E (mathematical constant) Guide, Meaning , Facts, Information and Description
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
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2 Properties 3 History 4 Humorous use of e 5 External links 6 Reference |
The three most common definitions of e are the following.
Definitions
A proof of the equivalence of these definitions is available here.
Many growth or decay processes can be modeled with an exponential function.
The exponential function 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 , for any ).
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:
Properties
The special case with x = π is known as Euler's identity:
The infinite continued fraction expansion of e contains an interesting pattern that can be written as follows:
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.
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 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, there was an even more difficult problem to solve.
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