E (mathematical constant)

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The mathematical constant (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. It is approximately equal to

e ≈ 2.71828 18284 59045 23536 02874 .

It is equal to exp(1) where exp is the exponential function and therefore it is the limit

and can also be written in a number of ways as an infinite series, the most important of which is:

Here stands for the factorial of . A proof of the equivalence of these definitions is given below.

The number e is relevant because one can show that the exponential function exp(x) can be written as ; the exponential function is important because it is, up to multiplication by a scalar, the unique function which is its own derivative and is hence commonly used to model growth or decay processes.

The number e is known to be irrational and even transcendental. It was the first number to be proved transcendental without having been specifically constructed; the proof was given by Charles Hermite in 1873. It is conjectured to be normal. It features (along with a few other fundamental constants) in Euler's identity:

described by Richard Feynman as "the most remarkable formula in mathematics"!

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

Table of contents

1 History
2 Proof of equivalence of two definitions
3 External links
4 Reference

History

The first references of the constant was published in 1618 in the table of an appendix of a work on logarithms by John Napier. However, it did not contain the constant, but only a list of natural logarithms thereof. 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 limit of the following equation.

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.

Proof of equivalence of two definitions

The following proof demonstrates the equivalence of the infinite series expansion given for e above, and the limit of the equation studied by Bernouilli.

Define

By the binomial theorem,

so that

Here, we must use limsup's, because we don't yet know that t_n actually converges. Now, for the other direction, note that by the above expression of t_n, if 2 ≤ m ≤ n, we have

Fix m, and let n approach infinity. We get

(again, we must use liminf's because we don't yet know that t_n converges). Now, take the above inequality, let m approach infinity, and put it together with the other inequality. This becomes

This completes the proof. Q.E.D

External links

Reference

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