# e (mathematical constant)

The number e, also known as Euler's number, is a mathematical constant approximately equal to 2.71828 which can be characterized in many ways. It is the base of the natural logarithms. It is the limit of (1 + 1/n)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

${\displaystyle e=\sum \limits _{n=0}^{\infty }{\frac {1}{n!}}=1+{\frac {1}{1}}+{\frac {1}{1\cdot 2}}+{\frac {1}{1\cdot 2\cdot 3}}+\cdots .}$

It is also the unique positive number a such that the graph of the function y = ax has a slope of 1 at x = 0.

The (natural) exponential function f(x) = ex is the unique function f that equals its own derivative and satisfies the equation f(0) = 1; hence one can also define e as f(1). 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 equals one (see image). There are various other characterizations.

e is sometimes called Euler's number (not to be confused with Euler's constant ${\displaystyle \gamma }$), after the Swiss mathematician Leonhard Euler, or Napier's constant, after John Napier.[1] The constant was discovered by the Swiss mathematician Jacob Bernoulli while studying compound interest.[2][3]

The number e is of great importance in mathematics,[4][page needed] alongside 0, 1, π, and i. All five appear in one formulation of Euler's identity, and play important and recurring roles across mathematics.[5][6] Like the constant π, e is irrational (that is, it cannot be represented as a ratio of integers) and transcendental (that is, it is not a root of any non-zero polynomial with rational coefficients).[1] To 50 decimal places the value of e is:

2.71828182845904523536028747135266249775724709369995... (sequence A001113 in the OEIS).