e is the unique number
a, such that the value of the
derivative (the slope of the tangent line) of the exponential
function
f (
x) =
a^{x} (blue
curve) at the point
x = 0 is exactly 1.
For comparison, functions 2^{x} (dotted
curve) and 4^{x} (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 the function
f(
x)
=
e^{x} at the point
x = 0 is
exactly 1. The function
e^{x} so defined is called
the
exponential function, and
its
inverse is the
natural logarithm, or logarithm to
base e. The number
e is also commonly
defined as the base of the
natural logarithm (using an
integral to
define the latter), as the
limit
of a certain
sequence, or as
the sum of a certain
series
(see the
alternative
characterizations, below).
e is one of the most important numbers in mathematics,
alongside the additive and multiplicative identities
0 and
1, the constant
π, and the
imaginary
unit i. (These are the five constants appearing in one
formulation of
Euler's
identity.)
The number
e is 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
irrational; it is not a ratio of integers.
Furthermore, it is
transcendental; it is not a root of
any non-zero polynomial with rational coefficients. The
numerical value of
e truncated to 20
decimal places is
- .
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 "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):
- \lim_{n\to\infty} \left(1+\frac{1}{n}\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.
Applications
The compound-interest problem
Jacob Bernoulli discovered this
constant by studying a question about
compound interest.
One 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
$1.00×1.25
^{4} = $2.4414…, and compounding
monthly yields
$1.00×(1.0833…)
^{12} = $2.613035….
Bernoulli noticed that this sequence approaches a limit (the
force of
interest) 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
e^{R} dollars with continuous
compounding.
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 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;
- \binom{10^6}{k}
\left(10^{-6}\right)^k(1-10^{-6})^{10^6-k}.
In particular, the probability of winning zero times (
k=0)
is
- \left(1-\frac{1}{10^6}\right)^{10^6}.
This is very close to the following limit for 1/
e:
- \frac{1}{e} = \lim_{n\to\infty}
\left(1-\frac{1}{n}\right)^n.
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. 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 labeled 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:
- p_n =
1-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+\cdots+(-1)^n\frac{1}{n!}.
As the number
n of guests tends to infinity,
p_{n} 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.
Asymptotics
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:
- n! \sim \sqrt{2\pi n}\, \frac{n^n}{e^n}.
A particular consequence of this is
- e = \lim_{n\to\infty} \frac{n}{\sqrt[n]{n!}}.
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. A general exponential function
y=
a^{x} has derivative given as
the
limit:
- \frac{d}{dx}a^x=\lim_{h\to 0}\frac{a^{x+h}-a^x}{h}=\lim_{h\to
0}\frac{a^{x}a^{h}-a^x}{h}=a^x\left(\lim_{h\to
0}\frac{a^h-1}{h}\right).
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:
- \frac{d}{dx}e^x = e^x.
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. Considering the definition of
the derivative of
log_{a}x as the limit:
- \frac{d}{dx}\log_a x = \lim_{h\to
0}\frac{\log_a(x+h)-\log_a(x)}{h}=\frac{1}{x}\left(\lim_{u\to
0}\frac{1}{u}\log_a(1+u)\right),
where the substitution
u =
h/
x was made
in the last step. The last limit appearing in this calculation is
again an undetermined limit which depends only on the base
a, and if that base is
e, the limit is one. So
symbolically,
- \frac{d}{dx}\log_e x=\frac{1}{x}.
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
a^{x} to
a^{x}. 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.
Alternative characterizations
The area under the graph
y =
1/
x is equal to 1 over the interval 1 ≤
x ≤
e.
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
- \frac{d}{dt}e^t = e^t.
2. The number
e is the unique positive real number such
that
- \frac{d}{dt} \log_e t = \frac{1}{t}.
The following three characterizations can be
proven equivalent:
3. The number
e is the
limit
- e = \lim_{n\to\infty} \left( 1 + \frac{1}{n} \right)^n
Similarly:
- e = \lim_{x\to 0} \left( 1 + x \right)^{1/x}
4. The number
e is the sum of the
infinite series
- e = \sum_{n = 0}^\infty \frac{1}{n!} = \frac{1}{0!} +
\frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} +
\cdots
where
n! is the
factorial of
n.
5. The number
e is the unique positive real number such
that
- \int_{1}^{e} \frac{1}{t} \, dt = {1}.
Properties
Calculus
As in the motivation, the
exponential function
e^{x} is important in part because it is
the unique nontrivial function (up to multiplication by a constant)
which is its own
derivative
- \frac{d}{dx}e^x=e^x
and therefore its own
antiderivative
as well:
- e^x= \int_{-\infty}^x e^t\,dt
- := \int_{-\infty}^0 e^t\,dt + \int_{0}^x e^t\,dt
- :\qquad= 1 + \int_{0}^x e^t\,dt.
Exponential-like functions
The
global maximum for the
function
- f(x) = \sqrt[x]{x}
occurs at
x =
e. Similarly,
x =
1/
e is where the
global
minimum occurs for the function
- f(x) = x^x\,
defined for positive
x. More generally, x = e^{-1/n} is
where the global minimum occurs for the function
- \!\ f(x) = x^{x^n}
for any
n > 0. The infinite
tetration
- x^{x^{x^{\cdot^{\cdot^{\cdot}}}}} or ^{∞}x
converges if and only if
e^{−e} ≤
x ≤
e^{1/e} (or approximately
between 0.0660 and 1.4447), due to a theorem of
Leonhard Euler.
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.
e is conjectured to be
normal.
Complex numbers
The
exponential function
e^{x} may be written as a
Taylor series
- e^{x} = 1 + {x \over 1!} + {x^{2} \over 2!} + {x^{3} \over 3!}
+ \cdots = \sum_{n=0}^{\infty} \frac{x^n}{n!}
Because this series keeps many important properties for
e^{x} even when
x is
complex, it is commonly used to extend the
definition of
e^{x} to the complex
numbers. This, with the Taylor series for
sin and cos x, allows one to
derive
Euler's formula:
- e^{ix} = \cos x + i\sin x,\,\!
which holds for all
x. The special case with
x =
π is known as
Euler's
identity:
- e^{i\pi}+1 =0 .\,\!
Consequently,
- e^{i\pi}=-1,\,\!
from which it follows that, in the
principal branch of the logarithm,
- \log_e (-1) = i\pi.\,\!
Furthermore, using the laws for exponentiation,
- (\cos x + i\sin x)^n = \left(e^{ix}\right)^n = e^{inx} = \cos
(nx) + i \sin (nx),
which is
de Moivre's
formula.
The case,
- \cos (x) + i \sin (x)\,\!
is commonly referred to as Cis(x).
Differential equations
The general function
- y(x) = ce^x\,
is the solution to the differential equation:
- y' = y.\,
Representations
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
- \lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n,
given above, as well as the series
- e=\sum_{n=0}^\infty \frac{1}{n!}
given by evaluating the above
power
series for
e^{x} at
x=1.
Still other less common representations are also available. For
instance,
e can be represented as an infinite simple
continued fraction:
- e=2+
\cfrac{1}{
1+\cfrac{1}{
{\mathbf 2}+\cfrac{1}{
1+\cfrac{1}{
1+\cfrac{1}{
{\mathbf 4}+\cfrac{1}{
1+\cfrac{1}{
1+\ddots
}
}
}
}
}
}
}
Or, in a more compact form :
- e =
2; 1, \textbf{2}, 1, 1, \textbf{4}, 1, 1, \textbf{6}, 1, 1,
\textbf{8}, 1, 1, \ldots, \textbf{2n}, 1, 1, \ldots, \,
which can be written more harmoniously by allowing zero:
- e =
1 , \textbf{0} , 1 , 1, \textbf{2}, 1, 1, \textbf{4}, 1 , 1 ,
\textbf{6}, 1, 1, \textbf{8}, 1, 1, \ldots, \,
whose convergence can be tripled by allowing just one decimal
number:
- e =
1 , \textbf{0.5} , 12 , 5 , 28 , 9 , 44 , 13 , 60 , 17 , \ldots ,
\textbf{4} , \textbf{4n+1} , \ldots. \,
Many other series, sequence, continued fraction, and infinite
product representations of
e have also been
developed.
Stochastic representations
In addition to the deterministic analytical expressions for
representation of
e, as described above, there are some
stochastic protocols for estimation of
e. In one such
protocol, random samples X_1, X_2, ..., X_n of size n from the
uniform
distribution on (0, 1) are used to approximate
e.
If
- U= \min { \left \{ n \mid X_1+X_2+...+X_n > 1 \right \}
},
then the expectation of
U is
e: E(U) = e. Thus
sample averages of
U variables will approximate
e.
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.
Number of known decimal digits of
e
Date |
Decimal digits |
Computation performed by |
1748 |
18 |
Leonhard Euler |
1853 |
137 |
William Shanks |
1871 |
205 |
William Shanks |
1884 |
346 |
J. Marcus Boorman |
1946 |
808 |
Unknown |
1949 |
2,010 |
John von
Neumann (on the ENIAC) |
1961 |
100,265 |
Daniel Shanks & John Wrench |
1978 |
116,000 |
Stephen Gary Wozniak (on
the Apple II) |
1994 |
10,000,000 |
Robert Nemiroff & Jerry Bonnell |
1997 May |
18,199,978 |
Patrick Demichel |
1997 August |
20,000,000 |
Birger Seifert |
1997 September |
50,000,817 |
Patrick Demichel |
1999 February |
200,000,579 |
Sebastian Wedeniwski |
1999 October |
869,894,101 |
Sebastian Wedeniwski |
1999 November 21 |
1,250,000,000 |
Xavier Gourdon |
2000 July 10 |
2,147,483,648 |
Shigeru Kondo & Xavier Gourdon |
2000 July 16 |
3,221,225,472 |
Colin Martin & Xavier Gourdon |
2000 August 2 |
6,442,450,944 |
Shigeru Kondo & Xavier Gourdon |
2000 August 16 |
12,884,901,000 |
Shigeru Kondo & Xavier Gourdon |
2003 August 21 |
25,100,000,000 |
Shigeru Kondo & Xavier Gourdon |
2003 September 18 |
50,100,000,000 |
Shigeru Kondo & Xavier Gourdon |
2007 April 27 |
100,000,000,000 |
Shigeru Kondo & Steve Pagliarulo |
2009 May 6 |
200,000,000,000 |
Shigeru Kondo & Steve Pagliarulo |
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 billboard 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 (now defunct). Solving
this problem and visiting the advertised 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. The first 10-digit prime in
e is
7427466391, which starts as late as at the 99th digit. (A random
stream of digits has a 98.4% chance of starting a 10-digit prime
sooner.)
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.
Notes
References
- Maor, Eli; e: The Story of a Number, ISBN
0-691-05854-7
External links