The mathematical constant A mathematical constant is a quantity, usually a real number or a complex number, that arises naturally in mathematics and does not change. Unlike physical constants, mathematical constants are defined independently of any physical measurement.
For example, up to multiplication with nonzero complex numbers, there is a unique holomorphic function
..... Click the link for more information. e (occasionally called Euler's number after the Swiss mathematician Leonhard Euler Leonhard Euler [oi'lər] (April 15, 1707 - September 18, 1783) was a Swiss mathematician and physicist. He is considered to be one of the greatest mathematicians who ever lived. Leonhard Euler was the first to use the term "function" (defined by Leibniz - 1694) to describe an expression involving various arguments; ie:
..... Click the link for more information. , or Napier's constant in honor of the Scottish mathematician A mathematician is a person whose area of study and research is mathematics.
Roles
Mathematicians not only study, but also research, and this must be given prominent mention here, because a misconception that everything in mathematics is already known is widespread among persons not learned in that field. In fact, the publication of new discoveries in mathematics continues at an immense rate in hundreds of scientific journals, many of them devoted to mathematics and many devoted to subjects to which mathematics is applied (such as theoretical computer science and theoretical physics).
..... Click the link for more information. John Napier John Napier (1550¨CApril 4, 1617) was a Scottish mathematician and astrologer. He is most remembered as the inventor of natural logarithms, of Napier's bones or Napier's rods and for popularizing the decimal point. He was born in Merchiston Tower, Edinburgh. Although he did not invent the natural logarithm function, it is sometimes known as the
..... Click the link for more information. who introduced logarithms 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. This function is written as n = logb x. Logarithms tell how many times a number
..... Click the link for more information. ) is the base of the natural logarithm The natural logarithm is the logarithm to the base e, where e is approximately equal to 2.71828... (no exact fraction can be given, as e is an irrational number just like pi). The natural logarithm is defined for all positive real numbers x and can also be defined for non-zero complex numbers as will be explained below. Although this function was not introduced by Napier, it is sometimes known as the
..... Click the link for more information. function. Its approximate value is:


e ¡Ö 2.71828 18284 59045 23536 02874 7135


Alongside the number ¦Ð The mathematical constant ¦Ð represents the ratio of a circle's circumference to its diameter and is commonly used in mathematics, physics, and engineering. ¦Ð is the lowercase Greek letter equivalent to "p" in the Roman alphabet; its name is "pi" (pronounced pie), and this spelling can be used when the Greek letter is not available
..... Click the link for more information. and the imaginary unit i In mathematics, the imaginary unit i (sometimes also represented by j, but in this article i will be used exclusively) allows the real number system to be extended to the complex number system . Its precise definition is dependent upon the particular method of extension.
..... Click the link for more information. , e is one of the most important mathematical constants. It has a number of equivalent definitions; some of them are given below.

Definitions

The three most common definitions of e are the following.


1. Define e by the following limit 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 sequence's elements, as their index approaches infinity. Limits are used in calculus and other branches of mathematical analysis to define derivatives and continuity.
..... Click the link for more information. .



:



2. Define e as the sum of the following infinite series In mathematics, a series is the sum of a sequence of terms. That is, a series is a list of numbers with addition operations between them, e.g,

1 + 2 + 3 + 4 + 5 + ...

which may or may not be meaningful.
In most cases of interest the terms of the sequence are produced according to a certain rule, e.g., by a formula, by an algorithm, by a sequence of measurements, or even by a random number generator.
..... Click the link for more information. .



:



: where n! is the factorial
This article is not about factorial experiments.

In mathematics, the factorial of a natural number n is the product of the positive integers less than or equal to n. This is written as n! and pronounced "n factorial". The notation n
..... Click the link for more information. of n.



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



:


These different definitions have been proven In mathematics, the exponential function can be characterized in many ways. The following three characterizations (definitions) are most common. This article discusses why each characterization makes sense, and why the characterizations are independent of and equivalent to each other. As a special case of these considerations, we will see that the three most common definitions given for the mathematical constant
..... Click the link for more information. to be equal.

Properties

Many growth In mathematics, a quantity that grows exponentially is one that grows at a rate proportional to its size. This does not mean merely that for any exponentially growing quantity, the larger the quantity gets, the faster it grows. It implies that the relationship between the size of the dependent variable and its rate of growth is governed by a strict law, of the simplest kind: direct proportion. It is proved in calculus that this law requires that the quantity is given by the exponential function, if we use the correct time scale. This explains the name.
..... Click the link for more information. or decay A quantity is said to be subject to exponential decay if it decreases at a rate proportional to its value. Symbolically, this can be expressed as the following differential equation, where N is the quantity and ¦Ë is a positive number called the decay constant:


..... Click the link for more information. processes can be modeled with an exponential function. The exponential function The exponential function is one of the most important functions in mathematics. It is written as exp(x) or ex, where e is the base of the natural logarithm.
As a function of the real variable x, the graph of e
..... Click the link for more information. is important because it is the unique function (up to multiplication by a constant) which is its own derivative In mathematics, the derivative of a function is one of the two central concepts of calculus. (The other one is the antiderivative, the inverse of the derivative.)
The derivative of a function at a point measures the rate at which the function's value changes as the function's argument changes. That is, a derivative provides a mathematical formulation of the notion of
..... Click the link for more information. :





e is known to be both irrational In mathematics, the series expansion of the number e

can be used to prove that e is irrational.
Suppose e = a/b, for some positive integers a and b. Consider the number


We will show that x
..... Click the link for more information. and transcendental In mathematics, the Lindemann-Weierstrass theorem states that if ¦Á1,...,¦Án 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
..... Click the link for more information. . It was the first number to be proved transcendental without having been specifically constructed for this purpose (compare Liouville number 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.

..... Click the link for more information. ); the proof was given by Charles Hermite 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. Hermite polynomials, Hermite normal form, Hermitian operators, and cubic Hermite splines are named in his honor.
..... Click the link for more information. in 1873 1873 was a common year starting on Wednesday (see link for calendar).
EventsJanuary - April
January 17 - Indian Wars: First Battle of the Stronghold during the Modoc War.
February 11 - Spanish Cortes deposes King Amadeus I and proclaims the First Spanish Republic.


..... Click the link for more information. . It is conjectured to be normal
A different topic is treated in the article titled normal number (computing).

In mathematics, a normal number is, roughly speaking, a real number whose digits show a random distribution with all digits being equally likely. "Digits" refers to the finitely many digits before the point (the integer part) and the infinite sequence of digits after the point (the fractional part).
..... Click the link for more information. . It features in Euler's Formula This article is about the Euler's formula in complex analysis. For Euler's formula in algebraic topology, see Euler characteristic.
--------------------------------------------------------------------------------

Euler's formula, named after Leonhard Euler, is a mathematical formula in the subfield of complex analysis that shows a deep relationship between the trigonometric functions and the complex exponential function. (Euler's identity is a special case of the Euler formula).
..... Click the link for more information. , one of the most important identities in mathematics:





The special case with x = ¦Ð is known as Euler's identity In mathematics, Euler's identity is the following equation:


sometimes expressed as:
presumably in order to use the fundamental numbers 0 and 1 (see below).
The equation appears in Leonhard Euler's Introduction, published in Lausanne in 1748. In this equation,
..... Click the link for more information. :





described by Richard Feynman Richard Phillips Feynman (May 11, 1918¨CFebruary 15, 1988) (surname pronounced FINE-man;
..... Click the link for more information. as Euler's jewel.

The infinite continued fraction In mathematics, a continued fraction is an expression such as


where a0 is some integer and all the other numbers an are positive integers. Longer expressions are defined analogously. If the numerators are allowed to differ from unity, the resulting expression is a generalized continued fraction. For clarity,
..... Click the link for more information. expansion of e contains an interesting pattern (sequence in OEIS The On-Line Encyclopedia of Integer Sequences (OEIS) is a web-based searchable database of integer sequences. It is considered one of the major resources in mathematics.
The databaseThe Encyclopedia is a database recording information on integer sequences that are of interest in mathematics. The database contains over 100,000 sequences as of November 2004. Each entry contains the leading terms of the sequence, keywords, mathematical motivations, literature links, and more.
..... Click the link for more information. ) that can be written as follows:





History

The first references to the constant were published in 1618 EventsMarch 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).
The margraves of Brandenburg is granted Polish approval to inherit Ducal Prussia.
The Synod of Dort is convened.


..... Click the link for more information. 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 William Oughtred (March 5, 1575 ¨C June 30, 1660) was an English mathematician. He is credited as the inventor of the slide rule in 1622, and introduced the "¡Á" symbol for multiplication as well as the abbreviations "sin" and "cos" for the sine and cosine functions.
Oughtred was born
..... Click the link for more information. . The first indication of e as a constant was discovered by Jacob Bernoulli Jakob Bernoulli (Basel, December 27, 1654 - August 16, 1705), also known as Jacob, Jacques or James Bernoulli was a Swiss mathematician and scientist and the older brother of Johann Bernoulli.
Jakob Bernoulli met Robert Boyle and Robert Hooke on a trip to England in 1676, after which he devoted his life to science and mathematics. He lectured at the University of Basel from 1682, becoming Professor of Mathematics in 1687.
..... Click the link for more information. , trying to find the value of the following expression.





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.

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.


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Note added at 17 mins (2005-07-15 20:55:27 GMT)
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The exponent required to produce a given number:

http://www.thefreedictionary.com/natural logarithm

The logarithm of a number y with respect to a base b is the exponent to which we have to raise b to obtain y.