Euler's number

The number , also known as Euler's number, is a mathematical constant approximately equal to 2.71828 that can be characterized in many ways. It is the base of the natural logarithms. It is the limit of as approaches infinity, an expression that arises in the study of compound interest. It can also be calculated as the sum of the infinite series
$e = \sum\limits_^ \frac = 1 + \frac + \frac + \frac + \cdots.$

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

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

The number is sometimes called Euler's number (not to be confused with Euler's constant $\gamma$)after the Swiss mathematician Leonhard Euleror Napier's constantafter John Napier. The constant was discovered by the Swiss mathematician Jacob Bernoulli while studying compound interest.Extract of page 166
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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 (it cannot be represented as a ratio of integers) and transcendental (it is not a root of any non-zero polynomial with rational coefficients). To 50 decimal places, the value of is:

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 $e$. It is assumed that the table was written by William Oughtred.

The discovery of the constant itself is credited to Jacob Bernoulli in 1683,Jacob Bernoulli considered the problem of continuous compounding of interest, which led to a series expression for ''e''. 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 he be owed 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 ''a''=''b'', debebitur plu quam 2½''a'' & minus quam 3''a''."'' ( … which our series geometric seriesis larger han … if ''a''=''b'', he lenderwill be owed more than 2½''a'' and less than 3''a''.) If ''a''=''b'', the geometric series reduces to the series for ''a'' × ''e'', so 2.5 < ''e'' < 3. (** 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.
the following expression (which is equal to ):
$\lim_ \left( 1 + \frac \right)^n.$

The first known use of the constant, represented by the letter , was in correspondence from Gottfried Leibniz to Christiaan Huygens in 1690 and 1691. Leonhard Euler introduced the letter as the base for natural logarithms, writing in a letter to Christian Goldbach on 25 November 1731. Euler 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...") while the first appearance of in publication was in Euler's '' Mechanica'' (1736). Although some researchers used the letter in the subsequent years, the letter was more common and eventually became standard.

In mathematics, the most common typographical convention is to typeset the constant as "", in italics, although sometimes "e" in roman is used. However, the ISO 80000-2:2019 standard recommends typesetting constants in an upright style.

Applications

Compound interest

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

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.

(Note here that 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, 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 . For , 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 ''n'' chance of winning. 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 times out of ''n'' trials is:
:$\binom \left(\frac\right)^k\left(1 - \frac\right)^.$

In particular, the probability of winning zero times () is
:$\left(1 - \frac\right)^.$

The limit of the above expression, as ''n'' tends to infinity, is precisely .

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
:$\phi(x) = \frac e^.$

The constraint of unit variance (and thus also unit standard deviation) results in the in the exponent, and the constraint of unit total area under the curve $\phi(x)$ results in the factor $\textstyle 1/\sqrt$. ^{roof/sup> 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 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. This probability, denoted by $p_n\!$, is:

:$p_n = 1 - \frac + \frac - \frac + \cdots + \frac = \sum_^n \frac.$

As the number of guests 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 gleaned from an event occurring with probability $1/x$, 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:
$e = \lim_ \frac .$

In 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:

:$\begin \fraca^x &= \lim_\frac = \lim_\frac \\ &= a^x \cdot \left(\lim_\frac\right). \end$

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, and is 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 .

Alternative characterizations

Other characterizations of 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 number is the unique positive real number such that $\frace^t = e^t$.
# The number is the unique positive real number such that $\frac \log_e t = \frac$.

The following four characterizations can be proved to be equivalent:

Properties

Calculus

As in the motivation, the exponential function is important in part because it is the unique nontrivial function that is its own derivative (up to multiplication by a constant):

:$\frace^x = e^x$

and therefore its own antiderivative as well:

:$\int e^x\,dx = e^x + C .$

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 .

The value of this maximum is .

For proof, the inequality $e^y \ge y + 1$, from above, evaluated at $y = (x - e)/e$ and simplifying gives $e^ \ge x$. So $e^ \ge x^$ for all positive ''x''.

Similarly, is where the global minimum occurs for the function

:$f(x) = x^x$

defined for positive . More generally, for the function

:$f(x) = x^$

the global maximum for positive occurs at for any ; and the global minimum occurs at for any .

The infinite tetration

:$x^$ or $x$

converges if and only if (or approximately between 0.0660 and 1.4447), due to a theorem of Leonhard Euler.

Number theory

The real number is irrational. Euler proved this by showing that its simple continued fraction expansion is infinite. (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-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 1873.

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).

It is conjectured that is not a Kontsevich-Zagier period.

Complex numbers

The exponential function may be written as a Taylor series

:$e^ = 1 + + + + \cdots = \sum_^ \frac$

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:

:$e^ = \cos x + i\sin x ,$

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

:$e^ + 1 = 0 ,$

from which it follows that, in the principal branch of the logarithm,

:$\ln (-1) = i\pi .$

Furthermore, using the laws for exponentiation,

:$(\cos x + i\sin x)^n = \left(e^\right)^n = e^ = \cos (nx) + i \sin (nx) ,$

which is de Moivre's formula.

The expression

:$\cos x + i \sin x$

is sometimes referred to as .

The expressions of and in terms of the exponential function can be deduced:

:$\sin x = \frac , \qquad \cos x = \frac.$

Differential equations

The family of functions

:$y(x) = Ce^x,$

where is any real number, is the solution to the differential equation

:$y' = y .$

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. Two of these representations, often used in introductory calculus courses, are the limit
:$e=\lim_\left(1 + \frac\right)^n,$

given above, and the series
:$e=\sum_^\infty \frac$

obtained by evaluating at the above power series representation of .

Less common is the continued fraction

:
e = ; 1, 2, 1, 1, 4, 1, 1, 6, 1, ..., 1, 2n, 1, ...

which written out looks like

:$e = 2 + \cfrac .$

This continued fraction for converges three times as quickly:
:$e = 1 + \cfrac.$

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 Let be the least number such that the sum of the first observations exceeds 1:

:$V = \min\left\.$

Then the expected value of is : .

Known digits

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

Since around 2010, the proliferation of modern high-speed desktop computers has made it feasible for most amateurs to compute trillions of digits of within acceptable amounts of time. On Dec 5, 2020, a record-setting calculation was made, giving to 31,415,926,535,897 (approximately ) 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 function $p(a,b)$ 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 digits of . This method uses binary splitting to compute with fewer single-digit arithmetic operations and 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 .

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 ".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 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 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.

References

Further reading

* Maor, Eli; '': The Story of a Number'',

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

External links

*