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e (mathematical constant)

e is the base of natural logarithms and a fundamental mathematical constant (~2.71828) that arises in calculus, exponential growth, compound interest, complex analysis and many areas of mathematics.

Author: Leandro Alegsa Creation date: November 13, 2022 Last updated: June 24, 2026

Overview

e is a fundamental mathematical constant approximately equal to 2.718281828459045... It is the base of the natural logarithm and appears naturally where continuous growth, rates of change and limits are involved. The symbol e is commonly called Euler's number after the Swiss mathematician Leonhard Euler, who popularized the notation; it is also sometimes referred to as Napier's constant in recognition of Scottish mathematician John Napier, an early pioneer of logarithms.

Definitions and characterizations

Several equivalent definitions show why e is central in analysis and applied mathematics. Common formulations include:

the limit definition: e = lim_{n→∞} (1 + 1/n)^n;
the infinite series: e = Σ_{k=0}^∞ 1/k! (the sum of reciprocals of factorials); and
a unique base for which the derivative of the exponential function equals itself — that is, d/dx (e^x) = e^x, which underlies many properties of exponential functions.

History and discovery

The constant began to appear in calculations of compound interest: in the late 17th century Jacob Bernoulli studied the behavior of (1 + 1/n)^n as n grows and observed the limiting value now recognized as e. Later work by Napier, Euler and others clarified the connection between logarithms, exponential functions and e. Euler in particular helped develop its analytic properties and adopt the convenient notation that is standard today.

Mathematical properties

e has several important algebraic and analytic properties. It is known to be irrational, meaning it cannot be expressed as a finite ratio of integers, and it is transcendental, so it is not a root of any non-zero polynomial with rational coefficients (that is, no finite-degree polynomial has e as a solution). In complex analysis e combines with the constants π and i in the compact relation known as Euler's identity: e^{iπ} + 1 = 0, which links five fundamental numbers in a single formula.

Uses and examples

Because of its defining limit and differential properties, e appears across mathematics and the sciences. It models continuous compound interest, describes solutions of many ordinary differential equations that express growth and decay, appears in probability theory (for example in limiting distributions), and underpins the Gaussian (normal) distribution through the exponential of a quadratic. Its self-replicating derivative makes e^x the natural building block for linear systems and for describing processes that change at rates proportional to their current size.

Computation and notable facts

Numerically e begins 2.718281828459045... and its digits can be extended arbitrarily using series or iterative algorithms; truncated lists of its decimal places are commonly tabulated for reference. Various continued fraction and product representations exist, and high-precision computation of e is routine with modern software. While e shares some philosophical status with constants such as π and i, its primary significance lies in analysis, where the natural logarithm and exponential functions form a concise and powerful language for many branches of mathematics.

For further reading on logarithms, exponentials and historical development see general references on the natural logarithm, biographies of Leonhard Euler and John Napier, and surveys of constants in mathematics.

Related topics and quick links: Euler, Euler's identity, irrational numbers, transcendental numbers, and the role of exponential functions in applied models. Also see material on ratios and ratio concepts and integers as they relate to series and limits, and foundational definitions such as polynomial equations.

Definition

The number was defined by Leonhard Euler by the following series:

${\begin{aligned}e&=1+{\frac {1}{1}}+{\frac {1}{1\cdot 2}}+{\frac {1}{1\cdot 2\cdot 3}}+{\frac {1}{1\cdot 2\cdot 3\cdot 4}}+\dotsb \\&={\frac {1}{0!}}+{\frac {1}{1!}}+{\frac {1}{2!}}+{\frac {1}{3!}}+{\frac {1}{4!}}+\dotsb \\&=\sum _{k=0}^{\infty }{\frac {1}{k!}}\\\end{aligned}}$

For $k \in \N_0$ , the factorial of , so in the case k>0 product k ! $k!=1\cdot 2\cdot \ldots \cdot k$ of the natural numbers from to while 0!:=1 is defined.

As Euler already proved, Euler's number also obtained as a functional limit:

$e=\lim _{t\to \infty \atop t\in \mathbb {R} }\left(1+{\frac {1}{t}}\right)^{t}$ ,

which means in particular that it can also be used as a limit of the sequence $(a_{n})_{n\in \mathbb {N} }$ with $a_{n}:=\left(1+{\frac {1}{n}}\right)^{n}$ :

$e = \lim_{n\to\infty} \left(1+\frac{1}{n}\right)^n$ .

This is based on the fact that

$e=\exp(1)=e^{1}$

is the function value of the exponential function (or also " -function") at the place The above series representation of results in this context from evaluating the Taylor series of the exponential function around the development point $0$ point

An alternative approach to the definition of Euler's number is that via interval nesting, for instance in the way it is presented in Theory and Application of Infinite Series by Konrad Knopp. According to this, for all $n\in \mathbb {N}$ :

$\left(1+{\frac {1}{n}}\right)^{n}<e<\left(1+{\frac {1}{n}}\right)^{n+1}$

Calculator, spreadsheet and programme

Euler had calculated 23 decimal places:

e = 2,71828 18284 59045 23536 028

Calculators and spreadsheet programs use 8 to 16 decimal places:

e = 2.71828183 (8 decimal places) e = 2.7182818284 (10 decimal places, Excel) e = 2.7182818284590452 (16 decimal places)

Spreadsheets use the constant: EXP(1)

In programmes, Euler's number is usually defined as a constant: const long double euler = 2.7182818284590452353602874713526625 (34 decimal places)

The prehistory before Euler

The history of Euler's number begins as early as the 16th century with three problem areas in which a number appears that mathematicians were approaching at the time and which was later called

As the basis of logarithms in the logarithm tables of John Napier and Jost Bürgi. Both had developed their tables independently, taking an idea from Michael Stifel and using results from Stifel and other 16th century mathematicians. Bürgi published his "Arithmetic and Geometric Progression Tabulas" in 1620. As the basis of his logarithmic system, Bürgi apparently instinctively uses a number close to . Napier published his "Mirifici logarithmorum canonis descriptio" in 1614, using a base proportional to Napier and Bürgi wanted to use the logarithm tables to reduce multiplications to additions in order to make extensive calculations simpler and less time-consuming.
As the limit value of a sequence in compound interest calculation. In 1669, Jacob Bernoulli posed the task: "Let a sum of money be invested at interest, that at the individual moments a proportional part of the annual interest is added to the capital". Today we call this proportional addition of interest "steady interest". Bernoulli asks whether arbitrarily large multiples of the initial sum can be achieved by contracts in which the individual instants become shorter and shorter and arrives at a number as a solution, which we know today as Euler's number
As an infinite series (area of the hyperbola of Apollonios of Perge). The question (in today's language) was how far an area under the hyperbola $xy=1$ from the right, which is as large as the area of the unit square. The Flemish mathematician Grégoire de Saint-Vincent (Latinised Gregorius a Sancto Vincentino) developed a function to solve this, which we now call the natural logarithm and $\ln$ denote by He discovered interesting properties, including an equation that we now call the functional equation of the logarithm, which Napier and Bürgi also used to construct and in the use of their logarithm tables. It is not certain whether he was aware that the base of this logarithm is the number that was later called This only became apparent after the publication of his work. At the latest, his pupil and co-author Alphonse Antonio de Sarasa represented the connection by a logarithm function. In an essay dealing with the dissemination of Saint-Vincent's ideas by de Sarasa, it is stated that "the relation between logarithms and the hyperbola was found in all properties by Saint-Vincent, except in name." It then became clear through work by Newton and Euler that is the base. Leibniz was apparently the first to use a letter for this number. In his correspondence with Christiaan Huygens of 1690/1, he used the letter b as the base of a power.

Origin of the symbol e

The earliest document showing Leonhard Euler's use of the letter for this number is considered to be a letter from Euler to Christian Goldbach dated 25 November 1731. Even earlier, in 1727 or 1728, Euler began to use the letter the article "Meditatio in experimenta explosione tormentorum nuper instituta" on explosive forces in cannons, which, however, was not published until 1862. Euler's work Mechanica sive motus scientia analytice exposita, II from 1736 is considered the next confirmed source for the use of this letter. In the Introductio in Analysin Infinitorum, published in 1748, Euler takes up this designation again.

There is no evidence that this choice of the letter was made in reference to his name. It is also unclear whether he did this in reference to the exponential function or out of practical considerations of differentiation from the much-used letters a, b, c or d. Although other designations were also in use, such as c in d'Alembert's Histoire de l'Académie, has prevailed.

In the formula set, not italicised according to DIN 1338 and ISO 80000-2 in order to distinguish the number from a variable. However, the italic notation is also common.

Properties

Euler's number is a transcendental (proof according to Charles Hermite, 1873) and thus irrational number (proof with continued fractions for e^2 and thus already in 1737 by Euler, proof in the proof archive or article). It can therefore (like the circle number π $\pi$ according to Ferdinand von Lindemann 1882) not be represented as a fraction of two natural numbers (not even as a solution of an algebraic equation) and consequently has an infinite non-periodic decimal fraction expansion. The irrationality measure of is 2 and thus as small as possible for an irrational number, in particular is not Liouvillian. It is not known whether is normal to any base.

In the Eulerian identity

$e^{\mathrm i\cdot\pi} = -1$

fundamental mathematical constants are put into context: The integer 1, Euler's number , the imaginary unit $\mathrm {i}$ of the complex numbers and the circle number π $\pi$ .

Euler's number also appears in the asymptotic estimation of the factorial (see Stirling formula):

${\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}\leq n!\leq {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}\cdot e^{{{\frac {1}{12n}}}}$

The Cauchy product formula for the two (in each case absolutely convergent) series and the binomial theorem result in

$\sum _{k=0}^{\infty }{\frac {1}{k!}}\cdot \sum _{k=0}^{\infty }{\frac {(-1)^{k}}{k!}}=\sum _{k=0}^{\infty }\sum _{j=0}^{k}{\frac {1}{j!}}{\frac {(-1)^{k-j}}{(k-j)!}}=\sum _{k=0}^{\infty }{\frac {1}{k!}}\sum _{j=0}^{k}{\binom {k}{j}}1^{j}(-1)^{k-j}=\sum _{k=0}^{\infty }(1+(-1))^{k}=\sum _{k=0}^{\infty }0^{k}=1=e\cdot e^{-1}$

and it follows immediately:

$e^{-1}={\frac {1}{e}}=\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{k!}}$

Geometric interpretation

Integral calculus provides a geometric interpretation of Euler's number. According to this, is that uniquely determined number b > 1 for which the content of the area below the function graph of the real reciprocal function $y={\tfrac {1}{x}}$ in the interval is $[1,b]$ exactly equal to }:

$\int _{1}^{e}{\frac {1}{x}}\,\mathrm {d} x=1$

Further representations for Euler's number

Euler's number can also be expressed by

$e = \lim_{n\to\infty} \frac{n}{\sqrt[n]{n!}}$

or by the limit value of the quotient of faculty and sub-faculty:

$e = \lim_{n\to\infty} \frac{n!}{!n}.$

A connection to the distribution of prime numbers is established via the formulas

$e = \lim_{n \to \infty} (\sqrt[n]{n})^{\pi(n)}$

$e = \lim_{n \to \infty} \sqrt[n]{n\#}$

where π is $\pi(n)$ the prime function and the symbol $n\#$ of the number

Also of more exotic appeal than practical importance is the Catalan representation

$e=\sqrt[1]{\frac{2}{1}}\cdot\sqrt[2]{\frac{4}{3}}\cdot\sqrt[4]{\frac{6\cdot 8}{5\cdot 7}}\cdot\sqrt[8]{\frac{10\cdot 12\cdot 14\cdot 16}{9\cdot 11\cdot 13\cdot 15}}\cdots$

Continued fraction developments

In connection with the number there have been a large number of continued fractions for and quantities derivable from at least since the publication of Leonhard Euler's Introductio in Analysin Infinitorum in 1748.

Thus Euler found the following classical identity for

$(1){\begin{aligned}e&=[2;1,2,1,1,4,1,1,6,1,1,8,1,1,10,1,\dotsc ]\\&=2+{\cfrac {1}{1+{\cfrac {1}{2+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{4+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{6+\dotsb }}}}}}}}}}}}}}}}\end{aligned}}$ (sequence A003417 in OEIS)

The identity (1) obviously has a regular pattern that continues to infinity. It reproduces a regular continued fraction, which was derived by Euler from the following:

$(2){\begin{aligned}{\frac {e+1}{e-1}}&=[2;6,10,14,\dotsc ]\\&={2+{\cfrac {1}{6+{\cfrac {1}{10+{\cfrac {1}{14+{\cfrac {1}{\;\,\ddots }}}}}}}}}\\&\approx 2{,}1639534137386\end{aligned}}$ (sequence A016825 in OEIS)

This continued fraction is in turn a special case of the following with k=2 :

$(3){\begin{aligned}{\coth {\frac {1}{k}}}&={\frac {e^{\frac {2}{k}}+1}{e^{\frac {2}{k}}-1}}\\&=[k;3k,5k,7k,\dots ]\\&={k+{\cfrac {1}{3k+{\cfrac {1}{5k+{\cfrac {1}{7k+{\cfrac {1}{\;\,\ddots }}}}}}}}}\\\end{aligned}}$ $( k = 1,2,3,\dots )$

Another classical continued fraction development, which is, however, not regular, also comes from Euler:

$(4){\begin{aligned}{\frac {1}{e-1}}&={0+{\cfrac {1}{1+{\cfrac {2}{2+{\cfrac {3}{3+{\cfrac {4}{\;\,\ddots }}}}}}}}}\approx 0{,}58197670686932\end{aligned}}$ (sequence A073333 in OEIS)

Another continued fraction development of Euler's number goes back to Euler and Ernesto Cesàro, which is of a different pattern than in (1):

$(5){\begin{aligned}e&=2+{\cfrac {1}{1+{\cfrac {1}{2+{\cfrac {2}{3+{\cfrac {3}{4+{\cfrac {4}{5+{\cfrac {5}{6+{\cfrac {6}{7+{\cfrac {7}{8+\dotsb }}}}}}}}}}}}}}}}\end{aligned}}$

In connection with Euler's number, a large number of general chain-breaking functional equations also exist. For example, Oskar Perron mentions as one of several the following generally valid representation of the -function:

$(6){\begin{aligned}{e^{z}}&=1+{\cfrac {z}{1-{\cfrac {1z}{2+z-{\cfrac {2z}{3+z-{\cfrac {3z}{4+z-{\cfrac {4z}{5+z-{\cfrac {5z}{6+z-{\cfrac {6z}{7+z-{\cfrac {7z}{8+z-\dotsb }}}}}}}}}}}}}}}}\end{aligned}}$ $(z\in \mathbb {C} )$

Another example of this is the development of the tangent hyperbolicus originating from Johann Heinrich Lambert, which is counted among the Lambertian continued fractions:

$(7){\begin{aligned}{\tanh z}&={\frac {e^{z}-e^{-z}}{e^{z}+e^{-z}}}\\&={\frac {e^{2z}-1}{e^{2z}+1}}\\&=0+{\cfrac {z}{1+{\cfrac {z^{2}}{3+{\cfrac {z^{2}}{5+{\cfrac {z^{2}}{7+{\cfrac {z^{2}}{9+{\cfrac {z^{2}}{11+{\cfrac {z^{2}}{13+{\cfrac {z^{2}}{15+\dotsb }}}}}}}}}}}}}}}}\end{aligned}}$ $\left(z\in \mathbb {C} \setminus \left\{{\frac {\mathrm {i} \pi }{2}}+k\pi \colon k=0,1,2,3,\dotsc \right\}\right)$

It was only in 2019, with the help of a computer program named after Srinivasa Ramanujan as the Ramanujan machine, ultimately based on a trial-and-error method, that a team led by Gal Raayoni at the Technion found another and previously unknown continued fraction development for Euler's number. In contrast to all previously known continued fraction evolutions, which all ascend from any integer number smaller than Euler's number, this is the first one that descends from the integer 3, an integer larger than Euler's number. mere finding of a (single) such descending continued fraction from an integer greater than Euler's number s $(3>e)$ that there are infinitely many such descending continued fractions from integers n {\displaystyle > e $n>e$ also lead to Euler's number.

$(8){\begin{aligned}e&=3+{\cfrac {-1}{4+{\cfrac {-2}{5+{\cfrac {-3}{6+{\cfrac {-4}{7+{\cfrac {-5}{8+\dotsb }}}}}}}}}}\end{aligned}}$

Illustrative interpretations of Euler's number

Compound interest

The following example not only makes the calculation of Euler's number more vivid, but it also describes the history of the discovery of Euler's number: its first digits were found by Jakob I Bernoulli while investigating compound interest.

The limit value of the first formula can be interpreted as follows: Someone deposits one euro in the bank on 1 January. The bank guarantees him a current interest rate $z=100\,\%$ per year. How big is his balance on 1 January next year if he invests the interest on the same terms?

According to the compound interest formula, the initial capital $K_{0}$ after interest payments with interest rate becomes the capital

$K_{n}=K_{0}(1+z)^{n}.$

In this example, K_0 = 1 and $z = 100\,\% = 1$ if the interest surcharge is annual, or z=1/n if the interest is added times a year, i.e. if the interest is paid during the year.

With annual surcharge would be

$K_{1}=1\cdot (1+1)^{1}=2{,}00.$

With semi-annual surcharge one has $z={\frac {1}{2}}$ ,

$K_{2}=1\cdot \left(1+{\frac {1}{2}}\right)^{2}=2{,}25$

so already a little more. With daily interest you get $(z=1/365)$

$K_{365}=1\cdot \left(1+{\frac {1}{365}}\right)^{365}=2{,}714567.$

If the interest is continuous at each instant, becomes infinite, and one gets the first formula for given above.

Probability calculation

is also frequently encountered in probability theory: For example, assume that a baker adds a sultana to the dough for each bun and kneads it well. Then statistically every -th bun does not contain a sultana. The probability that in buns none of the sultanas is in a fixed chosen one, gives in the limit for $n\to \infty$ (37% rule):

$p = \lim_{n\to\infty}\left(\frac {n-1}{n}\right)^n = \lim_{n\to\infty}\left(1-\frac {1}{n}\right)^n = \frac{1}{e}.$

Letters and the corresponding envelopes with the addresses are written independently of each other. Then, without looking, i.e. purely by chance, the letters are put into the envelopes. What is the probability that no letter is in the right envelope? Euler solved this problem and published it in 1751 in the essay "Calcul de la probabilité dans le jeu de rencontre." It is remarkable that from a number of 7 letters the probability almost does not change. It is very well $1/e=0{,}367879\dots \approx 36{,}7879\,\%$ approximated by , the limit of probabilities as the number of letters becomes larger and larger.

A hunter has only one shot at his disposal. He is supposed to shoot the largest pigeon from a flock of pigeons whose number knows, which fly past him in random order. Which strategy maximises his chances of hitting the largest pigeon? This pigeon problem was formulated by the American mathematician Herbert Robbins (* 1915). The same decision problem also exists for hiring the best employee given n applicants (secretary problem) and similar dressings. Solution : The optimal strategy is to first select pigeons $(k<n)$ , and then shoot at the next pigeon that is larger than all those that have flown by so far, or at the very last one if no larger one has flown by by then. The probability of catching the largest pigeon with this optimal strategy is about 1/e independent of n, which should not be too small, however. If we 1/e choose $k/n$ as an estimate for then it follows : $k\approx 1/e*n$ . So, if there are 27 pigeons, one should first let 10 fly by. It is remarkable that in about 2/3 all cases one does not obtain the desired optimal solution.

In the Poisson, exponential and normal distributions, is used among other quantities to describe the distribution.

Characterisation of Euler's number according to Steiner

In the fortieth volume of Crelle's Journal from 1850, the Swiss mathematician Jakob Steiner gives a characterisation of Euler's number according to which can be understood as the solution of an extreme value problem. Steiner showed that the number can be characterised as the uniquely determined positive real number that yields the largest root when taking the root with itself. Steiner literally wrote: "If every number is radicated by itself, then the number e grants the largest root.

Steiner deals here with the question of whether, for the function of

$f\colon (0,\infty )\to (0,\infty ),\;x\mapsto f(x)={\sqrt[{x}]{x}}=x^{\frac {1}{x}}$

the global maximum exists and how it is to be determined. His statement is that it exists and that it is assumed in and only in $x_\mathrm{max} = e$ .

In his book Triumph der Mathematik, Heinrich Dörrie gives an elementary solution to this extreme value problem. His approach starts from the following true statement about the real exponential function:

$\forall y\in \mathbb {R} \setminus \{0\}\colon e^{y}>1+y$

After the substitution $y={\frac {x-e}{e}}$ follows for all real numbers $x\neq e$

$e^{{{\frac {x-e}{e}}}}>1+{\frac {x-e}{e}},$

further by means of simple transformations

$e^{{{\frac {x}{e}}}}>x$

and finally for all positive $x\neq e$ by erasing

${\sqrt[ {e}]{e}}>{\sqrt[ {x}]{x}}.$

Fractional approximations

For the number and quantities derived from it, there are various approximate representations using fractions. Charles Hermite found the following approximations:

$e\approx {\frac {58291}{21444}}\approx 2{,}718289498$

$e^{2}\approx {\frac {158452}{21444}}\approx 7{,}38910651$

Here the former fraction deviates from by less than 0.0003 per cent.

The optimal fractional approximation in the three-digit number range, i.e. the optimal fractional approximation $e \approx \frac{Z_0}{N_0}$ with N_0, Z_0 < 1000 , is

$e \approx \frac{878}{323} \approx 2{,}718266254$ .

However, this approximation is not the best fraction approximation in the sense of the requirement that the denominator should have at most three digits. The best fraction approximation in this sense results as the 9th approximation fraction of the continued fraction development of Euler's number:

$e \approx \frac{1457}{536} \approx 2{,}71828358 \dots$

From the approximate fractions of the continued fraction expansions belonging to (see above), fraction approximations of arbitrary precision for and quantities derived from it are obtained. With these, one can very efficiently find best approximations of Euler's number in any number range. Thus, for example, in the five-digit number range, the best fractional approximation obtains

$e \approx \frac{49171}{18089} \approx 2{,}718281828735 \dots$ ,

which shows that the fractional approximation found by Charles Hermite for Euler's number in the five-digit number range was not yet optimal.

In the same way, C. D. Olds, for example, has shown that by approximating

$\frac {e - 1}{2} \approx \frac{342762}{398959}$

for Euler's number a further improvement, namely

$e \approx \frac{1084483}{398959} \approx 2{,}7182818284585 \dots$ ,

is to be achieved.

Overall, the sequence of the best approximate fractions of Euler's number resulting from its regular continued fraction representation begins as follows:

${\frac {p_{0}}{q_{0}}}=[2]={\frac {2}{1}}$

${\frac {p_{1}}{q_{1}}}=[2;1]={\frac {3}{1}}$

${\frac {p_{2}}{q_{2}}}=[2;1,2]={\frac {8}{3}}$

${\frac {p_{3}}{q_{3}}}=[2;1,2,1]={\frac {11}{4}}$

${\frac {p_{4}}{q_{4}}}=[2;1,2,1,1]={\frac {19}{7}}$

${\frac {p_{5}}{q_{5}}}=[2;1,2,1,1,4]={\frac {87}{32}}$

${\frac {p_{6}}{q_{6}}}=[2;1,2,1,1,4,1]={\frac {106}{39}}$

${\frac {p_{7}}{q_{7}}}=[2;1,2,1,1,4,1,1]={\frac {193}{71}}$

${\frac {p_{8}}{q_{8}}}=[2;1,2,1,1,4,1,1,6]={\frac {1264}{465}}$

${\frac {p_{9}}{q_{9}}}=[2;1,2,1,1,4,1,1,6,1]={\frac {1457}{536}}$

${\frac {p_{10}}{q_{10}}}=[2;1,2,1,1,4,1,1,6,1,1]={\frac {2721}{1001}}$

${\frac {p_{11}}{q_{11}}}=[2;1,2,1,1,4,1,1,6,1,1,8]={\frac {23225}{8544}}$

$\dots$

${\frac {p_{20}}{q_{20}}}=[2;1,2,1,1,4,1,1,6,1,1,8,1,1,10,1,1,12,1,1,14]={\frac {410105312}{150869313}}$

$\dots$

Calculation of the decimal places

To calculate the decimal places, the row representation is usually used.

$e=\sum _{k=0}^{\infty }{\frac {1}{k!}}={\frac {1}{0!}}+{\frac {1}{1!}}+{\frac {1}{2!}}+{\frac {1}{3!}}+{\frac {1}{4!}}+\dotsb =1+1+{\frac {1}{2}}+{\frac {1}{6}}+{\frac {1}{24}}+\dotsb$

which converges quickly. Long-number arithmetic is important in the implementation so that the rounding errors do not falsify the result. A method that is also based on this formula, but does not require an elaborate implementation, is the droplet algorithm for calculating the decimal places of which A. H. J. Sale found.

Development of the number of known decimal places of
Date	Number	Mathematician
1748	23	Leonhard Euler
1853	137	William Shanks
1871	205	William Shanks
1884	346	J. Marcus Boorman
1946	808	?
1949	2.010	John von Neumann (calculated on the ENIAC)
1961	100.265	Daniel Shanks and John Wrench
1981	116.000	Steve Wozniak (calculated with the help of an Apple II)
1994	10.000.000	Robert Nemiroff and Jerry Bonnell
May 1997	18.199.978	Patrick Demichel
August 1997	20.000.000	Birger Seifert
September 1997	50.000.817	Patrick Demichel
February 1999	200.000.579	Sebastian Wedeniwski
October 1999	869.894.101	Sebastian Wedeniwski
21 November 1999	1.250.000.000	Xavier Gourdon
10 July 2000	2.147.483.648	Shigeru Kondo and Xavier Gourdon
16 July 2000	3.221.225.472	Colin Martin and Xavier Gourdon
2 August 2000	6.442.450.944	Shigeru Kondo and Xavier Gourdon
16 August 2000	12.884.901.000	Shigeru Kondo and Xavier Gourdon
21 August 2003	25.100.000.000	Shigeru Kondo and Xavier Gourdon
18 September 2003	50.100.000.000	Shigeru Kondo and Xavier Gourdon
27 April 2007	100.000.000.000	Shigeru Kondo and Steve Pagliarulo
6 May 2009	200.000.000.000	Shigeru Kondo and Steve Pagliarulo
20 February 2010	500.000.000.000	Alexander Yee
5 July 2010	1.000.000.000.000	Shigeru Kondo
24 June 2015	1.400.000.000.000	Ellie Hebert
14 February 2016	1.500.000.000.000	Ron Watkins
29 May 2016	2.500.000.000.000	"yoyo" - unverified calculation
29 August 2016	5.000.000.000.000	Ron Watkins
3 January 2019	8.000.000.000.000	Gerald Hofmann
11 July 2020	12.000.000.000.000	David Christle
22 November 2020	31.415.926.535.897	David Christle

Euler's number in the media

There are many mathematical references in the television series The Simpsons and its sequel Futurama, some of which have to do with Euler's number and Euler.

In 1995, in the television series The X-Files, the series of numbers 2-7-1-8-2-8 granted two FBI agents access to a secret archive. There, they were not talking about Euler's number, but about Napier's constants.

Questions and Answers

Q: What is the number e?

A: The number e is a mathematical constant that is the base of natural logarithm and has a value of approximately 2.71828.

Q: Who is Euler and why is e sometimes called Euler's number?

A: Euler was a Swiss mathematician and e is sometimes called Euler's number after him because he made important contributions to its study.

Q: Who is Napier and why is e sometimes called Napier's constant?

A: Napier was a Scottish mathematician who introduced logarithms, and e is sometimes called Napier's constant in his honor.

Q: Is e an important mathematical constant?

A: Yes, e is an important mathematical constant that is equally important as π and i.

Q: What kind of number is e?

A: e is an irrational number that cannot be represented as a ratio of integers and is also transcendental (not a root of any non-zero polynomial with rational coefficients).

Q: Why is the number e important in mathematics?

A: The number e is important in mathematics because it has great significance for exponential functions, and it is part of a group of five important mathematical constants that appear in one formulation of Euler's identity.

Q: Who discovered the number e and when?

A: The number e was discovered by the Swiss mathematician Jacob Bernoulli in 1683 while he was studying compound interest.