11/30/13 e (mathematical constant) - Wikipedia, the free encyclopedia

en.wikipedia.org/wiki/ℯ 1/11

Functions f(x) = ax are shown for several

values of a. e is the unique value of a,

such that the derivative of f(x) = ax at the

point x = 0 is equal to 1. The blue curve

illustrates this case, ex. For comparison,

functions 2x (dotted curve) and 4x (dashed

curve) are shown; they are not tangent to

the line of slope 1 and y-intercept 1 (red).

Part of a series of articles on

The mathematical constante

Natural logarithm · Exponential function

Applications in: compound interest ·

Euler's identity & Euler's formula · half-lives& exponential growth/decay

e (mathematical constant)From Wikipedia, the free encyclopedia

(Redirected from ℯ)

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

as 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]

The constant can be defined in many ways; for example, e is the unique realnumber such that the value of the derivative (slope of the tangent line) of the

function f(x) = ex at the point x = 0 is equal to 1.[3] The function ex so definedis called the exponential function, and its inverse is the natural logarithm, orlogarithm to base e. The natural logarithm of a positive number k can also bedefined 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 morealternative 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 calledsimply Euler's constant. The number e is also known as Napier's constant, but

Euler's choice of this symbol 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. Allfive of these numbers play important and recurring roles across mathematics, andare the five constants appearing in one formulation of Euler's identity. Like theconstant 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 withrational coefficients. The numerical value of e truncated to 50 decimal places is

2.71828 18284 59045 23536 02874 71352 66249 77572 47093 69995... (sequence A001113 in OEIS).

Contents

1 History

2 Applications2.1 Compound interest2.2 Bernoulli trials

2.3 Derangements2.4 Asymptotics

2.5 Standard normal distribution3 e in calculus

3.1 Alternative characterizations

4 Properties4.1 Calculus

4.2 Exponential-like functions4.3 Number theory4.4 Complex numbers

4.5 Differential equations

11/30/13 e (mathematical constant) - Wikipedia, the free encyclopedia

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Defining e: proof that e is irrational ·

representations of e · Lindemann–Weierstrass theorem

People John Napier · Leonhard Euler

Schanuel's conjecture

The effect of earning 20% annual interest on an initial $1,000

investment at various compounding frequencies

5 Representations5.1 Stochastic representations

5.2 Known digits6 In computer culture7 Notes

8 Further reading9 External links

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 thetable was written by William Oughtred. The discovery of the constant itself is credited to Jacob Bernoulli, who attempted to findthe 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 ChristiaanHuygens 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.[7] Euler started to use the letter e for the constant in 1727 or 1728, in an unpublished

paper on explosive forces in cannons,[8] and the first appearance of e in a publication was Euler's Mechanica (1736). While in thesubsequent years some researchers used the letter c, e was more common and eventually became the standard.

Applications

Compound interest

Jacob Bernoulli discovered this constant by studying a question

about compound interest:[6]

An account starts with $1.00 and pays 100 percent interestper 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 foreach 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... Ifthere are n compounding intervals, the interest for each intervalwill 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 compoundingintervals. Compounding weekly (n = 52) yields $2.692597..., while compounding daily (n = 365) yields $2.714567..., just twocents more. The limit as n grows large is the number that came to be known as e; with continuous compounding, the account

11/30/13 e (mathematical constant) - Wikipedia, the free encyclopedia

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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 eRt 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 exponentialgrowth. Suppose that a gambler plays a slot machine that pays out with a probability of one in n and plays it n times. Then, forlarge n (such as a million) the probability that the gambler will lose every bet is (approximately) 1/e. For n = 20 it is already1/2.72.

This is an example of a Bernoulli trials process. Each time the gambler plays the slots, there is a one in one million chance ofwinning. Playing one million times is modelled by the binomial distribution, which is closely related to the binomial theorem. Theprobability 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:

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:[9] n guests are invited to a party, and at the door each guest checks his hatwith the butler who then places them into n boxes, each labelled with the name of one guest. But the butler does not know theidentities of the guests, and so he puts the hats into boxes selected at random. The problem of de Montmort is to find theprobability 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 n!/e rounded to the nearest integer, for every positive n.[10]

Asymptotics

The number e occurs naturally in connection with many problems involving asymptotics. A prominent example is Stirling's formulafor the asymptotics of the factorial function, in which both the numbers e and π enter:

A particular consequence of this is

11/30/13 e (mathematical constant) - Wikipedia, the free encyclopedia

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The natural log at (x-axis) e, ln(e),

is equal to 1

.

Standard normal distribution

(from Normal distribution)

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

The factor in this expression ensures that the total area under the curve ϕ(x) is equal to one[proof]. The 12 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 ; and has inflection points at +1 and −1.

e in calculus

The principal motivation for introducing the number e, particularly in calculus, is to

perform differential and integral calculus with exponential functions and logarithms.[11] A

general exponential function y = ax has derivative given as the limit:

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 toone, and so e is symbolically defined by the equation:

Consequently, the exponential function with base e is particularly suited to doing calculus. Choosing e, as opposed to some othernumber, as the base of the exponential function makes calculations involving the derivative much simpler.

Another motivation comes from considering the base-a logarithm.[12] Considering the definition of the derivative of loga x as the

limit:

where the substitution u = h/x was made in the last step. The last limit appearing in this calculation is again an undetermined limitthat depends only on the base a, and if that base is e, the limit is one. So symbolically,

11/30/13 e (mathematical constant) - Wikipedia, the free encyclopedia

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The area between the x-axis and the

graph y = 1/x, between x = 1 and

x = e is 1.

The logarithm in this special base is called the natural logarithm and is represented as ln; it behaves well under differentiation sincethere 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, 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, onearrives 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

See also: Representations of e

Other characterizations of e are also possible: one is as the limit of a sequence, anotheris as the sum of an infinite series, and still others rely on integral calculus. So far, thefollowing 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 three characterizations can be proven equivalent:

3. The number e is the limit

Similarly:

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

Properties

11/30/13 e (mathematical constant) - Wikipedia, the free encyclopedia

en.wikipedia.org/wiki/ℯ 6/11

The global maximum of occurs at

x = e.

Calculus

As in the motivation, the exponential function ex is important in part because it is the unique nontrivial function (up to multiplicationby a constant) which is its own derivative

and therefore its own antiderivative as well:

Exponential-like functions

See also: Steiner's problem

The global maximum for the function

occurs at x = e. Similarly, x = 1/e is where the global minimum occurs for thefunction

defined for positive x. More generally, x = e−1/n is where the global minimumoccurs for the function

for any n > 0. The infinite tetration

or ∞

converges if and only if e−e ≤ x ≤ e1/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.[13] (See alsoFourier's proof that e is irrational.)

Furthermore, by the Lindemann–Weierstrass theorem, e is transcendental, meaning that it is not a solution of any non-constantpolynomial equation with rational coefficients. It was the first number to be proved transcendental without having been specificallyconstructed 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 uniformlydistributed (occur with equal probability in any sequence of given length).

Complex numbers

The exponential function ex may be written as a Taylor series

11/30/13 e (mathematical constant) - Wikipedia, the free encyclopedia

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Because this series keeps many important properties for ex even when x is complex, it is commonly used to extend the definition

of ex to the complex numbers. This, with the Taylor series for sin and cos x, allows one to derive Euler's formula:

which holds for all x. The special case with x = π is Euler's identity:

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

Furthermore, using the laws for exponentiation,

which is de Moivre's formula.

The expression

is sometimes referred to as cis(x).

Differential equations

The general function

is the solution to the differential equation:

Representations

Main article: List of 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 continuedfraction, 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

11/30/13 e (mathematical constant) - Wikipedia, the free encyclopedia

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given by evaluating the above power series for ex at x = 1.

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

[14]

which written out looks like

This continued fraction for e converges three times as quickly:

which written out looks like

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 suchapproach begins with an infinite sequence of independent random variables X1, X2..., 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:

Then the expected value of V is e: E(V) = e.[15][16]

11/30/13 e (mathematical constant) - Wikipedia, the free encyclopedia

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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.[17][18]

Number of known decimal digits of e

Date Decimal digits Computation performed by

1748 23 Leonhard Euler[19]

1853 137 William Shanks

1871 205 William Shanks

1884 346 J. Marcus Boorman

1949 2,010 John von Neumann (on the ENIAC)

1961 100,265 Daniel Shanks and John Wrench[20]

1978 116,000 Steve Wozniak on the Apple II[21]

1994 April 1 1,000,000 Robert Nemiroff & Jerry Bonnell [22]

1999 November 21 1,250,000,000 Xavier Gourdon [23]

2000 July 16 3,221,225,472 Colin Martin & Xavier Gourdon [24]

2003 September 18 50,100,000,000 Shigeru Kondo & Xavier Gourdon [25]

2007 April 27 100,000,000,000 Shigeru Kondo & Steve Pagliarulo [26]

2009 May 6 200,000,000,000 Rajesh Bohara & Steve Pagliarulo [26]

2010 July 5 1,000,000,000,000 Shigeru Kondo & Alexander J. Yee [27]

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 announcedits intention to raise $2,718,281,828, which is e billion dollars to the nearest dollar. Google was also responsible for a

billboard[28] 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 website (now defunct) 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.[29] The first 10-digit prime in e is 7427466391, which starts as late as at the 99th digit.[30]

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.[31]

Notes

1. ^ Oxford English Dictionary, 2nd ed.: natural logarithm (http://oxforddictionaries.com/definition/english/natural%2Blogarithm)

2. ^ Encyclopedic Dictionary of Mathematics 142.D

3. ^ Jerrold E. Marsden, Alan Weinstein (1985). Calculus (http://books.google.com/?id=KVnbZ0osbAkC&printsec=frontcover).Springer. ISBN 0-387-90974-5.

4. ^ Sondow, Jonathan. "e" (http://mathworld.wolfram.com/e.html). Wolfram Mathworld. Wolfram Research. Retrieved 10 May2011.

5. ^ Howard Whitley Eves (1969). An Introduction to the History of Mathematics. Holt, Rinehart & Winston. ISBN 0-03-029558-0.

6. ̂a b O'Connor, J J; Robertson, E F. "The number e" (http://www-history.mcs.st-and.ac.uk/HistTopics/e.html). MacTutor History

11/30/13 e (mathematical constant) - Wikipedia, the free encyclopedia

en.wikipedia.org/wiki/ℯ 10/11

6. ̂a b O'Connor, J J; Robertson, E F. "The number e" (http://www-history.mcs.st-and.ac.uk/HistTopics/e.html). MacTutor Historyof Mathematics.

7. ^ Remmert, Reinhold (1991). Theory of Complex Functions. Springer-Verlag. p. 136. ISBN 0-387-97195-5

8. ^ Euler, Meditatio in experimenta explosione tormentorum nuper instituta(http://www.math.dartmouth.edu/~euler/pages/E853.html).

9. ^ Grinstead, C.M. and Snell, J.L.Introduction to probability theory(http://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/book.html) (published online under theGFDL), p. 85.

10. ^ Knuth (1997) The Art of Computer Programming Volume I, Addison-Wesley, p. 183 ISBN 0-201-03801-3.

11. ^ Kline, M. (1998) Calculus: An intuitive and physical approach, section 12.3 "The Derived Functions of Logarithmic Functions."(http://books.google.co.jp/books?id=YdjK_rD7BEkC&pg=PA337), pp. 337 ff, Courier Dover Publications, 1998, ISBN 0-486-40453-6

12. ^ This is the approach taken by Kline (1998).

13. ^ Sandifer, Ed (Feb. 2006). "How Euler Did It: Who proved e is Irrational?"

(http://www.maa.org/editorial/euler/How%20Euler%20Did%20It%2028%20e%20is%20irrational.pdf). MAA Online. Retrieved2010-06-18.

14. ^ Hofstadter, D. R., "Fluid Concepts and Creative Analogies: Computer Models of the Fundamental Mechanisms of Thought"Basic Books (1995) ISBN 0-7139-9155-0

15. ^ Russell, K. G. (1991) Estimating the Value of e by Simulation (http://links.jstor.org/sici?sici=0003-1305%28199102%2945%3A1%3C66%3AETVOEB%3E2.0.CO%3B2-U) The American Statistician, Vol. 45, No. 1. (Feb., 1991),pp. 66–68.

16. ^ Dinov, ID (2007) Estimating e using SOCR simulation(http://wiki.stat.ucla.edu/socr/index.php/SOCR_EduMaterials_Activities_LawOfLargeNumbers#Estimating_e_using_SOCR_simulation), SOCR Hands-on Activities (retrieved December 26, 2007).

17. ^ Sebah, P. and Gourdon, X.; The constant e and its computation (http://numbers.computation.free.fr/Constants/E/e.html)

18. ^ Gourdon, X.; Reported large computations with PiFast(http://numbers.computation.free.fr/Constants/PiProgram/computations.html)

19. ^ Introductio in analysin infinitorum p. 90 (http://books.google.de/books?id=jQ1bAAAAQAAJ&pg=PA90)

20. ^ Daniel Shanks and John W Wrench (1962). "Calculation of Pi to 100,000 Decimals" (http://www.ams.org/journals/mcom/1962-

16-077/S0025-5718-1962-0136051-9/S0025-5718-1962-0136051-9.pdf). Mathematics of Computation 16 (77): 76–99 (78). "Wehave computed e on a 7090 to 100,265D by the obvious program"

21. ^ Wozniak, Steve (June 1981). "The Impossible Dream: Computing e to 116 Places with a Personal Computer"(http://archive.org/stream/byte-magazine-1981-06/1981_06_BYTE_06-06_Operating_Systems#page/n393/mode/2up). BYTE.p. 392. Retrieved 18 October 2013.

22. ^ Email from Robert Nemiroff and Jerry Bonnell – The Number e to 1 Million Digits (http://apod.nasa.gov/htmltest/gifcity/e.1mil).None. Retrieved on 2012-02-24.

23. ^ Email from Xavier Gourdon to Simon Plouffe – I have made a new e computation (with verification): 1,250,000,000 digits(http://web.archive.org/web/20021223163426/http://pi.lacim.uqam.ca/piDATA/expof1.txt). None. Retrieved on 2012-02-24.

24. ^ PiHacks message 177 – E to 3,221,225,472 D (http://groups.yahoo.com/group/pi-hacks/message/177). Groups.yahoo.com.Retrieved on 2012-02-24.

25. ^ PiHacks message 1071 – Two new records: 50 billions for E and 25 billions for pi (http://groups.yahoo.com/group/pi-hacks/message/1071). Groups.yahoo.com. Retrieved on 2012-02-24.

26. ̂a b English Version of PI WORLD (http://web.archive.org/web/20021221061853/http://ja0hxv.calico.jp/pai/eevalue.html).Ja0hxv.calico.jp. Retrieved on 2012-02-24.

27. ^ A list of notable large computations of e (http://www.numberworld.org/digits/E/). Numberworld.org. Last updated: March 7,2011. Retrieved on 2012-02-24.

28. ^ First 10-digit prime found in consecutive digits of e} (http://braintags.com/archives/2004/07/first-10digit-prime-found-in-

consecutive-digits-of-e/). Brain Tags. Retrieved on 2012-02-24.

29. ^ Shea, Andrea. "Google Entices Job-Searchers with Math Puzzle" (http://www.npr.org/templates/story/story.php?storyId=3916173). NPR. Retrieved 2007-06-09.

30. ^ Kazmierczak, Marcus (2004-07-29). "Google Billboard" (http://mkaz.com/math/google-billboard). mkaz.com. Retrieved 2007-06-09.

31. ^ Knuth, Donald. "The Future of TeX and Metafont" (http://www.tex.ac.uk/tex-archive/digests/tex-mag/v5.n1). TeX Mag 5 (1).

Further reading

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

11/30/13 e (mathematical constant) - Wikipedia, the free encyclopedia

en.wikipedia.org/wiki/ℯ 11/11

Commentary on Endnote 10 (http://www.johnderbyshire.com/Books/Prime/Blog/page.html#endnote10) of the book PrimeObsession for another stochastic representation

External links

An Intuitive Guide To Exponential Functions &e (http://betterexplained.com/articles/an-intuitive-guide-to-exponential-

functions-e/) for the non-mathematician

The number e to 1 million places (http://gutenberg.org/ebooks/127) and 2 and 5 million places (link obsolete)

(http://apod.nasa.gov/htmltest/gifcity/e.2mil)

e Approximations (http://mathworld.wolfram.com/eApproximations.html) – Wolfram MathWorld

Earliest Uses of Symbols for Constants (http://jeff560.tripod.com/constants.html) Jan. 13, 2008

"The story of e" (http://www.gresham.ac.uk/lectures-and-events/the-story-of-e), by Robin Wilson at Gresham College, 28

February 2007 (available for audio and video download)

e Search Engine (http://www.subidiom.com/e) 2 billion searchable digits of e, π and √2

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