e (mathematics).

e (mathematics). e (mathematics), in mathematics, number of great importance, comparable only to p (the ratio of the circumference of a circle to its diameter) in the wide variety of its applications. The number e is most commonly defined as the limit of the expression (1 + 1/n )n as n becomes large without bound. Some values of this expression for increasing values of n are included in the accompanying table. An examination of the right-hand column of the above table will show that the value of the expression becomes closer and closer to a limiting value. This limiting value is approximately 2.7182818285. The value of e may also be determined by computing the limit of certain infinite series. One example of such a series is Unlike p, e has no simple geometric interpretation. Like p, it is a transcendental number; that is, it is not the root of any polynomial equation a0xn + a1xn-1 + ... + an = 0 with integral coefficients. The number e forms the base of natural, or Napierian, logarithms (see Logarithm). It appears in the so-called exponential function, ex, the only function having a rate of growth equal to its size (in the language of calculus, the only function having a derivative equal to itself), and therefore the fundamental function for equations describing growth and many other processes of change. In geometry, e is a necessary component of the formulas for many curves, such as the catenary, the shape assumed by a cord suspended from its extremities. In the study of Imaginary Numbers, e appears in the extraordinary equation eip = -1, in which i is the square root of -1. The number e appears constantly in the theory of probability. For example, if many letters are written and the corresponding envelopes addressed, and the letters are then fitted at random into the envelopes, the probability that every letter will go into a wrong envelope is extremely close to e-1. The number e also appears in formulas for calculating compound interest. Even in pure theory of numbers, e crops up. Thus, the number of prime numbers in the first N numbers (if N is extremely large) is given by the expression N/log N in which log N is the natural logarithm of N and, therefore, a function of e. See also Pi. Contributed By: James Singer Microsoft ® Encarta ® 2009. © 1993-2008 Microsoft Corporation. All rights reserved.