E (mathematical Constant)
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E Mathematical Constant
Graph of the equation Here, e is the unique number larger than 1 that makes the shaded area equal to 1.

The number e is a mathematical constant, approximately equal to 2.71828, which appears in many different settings throughout mathematics. It was discovered by the Swiss mathematician Jacob Bernoulli while studying compound interest,[1] where e arises as the limit of (1 + 1/n)n as n approaches infinity. The number e can also be calculated as the sum of the infinite series[2]

Also called Euler's number after the Swiss mathematician Leonhard Euler, e is different from ?, the Euler-Mascheroni constant, which is sometimes called simply Euler's constant. Occasionally, the number e is termed Napier's constant, but Euler's choice of the symbol e is said to have been retained in his honor.[3][better source needed]

The number e is of eminent importance in mathematics,[4] alongside 0, 1, ? 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 ?, e is irrational: it is not a ratio of integers. Also like ?, 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 .

The function f(x)=ex is called the (natural) exponential function, and is the unique exponential function of type ax equal to its own derivative (f(x) = f(x) = ex). This is easily spotted at x = 0, where ax = 1 for any value of a, but a = e is the only positive number such that the slope at x = 0 of the graph of the function y = ax also equals 1.[5]

The natural logarithm, or logarithm to base e, is the inverse function to the natural exponential function. The natural logarithm of a number k > 1 can also 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). In this view e is the unique number whose natural logarithm is equal to one.[6]


The first references to the constant were published in 1618 in the table of an appendix of a work on logarithms by John Napier.[1] 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, the letter e was more common and eventually became the standard.[]

The constant has been historically typeset as "e", in italics, although the ISO 80000-2:2009 standard recommends typesetting constants in an upright style.


Compound interest


Jacob Bernoulli discovered this constant in 1683 by studying a question about compound interest:[1]




Bernoulli trials


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.







Optimal planning problems

A stick of length L is broken into n equal parts. The value of n that maximizes the product of the lengths is then either[15]

The stated result follows because the maximum value of occurs at (Steiner's problem, discussed below). The quantity is a measure of information gleaned from an event occurring with probability , so that essentially the same optimal division appears in optimal planning problems like the secretary problem.



A particular consequence of this is


Standard normal distribution


The factor in this expression ensures that the total area under the curve ?(x) is equal to one.[proof] The 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

The functions are shown for (dotted), (blue), and (dashed). They all pass through the point but only is there tangent to the red line of slope
The value of the natural log function for argument e, i.e. ln(e), equals 1.

The principal motivation for introducing the number e, particularly in calculus, is to perform differential and integral calculus with exponential functions and logarithms.[16] A general exponential has a derivative, given by a limit:

The parenthesized limit on the right is independent of the it depends only on the When the base is set this limit is equal and so e is symbolically defined by the equation:

Consequently, the exponential function with base e is particularly suited to doing calculus. 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 derivative of the base-a logarithm,[17] i.e., of logax for x > 0:

where the substitution u = h/x was made. The a-logarithm of e is 1, if a equals e. So symbolically,

The logarithm with this special base is called the natural logarithm and is denoted 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 such special numbers a. One way is to set the derivative of the exponential function ax equal 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. It turns out that these two solutions for a are actually the same, the number e.

Alternative characterizations



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

The following four 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.
  4. If f(t) is an exponential function, then the quantity is a constant, sometimes called the time constant (it is the reciprocal of the exponential growth constant or decay constant). The time constant is the time it takes for the exponential function to increase by a factor of e: .




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


Exponential-like functions

The global maximum of occurs at x = e.

Steiner's problem asks to find the global maximum 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 e-ex e1/e (or approximately between 0.0660 and 1.4447), due to a theorem of Leonhard Euler.[20]

Number theory




Complex numbers

The exponential function ex may be written as a Taylor series


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


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:



given above, as well as the series


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


which written out looks like

This continued fraction for e converges three times as quickly:

which written out looks like


Stochastic representations



Known digits


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





  2. ^ Encyclopedic Dictionary of Mathematics 142.D
  3. ^ Sondow, Jonathan. "e". Wolfram Mathworld. Wolfram Research. Retrieved 2011. 
  6. ^ Oxford English Dictionary, 2nd ed.: natural logarithm
  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) ... )
  15. ^ Steven Finch (2003). Mathematical constants. Cambridge University Press. p. 14. 
  17. ^ This is the approach taken by Kline (1998).
  18. ^ Dorrie, Heinrich (1965). 100 Great Problems of Elementary Mathematics. Dover. pp. 44-48. 
  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 . 
  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, ... )
  29. ^ William Shanks, Contributions to Mathematics, ... (London, England: G. Bell, 1853), page 89.
  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 
  34. ^ Alexander Yee. "e". 
  37. ^ Kazmierczak, Marcus (2004-07-29). "Google Billboard". mkaz.com. Retrieved . 
  38. ^ Knuth, Donald (1990-10-03). "The Future of TeX and Metafont" (PDF). TeX Mag. 5 (1): 145. Retrieved . 

Further reading

External links




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