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The mathematical constant e is the base of the natural logarithm. It is 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. (e is not to be confused with γ – the Euler-Mascheroni constant – which is itself sometimes called Euler's constant.) e is one of the most important numbers in mathematics, alongside the additive and multiplicative identities 0 and 1, the imaginary unit i, and π, the circumference to diameter ratio for any circle. It has a number of equivalent definitions; some of them are given below. To the 20th decimal place: e ≈ 2.71828 18284 59045 23536
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: ight)^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, e was more common and eventually became the standard. The exact reasons for the use of the letter e are unknown, but it may be because it is the first letter of the word exponential. Another possibility is that Euler used it because it was the first vowel after a, which he was already using for another number, but his reason for using vowels is unknown. It is unlikely that Euler chose the letter because it is his last initial, since he was a very modest man, and tried to give proper credit to the work of others. Definitions The three most common definitions of e are listed below. where n! is the factorial of n. (that is, the number e such that area under the hyperbola from 1 to e is equal to 1). These definitions can be proved to be equivalent. Properties The exponential function f(x) = ex is important in part because it is the unique nontrivial function (up to multiplication by a constant) which is its own derivative, and therefore, its own primitive: and , where C is the arbitrary constant of integration. It is known that e is irrational (proof). In fact, if m/n is any rational number with n > 1, then (see Sondow, 2006) ight | > rac ,,,! where S(n) is the smallest positive integer such that n divides S(n)!. It is also known that e is transcendental (proof). It was the first number to be proved transcendental without having been specifically constructed for this purpose (cf. 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 formulas in mathematics: described by Richard Feynman (p. I-22-10) as "... the most remarkable formula in mathematics ..., our jewel". The special case with x = π is known as Euler's identity: The following is an infinite simple continued fraction expansion of e (sequence in OEIS): The following is an infinite generalized continued fraction expansion of e: The number e is also equal to the sum of the following infinite series: The number e is also given by several infinite product forms including Pippenger's product ight )^ left ( rac; rac ight )^ left ( rac; rac; rac; rac ight )^ cdots and Guillera's product (see Sondow, 2005) ight )^ left ( rac ight )^ left ( rac ight )^ left ( rac ight )^ cdots , where the nth factor is the nth root of the product as well as the infinite product The number e is equal to the limit of several infinite sequences: ight )^ and (both by Stirling's formula). The symmetric limit, may be obtained by manipulation of the basic limit definition of e. Another limit is ( S. M. Ruiz 1997 ) where is the nth prime and is the primorial of the nth prime. It was shown by Euler that the infinite tetration converges only if The number e is the global maximum of the function The value of this function at e is More generally, is the global maximum of the function Non-mathematical uses of e One of the most famous mathematical constants, e is also frequently referenced outside of mathematics. Some examples are: Notes O'Connor, "The number e" | ||||||||
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