E mathematical constant

The title given to this article is incorrect due to . The correct title is e (mathematical constant).

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. It is approximately equal to

e \approx 2.71828 18284 59045 23536 02874...

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.

Contents

Definitions

The three most common definitions of e are the following.

1. Define e by the following limit.
e = \lim_{n\to\infty} \left(1+\frac{1}{n}\right)^n
2. Define e as the sum of the following infinite series.
e = \sum_{n=0}^\infty {1 \over n!} = {1 \over 0!} + {1 \over 1!} + {1 \over 2!} + {1 \over 3!} + {1 \over 4!} + \cdots
where
n! is the factorial of n.
3. Define e to be the unique number x > 0 such that
\ln{x} = \int_{1}^{x} \frac{dt}{t} = {1}

A proof of the equivalence of these definitions is available here.

Properties

Many growth or decay processes can be modeled with an exponential function. The exponential function ex is important because it is the unique function which is its own derivative (up to a constant factor; the most general function that is its own derivative is kex, for any k).

The number 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 (c.f. 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:

e^{ix} = \cos(x) + i\sin(x) \,\!

The special case with x = π is known as Euler's identity:

e^{i\pi}=-1 \,\!

described by Richard Feynman as "Euler's Jewel".

The infinite continued fraction expansion of e contains an interesting pattern that can be written as follows:

e = [1; 0, 1, 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10, \ldots] \,

Proofs

See the following articles for proofs of properties of e:

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.

\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 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 is used nowadays as the standard symbol for the constant.

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.

Humorous use 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 integer.

Google was also the culprit of a mysterious billboard (http://mattwalsh.com/tAskFactMaster.Com/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 surprisingly starts as soon as the 101st digit), and visiting the web site advertised (http://www.7427466391.com), there was an even more difficult problem to solve.

External links

Reference

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



ca:Nombre e de:Eulersche Zahl es:Número e fr:Base naturelle des logarithmes ko:E (수학상수) he:E (קבוע מתמטי) la:Numerus Euleri nl:E (wiskundige constante) ja:ネピア数 pl:Podstawa logarytmu naturalnego ru:E (математическая константа) sl:E (matematična konstanta) fi:Neperin luku zh:纳皮尔常数

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