>\frac{d}{dx}e^x = e^x.</math>

Consequently, the exponential function with base Шаблон:Mvar is particularly suited to doing calculus. Шаблон:Nowrap (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-Шаблон:Mvar logarithm (i.e., Шаблон:Math),Шаблон:R for Шаблон:Math:

<math>\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}</math>

where the substitution Шаблон:Math was made. The base-Шаблон:Mvar logarithm of Шаблон:Mvar is 1, if Шаблон:Mvar equals Шаблон:Mvar. So symbolically,

<math>\frac{d}{dx}\log_e x = \frac{1}{x}.</math>

The logarithm with this special base is called the natural logarithm, and is denoted as Шаблон:Math; 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 Шаблон:Mvar. One way is to set the derivative of the exponential function Шаблон:Math equal to Шаблон:Math, and solve for Шаблон:Mvar. The other way is to set the derivative of the base Шаблон:Mvar logarithm to Шаблон:Math and solve for Шаблон:Mvar. In each case, one arrives at a convenient choice of base for doing calculus. It turns out that these two solutions for Шаблон:Mvar are actually the same: the number Шаблон:Mvar.

Файл:Area under rectangular hyperbola.svg

The five colored regions are of equal area, and define units of hyperbolic angle along the Шаблон:Nowrap

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: <math display="block">e^x = \sum_{n=0}^\infty \frac{x^n}{n!}.</math> Setting <math>x = 1</math> recovers the definition of Шаблон:Mvar as the sum of an infinite series.

The natural logarithm function can be defined as the integral from 1 to <math>x</math> of <math>1/t</math>, and the exponential function can then be defined as the inverse function of the natural logarithm. The number Шаблон:Mvar is the value of the exponential function evaluated at <math>x = 1</math>, or equivalently, the number whose natural logarithm is 1. It follows that Шаблон:Mvar is the unique positive real number such that <math display="block">\int_1^e \frac{1}{t} \, dt = 1.</math>

Because Шаблон:Math is the unique function (up to multiplication by a constant Шаблон:Mvar) that is equal to its own derivative,

<math display="block">\frac{d}{dx}Ke^x = Ke^x,</math>

it is therefore its own antiderivative as well:

<math display="block">\int Ke^x\,dx = Ke^x + C .</math>

Equivalently, the family of functions

<math display="block">y(x) = Ke^x</math>

where Шаблон:Mvar is any real or complex number, is the full solution to the differential equation

<math display="block">y' = y .</math>

Inequalities

Файл:Exponentials vs x+1.pdf

Exponential functions Шаблон:Math and Шаблон:Math intersect the graph of Шаблон:Math, respectively, at Шаблон:Math and Шаблон:Math. The number Шаблон:Mvar is the unique base such that Шаблон:Math intersects only at Шаблон:Math. We may infer that Шаблон:Mvar lies between 2 and 4.

The number Шаблон:Mvar is the unique real number such that <math display="block">\left(1 + \frac{1}{x}\right)^x < e < \left(1 + \frac{1}{x}\right)^{x+1}</math> for all positive Шаблон:Mvar.

Also, we have the inequality <math display="block">e^x \ge x + 1</math> for all real Шаблон:Mvar, with equality if and only if Шаблон:Math. Furthermore, Шаблон:Mvar is the unique base of the exponential for which the inequality Шаблон:Math holds for all Шаблон:Mvar. This is a limiting case of Bernoulli's inequality.

Exponential-like functions

Файл:Xth root of x.svg

The global maximum of Шаблон:Math Шаблон:Nowrap

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

<math display="block"> f(x) = x^\frac{1}{x} .</math>

This maximum occurs precisely at Шаблон:Math. (One can check that the derivative of Шаблон:Math is zero only for this value of Шаблон:Mvar.)

Similarly, Шаблон:Math is where the global minimum occurs for the function

<math display="block"> f(x) = x^x .</math>

The infinite tetration

<math> x^{x^{x^{\cdot^{\cdot^{\cdot}}}}} </math> or <math>{^\infty}x</math>

converges if and only if Шаблон:Math, shown by a theorem of Leonhard Euler.

Number theory

The real number Шаблон:Mvar is irrational. Euler proved this by showing that its simple continued fraction expansion does not terminate. (See also Fourier's [[proof that e is irrational|proof that Шаблон:Mvar is irrational]].)

Furthermore, by the Lindemann–Weierstrass theorem, Шаблон:Mvar 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.

It is conjectured that Шаблон:Mvar is normal, meaning that when Шаблон:Mvar 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 Шаблон:Pi is a period, but it is conjectured that Шаблон:Mvar is not.

Complex numbers

The exponential function Шаблон:Math may be written as a Taylor series

<math display="block"> e^{x} = 1 + {x \over 1!} + {x^{2} \over 2!} + {x^{3} \over 3!} + \cdots = \sum_{n=0}^{\infty} \frac{x^n}{n!}.</math>

Because this series is convergent for every complex value of Шаблон:Mvar, it is commonly used to extend the definition of Шаблон:Math to the complex numbers. This, with the Taylor series for [[trigonometric functions|Шаблон:Math and Шаблон:Math]], allows one to derive Euler's formula:

<math display="block">e^{ix} = \cos x + i\sin x ,</math>

which holds for every complex Шаблон:Mvar. The special case with Шаблон:Math is Euler's identity:

<math display="block">e^{i\pi} + 1 = 0 ,</math> 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 Шаблон:Pi is transcendental, which implies the impossibility of squaring the circle. Moreover, the identity implies that, in the principal branch of the logarithm,

<math display="block">\ln (-1) = i\pi .</math>

Furthermore, using the laws for exponentiation,

<math display="block">(\cos x + i\sin x)^n = \left(e^{ix}\right)^n = e^{inx} = \cos (nx) + i \sin (nx) ,</math>

which is de Moivre's formula.

The expressions of Шаблон:Math and Шаблон:Math in terms of the exponential function can be deduced from the Taylor series: <math display="block">

 \cos x = \frac{e^{ix} + e^{-ix}}{2} , \qquad
 \sin x = \frac{e^{ix} - e^{-ix}}{2i}.

</math>

The expression <math display=inline>\cos x + i \sin x</math> is sometimes abbreviated as Шаблон:Math.

Representations

The number Шаблон:Mvar 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 continued fraction

<math>

 e = [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, ..., 1, 2n, 1, ...],

</math>

which written out looks like

<math>e = 2 +

\cfrac{1}

  {1 + \cfrac{1}
     {2 + \cfrac{1}
        {1 + \cfrac{1}
           {1 + \cfrac{1}
              {4 + \cfrac{1}
           {1 + \cfrac{1}
              {1 + \ddots}
                 }
              }
           }
        }
     }
  }

. </math>

The following infinite product evaluates to Шаблон:Mvar: <math display="block">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.</math>

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

Stochastic representations

In addition to exact analytical expressions for representation of Шаблон:Mvar, there are stochastic techniques for estimating Шаблон:Mvar. One such approach begins with an infinite sequence of independent random variables Шаблон:Math, Шаблон:Math..., drawn from the uniform distribution on [0, 1]. Let Шаблон:Mvar be the least number Шаблон:Mvar such that the sum of the first Шаблон:Mvar observations exceeds 1:

<math>V = \min\left\{ n \mid X_1 + X_2 + \cdots + X_n > 1 \right\}.</math>

Then the expected value of Шаблон:Mvar is Шаблон:Mvar: Шаблон:Math.

Known digits

The number of known digits of Шаблон:Mvar has increased substantially during the last decades. This is due both to the increased performance of computers and to algorithmic improvements.

Number of known decimal digits of Шаблон:Mvar

Date	Decimal digits	Computation performed by
1690	1	Jacob Bernoulli
1714	13	Roger Cotes
1748	23	Leonhard Euler
1853	137	William Shanks
1871	205	William Shanks
1884	346	J. Marcus Boorman
1949	2,010	John von Neumann (on the ENIAC)
1961	100,265	Daniel Shanks and John Wrench
1978	116,000	Steve Wozniak on the Apple II

Since around 2010, the proliferation of modern high-speed desktop computers has made it feasible for amateurs to compute trillions of digits of Шаблон:Mvar within acceptable amounts of time. On Dec 5, 2020, a record-setting calculation was made, giving Шаблон:Mvar to 31,415,926,535,897 (approximately Шаблон:MvarШаблон:X10^) digits.

Computing the digits

One way to compute the digits of Шаблон:Mvar is with the series <math display=block>e=\sum_{k=0}^\infty \frac{1}{k!}.</math>

A faster method involves two recursive functions <math>p(a,b)</math> and <math>q(a,b)</math>. The functions are defined as <math display=block>\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}</math>

The expression <math display=block>1+\frac{p(0,n)}{q(0,n)}</math> produces the Шаблон:Mvarth partial sum of the series above. This method uses binary splitting to compute Шаблон:Mvar 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

During the emergence of internet culture, individuals and organizations sometimes paid homage to the number Шаблон:Mvar.

In an early example, the computer scientist Donald Knuth let the version numbers of his program Metafont approach Шаблон:Mvar. 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 Шаблон:Mvar 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 Шаблон:Mvar}.com". The first 10-digit prime in Шаблон:Mvar 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 in 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 Шаблон:Mvar 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.

Further reading

• Maor, Eli; Шаблон:Mvar: The Story of a Number, Шаблон:Isbn
• Commentary on Endnote 10 of the book Prime Obsession for another stochastic representation
• Шаблон:Cite journal

External links

• The number Шаблон:Mvar to 1 million places and NASA.gov 2 and 5 million places
• Шаблон:Mvar Approximations – Wolfram MathWorld
• Earliest Uses of Symbols for Constants Jan. 13, 2008
• "The story of Шаблон:Mvar", by Robin Wilson at Gresham College, 28 February 2007 (available for audio and video download)
• Шаблон:Mvar Search Engine 2 billion searchable digits of Шаблон:Mvar, Шаблон:Pi and Шаблон:Radic