E (mathematical constant)

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  Part of a series of articles on
The mathematical constant e

Natural logarithm · Exponential function

Applications in: compound interest · Euler's identity & Euler's formula  · half-lives & exponential growth/decay

Defining e: proof that e is irrational  · representations of e · Lindemann–Weierstrass theorem

People John Napier  · Leonhard Euler

Schanuel's conjecture

The mathematical constant 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 is equal to 1.[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 also commonly defined as the base of the natural logarithm (using an integral to define the latter), as the limit of a certain sequence, or as the sum of a certain series (see the alternative characterizations, below).

The number e is 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 of eminent importance in mathematics,[2] alongside 0, 1, π and i. Besides being abstract objects, 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.

The number e is irrational; it is not a ratio of integers. Furthermore, it 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

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