E (mathematical constant)

Part of a series of articles on the
mathematical constant e
Properties
Natural logarithm Exponential function
Applications
compound interest Euler's identity Euler's formula half-lives exponential growth and decay
Defining e
proof that e is irrational representations of e Lindemann–Weierstrass theorem
People
John Napier Leonhard Euler
Related topics
Schanuel's conjecture
v t e

The number e is a mathematical constant approximately equal to 2.71828 that can be characterized in many ways. It is the base of the natural logarithm function. It is the limit of ${\displaystyle (1+1/n)^{n}}$ as n tends to infinity, an expression that arises in the computation of compound interest. It is the value at 1 of the (natural) exponential function, commonly denoted ${\displaystyle e^{x}.}$ It is also the sum of the infinite series

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

The number e is sometimes called Euler's number, after the Swiss mathematician Leonhard Euler, though this can invite confusion with Euler numbers, or with Euler's constant, a different constant typically denoted ${\displaystyle \gamma }$. Alternatively, e can be called Napier's constant after John Napier.^[2][3] The constant was discovered by the Swiss mathematician Jacob Bernoulli while studying compound interest.^[4][5]

The number e is of great importance in mathematics,^[6] 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.^[7][8] Like the constant π, e is irrational, meaning that it cannot be represented as a ratio of integers, and moreover it is transcendental, meaning that it is not a root of any non-zero polynomial with rational coefficients.^[3] To 40 decimal places, the value of e is:^[1]