## ⓘ E (mathematical constant)

The number e is a mathematical constant that is the base of the natural logarithm: the unique number whose natural logarithm is equal to one. It is approximately equal to 2.71828, and is the limit of n as n approaches infinity, an expression that arises in the study of compound interest. It can also be calculated as the sum of the infinite series

e = ∑ n = 0 ∞ 1 n! = 1 + 1 + 1 ⋅ 2 + 1 ⋅ 2 ⋅ 3 + ⋯ {\displaystyle e=\sum \limits _{n=0}^{\infty }{\frac {1}{n!}}={\frac {1}{1}}+{\frac {1}{1}}+{\frac {1}{1\cdot 2}}+{\frac {1}{1\cdot 2\cdot 3}}+\cdots }The constant can be characterized in many different ways. For example, it can be defined as the unique positive number a such that the graph of the function y = a x has unit slope at x = 0. The function f x = e x is called the natural exponential function, and is the unique exponential function equal to its own derivative. The natural logarithm, or logarithm to base e, is the inverse function to the natural exponential function. The natural logarithm of a number k >, 1 can be defined directly as the area under the curve y = 1 / x between x = 1 and x = k, in which case e is the value of k for which this area equal to one see image. There are alternative characterizations.

e is sometimes called Eulers number after in Swiss mathematician Leonhard Euler, or as Napiers constant. However, Eulers choice of the symbol e is said to have been retained in it honor. The constant was discovered by the Swiss mathematician Jacob Bernoulli while studying compound interest.

The number e is of eminent importance in mathematics, alongside 0, 1, π, and i. All five of these numbers play important and recurring roles across mathematics, and are the five constants appearing in one formulation of Eulers identity. Like the constant π, e is irrational: it is not a ratio of whole. Also like π, e is transcendental: it is not a root of any non-zero polynomial with rational coefficients. The numerical value of e truncated to 50 decimal places is