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e (mathematical constant)

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The graph is , and e is the unique number that makes the shaded area equal to 1.

The number e is a mathematical constant that is the base of the natural logarithm: the unique number whose natural logarithm is equal to one. It is approximately equal to 2.71828,[1] and is the limit of (1 + 1/n)n as n approaches infinity, an expression that arises in the study of compound interest. It can also be calculated as the sum of the infinite series[2]

The constant can be characterized in many different ways. For example, e can be defined as the unique positive number a such that the graph of the function y = ax has unit slope at x = 0.[3] The function f(x) = ex is called the (natural) exponential function. The natural logarithm, or logarithm to base e, is the inverse function to the natural exponential function. The natural logarithm of a positive number k can be defined directly as the area under the curve y = 1/x between x = 1 and x = k, in which case e is the value of k for which this area equals one (see image). There are alternative characterizations.

Sometimes called Euler's number after the Swiss mathematician Leonhard Euler, e is not to be confused with γ, the Euler–Mascheroni constant, sometimes called simply Euler's constant. The number e is also known as Napier's constant, but Euler's choice of the symbol e is said to have been retained in his honor.[4] The constant was discovered by the Swiss mathematician Jacob Bernoulli while studying compound interest.[5]

The number e is of eminent importance in mathematics,[6] alongside 0, 1, π and i. All five of these numbers play important and recurring roles across mathematics, and are the five constants appearing in one formulation of Euler's identity. Like the constant π, e is irrational: it is not a ratio of integers. Also like π, e is transcendental: it is not a root of any non-zero polynomial with rational coefficients. The numerical value of e truncated to 50 decimal places is

2.71828182845904523536028747135266249775724709369995... (sequence A001113 in the OEIS).


The first references to the constant were published in 1618 in the table of an appendix of a work on logarithms by John Napier.[5] However, this did not contain the constant itself, but simply a list of logarithms calculated from the constant. It is assumed that the table was written by William Oughtred. The discovery of the constant itself is credited to Jacob Bernoulli in 1683,[7][8] who attempted to find the value of the following expression (which is in fact e):

The first known use of the constant, represented by the letter b, was in correspondence from Gottfried Leibniz to Christiaan Huygens in 1690 and 1691. Leonhard Euler introduced the letter e as the base for natural logarithms, writing in a letter to Christian Goldbach of 25 November 1731.[9][10] Euler started to use the letter e for the constant in 1727 or 1728, in an unpublished paper on explosive forces in cannons,[11] and the first appearance of e in a publication was in Euler's Mechanica (1736).[12] While in the subsequent years some researchers used the letter c, e was more common and eventually became the standard.


Compound interest

The effect of earning 20% annual interest on an initial $1,000 investment at various compounding frequencies

Jacob Bernoulli discovered this constant in 1683 by studying a question about compound interest:[5]

An account starts with $1.00 and pays 100 percent interest per year. If the interest is credited once, at the end of the year, the value of the account at year-end will be $2.00. What happens if the interest is computed and credited more frequently during the year?

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 $1.00×1.52 = $2.25 at the end of the year. Compounding quarterly yields $1.00×1.254 = $2.4414..., and compounding monthly yields $1.00×(1+1/12)12 = $2.613035... If there are n compounding intervals, the interest for each interval will be 100%/n and the value at the end of the year will be $1.00×(1 + 1/n)n.

Bernoulli noticed that this sequence approaches a limit (the force of interest) with larger n and, thus, smaller compounding intervals. Compounding weekly (n = 52) yields $2.692597..., while compounding daily (n = 365) yields $2.714567..., just two cents more. The limit as n grows large is the number that came to be known as e; with continuous compounding, the account value will reach $2.7182818.... More generally, an account that starts at $1 and offers an annual interest rate of R will, after t years, yield eRt dollars with continuous compounding. (Here R is the decimal equivalent of the rate of interest expressed as a percentage, so for 5% interest, R = 5/100 = 0.05)

Bernoulli trials

Graphs of probability P of not observing independent events each of probability 1/n after n Bernoulli trials, and 1 - P  vs n ; it can be observed that as n increases, the probability of a 1/n-chance event never appearing after n tries rapidly converges to 1/e.

The number e itself also has applications to probability theory, where it arises in a way not obviously related to exponential growth. Suppose that a gambler plays a slot machine that pays out with a probability of one in n and plays it n times. Then, for large n (such as a million) the probability that the gambler will lose every bet is approximately 1/e. For n = 20 it is already approximately 1/2.79.

This is an example of a Bernoulli trials process. Each time the gambler plays the slots, there is a one in one million chance of winning. Playing one million times is modelled by the binomial distribution, which is closely related to the binomial theorem. The probability of winning k times out of a million trials is:

In particular, the probability of winning zero times (k = 0) is

This is very close to the following limit for 1/e:


Another application of e, also discovered in part by Jacob Bernoulli along with Pierre Raymond de Montmort, is in the problem of derangements, also known as the hat check problem:[13] n guests are invited to a party, and at the door each guest checks his hat with the butler who then places them into n boxes, each labelled with the name of one guest. But the butler does not know 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. The answer is:

As the number n of guests tends to infinity, pn approaches 1/e. 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 n!/e rounded to the nearest integer, for every positive n.[14]


The number e occurs naturally in connection with many problems involving asymptotics. A prominent example is Stirling's formula for the asymptotics of the factorial function, in which both the numbers e and π enter:

A particular consequence of this is


Standard normal distribution

The simplest case of a normal distribution is known as the standard normal distribution, described by this probability density function:

The factor in this expression ensures that the total area under the curve ϕ(x) is equal to one.[proof] The 1/2 in the exponent ensures that the distribution has unit variance (and therefore also unit standard deviation). This function is symmetric around x = 0, where it attains its maximum value ; and has inflection points at +1 and −1.

In calculus

Function f(x) = ax for several values of a. e is the value of a such that the gradient of f(x) = ax at x = 0 equals 1. This is the blue curve, ex. Functions 2x (dotted curve) and 4x (dashed curve) are also shown; they are not tangent to the line of slope 1 (red).
The natural log at (x-axis) e, ln(e), is equal to 1

The principal motivation for introducing the number e, particularly in calculus, is to perform differential and integral calculus with exponential functions and logarithms.[15] A general exponential function y = ax has derivative given as the limit:

The limit on the far right is independent of the variable x: it depends only on the base a. When the base is e, this limit is equal to 1, and so e is symbolically defined by the equation:

Consequently, the exponential function with base e is particularly suited to doing calculus. Choosing e, as opposed to some other number, as the base of the exponential function makes calculations involving the derivative much simpler.

Another motivation comes from considering the base-a logarithm.[16] Considering the definition of the derivative of loga x as the limit:

where the substitution u = h/x was made in the last step. The last limit appearing in this calculation is again an undetermined limit that depends only on the base a, and if that base is e, the limit is equal to 1. So symbolically,

The logarithm in this special base is called the natural logarithm and is represented as ln; it behaves well under differentiation since there is no undetermined limit to carry through the calculations.

There are thus two ways in which to select a special number a = e. One way is to set the derivative of the exponential function ax to ax, and solve for a. The other way is to set the derivative of the base a logarithm to 1/x and solve for a. In each case, one arrives at a convenient choice of base for doing calculus. In fact, these two solutions for a are actually the same, the number e.

Alternative characterizations

The five shaded regions are of equal area, and define units of hyperbolic angle along the hyperbola .

Other characterizations of e 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:

  1. The number e is the unique positive real number such that
  2. The number e is the unique positive real number such that

The following three characterizations can be proven equivalent:

  1. The number e is the limit


  2. The number e is the sum of the infinite series
    where n! is the factorial of n.
  3. The number e is the unique positive real number such that[17]



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

and therefore its own antiderivative as well:


The number e is the unique real number such that

for all positive x.[18]

Also, we have the inequality

for all real x, with equality if and only if x = 0. Furthermore, e is the unique base of the exponential for which the inequality axx + 1 holds for all x.

Exponential-like functions

The global maximum of occurs at x = e.

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

This maximum occurs precisely at x = e. For proof, the inequality , from above, evaluated at and simplifying gives . So for all positive x.[19]

Similarly, x = 1/e is where the global minimum occurs for the function

defined for positive x. More generally, for the function

the global maximum for positive x occurs at x = 1/e for any n < 0; and the global minimum occurs at x = e−1/n for any n > 0.

The infinite tetration


converges if and only if eexe1/e (or approximately between 0.0660 and 1.4447), due to a theorem of Leonhard Euler.[20]

Number theory

The real number e is irrational. Euler proved this by showing that its simple continued fraction expansion is infinite.[21] (See also Fourier's proof that e is irrational.)

Furthermore, by the Lindemann–Weierstrass theorem, e 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 e is normal, meaning that when e is expressed in any base the possible digits in that base are uniformly distributed (occur with equal probability in any sequence of given length).

Complex numbers

The exponential function ex may be written as a Taylor series

Because this series keeps many important properties for ex even when x is complex, it is commonly used to extend the definition of ex to the complex numbers. This, with the Taylor series for sin and cos x, allows one to derive Euler's formula:

which holds for all x. The special case with x = π is Euler's identity:

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

Furthermore, using the laws for exponentiation,

which is de Moivre's formula.

The expression

is sometimes referred to as cis(x).

Differential equations

The general function

is the solution to the differential equation:


The number e can be represented as a real number in a variety of ways: as an infinite series, an infinite product, a continued fraction, or a limit of a sequence. The chief among these representations, particularly in introductory calculus courses is the limit

given above, as well as the series

given by evaluating the above power series for ex at x = 1.

Less common is the continued fraction (sequence A003417 in the OEIS).


which written out looks like

This continued fraction for e converges three times as quickly:

which written out looks like

Many other series, sequence, continued fraction, and infinite product representations of e have been developed.

Stochastic representations

In addition to exact analytical expressions for representation of e, there are stochastic techniques for estimating e. One such approach begins with an infinite sequence of independent random variables X1, X2..., drawn from the uniform distribution on [0, 1]. Let V be the least number n such that the sum of the first n observations exceeds 1:

Then the expected value of V is e: E(V) = e.[23][24]

Known digits

The number of known digits of e has increased substantially during the last decades. This is due both to the increased performance of computers and to algorithmic improvements.[25][26]

Number of known decimal digits of e
Date Decimal digits Computation performed by
1690 1 Jacob Bernoulli[7]
1714 13 Roger Cotes[27]
1748 23 Leonhard Euler[28]
1853 137 William Shanks[29]
1871 205 William Shanks[30]
1884 346 J. Marcus Boorman[31]
1949 2,010 John von Neumann (on the ENIAC)
1961 100,265 Daniel Shanks and John Wrench[32]
1978 116,000 Steve Wozniak on the Apple II[33]

Since that time, the proliferation of modern high-speed desktop computers has made it possible for amateurs, with the right hardware, to compute trillions of digits of e.[34]

In computer culture

In contemporary internet culture, individuals and organizations frequently pay homage to the number e.

For instance, in 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, which is e billion dollars rounded to the nearest dollar. Google was also responsible for a billboard[35] 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 e}.com". Solving this problem and visiting the advertised (now defunct) web site led to an even more difficult problem to solve, which in turn led to Google Labs where the visitor was invited to submit a resume.[36] The first 10-digit prime in e is 7427466391, which starts at the 99th digit.[37]

In another instance, the computer scientist Donald Knuth let the version numbers of his program Metafont approach e. The versions are 2, 2.7, 2.71, 2.718, and so forth.[38]


  1. ^ Oxford English Dictionary, 2nd ed.: natural logarithm
  2. ^ Encyclopedic Dictionary of Mathematics 142.D
  3. ^ Jerrold E. Marsden, Alan Weinstein (1985). Calculus. Springer. ISBN 0-387-90974-5. 
  4. ^ Sondow, Jonathan. "e". Wolfram Mathworld. Wolfram Research. Retrieved 10 May 2011. 
  5. ^ a b c O'Connor, J J; Robertson, E F. "The number e". MacTutor History of Mathematics. 
  6. ^ Howard Whitley Eves (1969). An Introduction to the History of Mathematics. Holt, Rinehart & Winston. ISBN 0-03-029558-0. 
  7. ^ a b 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. 219–223. 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 he be owed [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 a=b, debebitur plu quam 2½a & minus quam 3a." ( … which our series [a geometric series] is larger [than]. … if a=b, [the lender] will be owed more than 2½a and less than 3a.) 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 of page 314.)
  8. ^ Carl Boyer; Uta Merzbach (1991). A History of Mathematics (2nd ed.). Wiley. p. 419. 
  9. ^ 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 page 58. From page 58: " … ( e denotat hic numerum, cujus logarithmus hyperbolicus est = 1), … " ( … (e denotes that number whose hyperbolic [i.e., natural] logarithm is equal to 1) … )
  10. ^ Remmert, Reinhold (1991). Theory of Complex Functions. Springer-Verlag. p. 136. ISBN 0-387-97195-5 
  11. ^ Euler, Meditatio in experimenta explosione tormentorum nuper instituta.
  12. ^ 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 e denotat numerum, cuius logarithmus hyperbolicus est 1. (So it [i.e., c, the speed] will be or , where e denotes the number whose hyperbolic [i.e., natural] logarithm is 1.)
  13. ^ Grinstead, C.M. and Snell, J.L.Introduction to probability theory (published online under the GFDL), p. 85.
  14. ^ Knuth (1997) The Art of Computer Programming Volume I, Addison-Wesley, p. 183 ISBN 0-201-03801-3.
  15. ^ Kline, M. (1998) Calculus: An intuitive and physical approach, section 12.3 "The Derived Functions of Logarithmic Functions.", pp. 337 ff, Courier Dover Publications, 1998, ISBN 0-486-40453-6
  16. ^ This is the approach taken by Kline (1998).
  17. ^ More generally, by the substitution
  18. ^ Dorrie, Heinrich (1965). 100 Great Problems of Elementary Mathematics. Dover. pp. 44–48. 
  19. ^ Dorrie, Heinrich (1965). 100 Great Problems of Elementary Mathematics. Dover. p. 359. 
  20. ^ 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)
  21. ^ Sandifer, Ed (Feb 2006). "How Euler Did It: Who proved e is Irrational?" (PDF). MAA Online. Archived from the original (PDF) on 2014-02-23. Retrieved 2010-06-18. 
  22. ^ Hofstadter, D. R., "Fluid Concepts and Creative Analogies: Computer Models of the Fundamental Mechanisms of Thought" Basic Books (1995) ISBN 0-7139-9155-0
  23. ^ Russell, K. G. (1991) Estimating the Value of e by Simulation The American Statistician, Vol. 45, No. 1. (Feb., 1991), pp. 66–68.
  24. ^ Dinov, ID (2007) Estimating e using SOCR simulation, SOCR Hands-on Activities (retrieved December 26, 2007).
  25. ^ Sebah, P. and Gourdon, X.; The constant e and its computation
  26. ^ Gourdon, X.; Reported large computations with PiFast
  27. ^ Roger 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, … )
  28. ^ Leonhard Euler, Introductio in Analysin Infinitorum (Lausanne, Switzerland: Marc Michel Bousquet & Co., 1748), volume 1, page 90.
  29. ^ William Shanks, Contributions to Mathematics, … (London, England: G. Bell, 1853), page 89.
  30. ^ William Shanks (1871) "On the numerical values of e, loge 2, loge 3, loge 5, and loge 10, also on the numerical value of M the modulus of the common system of logarithms, all to 205 decimals," Proceedings of the Royal Society of London, 20 : 27-29.
  31. ^ J. Marcus Boorman (October 1884) "Computation of the Naperian base," Mathematical Magazine, 1 (12) : 204-205.
  32. ^ Daniel Shanks and John W Wrench (1962). "Calculation of Pi to 100,000 Decimals" (PDF). Mathematics of Computation. 16 (77): 76–99 (78). doi:10.2307/2003813. We have computed e on a 7090 to 100,265D by the obvious program 
  33. ^ Wozniak, Steve (June 1981). "The Impossible Dream: Computing e to 116,000 Places with a Personal Computer". BYTE. p. 392. Retrieved 18 October 2013. 
  34. ^ Alexander Yee. "e". 
  35. ^ First 10-digit prime found in consecutive digits of e}. Brain Tags. Retrieved on 2012-02-24.
  36. ^ Shea, Andrea. "Google Entices Job-Searchers with Math Puzzle". NPR. Retrieved 2007-06-09. 
  37. ^ Kazmierczak, Marcus (2004-07-29). "Google Billboard". Retrieved 2007-06-09. 
  38. ^ Knuth, Donald (1990-10-03). "The Future of TeX and Metafont" (PDF). TeX Mag. 5 (1): 145. Retrieved 2017-02-17. 

Further reading

  • Maor, Eli; e: The Story of a Number, ISBN 0-691-05854-7
  • Commentary on Endnote 10 of the book Prime Obsession for another stochastic representation
  • McCartin, Brian J. (2006). "e: The Master of All" (PDF). The Mathematical Intelligencer. 28 (2): 10–21. doi:10.1007/bf02987150. 

External links

  • The number e to 1 million places and 2 and 5 million places (link obsolete)
  • e Approximations – Wolfram MathWorld
  • Earliest Uses of Symbols for Constants Jan. 13, 2008
  • "The story of e", by Robin Wilson at Gresham College, 28 February 2007 (available for audio and video download)
  • e Search Engine 2 billion searchable digits of e, π and √2
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