e (mathematical constant)

The mathematical constant e is the unique real number such that the value of the derivative (slope of the tangent line) of f(x) = ex at the point x = 0 is exactly 1. The function ex so defined is called the exponential function, and its inverse is the natural logarithm, or logarithm to base e.

The number e is one of the most important in mathematics, alongside the additive and multiplicative identities 0 and 1, the imaginary unit i, and π, the circumference to diameter ratio for any circle in a plane. It has a number of equivalent definitions; some of them are given below.

The number e is occasionally called Euler's number after the Swiss mathematician Leonhard Euler, or Napier's constant in honor of the Scottish mathematician John Napier who introduced logarithms. (e is not to be confused with γ – the Euler–Mascheroni constant, sometimes called simply Euler's constant.)

Since e is transcendental, and therefore irrational, its value cannot be given exactly as a finite or eventually repeating decimal. The numerical value of e truncated to 20 decimal places is: