e

e, in mathematics, number of great importance, comparable only to p (pi) 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, as n increases, the numerical 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

The number e forms the base of natural, or Napierian, logarithms. It appears in the exponential function, e^x, 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 e^ip = -1, in which i is the square root of -1.

The number e constantly appears 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 and even in pure number theory. Thus, the number of prime numbers in the first N numbers (if N is extremely large) is given by the expression N/ln N in which ln N is the natural logarithm of N and, therefore, a function of e.