E (mathematical constant)
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.
e 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.
Encyclopedia
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.
e 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
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:
-
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 . 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
The three most common definitions of
e are listed below.
- The limit
- The sum of the infinite series
- where n! is the factorial of n.
- The unique real number e > 0 such that
- .
These definitions can be proved to be equivalent.
Properties
The
exponential function f =
ex is important in part because it is the unique nontrivial function which is its own
derivative, and therefore, its own
primitive:
and
, where
C is the arbitrary constant of integration.
It is known that
e is irrational . In fact, if
m/
n is any rational number with
n > 1, then
where
S is the smallest positive integer such that
n divides
S!.
It is also known that
e is transcendental . It was the first number to be proved transcendental without having been specifically constructed for this purpose ; 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:
described by
Richard Feynman as "[...] the most remarkable formula in mathematics [...], our jewel".
The special case with
x = π is known as
Euler's identity:
The following is an infinite simple continued fraction expansion of
e :
The following is an infinite generalized continued fraction expansion of
e:
The number
e is also equal to the sum of the following infinite series:The number
e is also given by several infinite product forms including Pippenger's product
and Guillera's product
where the
nth factor is the
nth root of the product
as well as the infinite product
The number
e is equal to the limit of several infinite sequences:
and
.
The symmetric limit,
may be obtained by manipulation of the basic limit definition of
e. Another limit is
where is the
nth prime and is the
primorial of the
nth prime.
It was shown by Euler that the infinite
tetrationconverges only if
The number
e is the
global maximum of the function
The value of this function at
e is
Non-mathematical uses of e
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, Seattle, Washington, and Austin, Texas which read .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.
- The famous computer scientist Donald Knuth let the version numbers of his program METAFONT approach e .
References
- Maor, Eli; e: The Story of a Number, ISBN 0-691-05854-7
- O'Connor, J.J., and Roberson, E.F.; The MacTutor History of Mathematics archive: ; University of St Andrews Scotland
- , Amer. Math. Monthly 112 729-734.
- Jonathan Sondow, "A Geometric Proof that e Is Irrational and a New Measure of Its Irrationality," Amer. Math. Monthly 113 637-641
Notes
O'Connor, "The number
e"
External links
- and or
-
- - Keith Tognetti, University of Wollongong, NSW, Australia
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