The number e, also known as Euler's number, is a mathematical constant approximately equal to 2.71828 that can be characterized in many ways. It is the base of 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

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

The (natural) exponential function f(x) = e^x 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 1 (see image). There are various other characterizations.

The number 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. The constant was discovered by the Swiss mathematician Jacob Bernoulli while studying compound interest.

The number e is of great importance in mathematics, alongside 0, 1, π, and i. All five appear in one formulation of Euler's identity ${\displaystyle e^{i\pi }+1=0}$ and play important and recurring roles across mathematics. Like the constant π, e is irrational (it cannot be represented as a ratio of integers) and transcendental (it is not a root of any non-zero polynomial with rational coefficients). To 50 decimal places, the value of e is: