E (mathematical constant)

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

The number Template:Mvar is a mathematical constant approximately equal to 2.71828 that is the base of the natural logarithm and exponential function. It is sometimes called Euler's number, after the Swiss mathematician Leonhard Euler, though this can invite confusion with Euler numbers, or with Euler's constant, a different constant typically denoted $\gamma$. Alternatively, Template:Mvar can be called Napier's constant after John Napier. The Swiss mathematician Jacob Bernoulli discovered the constant while studying compound interest.

The number Template:Mvar is of great importance in mathematics, alongside 0, 1, [[Pi|Template:Pi]], and Template:Mvar. All five appear in one formulation of Euler's identity $e^{i\pi }+1=0$ and play important and recurring roles across mathematics. Like the constant Template:Pi, Template: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 30 decimal places, the value of Template:Mvar is: Template:Block indent

Definitions

The number Template:Mvar is the limit $$\underset{n\to \mathrm{\infty }}{\lim }{(1+\frac{1}{n})}^n,$$ an expression that arises in the computation of compound interest.

It is the sum of the infinite series $$e=\sum _{n=0}^{\mathrm{\infty }}\frac{1}{n!}=1+\frac{1}{1}+\frac{1}{1\cdot 2}+\frac{1}{1\cdot 2\cdot 3}+\cdots .$$

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

One has $$e=\exp (1),$$ where $\exp$ is the (natural) exponential function, the unique function that equals its own derivative and satisfies the equation $\exp (0)=1.$ Since the exponential function is commonly denoted as $x\mapsto e^x,$ one has also $$e=e^1.$$

The logarithm of base Template:Mvar can be defined as the inverse function of the function $x\mapsto b^x.$ Since $b=b^1,$ one has ${\log }_{b}b=1.$ The equation $e=e^1$ implies therefore that Template:Mvar is the base of the natural logarithm.

The number Template:Mvar can also be characterized in terms of an integral: $$\underset{1}{\overset{e}{\int }}\frac{dx}{x}=1.$$

For other characterizations, see Template:Slink.

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. 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 Template:Mvar, but he did not recognize Template: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 Template:Mvar occurs as the limit $$\underset{n\to \mathrm{\infty }}{\lim }{(1+\frac{1}{n})}^n,$$ where Template:Mvar 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 Template:Mvar by Gottfried Leibniz in letters to Christiaan Huygens in 1690 and 1691.

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

Euler proved that Template:Mvar is the sum of the infinite series $$e=\sum _{n=0}^{\mathrm{\infty }}\frac{1}{n!}=\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+\cdots ,$$ where Template:Math is the factorial of Template:Mvar. The equivalence of the two characterizations using the limit and the infinite series can be proved via the binomial theorem.

Applications

Compound interest

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

Jacob Bernoulli discovered this constant in 1683, while studying a question about compound interest: Template: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 Template:Nowrap at the end of the year. Compounding quarterly yields Template:Nowrap, and compounding monthly yields Template:Nowrap. If there are Template:Mvar compounding intervals, the interest for each interval will be Template:Math and the value at the end of the year will be $1.00 × Template:Math.

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

Bernoulli trials

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

The number Template: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 Template:Mvar and plays it Template:Mvar times. As Template:Mvar increases, the probability that gambler will lose all Template:Mvar bets approaches Template:Math. For Template: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 Template:Mvar chance of winning. Playing Template:Mvar times is modeled by the binomial distribution, which is closely related to the binomial theorem and Pascal's triangle. The probability of winning Template:Mvar times out of Template:Mvar trials is:

$\mathrm{Pr}[k\mathrm{w}\mathrm{i}\mathrm{n}\mathrm{s}\mathrm{o}\mathrm{f}n]=(\genfrac{}{}{0ex}{}{n}{k}){\left(\frac{1}{n}\right)}^k{(1-\frac{1}{n})}^{n-k}.$

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

$\mathrm{Pr}[0\mathrm{w}\mathrm{i}\mathrm{n}\mathrm{s}\mathrm{o}\mathrm{f}n]={(1-\frac{1}{n})}^n.$

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

Exponential growth and decay

Template: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 Template:Mvar 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 Template:Mvar, Template:Mvar is the growth constant, and $\tau$ is the time it takes the quantity to grow by a factor of Template:Mvar.

Standard normal distribution

The normal distribution with zero mean and unit standard deviation is known as the standard normal distribution,Template:R given by the probability density function $$\varphi (x)=\frac{1}{\sqrt{2\pi }}{e}^{-\frac{1}{2}{x}^2}.$$

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

Derangements

Template:Main Another application of Template: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: Template: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 Template: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 $p_n$, 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 Template:Mvar tends to infinity, Template:Math approaches Template: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 Template:Math rounded to the nearest integer, for every positive Template:Mvar.

Optimal planning problems

The maximum value of $\sqrt[x]{x}$ occurs at $x=e$. Equivalently, for any value of the base Template:Math, 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 Template:Mvar that is broken into Template:Mvar equal parts. The value of Template:Mvar that maximizes the product of the lengths is then either

$n=\lfloor \frac{L}{e}\rfloor$ or $\lceil \frac{L}{e}\rceil .$

The quantity $x^{-1}{\log }_{b}x$ is also a measure of information gleaned from an event occurring with probability $1/x$ (approximately $36.8\mathrm{\%}$ when $x=e$), so that essentially the same optimal division appears in optimal planning problems like the secretary problem.

Asymptotics

The number Template: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 Template:Mvar and [[pi|Template:Pi]] appear: $$n!\sim \sqrt{2\pi n}{\left(\frac{n}{e}\right)}^n.$$

As a consequence, $$e=\underset{n\to \mathrm{\infty }}{\lim }\frac{n}{\sqrt[n]{n!}}.$$

Properties

Calculus

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

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

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

$\begin{array}{cc}\frac{d}{dx}{a}^x& =\underset{h\to 0}{\lim }\frac{a^{x+h}-a^x}{h}=\underset{h\to 0}{\lim }\frac{a^x{a}^h-a^x}{h}\\ & =a^x\cdot \left(\underset{h\to 0}{\lim }\frac{a^h-1}{h}\right).\end{array}$

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

$\frac{d}{dx}{e}^x=e^x.$

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

$\begin{array}{cc}\frac{d}{dx}{\log }_{a}x& =\underset{h\to 0}{\lim }\frac{{\log }_a(x+h)-{\log }_a(x)}{h}\\ & =\underset{h\to 0}{\lim }\frac{{\log }_a(1+h/x)}{x\cdot h/x}\\ & =\frac{1}{x}{\log }_a(\underset{u\to 0}{\lim }(1+u{)}^{\frac{1}{u}})\\ & =\frac{1}{x}{\log }_{a}e,\end{array}$

where the substitution Template:Math was made. The base-Template:Mvar logarithm of Template:Mvar is 1, if Template:Mvar equals Template:Mvar. 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 Template: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 Template:Mvar. One way is to set the derivative of the exponential function Template:Math equal to Template:Math, and solve for Template:Mvar. The other way is to set the derivative of the base Template:Mvar logarithm to Template:Math and solve for Template:Mvar. In each case, one arrives at a convenient choice of base for doing calculus. It turns out that these two solutions for Template:Mvar are actually the same: the number Template:Mvar.

The five colored regions are of equal area, and define units of hyperbolic angle along the Template: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: $$e^x=\sum _{n=0}^{\mathrm{\infty }}\frac{x^n}{n!}.$$ Setting $x=1$ recovers the definition of Template:Mvar 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 Template:Mvar is the value of the exponential function evaluated at $x=1$, or equivalently, the number whose natural logarithm is 1. It follows that Template:Mvar is the unique positive real number such that $$\underset{1}{\overset{e}{\int }}\frac{1}{t}dt=1.$$

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

$$\frac{d}{dx}K{e}^x=K{e}^x,$$

it is therefore its own antiderivative as well:

$$\int K{e}^{x}dx=K{e}^x+C.$$

Equivalently, the family of functions

$$y(x)=K{e}^x$$

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

$$y^{\prime }=y.$$

Inequalities

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

The number Template:Mvar is the unique real number such that $${(1+\frac{1}{x})}^x<e<{(1+\frac{1}{x})}^{x+1}$$ for all positive Template:Mvar.

Also, we have the inequality $$e^x\ge x+1$$ for all real Template:Mvar, with equality if and only if Template:Math. Furthermore, Template:Mvar is the unique base of the exponential for which the inequality Template:Math holds for all Template:Mvar. This is a limiting case of Bernoulli's inequality.

Exponential-like functions

The global maximum of Template:Math Template:Nowrap

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

$$f(x)=x^{\frac{1}{x}}.$$

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

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

$$f(x)=x^x.$$

The infinite tetration

$x^{x^{x^{{\cdot }^{{\cdot }^{\cdot }}}}}$ or ${}^{\mathrm{\infty }}x$

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

Number theory

The real number Template: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 Template:Mvar is irrational]].)

Furthermore, by the Lindemann–Weierstrass theorem, Template: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. The number Template:Mvar 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 Template:Mvar and Template:Mvar are algebraically independent. This would be resolved by Schanuel's conjecture – a currently unproven generalization of the Lindemann–Weierstrass theorem.

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

Complex numbers

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

$$e^x=1+\frac{x}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots =\sum _{n=0}^{\mathrm{\infty }}\frac{x^n}{n!}.$$

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

$$e^{ix}=\cos x+i\sin x,$$

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

$$e^{i\pi }+1=0,$$ 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 Template:Pi is transcendental, which implies the impossibility of squaring the circle. Moreover, the identity implies that, in the principal branch of the logarithm,

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

Furthermore, using the laws for exponentiation,

$$(\cos x+i\sin x{)}^n={\left(e^{ix}\right)}^n=e^{inx}=\cos nx+i\sin nx$$

for any integer Template:Mvar, which is de Moivre's formula.

The expressions of Template:Math and Template:Math in terms of the exponential function can be deduced from the Taylor series: $$\cos x=\frac{e^{ix}+e^{-ix}}{2},\sin x=\frac{e^{ix}-e^{-ix}}{2i}.$$

The expression $\cos x+i\sin x$ is sometimes abbreviated as Template:Math.

Representations

The number Template: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 simple continued fraction

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

which written out looks like

$e=2+\frac{1}{1+\frac{1}{2+\frac{1}{1+\frac{1}{1+\frac{1}{4+\frac{1}{1+\frac{1}{1+\ddots }}}}}}}.$

The following infinite product evaluates to Template:Mvar: $$e=\frac{2}{1}{\left(\frac{4}{3}\right)}^{1/2}{\left(\frac{6\cdot 8}{5\cdot 7}\right)}^{1/4}{\left(\frac{10\cdot 12\cdot 14\cdot 16}{9\cdot 11\cdot 13\cdot 15}\right)}^{1/8}\cdots .$$

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

Stochastic representations

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

$V=\min \{n\mid X_1+X_2+\cdots +X_n>1\}.$

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

Known digits

The number of known digits of Template:Mvar has increased substantially since the introduction of the computer, due both to increasing performance of computers and to algorithmic improvements.

Number of known decimal digits of Template: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 Template:Mvar within acceptable amounts of time. On Dec 5, 2020, a record-setting calculation was made, giving Template:Mvar to 31,415,926,535,897 (approximately Template:MvarTemplate:X10^) digits.

Computing the digits

One way to compute the digits of Template:Mvar is with the series $$e=\sum _{k=0}^{\mathrm{\infty }}\frac{1}{k!}.$$

A faster method involves two recursive functions $p(a,b)$ and $q(a,b)$. The functions are defined as $$(\genfrac{}{}{0ex}{}{p(a,b)}{q(a,b)})=\{\begin{array}{cc}(\genfrac{}{}{0ex}{}{1}{b}),& \text{if }b=a+1\text{,}\\ (\genfrac{}{}{0ex}{}{p(a,m)q(m,b)+p(m,b)}{q(a,m)q(m,b)}),& \text{otherwise, where }m=\lfloor (a+b)/2\rfloor .\end{array}$$

The expression $$1+\frac{p(0,n)}{q(0,n)}$$ produces the Template:Mvarth partial sum of the series above. This method uses binary splitting to compute Template: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 Template:Mvar.

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

The last release of the official Python 2 interpreter has version number 2.7.18, a reference to e.

Further reading

• Template:Cite book
• Commentary on Endnote 10 of the book Prime Obsession for another stochastic representation
• Template:Cite journal

External links

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