The mathematical constant e (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) is the base of the natural logarithm function. Its approximate value is: A mathematical constant is a quantity, usually a real number or a complex number, that arises naturally in mathematics and does not change. ... Leonhard Euler aged 49 (oil painting by Emanuel Handmann, 1756) Leonhard Euler [oilər] (April 15, 1707 - September 18, 1783) was a Swiss mathematician and physicist. ... A mathematician is a person whose area of study and research is mathematics. ... John Napier John Napier (1550–April 4, 1617) was a Scottish mathematician and astrologer. ... In mathematics, a logarithm of x with base b may be defined as the following: for the equation bn = x, the logarithm is a function which gives n. ... The natural logarithm is the logarithm to the base e, where e is approximately equal to 2. ...

e ≈ 2.71828 18284 59045 23536 02874 7135

Alongside the number π and the imaginary unit i, e is one of the most important mathematical constants. It has a number of equivalent definitions; some of them are given below. The mathematical constant π represents the ratio of a circles circumference to its diameter and is commonly used in mathematics, physics, and engineering. ... In mathematics, the imaginary unit i allows the real number system to be extended to the complex number system . ...

Definitions

The three most common definitions of e are the following.

1. Define e by the following limit.

2. Define e as the sum of the following infinite series.

where n! is the factorial of n.

3. Define e to be the unique number x > 0 such that

These different definitions have been proven to be equal. In mathematics, the concept of a limit is used to describe the behavior of a function, as its argument gets close to either some point, or infinity; or the behavior of a sequences elements, as their index approaches infinity. ... In mathematics, a series is a sum of a sequence of terms. ... In mathematics, the factorial of a natural number n is the product of the positive integers less than or equal to n. ... In mathematics, the exponential function can be characterized in many ways. ...

Properties

Many growth or decay processes can be modeled with an exponential function. The exponential function e^x is important because it is the unique function (up to multiplication by a constant) which is its own derivative: In mathematics, a quantity that grows exponentially is one that grows at a rate proportional to its size. ... A quantity is said to be subject to exponential decay if it decreases at a rate proportional to its value. ... The exponential function is one of the most important functions in mathematics. ... In mathematics, the derivative of a function is one of the two central concepts of calculus. ...

e is known to be both irrational and transcendental. It was the first number to be proved transcendental without having been specifically constructed for this purpose (compare Liouville number); the proof was given by Charles Hermite in 1873. It is conjectured to be normal. It features in Euler's Formula, one of the most important identities in mathematics: In mathematics, the series expansion of the number e can be used to prove that e is irrational. ... In mathematics, the Lindemann-Weierstrass theorem states that if α1,...,αn are algebraic numbers which are linearly independent over the rational numbers, then are algebraically independent over the algebraic numbers; in other words the set has transcendence degree n over . ... In number theory, a Liouville number is a real number x with the property that, for any positive integer n, there exist integers p and q with q > 1 and such that 0 < |x - p/q| < 1/qn. ... Charles Hermite (pronounced air meet) (December 24, 1822 - January 14, 1901) was a French mathematician who did research on number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra. ... 1873 was a common year starting on Wednesday (see link for calendar). ... In mathematics, a normal number is, roughly speaking, a real number whose digits show a random distribution with all digits being equally likely. ... This article is about the Eulers formula in complex analysis. ...

The special case with x = π is known as Euler's identity: In mathematics, Eulers identity is the following equation: sometimes expressed as: presumably in order to use the fundamental numbers 0 and 1 (see below). ...

described by Richard Feynman as Euler's jewel. Richard Phillips Feynman (May 11, 1918 – February 15, 1988) (surname pronounced FINE-man; in IPA) was one of the most influential American physicists of the 20th century, expanding greatly the theory of quantum electrodynamics. ...

The infinite continued fraction expansion of e contains an interesting pattern (sequence A005131 in OEIS) that can be written as follows: In mathematics, a continued fraction is an expression such as where a0 is some integer and all the other numbers an are positive integers. ... The On-Line Encyclopedia of Integer Sequences (OEIS) is a web-based searchable database of integer sequences. ...

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 first indication of e as a constant was discovered by Jacob Bernoulli, trying to find the value of the following expression. Events March 8 - Johannes Kepler discovers the third law of planetary motion (he soon rejects the idea after some initial calculations were made but on May 15 confirms the discovery). ... William Oughtred (March 5, 1575 – June 30, 1660) was an English mathematician. ... Jakob Bernoulli. ...

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, the use of e was more common and eventually became the standard. Gottfried Leibniz Gottfried Wilhelm von Leibniz (also Leibnitz) (Leipzig July 1 (June 21 O.S.), 1646 – November 14, 1716 in Hannover) was a German philosopher, scientist, mathematician, diplomat, librarian, and lawyer of Sorb descent. ... Christiaan Huygens Christiaan Huygens (pronounced in English ( IPA): ; in Dutch: ) ( April 14, 1629– July 8, 1695), was a Dutch mathematician and physicist; born in The Hague as the son of Constantijn Huygens. ... Events Giovanni Domenico Cassini observes differential rotation within Jupiters atmosphere. ... Events March 5 - French troops under Marshal Louis-Francois de Boufflers besiege the Spanish-held town of Mons March 29 - Siege of Mons ends to the city’s surrender October 3 - Treaty of Limerick which guaranteed civil rights to catholics was signed. ... Events June 11 - George, Prince of Wales becomes King George II of Great Britain. ... Events January 26 - Stanislaus I of Poland abdicates his throne. ...

The exact reasons for the use of e are unknown, but it may be because the letter e is the first letter of the word exponential. Another view is that the letters a, b, c, and d were already frequently used for other purposes, and e was the first available letter. It is unlikely that Euler choose the letter because it is his first initial, since he was a very modest man, always trying to give proper credit to the work of others. The term exponential may refer to any of several topics in mathematics: Exponential distribution Exponential function Exponential growth, exponential decay Exponential time Matrix exponential Exponential map (in differential geometry) All relate in some fashion to exponents. ...

Non-mathematical uses of e

In the IPO filing for Google Inc., in 2004, rather than a typical round-number amount of money, the company announced its intention to raise $2,718,281,828, which is, of course, e billion dollars to the nearest dollar. Wikipedia does not yet have an article with this exact name. ... This page is about Google Inc. ... 2004 is a leap year starting on Thursday of the Gregorian calendar. ... The United States dollar is the official currency of the United States. ...

Google was also the culprit of a mysterious billboard (http://mattwalsh.com/twiki/pub/Main/GoogleBillboardContestFindingPrimesInE/IMG_0742.JPG) that appeared in the heart of Silicon Valley, and later in Cambridge, Massachusetts, which read {first 10-digit prime found in consecutive digits of e}.com. Once solving this problem (the first 10-digit prime in e is 7427466391, which starts at the 101st digit), and visiting the web site advertised, there was an even more difficult problem to solve. Silicon Valley is a commonly used nickname for the southern part of the San Francisco Bay Area in northern California, USA, originally referring to the concentration of silicon chip innovators and manufacturers, but eventually becoming a metaphor for the entire concentration of high tech businesses. ... Harvard Square, May 2000 Cambridge is a city in the Greater Boston area in Massachusetts, United States. ...

Donald Knuth, the famous computer scientist let the version-numbers of his book METAFONT approach e (i.e the versions are 2, 2.7, 2.71, 2.718, etc.) Donald Knuth Donald Ervin Knuth (born January 10, 1938) (Chinese name: 高德纳, pinyin: Gāo Dénà) is a renowned computer scientist and professor emeritus at Stanford University. ... Wikibooks Wikiversity has more about this subject: School of Computer Science Open Directory Project: Computer Science Collection of Computer Science Bibliographies Belief that title science in computer science is inappropriate Categories: Computer science | Academic disciplines ... METAFONT is a programming language used to produce rasterized outline fonts. ...

External links

• The number e (history) (http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/e.html)
• Earliest Uses of Symbols for Constants (http://members.aol.com/jeff570/constants.html)

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Reference

• Eli Maor, e: The Story of a Number, ISBN 0691058547