The mathematical constant e is the base of the natural logarithm. It is occasionally called Euler's number after the Swiss mathematician Leonhard Euler, or Napier's constant in honor of the Scottish mathematician John Napier who introduced logarithms. (Note: This constant is not to be confused with γ, the Euler-Mascheroni constant, which is itself sometimes called Euler's constant.) It is one of the most important numbers in mathematics, alongside the additive and multiplicative identities 0 and 1, the imaginary unit i, and π, the circumference to diameter ratio for any circle. It has a number of equivalent definitions; some of them are given below. To the 20th decimal place:

e ≈ 2.71828 18284 59045 23536

History

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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 natural logarithms calculated from the constant. It is assumed that the table was written by William Oughtred. The first indication of e as a constant was discovered by Jacob Bernoulli, trying to find the value of the following expression:

$\backslash lim\_\{n\backslash to\backslash infty\}\; \backslash left(1+\backslash frac\{1\}\{n\}\backslash right)^n$

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 started to use the letter e for the constant in 1727, and the first use of e in a publication was Euler's Mechanica (1736). While in the subsequent years some researchers used the letter c, e was more common and eventually became the standard.

The exact reasons for the use of the letter e are unknown, but it may be because it is the first letter of the word exponential. Another possibility is that Euler used it because it was the first vowel after a, which he was already using for another number, but his reason for using vowels is unknown. It is unlikely that Euler chose the letter because it is his last initial, since he was a very modest man, and tried to give proper credit to the work of others.

Definitions

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The three most common definitions of e are listed below.

1. The limit

   $e\; =\; \backslash lim\_\{n\backslash to\backslash infty\}\; \backslash left(\; 1\; +\; \backslash frac\{1\}\{n\}\; \backslash right)^n$

2. The sum of the infinite series

   $e\; =\; \backslash sum\_\{n\; =\; 0\}^\backslash infty\; \backslash frac\{1\}\{n!\}\; =\; \backslash frac\{1\}\{0!\}\; +\; \backslash frac\{1\}\{1!\}\; +\; \backslash frac\{1\}\{2!\}\; +\; \backslash frac\{1\}\{3!\}\; +\; \backslash frac\{1\}\{4!\}\; +\; \backslash cdots$

   where n! is the factorial of n.

3. The unique real number e > 0 such that

   $\backslash int\_\{1\}^\{e\}\; \backslash frac\{1\}\{t\}\; \backslash ,\; dt\; =\; \{1\}$

   (that is, the number e such that area under the hyperbola $f(t)=1/t$ from 1 to e is equal to 1).

These definitions can be proved to be equivalent.

Properties

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The exponential function f(x)=e^x is important because it is the unique nontrivial function (up to multiplication by a constant) which is its own derivative, and therefore, its own primitive:

$\backslash frac\{d\}\{dx\}e^x=e^x$ and

$\backslash int\; e^x\backslash ,dx=e^x\; +\; C$, where C is the arbitrary constant of integration.

It is known that e is irrational (proof) and transcendental (proof). It was the first number to be proved transcendental without having been specifically constructed for this purpose (cf. Liouville number); the proof was given by Charles Hermite in 1873. It is conjectured to be normal. It features in Euler's formula, one of the most important formulas in mathematics:

$e^\{ix\}\; =\; \backslash cos(x)\; +\; i\backslash sin(x),\backslash ,\backslash !$

described by Richard Feynman (p. I-22-10) as "the most remarkable formula in mathematics [..., our jewel".

The special case with x = π is known as Euler's identity:

$e^\{i\backslash pi\}+1\; =0\; .\backslash ,\backslash !$

The following is an infinite simple continued fraction expansion of e (sequence A005131 in OEIS):

$e\; =\backslash textbf\{0\},\; 1,\; 1,\; \backslash textbf\{2\},\; 1,\; 1,\; \backslash textbf\{4\},\; 1,\; 1,\; \backslash textbf\{6\},\; 1,\; 1,\; \backslash textbf\{8\},\; 1,\; \backslash ldots,1,\; \backslash textbf\{2n\},\; 1,\backslash ldots\backslash ,$

The following is an infinite generalized continued fraction expansion of e:

$e=\; 2+\backslash frac\{1\}\{1+\backslash frac\{1\}\{2+\backslash frac\{2\}\{3+\backslash frac\{3\}\{\backslash ddots\}\}\}\}.$

The number e is also equal to the sum of the following infinite series:

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

$e\; =\; \backslash left\backslash sum\_\{k=0\}^\backslash infty\; \backslash frac\{1-2k\}\{(2k)!\}\; \backslash right^\{-1\}$

$e\; =\; \backslash frac\{1\}\{2\}\; \backslash sum\_\{k=0\}^\backslash infty\; \backslash frac\{k+1\}\{k!\}$

$e\; =\; 2\; \backslash sum\_\{k=0\}^\backslash infty\; \backslash frac\{k+1\}\{(2k+1)!\}$

$e\; =\; \backslash sum\_\{k=0\}^\backslash infty\; \backslash frac\{3-4k^2\}\{(2k+1)!\}$

$e\; =\; \backslash sum\_\{k=0\}^\backslash infty\; \backslash frac\{(3k)^2+1\}\{(3k)!\}$

$e\; =\; \backslash left\backslash sum\_\{k=0\}^\backslash infty\; \backslash frac\{4k+3\}\{2^\{2k+1\}\backslash ,(2k+1)!\}\; \backslash right^2$

$e\; =\; \backslash frac\{-12\}\{\backslash pi^2\}\; \backslash left\backslash sum\_\{k=1\}^\backslash infty\; \backslash frac\{1\}\{k^2\}\; \backslash \; \backslash cos\; \backslash left\; (\; \backslash frac\{9\}\{k\backslash pi+\backslash sqrt\{k^2\backslash pi^2-9\}\}\; \backslash right\; )\; \backslash right^\{-1/3\}$

$e\; =\; \backslash sum\_\{k=1\}^\backslash infty\; \backslash frac\{k^2\}\{2(k!)\}$

The number e is also given by several infinite product forms including the Pippenger product

$e=\; 2\; \backslash left\; (\; \backslash frac\{2\}\{1\}\; \backslash right\; )^\{1/2\}\; \backslash left\; (\; \backslash frac\{2\}\{3\}\backslash ;\; \backslash frac\{4\}\{3\}\; \backslash right\; )^\{1/4\}\; \backslash left\; (\; \backslash frac\{4\}\{5\}\backslash ;\; \backslash frac\{6\}\{5\}\backslash ;\; \backslash frac\{6\}\{7\}\backslash ;\; \backslash frac\{8\}\{7\}\; \backslash right\; )^\{1/8\}\; \backslash cdots$

as well as,

$\backslash frac\{2\backslash cdot\; 2^\{(\backslash ln(2)-1)^2\}\; \backslash cdots\}\{2^\{\backslash ln(2)-1\}\backslash cdot\; 2^\{(\backslash ln(2)-1)^3\}\backslash cdots\; \}$

The number e is equal to the limit of several infinite sequences:

$e=\; \backslash lim\_\{n\; \backslash to\; \backslash infty\}\; n\backslash cdot\backslash left\; (\; \backslash frac\{\backslash sqrt\{2\; \backslash pi\; n\}\}\{n!\}\; \backslash right\; )^\{1/n\}$ and

$e=\backslash lim\_\{n\; \backslash to\; \backslash infty\}\; \backslash frac\{n\}\{\backslash sqrt*\{n!\}\}$ (both by Stirling's formula).

The symmetric limit,

$e=\backslash lim\_\{n\; \backslash to\; \backslash infty\}\; \backslash left\backslash frac\{(n+1)^\{n+1\}\}\{n^n\}-\; \backslash frac\{n^n\}\{(n-1)^\{n-1\}\}\; \backslash right$

may be obtained by manipulation of the basic limit definition of e. Another limit is

$e=\; \backslash lim\_\{n\; \backslash to\; \backslash infty\}(p\_n\; \backslash \#)^\{1/p\_n\}$

where $p\_n$ is the nth prime and $p\_n\; \backslash \#$ is the primorial of the nth prime.

It was shown by Euler that the infinite tetration

$x^\{x^\{\backslash cdot^\{\backslash cdot^\{\backslash cdot\}\}\}\},$

converges only if $e^\{-e\}\; \backslash le\; x\; \backslash le\; e^\{1/e\}.$

The number e is the global maximum of the function

$f(x)\; =\; x^\{1\; \backslash over\; x\}.$

The value of this function at e is

$f(e)\; \backslash approx\; 1.444667861...$

Non-mathematical uses of e

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One of the most famous mathematical constants, e is also frequently referenced outside of mathematics. Some examples are:

• 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 to the nearest dollar.

• Google was also responsible for a mysterious billboard that appeared in the heart of Silicon Valley, and later in Cambridge, Massachusetts, which read {first 10-digit prime found in consecutive digits of e}.com. Solving this problem and visiting the web site advertised led to an even more difficult problem to solve, which in turn leads to Google Labs where the visitor is invited to submit a resume. The first 10-digit prime in e is 7427466391, which surprisingly starts as late as at the 101st digit. [http://www.mkaz.com/math/google/

• The famous computer scientist Donald Knuth let the version numbers of his program METAFONT approach e (the versions are 2, 2.7, 2.71, 2.718, etc.).

References

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• Maor, Eli; e: The Story of a Number, ISBN 0691058547
• O'Connor, J.J., and Roberson, E.F.; The MacTutor History of Mathematics archive: "The number e"; University of St Andrews Scotland (2001)

Notes

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O'Connor, "The number e"

External links

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• The number e to 1 million places and 2, 5 or 10 million places
• Earliest Uses of Symbols for Constants
• e the EXPONENTIAL - the Magic Number of GROWTH - Keith Tognetti, University of Wollongong, NSW, Australia

Transcendental numbers | Mathematical constants | Exponentials | Logarithms | Famous numbers

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