e (mathematical constant)

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e is the unique number such that the value of the
derivative of f (x) = ex (blue curve) at the
point x = 0 is exactly 1. For comparison, functions
2x (dotted curve) and 4x (dashed curve) are
shown; they are not tangent to the line of slope 1 (red).

The mathematical
constant e is the unique real number such that the
value of the derivative (slope of the tangent line) of
f(x) = ex at the point x = 0 is
exactly 1. The function ex so defined is called the exponential
function, and its inverse is the natural logarithm, or
logarithm to base
e.

The number e is one of the most important numbers in mathematics,[1]
alongside the additive and multiplicative identities 0 and 1, the
imaginary unit i,
and π, the
circumference-to-diameter ratio for any circle in a plane. It has a number of
equivalent definitions; some of them are given below.

The number e 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 not
to be confused with γ – the Euler–Mascheroni
constant, sometimes called simply Euler's constant.)

Since e is transcendental,
and therefore irrational, its value
cannot be given exactly as a finite or eventually repeating decimal. The
numerical value of e truncated to 20 decimal places is:

2.71828 18284 59045 23536...

Part
of a series of articles on
The mathematical constant, e

Natural
logarithm

Applications in Compound
interest · Euler's
identity & Euler's
formula  · Half
lives & Exponential growth/decay

Defining e Proof
that e is irrational  · Representations
of e · Lindemann–Weierstrass
theorem

People John Napier
· Leonhard
Euler

Schanuel's
conjecture

List of numbers -
Irrational
numbers
ζ(3) - √2 - φ - √3 - √5 - α - e - π - δ

Binary

10.10110111111000010101...

Decimal

2.7182818284590452354...

Hexadecimal

2.B7E151628AED2A6B...

Continued
fraction

Note
that this continued fraction is not periodic.

Contents

1 History

2 Applications

2.1 The compound-interest
problem

2.2 Bernoulli
trials

2.3 Derangements

3 e in calculus

3.1 Alternative
characterizations

4 Properties

4.1 Calculus

4.2 Exponential-like
functions

4.3 Number
theory

4.4 Complex
numbers

5 Representations of
e

5.1 Known digits

6 e in computer
culture

7 Notes

8 References

9 External links

[edit]
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.[2]
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
"discovery" of the constant itself is credited to Jacob Bernoulli, 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 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 the first letter of his
surname, since he was a very modest man and tried to give proper credit to the
work of others.

[edit]
Applications

[edit]
The compound-interest problem

Jacob Bernoulli
discovered this constant by studying a question about compound interest.

One simple example is an account that starts with $1.00 and pays 100%
interest per year. If the interest is credited once, at the end of the year, the
value is $2.00; but if the interest is computed and added twice in the year, the
$1 is multiplied by 1.5 twice, yielding $1.00×1.5² = $2.25.
Compounding quarterly yields $1.00×1.254 = $2.4414…, and
compounding monthly yields
$1.00×(1.0833…)12 = $2.613035….

Bernoulli noticed that this sequence approaches a limit for more and smaller
compounding intervals. Compounding weekly yields $2.692597…, while compounding
daily yields $2.714567…, just two cents more. Using n as the number of
compounding intervals, with interest of 1/n in each interval, the limit
for large n 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 yields (1+R) dollars at
simple interest, will yield eR dollars with continuous
compounding.

[edit]
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 with a one in n probability and plays it n
times. Then, for large n (such as a million) the probability that the gambler
will win nothing at all is (approximately) 1/e.

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:

[edit]
Derangements

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.[3]
Here n guests are invited to a party, and at the door each guest checks
his hat with the butler who then places them into labelled boxes. But the butler
does not know the name of the guests, and so must put them into boxes selected
at random. The problem of de Montmort is: what is 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 is in the right box is exactly
n!/e, rounded to the nearest integer.[4]

[edit]
e in calculus

The natural log at 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.[5]
A general exponential function y=ax has
derivative given as the limit:

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

As a consequence, 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.[6]
Considering the definition of the derivative of logax
as the limit:

Once again, there is an undetermined limit which depends only on the base
a, and if that base is e, the limit is one. So symbolically,

The logarithm in this special base is called the natural logarithm
(often represented as "ln"), and it also 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. The other way is to set the
derivative of the base a logarithm to 1/x. In each case, one
arrives at a convenient choice of base for doing calculus. In fact, these two
bases are actually the same, the number e.

[edit]
Alternative characterizations

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 alternative characterizations can also be proven
to be equivalent:

3. The number e is the limit

The area under the graph y = 1/x is equal to 1
over the interval 1 ≤ x ≤ e.

4. The number e is the sum of the infinite series

where n! is the factorial of n.

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

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

6. The number e is the limit

[edit]
Properties

[edit]
Calculus

As in the motivation, the exponential
function f(x) = 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:

and

[edit]
Exponential-like functions

The number x=e is where the global maximum occurs for
the function

More generally,
is where the global maximum occurs for the function

The infinite tetration

converges only if
due to a theorem of Leonhard Euler.

ex is usually defined as

[edit]
Number theory

The real number e is irrational (see proof that e is
irrational), and furthermore is transcendental (Lindemann–Weierstrass
theorem). 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
to be normal.

[edit]
Complex numbers

It features in Euler's formula, an
important formula related to complex numbers:

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

from which it follows that

Furthermore, using the laws for exponentiation,

which is de Moivre's
formula.

[edit]
Representations of e

Main article: Representations of
e

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.

Still other less common representations are also available. For instance,
e can be represented as an infinite simple continued
fraction:

Or, in a more compact form (sequence A003417 in OEIS):

Which can be written more harmoniously by allowing zero:[7]

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

[edit]
Known digits

The number of known digits of e has increased dramatically during the
last decades. This is due both to the increase of performance of computers as
well as to algorithmic improvements.[8][9]

Number of known decimal digits of e

Date

Decimal digits

Computation performed by

1748

18[10]

Leonhard
Euler

1853

137

William
Shanks

1871

205

William
Shanks

1884

346

J. M. Boorman

1946

808

?

1949

2,010

John von Neumann
(on the ENIAC)

1961

100,265

Daniel Shanks & John W. Wrench

1994

10,000,000

Robert Nemiroff & 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

November 21 1999

1,250,000,000

Xavier Gourdon

July 10 2000

2,147,483,648

Shigeru Kondo & Xavier Gourdon

July 16 2000

3,221,225,472

Colin Martin & Xavier Gourdon

August 2 2000

6,442,450,944

Shigeru Kondo & Xavier Gourdon

August 16 2000

12,884,901,000

Shigeru Kondo & Xavier Gourdon

August 21 2003

25,100,000,000

Shigeru Kondo & Xavier Gourdon

September 18 2003

50,100,000,000

Shigeru Kondo & Xavier Gourdon

April 27 2007

100,000,000,000

Shigeru Kondo & Steve Pagliarulo

[edit]
e in computer culture

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

For example, 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[11]
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 web site 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.[12]
The first 10-digit prime in e is 7427466391, which starts at the 99th
digit.[13]
(A random stream of digits has a 98.4% chance of starting a 10-digit prime
sooner.)

In another instance, the eminent 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.

[edit]
Notes

^
Howard Whitley Eves (1969). An Introduction to the History of Mathematics. Holt,
Rinehart & Winston.

^
O'Connor, J.J., and Roberson, E.F.; The MacTutor History of Mathematics
archive: "The number e"; University of St Andrews Scotland
(2001)

^
Grinstead, C.M. and Snell, J.L. Introduction to probability theory (published online
under the GFDL), p.
85.

^
Knuth (1997) The Art of
Computer Programming Volume I, Addison-Wesley, p. 183.

^
See, for instance, Kline, M. (1998) Calculus: An intuitive and physical
approach, Dover, section 12.3 "The Derived Functions of Logarithmic
Functions."

^
This is the approach taken by Klein (1998).

^
Hofstadter, D. R., "Fluid Concepts and Creative Analogies: Computer Models of
the Fundamental Mechanisms of Thought" Basic Books (1995)

^
Sebah, P. and Gourdon, X.; The
constant e and its computation

^
Gourdon, X.; Reported large computations with PiFast

^
New Scientist 21st July 2007 p.40

^
Archive copy at the Internet
Archive Wayback Machine

^
Shea, Andrea. "Google Entices Job-Searchers with Math Puzzle", NPR.
Retrieved on 2007-06-09.

^
Kazmierczak, Marcus (2004-07-29). Math : Google Labs
Problems. mkaz.com. Retrieved on 2007-06-09.

[edit]
References

Maor, Eli; e: The Story of a Number, ISBN
0-691-05854-7

[edit]
External links

The number e to
1 million places and 2 and 5
million places

Earliest
Uses of Symbols for Constants

e the EXPONENTIAL -
the Magic Number of GROWTH - Keith Tognetti, University of Wollongong,
NSW, Australia

An Intuitive Guide To Exponential Functions & e

"The story of e", by Robin Wilson at Gresham College, 28
February 2007 (available for audio and video download)

Retrieved from "http://en.wikipedia.org/wiki/E_%28mathematical_constant%29"

Categories: Transcendental
numbers | Mathematical
constants | Exponentials
| Logarithms |
Numbers in
pop culture

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