The number e

The number e

The number e is an important mathematical constant that is the base of the natural logarithm. It is approximately equal to 2.71828,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

e=\displaystyle \sum \limits _{n=0}^{\infty }{\dfrac {1}{n!}}=1+{\frac {1}{1}}+{\frac {1}{1\cdot 2}}+{\frac {1}{1\cdot 2\cdot 3}}+\cdots

The constant can be defined in many ways. For example, e can be defined as the unique positive number a such that the graph of the function y = ax    has unit slope at x = 0.The function f(x) = ex    is called the exponential function, and its inverse is the natural logarithm, or logarithm to base e. The natural logarithm of a positive number k can also be defined 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 alternative 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 called simply Euler’s constant. The number e is also known as Napier’s constant, but Euler’s choice of the symbol e is said to have been retained in his honor.The constant was discovered by the Swiss mathematician Jacob Bernoulli while studying compound interest.

The number e is of eminent importance in mathematics,alongside 0, 1, π and i. All five of these numbers play important and recurring roles across mathematics, and are the five constants appearing in one formulation of Euler’s identity. Like the constant π, e is irrational: it is not a ratio of integers. Also like π, e is transcendental: it is not a root of any non-zero polynomial with rational coefficients. The numerical value of e truncated to 50 decimal places is

2.71828182845904523536028747135266249775724709369995… (sequence A001113 in the OEIS).

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 logarithms calculated from the constant. It is assumed that the table was written by William Oughtred. The discovery of the constant itself is credited to Jacob Bernoulli in 1683,who attempted to find the value of the following expression (which is in fact e):

 

\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 introduced the letter e as the base for natural logarithms, writing in a letter to Christian Goldbach of 25 November 1731.Euler started to use the letter e for the constant in 1727 or 1728, in an unpublished paper on explosive forces in cannons, and the first appearance 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.

In computer culture

In contemporary internet culture, individuals and organizations frequently pay homage to the number e.

For instance, 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 rounded to the nearest dollar. Google was also responsible for a 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”. Solving this problem and visiting the advertised (now defunct) web site 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. The first 10-digit prime in e is 7427466391, which starts at the 99th digit.

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.


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