The number e is an important mathematical constant that is the base of the natural logarithm. It is approximately equal to2.71828, and 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
The constant can be defined in many ways; for example, e is the unique real number such that the value of the derivative (slope of the tangent line) of the function f(x) = ex at the point x = 0 isequal to 1. The function ex so defined is called the exponential function, and its inverse is the natural logarithm,or logarithm to base e. The natural logarithm of a positive number k can also be defined directly as the area under thecurve y = 1/x between x = 1 and x = k, in which case, e is the number whose natural logarithm is 1. There are also more alternative characterizations.
Sometimes called Euler's number after the Swiss mathematician Leonhard Euler, e is not to be confused with γ—the Euler–Mascheroni constant, sometimes called simply Euler's constant. The number e is also known as Napier's constant, but Euler's choice of this symbol is said to have been retained in his honor. 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 Euler's identity. Like the constant π, e isirrational: it is not a ratio of integers; and it is transcendental: itis not a root of any non-zero polynomial with rational coefficients. The numerical value of e truncated to 50 decimal places is
- 2.71828182845904523536028747135266249775724709369995...(sequence A001113 in OEIS).