thumb|237px|right|Graph of the equation . Here, is the unique number larger than 1 that makes the shaded area under the curve equal to 1.
The number 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 . Alternatively, can be called Napier's constant after John Napier. The Swiss mathematician Jacob Bernoulli discovered the constant while studying compound interest. Extract of page 166
The number is of great importance in mathematics, alongside 0, 1, , and . All five appear in one formulation of Euler's identity and play important and recurring roles across mathematics. is irrational, meaning that it cannot be represented as a ratio of integers. Moreover, like the constant , 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 is:
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Definitions
The number is the limit
an expression that arises in the computation of compound interest.
It is the sum of the infinite series
It is the unique positive number such that the graph of the function has a slope of 1 at .
One has where is the (natural) exponential function, the unique function that equals its own derivative and satisfies the equation Therefore, is also the base of the natural logarithm, the inverse of the natural exponential function.
The number can also be characterized in terms of an integral:
For other characterizations, see .
History
The first references to this 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 . 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 , but he did not recognize 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 occurs as the limit
where represents the number of intervals in a year on which the compound interest is evaluated (for example, for monthly compounding).Jacob Bernoulli considered the problem of continuous compounding of interest, which led to a series expression for . 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 [a] sum of money [at] interest, let it accumulate, so that [at] every moment [it] were to receive [a] proportional part of [its] annual interest; how much would be owing [at the] end of [the] year?) Bernoulli constructs a power series to calculate the answer, and then writes: " … quæ nostra serie [mathematical expression for a geometric series] &c. major est. … si , debebitur plu quam & minus quam ." ( … which our series [a geometric series] is larger [than]. … if , [the lender] will be owed more than and less than .) If , the geometric series reduces to the series for , so . (** The reference is to a problem which Jacob Bernoulli posed and which appears in the Journal des Sçavans of 1685 at the bottom of page 314.)
The first symbol used for this constant was the letter by Gottfried Leibniz in letters to Christiaan Huygens in 1690 and 1691.
Leonhard 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...") and in a letter to Christian Goldbach on 25 November 1731.Lettre XV. Euler à Goldbach, dated November 25, 1731 in: P.H. Fuss, ed., Correspondance Mathématique et Physique de Quelques Célèbres Géomètres du XVIIIeme Siècle … (Mathematical and physical correspondence of some famous geometers of the 18th century), vol. 1, (St. Petersburg, Russia: 1843), pp. 56–60, see especially p. 58. From p. 58: " … ( e denotat hic numerum, cujus logarithmus hyperbolicus est = 1), … " ( … (e denotes that number whose hyperbolic [i.e., natural] logarithm is equal to 1) … ) The first appearance of in a printed publication was in Euler's Mechanica (1736).Leonhard Euler, Mechanica, sive Motus scientia analytice exposita (St. Petersburg (Petropoli), Russia: Academy of Sciences, 1736), vol. 1, Chapter 2, Corollary 11, paragraph 171, p. 68. From page 68: Erit enim seu ubi denotat numerum, cuius logarithmus hyperbolicus est 1. (So it [i.e., , the speed] will be or , where denotes the number whose hyperbolic [i.e., natural] logarithm is 1.) It is unknown why Euler chose the letter . p. 124. Although some researchers used the letter in the subsequent years, the letter was more common and eventually became standard.
Euler proved that is the sum of the infinite series
where is the factorial of . The equivalence of the two characterizations using the limit and the infinite series can be proved via the binomial theorem.
Applications
Compound interest
right|thumb|The effect of earning 20% annual interest on an investment at various compounding frequencies. The limiting curve on top is the graph , where is in dollars, in years, and 0.2 = 20%.
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. Here, is the decimal equivalent of the rate of interest expressed as a percentage, so for 5% interest, .<ref name="Gonick">{{cite book
| last = Gonick
| first = Larry
| author-link = Larry Gonick
| year = 2012
| title = The Cartoon Guide to Calculus
| publisher = William Morrow
| url = https://www.larrygonick.com/titles/science/cartoon-guide-to-calculus-2/
| isbn = 978-0-06-168909-3
| pages = 29–32
}}</ref>
Bernoulli trials
thumb|300px|Graphs of probability of observing independent events each of probability after Bernoulli trials, and vs ; it can be observed that as increases, the probability of a -chance event never appearing after n tries rapidly
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 , which is approximately 36.79%. For , this is already 1/2.789509... (approximately 35.85%).
This is an example of a Bernoulli trial process. Each time the gambler plays the slots, there is a one in chance of winning. Playing 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 trials is:
In particular, the probability of winning zero times () is
The limit of the above expression, as tends to infinity, is precisely .
Exponential growth and decay
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 is a common and convenient choice:
Here, denotes the initial value of the quantity , is the growth constant, and is the time it takes the quantity to grow by a factor of .
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
The constraint of unit standard deviation (and thus also unit variance) results in the in the exponent, and the constraint of unit total area under the curve results in the factor . This function is symmetric around , where it attains its maximum value , 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 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 , is:
As 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 occurs at . Equivalently, for any value of the base , it is the case that the maximum value of occurs at (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
or
The quantity is the contribution to the entropy gleaned from an event occurring with probability (approximately when ), 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:
As a consequence,
Properties
Calculus
thumb|right|The graphs of the functions are shown for (dotted), (blue), and (dashed). They all pass through the point , but the red line (which has slope ) is tangent to only there.
thumb|right|The value of the natural log function for argument , i.e. , equals
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:
<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 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:
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 :
<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 was made. The base- logarithm of is 1, if equals . So symbolically,
The logarithm with this special base is called the natural logarithm, and is usually 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 .
thumb|right|The five colored regions are of equal area, and define units of hyperbolic angle along the
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:
Setting recovers the definition of as the sum of an infinite series.
The natural logarithm function can be defined as the integral from 1 to of , and the exponential function can then be defined as the inverse function of the natural logarithm. The number is the value of the exponential function evaluated at , or equivalently, the number whose natural logarithm is 1. It follows that is the unique positive real number such that
Because is the unique function (up to multiplication by a constant ) that is equal to its own derivative,
it is therefore its own antiderivative as well:
Equivalently, the family of functions
where is any real or complex number, is the full solution to the differential equation
Inequalities
thumb|right|Exponential functions and intersect the graph of , respectively, at and . The number is the unique base such that intersects only at . We may infer that lies between 2 and 4.
The number is the unique real number such that
for all positive .
Also, there is the inequality
for all real , with equality if and only if . Furthermore, is the unique base of the exponential for which the inequality holds for all .A standard calculus exercise using the mean value theorem; see for example Apostol (1967) Calculus, § 6.17.41. This is a limiting case of Bernoulli's inequality.
Exponential-like functions
thumb|right|250px|The global maximum of
Steiner's problem asks to find the global maximum for the function
This maximum occurs precisely at . (One can check that the derivative of is zero only for this value of .)
Similarly, is where the global minimum occurs for the function
The infinite tetration
or
converges if and only if , shown by a theorem of Leonhard Euler.Euler, L. "De serie Lambertina Plurimisque eius insignibus proprietatibus." Acta Acad. Scient. Petropol. 2, 29–51, 1783. Reprinted in Euler, L. Opera Omnia, Series Prima, Vol. 6: Commentationes Algebraicae. Leipzig, Germany: Teubner, pp. 350–369, 1921. (facsimile)
Number theory
The real number is irrational. Euler proved this by showing that its simple continued fraction expansion does not terminate. (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-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 is one of only a few transcendental numbers for which the exact irrationality exponent is known (given by ).
An unsolved problem thus far is the question of whether or not the numbers and are algebraically independent. This would be resolved by Schanuel's conjecture – a currently unproven generalization of the Lindemann–Weierstrass theorem.
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).<ref>{{cite book|author-link=Davar Khoshnevisan |last=Khoshnevisan |first=Davar |chapter=Normal numbers are normal |year=2006 |title=Clay Mathematics Institute Annual Report 2006 |publisher=Clay Mathematics Institute |pages=15, 27–31 |chapter-url=http://www.claymath.org/library/annual_report/ar2006/06report_normalnumbers.pdf
}}</ref>
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 is a period, but it is conjectured that is not.
Complex numbers
The exponential function may be written as a Taylor series
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:
which holds for every complex . The special case with is Euler's identity:
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 is transcendental, which implies the impossibility of squaring the circle. Moreover, the identity implies that, in the principal branch of the logarithm,
Furthermore, using the laws for exponentiation,
for any integer , which is de Moivre's formula.
The expressions of and 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
is sometimes abbreviated as .
Entropy
The constant plays a distinguished role in the theory of entropy in probability theory and ergodic theory., §4.2. The basic idea is to consider a partition of a probability space into a finite number of measurable sets, , the entropy of which is the expected information gained regarding the probability distribution by performing a random sample (or "experiment"). The entropy of the partition is
The function is thus of fundamental importance, representing the amount of entropy contributed by a particular element of the partition, . This function is maximized when . What this means, concretely, is that the entropy contribution of the particular event is maximized when ; outcomes that are either too likely or too rare contribute less to the total entropy.
Using the natural logarithm gives entropy units in nats (as opposed, for example, to the use of the base-2 logarithm giving entropy in bits).
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. In addition to the limit and the series given above, there is also the simple 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 :
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 [0, 1]. Let be the least number such that the sum of the first observations exceeds 1:
Then the expected value of is : .Dinov, ID (2007) Estimating e using SOCR simulation; , SOCR Hands-on Activities (retrieved December 26, 2007).
Known digits
The number of known digits of has increased substantially since the introduction of the computer, due both to increasing performance of computers and to algorithmic improvements.Sebah, P. and Gourdon, X.; The constant and its computationGourdon, X.; Reported large computations with PiFast
{| class="wikitable" style="margin: 1em auto 1em auto"
|+ Number of known decimal digits of
! Date || Decimal digits || Computation performed by
|-
| 1690 ||align=right| 1 || Jacob Bernoulli
|-
| 1714 ||align=right| 13 || Roger CotesRoger Cotes (1714) "Logometria," Philosophical Transactions of the Royal Society of London, 29 (338) : 5–45; see especially the bottom of page 10. From page 10: "Porro eadem ratio est inter 2,718281828459 &c et 1, … " (Furthermore, by the same means, the ratio is between 2.718281828459… and 1, … )
|-
| 1748 ||align=right| 23 || Leonhard EulerLeonhard Euler, Introductio in Analysin Infinitorum (Lausanne, Switzerland: Marc Michel Bousquet & Co., 1748), volume 1, page 90.
|-
| 1853 ||align=right| 137 || William ShanksWilliam Shanks, Contributions to Mathematics, ... (London, England: G. Bell, 1853), page 89.
|-
| 1871 ||align=right| 205 || William ShanksWilliam Shanks (1871) "On the numerical values of , , , , and , also on the numerical value of the modulus of the common system of logarithms, all to 205 decimals," Proceedings of the Royal Society of London, 20 : 27–29.
|-
| 1884 ||align=right| 346 || J. Marcus BoormanJ. Marcus Boorman (October 1884) "Computation of the Naperian base," Mathematical Magazine, 1 (12) : 204–205.
|-
| 1949 ||align=right| 2,010 || John von Neumann (on the ENIAC)
|-
| 1961 ||align=right| 100,265 || Daniel Shanks and John Wrench
|-
| 1981 ||align=right| 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 within acceptable amounts of time. On December 24, 2023, a record-setting calculation was made by Jordan Ranous, giving to 35,000,000,000,000 digits.<ref>{{cite web
| title= y-cruncher - A Multi-Threaded Pi Program
| editor= Alexander Yee
| work= Numberworld
| date= 15 March 2025
| url=http://www.numberworld.org/y-cruncher/#Records
}}</ref>
Computing the digits
One way to compute the digits of is with the series
A faster method involves two recursive functions and . The functions are defined as
The expression produces the th partial sum of the series above. This method uses binary splitting to compute 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
Both individuals and organizations have paid homage in Internet culture.
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.<ref>{{Cite journal
| title= The Future of TeX and Metafont
| first= Donald
| last= Knuth
| author-link= Donald Knuth
| journal= TeX Mag
| volume= 5
| issue= 1
| page= 145
| date= 1990-10-03
| url= http://www.ntg.nl/maps/05/34.pdf
| access-date= 2017-02-17
}}</ref>
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 "{first 10-digit prime found in consecutive digits of }.com". The first 10-digit prime in is 7427466391, which starts at the 99th digit.<ref>{{cite web
| first= Marcus
| last= Kazmierczak
| title= Google Billboard
| publisher= mkaz.com
| date= 2004-07-29
| url= http://mkaz.com/math/google-billboard
| access-date= 2007-06-09
| archive-date= 2010-09-23
| archive-url= https://web.archive.org/web/20100923111259/http://mkaz.com/math/google-billboard/
| url-status= dead
}}</ref> Solving this problem and visiting the advertised (now defunct) website led to an even more difficult problem to solve, which consisted of 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.The first 10-digit prime in . Explore Portland Community. Retrieved on 2020-12-09.
Solving this second problem finally led to a Google Labs webpage where the visitor was invited to submit a résumé.<ref>{{cite news
| first= Andrea
| last= Shea
| title= Google Entices Job-Searchers with Math Puzzle
| work= NPR
| url= https://www.npr.org/templates/story/story.php?storyId=3916173
| access-date= 2007-06-09
}}</ref>
The last release of the official Python 2 interpreter has version number 2.7.18, a reference to e.
In computing
In scientific computing, the constant is often hard-coded. For example, the Python standard library includes math.e = 2.718281828459045, a floating-point approximation of . Despite this, it is generally more numerically stable and efficient to use the built-in exponential function—such as math.exp(x) in Python—rather than computing via pow(e, x), even when is an integer.
Most implementations of the exponential function use range reduction, lookup tables, and polynomial or rational approximations (such as Padé approximants or Taylor expansions) to achieve accurate results across a wide range of inputs. In contrast, general-purpose exponentiation functions—like pow—may involve additional intermediate computations, such as logarithms and multiplications, and may accumulate more rounding error, particularly when is used in floating-point form.
At very high precision, methods based on elliptic functions and fast convergence of the AGM and Newton's method can be used to compute the exponential function.<ref name="Brent1976">{{cite journal
| last = Brent
| first = Richard P.
| title = Fast Multiple-Precision Evaluation of Elementary Functions
| journal = Journal of the ACM
| volume = 23
| issue = 2
| year = 1976
| pages = 242–251
| doi = 10.1145/321879.321886
| jstor = 321886
}}</ref> The digit expansion of can then be obtained as Although this is asymptotically faster than other known methods for computing the exponential function, it is impractical because of high overhead cost.
Tools such as y-cruncher are optimized for computing many digits of individual constants like , and use the Taylor series for because it converges very rapidly, especially when combined with various optimizations. In particular, the method of binary splitting applies to computing the series for , as opposed to the series for , because the summands in the former series are simple rational numbers. This allows the complexity of computing digits of to be reduced to , asymptotically the same as AGM methods, but much cheaper in practice.
References
{{Reflist|30em|refs=
<!--<ref name="nordistr">{{cite book
| last = Bryc | first = Wlodzimierz
| year = 1995
| title = The Normal Distribution: Characterizations with Applications
| url = https://books.google.com/books?id=tyXjBwAAQBAJ&pg=PA23
| page = 23
| publisher = Springer-Varleg
}}</ref>-->
<ref name="openstax">{{cite book
| title = Statistics
| chapter-url = https://openstax.org/books/statistics/pages/6-1-the-standard-normal-distribution
| chapter = 6.1 The Standard Normal Distribution |first1=Barbara
| last1 = Illowsky
| first2 = Susan | last2 = Dean
| display-authors = etal
| publisher = OpenStax
| isbn = 978-1-951693-22-0
| year = 2023
}}</ref>
}}
Further reading
- Commentary on Endnote 10 of the book Prime Obsession for another stochastic representation
External links
- The number to 1 million places and NASA.gov 2 and 5 million places
- Approximations – Wolfram MathWorld
- Earliest Uses of Symbols for Constants Jan. 13, 2008
- "The story of ", by Robin Wilson at Gresham College, 28 February 2007 (available for audio and video download)
- Search Engine 2 billion searchable digits of , and √2