Tuesday, September 9, 2014

e (mathematical constant)

$e$ (mathematical constant)

From Wikipedia, the free encyclopedia

The number $e$ is an important mathematical constant that is the base of the natural logarithm. It is approximately equal to 2.71828,^[1] and is the limit of

(1 + 1/ n) n

n

approaches infinity, an expression that arises in the study of compound interest. It can also be calculated as the sum of the infinite series^[2]

e = \displaystyle\sum\limits_{n = 0}^{ \infty} \dfrac{1}{n!} = 1 + \frac{1}{1} + \frac{1}{1\cdot 2} + \frac{1}{1\cdot 2\cdot 3} + \cdots

The constant can be defined in many ways; for example,

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 equal to 1.^[3] The function

e x

so defined is called the exponential function, and its inverse is the natural logarithm, or logarithm to base

e

. The natural logarithm of a positive number

k

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 number whose natural logarithm is 1. There are also more alternative characterizations.

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 also known as Napier's constant, but Euler's choice of the symbol

e

is said to have been retained in his honor.^[4] The number

e

is of eminent importance in mathematics,^[5] alongside 0, 1,

π

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

π

e

is irrational: it is not a ratio of integers; and it 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... (sequence A001113 in OEIS).

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.[6] 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,[7] 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 introduced the letter

e

as the base for natural logarithms, writing in a letter to Christian Goldbach of 25 November 1731.^[8] Euler started to use the letter

e

for the constant in 1727 or 1728, in an unpublished paper on explosive forces in cannons,^[9] and the first appearance of

e

in a publication was Euler's Mechanica (1736).^[10] While in the subsequent years some researchers used the letter

c

e

was more common and eventually became the standard.

Applications

Compound interest

The effect of earning 20% annual interest on an initial $1,000 investment at various compounding frequencies

Jacob Bernoulli discovered this constant by studying a question about compound interest:^[6]

An account starts with $1.00 and pays 100 percent interest per year. If the interest is credited once, at the end of the year, the value of the account at year-end will be $2.00. What happens if the interest is computed and credited more frequently during the year?

If the interest is credited twice in the year, the interest rate for each 6 months will be 50%, so the initial $1 is multiplied by 1.5 twice, yielding $1.00×1.5² = $2.25 at the end of the year. Compounding quarterly yields $1.00×1.25⁴ = $2.4414..., and compounding monthly yields $1.00×(1+1/12)¹² = $2.613035... If there are

n

compounding intervals, the interest for each interval will be

100%/ n

and the value at the end of the year will be $1.00×

(1 + 1/ n) n

.

Bernoulli noticed that this sequence approaches a limit (the force of interest) with larger

n

and, thus, smaller compounding intervals. Compounding weekly (

n = 52

) yields $2.692597..., while compounding daily (

n = 365

) yields $2.714567..., just two cents more. The limit as

n

grows large 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 offers an annual interest rate of

R

will, after

t

years, yield

e Rt

dollars with continuous compounding. (Here

R

is a fraction, so for 5% interest,

R = 5/100 = 0.05

)

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.
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:^[11]

n

guests are invited to a party, and at the door each guest checks his hat with the butler who then places them into

n

boxes, each labelled with the name of one guest. But the butler does not know the identities of the guests, and so he puts the hats into boxes selected at random. The problem of de Montmort is to find 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+\frac{(-1)^n}{n!} = \sum_{k = 0}^n \frac{(-1)^k}{k!}.

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

n!/ e

rounded to the nearest integer, for every positive

n

.^[12]

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}\, \left(\frac{n}{e}\right)^n.

A particular consequence of this is

e = \lim_{n\to\infty} \frac{n}{\sqrt[n]{n!}}

Standard normal distribution

The simplest case of a normal distribution is known as the standard normal distribution, described by this probability density function:

\phi(x) = \frac{1}{\sqrt{2\pi}}\, e^{- \frac{\scriptscriptstyle 1}{\scriptscriptstyle 2} x^2}.

The factor

\scriptstyle\ 1/\sqrt{2\pi}

in this expression ensures that the total area under the curve ϕ(x) is equal to one^[proof]. The 1/2 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

\scriptstyle\ 1/\sqrt{2\pi}

; and has inflection points at +1 and −1.

$e$ in calculus

Function

f (x) = a x

for several values of

a

e

is the value of

a

such that the gradient of

f (x) = a x

x = 0

equals 1. This is the blue curve,

e x

. Functions

2 x

(dotted curve) and

4 x

(dashed curve) are also shown; they are not tangent to the line of slope 1 (red).

The natural log at (x-axis)

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.^[13] 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 far right is independent of the variable

x

: it depends only on the base

a

. When the base is

e

, this limit is equal to 1, 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.^[14] 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 that depends only on the base

a

, and if that base is

e

, the limit is equal to
1. So symbolically,

\frac{d}{dx}\log_e x=\frac{1}{x}.

The logarithm in this special base is called the natural logarithm and is represented 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 a special number

a = e

. One way is to set the derivative of the exponential function

a x

a x

, 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. In fact, these two solutions for

a

are actually the same, the number

e

Alternative characterizations

The area between the

x

-axis and the graph

y = 1/ x

, between

x = 1

and

x = e

is 1.

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)^{\frac{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:

\int e^x\,dx = e^x + C.

Exponential-like functions

The global maximum of

\sqrt[x]{x}

occurs at

x = e

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 \leq x \leq 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. Euler proved this by showing that its simple continued fraction expansion is infinite.^[15]

Furthermore, by the Lindemann–Weierstrass theorem,

e

is transcendental, meaning that it is not a solution of any non-constant polynomial equation with rational coefficients. 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 that

e

is normal, meaning that when

e

is expressed in any base the possible digits in that base are uniformly distributed (occur with equal probability in any sequence of given length).

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 Euler's identity:

e^{i\pi} + 1 = 0\,\!

from which it follows that, in the principal branch of the logarithm,

\ln (-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 expression

\cos x + i \sin x \,

is sometimes 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

x = 1

.

Less common is the continued fraction (sequence A003417 in OEIS).

e = [2;1,\mathbf 2,1,1,\mathbf 4,1,1,\mathbf 6,1,1,...,\mathbf {2n},1,1,...] = [1;\mathbf 0,1,1,\mathbf 2,1,1,\mathbf 4,1,1,...,\mathbf {2n},1,1,...],

^[16]

which written out looks like

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} } } } } } } = 1+ \cfrac{1} {\mathbf 0 + \cfrac{1} {1 + \cfrac{1} {1 + \cfrac{1} {\mathbf 2 + \cfrac{1} {1 + \cfrac{1} {1 + \cfrac{1} {\mathbf 4 + \cfrac{1} {1 + \cfrac{1} {1 + \ddots} } } } } } } } }.

This continued fraction for

e

converges three times as quickly:

e = [ 1 ; 0.5 , 12 , 5 , 28 , 9 , 44 , 13 , \ldots , 4(4n-1) , (4n+1) , \ldots ],

which written out looks like

e = 1+\cfrac{2}{1+\cfrac{1}{6+\cfrac{1}{10+\cfrac{1}{14+\cfrac{1}{18+\cfrac{1}{22+\cfrac{1}{26+\ddots\,}}}}}}}.

Many other series, sequence, continued fraction, and infinite product representations of

e

have been developed.

Stochastic representations

In addition to exact analytical expressions for representation of

e

, there are stochastic techniques for estimating

e

. One such approach begins with an infinite sequence of independent random variables

X 1

X 2

..., drawn from the uniform distribution on [0, 1]. Let

V

be the least number

n

such that the sum of the first

n

samples exceeds 1:

V = \min { \left \{ n \mid X_1+X_2+\cdots+X_n > 1 \right \} }.

Then the expected value of

V

e

E(V) = e

.^[17]^[18]

Known digits

The number of known digits of

e

has increased dramatically during the last decades. This is due both to the increased performance of computers and to algorithmic improvements.^[19]^[20]

**Number of known decimal digits of $e$**
Date	Decimal digits	Computation performed by
1690	1	Jacob Bernoulli^[7]
1714	13	Roger Cotes^[21]
1748	23	Leonhard Euler^[22]
1853	137	William Shanks^[23]
1871	205	William Shanks^[24]
1884	346	J. Marcus Boorman^[25]
1949	2,010	John von Neumann (on the ENIAC)
1961	100,265	Daniel Shanks and John Wrench^[26]
1978	116,000	Steve Wozniak on the Apple II^[27]
1994 April 1	1,000,000	Robert J. Nemiroff & Jerry Bonnell ^[28]
1999 November 21	1,250,000,000	Xavier Gourdon ^[29]
2000 July 16	3,221,225,472	Colin Martin & Xavier Gourdon ^[30]
2003 September 18	50,100,000,000	Shigeru Kondo & Xavier Gourdon ^[31]
2007 April 27	100,000,000,000	Shigeru Kondo & Steve Pagliarulo ^[32]
2009 May 6	200,000,000,000	Rajesh Bohara & Steve Pagliarulo ^[32]
2010 July 5	1,000,000,000,000	Shigeru Kondo & Alexander J. Yee ^[33]

In computer culture

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

e

.

For instance, 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^[34] 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 (now defunct) 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.^[35] The first 10-digit prime in

e

is 7427466391, which starts at the 99th digit.^[36]

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. Similarly, the version numbers of his TeX program approach

π

.[37]

Imaginary number

From Wikipedia, the free encyclopedia

An imaginary number is a number that can be written as a real number multiplied by the imaginary unit

i

,^{[note 1]} which is defined by its property

i 2 = -1

.^[1] The square of an imaginary number

bi

- b 2

. For example,

5 i

is an imaginary number, and its square is

-25

. Except for 0 (which is both real and imaginary^[2]), imaginary numbers produce negative real numbers when squared.

An imaginary number

bi

can be added to a real number

a

to form a complex number of the form

a + bi

, where the real numbers

a

and

b

are called, respectively, the real part and the imaginary part of the complex number.^{[note 2]} Imaginary numbers can therefore be thought of as complex numbers whose real part is zero. The name "imaginary number" was coined in the 17th century as a derogatory term, as such numbers were regarded by some as fictitious or useless. The term "imaginary number" now means simply a complex number with a real part equal to

0

, that is, a number of the form

bi

History

An illustration of the complex plane. The imaginary numbers are on the vertical coordinate axis.

Although Greek mathematician and engineer Heron of Alexandria is noted as the first to have conceived these numbers,^[3]^[4] Rafael Bombelli first set down the rules for multiplication of complex numbers in 1572. The concept had appeared in print earlier, for instance in work by Gerolamo Cardano. At the time, such numbers were poorly understood and regarded by some as fictitious or useless, much as zero and the negative numbers once were. Many other mathematicians were slow to adopt the use of imaginary numbers, including René Descartes, who wrote about them in his La Géométrie, where the term imaginary was used and meant to be derogatory.^[5] The use of imaginary numbers was not widely accepted until the work of Leonhard Euler (1707–1783) and Carl Friedrich Gauss (1777–1855). The geometric significance of complex numbers as points in a plane was first described by Caspar Wessel (1745–1818).^[6]

In 1843 a mathematical physicist, William Rowan Hamilton, extended the idea of an axis of imaginary numbers in the plane to a three-dimensional space of quaternion imaginaries.

With the development of quotient rings of polynomial rings, the concept behind an imaginary number became more substantial, but then one also finds other imaginary numbers such as the j of tessarines which has a square of

+1

. This idea first surfaced with the articles by James Cockle beginning in 1848.^[7]

Geometric interpretation

90-degree rotations in the complex plane

Geometrically, imaginary numbers are found on the vertical axis of the complex number plane, allowing them to be presented perpendicular to the real axis. One way of viewing imaginary numbers is to consider a standard number line, positively increasing in magnitude to the right, and negatively increasing in magnitude to the left. At 0 on this

x

-axis, a

y

-axis can be drawn with "positive" direction going up; "positive" imaginary numbers then increase in magnitude upwards, and "negative" imaginary numbers increase in magnitude downwards. This vertical axis is often called the "imaginary axis" and is denoted

i ℝ

\scriptstyle\mathbb{I}

, or

ℑ

.

In this representation, multiplication by

-1

corresponds to a rotation of 180 degrees about the origin. Multiplication by

i

corresponds to a 90-degree rotation in the "positive" direction (i.e., counterclockwise), and the equation

i 2 = -1

is interpreted as saying that if we apply two 90-degree rotations about the origin, the net result is a single 180-degree rotation. Note that a 90-degree rotation in the "negative" direction (i.e. clockwise) also satisfies this interpretation. This reflects the fact that

- i

also solves the equation

x 2 = -1

. In general, multiplying by a complex number is the same as rotating around the origin by the complex number's argument, followed by a scaling by its magnitude.

Multiplication of square roots

Care must be used in multiplying square roots of negative numbers. For example,^[8] the following reasoning is incorrect:

-1 = i^2 = \sqrt{-1}\sqrt{-1} = \sqrt{(-1)(-1)} = \sqrt{1} = 1

The fallacy is that the rule

\sqrt x \sqrt y = \sqrt xy

, where the principal value of the square root is taken in each instance, is generally valid only if

x

and

y

are suitably constrained.^{[note 3]} It is not possible to extend the definition of principal values to the square roots of all complex numbers in a way that preserves the validity of the multiplication rule. Hence

\sqrt -1

in such contexts should be regarded either as meaningless, or as a two-valued expression with the possible values

i

and

- i

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