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The mathematical constant e is the unique real number such that the function e^{x} has the same value as the slope of the tangent line, for all values of x. More generally, the only functions equal to their own derivatives are of the form Ce^{x}, where C is a constant. 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 representations of e, below).

The number 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.

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.)

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...

- $lim\_\{ntoinfty\}\; left(1+frac\{1\}\{n\}right)^n.$

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.

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.

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 $1.00×1.25^{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.

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;

- $binom\{10^6\}\{k\}\; left(10^\{-6\}right)^k(1-10^\{-6\})^\{10^6-k\}.$

- $left(1-frac\{1\}\{10^6\}right)^\{10^6\}.$

- $frac\{1\}\{e\}\; =\; lim\_\{ntoinfty\}\; left(1-frac\{1\}\{n\}right)^n.$

- $p\_n\; =\; 1-frac\{1\}\{1!\}+frac\{1\}\{2!\}-frac\{1\}\{3!\}+cdots+(-1)^nfrac\{1\}\{n!\}.$

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.

- $n!\; sim\; sqrt\{2pi\; n\},\; frac\{n^n\}\{e^n\}.$

- $e\; =\; lim\_\{ntoinfty\}\; frac\{n\}\{sqrt[n]\{n!\}\}.$

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:

- $frac\{d\}\{dx\}a^x=lim\_\{hto\; 0\}frac\{a^\{x+h\}-a^x\}\{h\}=lim\_\{hto\; 0\}frac\{a^\{x\}a^\{h\}-a^x\}\{h\}=a^xleft(lim\_\{hto\; 0\}frac\{a^h-1\}\{h\}right).$

- $frac\{d\}\{dx\}e^x\; =\; e^x.$

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:

- $frac\{d\}\{dx\}log\_a\; x\; =\; lim\_\{hto\; 0\}frac\{log\_a(x+h)-log\_a(x)\}\{h\}=frac\{1\}\{x\}left(lim\_\{uto\; 0\}frac\{1\}\{u\}log\_a(1+u)right).$

- $frac\{d\}\{dx\}log\_e\; x=frac\{1\}\{x\}.$

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.

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

- $e\; =\; lim\_\{ntoinfty\}\; left(1\; +\; frac\{1\}\{n\}\; right)^n$

Similarly:

- $e\; =\; lim\_\{nto\; 0\}\; left(1\; +\; n\; right)^\{frac\{1\}\{n\}\; \}$

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

- $e\; =\; sum\_\{n\; =\; 0\}^infty\; frac\{1\}\{n!\}\; =\; frac\{1\}\{0!\}\; +\; frac\{1\}\{1!\}\; +\; frac\{1\}\{2!\}\; +\; frac\{1\}\{3!\}\; +\; frac\{1\}\{4!\}\; +\; cdots$

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

- $int\_\{1\}^\{e\}\; frac\{1\}\{t\}\; ,\; dt\; =\; \{1\}$.

- $frac\{d\}\{dx\}e^x=e^x$

and therefore its own antiderivative as well:

- $e^x=\; int\_\{-infty\}^x\; e^t,dt$

- $=\; int\_\{-infty\}^0\; e^t,dt\; +\; int\_\{0\}^x\; e^t,dt$

- $qquad=\; 1\; +\; int\_\{0\}^x\; e^t,dt.$

- $f(x)\; =\; x^\{1/x\}.,$

More generally, x = ^{n}√e is where the global maximum occurs for the function

- $!\; f(x)\; =\; x^\{1/x^n\}.$

The infinite tetration

- $x^\{x^\{x^\{cdot^\{cdot^\{cdot\}\}\}\}\}$

converges only if e^{−e} ≤ x ≤ e^{1/e}, due to a theorem of Leonhard Euler.

The exponential function e^{x} may be written as a Taylor series

- $e^\{x\}\; =\; 1\; +\; \{x\; over\; 1!\}\; +\; \{x^\{2\}\; over\; 2!\}\; +\; \{x^\{3\}\; over\; 3!\}\; +\; cdots$

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:

- $e^\{ix\}\; =\; cos\; x\; +\; isin\; x,,!$

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

- $e^\{ipi\}+1\; =0\; .,!$

Consequently,

- $e^\{ipi\}=-1,,!$

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

- $log\_e\; (-1)\; =\; ipi.,!$

Furthermore, using the laws for exponentiation,

- $(cos\; x\; +\; isin\; x)^n\; =\; left(e^\{ix\}right)^n\; =\; e^\{inx\}\; =\; cos\; (nx)\; +\; i\; sin\; (nx),$

which is de Moivre's formula.

The case,

- $cos\; (x)\; +\; i\; sin\; (x),!$

is commonly referred to as Cis(x).

The general function

- $y(x)\; =\; ce^x,$

is the solution to the differential equation:

- $y\text{'}\; =\; y.,$

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

- $lim\_\{ntoinfty\}left(1+frac\{1\}\{n\}right)^n,$

- $e=sum\_\{n=0\}^infty\; frac\{1\}\{n!\}$

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

- $e=2+$

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Or, in a more compact form :

- $e\; =2;\; 1,\; textbf\{2\},\; 1,\; 1,\; textbf\{4\},\; 1,\; 1,\; textbf\{6\},\; 1,\; 1,\; textbf\{8\},\; 1,\; ldots,1,\; textbf\{2n\},\; 1,ldots,\; ,$

which can be written more harmoniously by allowing zero:

- $e\; =1\; ,\; textbf\{0\}\; ,\; 1\; ,\; 1,\; textbf\{2\},\; 1,\; 1,\; textbf\{4\},\; 1\; ,\; 1\; ,\; textbf\{6\},\; 1,\; ldots.\; ,$

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

- $U=\; min\; \{\; left\; \{\; n\; mid\; X\_1+X\_2+...+X\_n\; >\; 1\; right\; \}\; \},$

then the expectation of U is e: $E(U)\; =\; e$. Thus sample averages of U variables will approximate e.

Date | Decimal digits | Computation performed by |
---|---|---|

1748 | 18 | 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 |

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 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 leads to Google Labs where the visitor is 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 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.

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

- 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)
- Class Library for Numbers (part of the GiNaC distribution) includes example code for computing e to arbitrary precision.
- The SOCR resource provides a hands-on activity and an interactive Java applet (Uniform E-Estimate Experiment) for computing e using a simulation based on uniform distribution.

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Last updated on Friday October 10, 2008 at 20:22:47 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Friday October 10, 2008 at 20:22:47 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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