section_path: Definitions
section_name: Definitions
paragraph_text: Definitions
  
  The number e is the limit lim n → ∞ (1 + 1 n) n, {\displaystyle \lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)^{n},} an expression that arises in the computation of compound interest.
  
  It is the sum of the infinite series e = ∑ n = 0 ∞ 1 n ! = 1 + 1 1 + 1 1 ⋅ 2 + 1 1 ⋅ 2 ⋅ 3 + ⋯. {\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.}
  
  It is the unique positive number a such that the graph of the function y = a has a slope of 1 at x = 0.
  
  One has e = exp (1), {\displaystyle e=\exp(1),} where exp {\displaystyle \exp } is the (natural) exponential function, the unique function that equals its own derivative and satisfies the equation exp (0) = 1. {\displaystyle \exp(0)=1.} Since the exponential function is commonly denoted as x ↦ e x, {\displaystyle x\mapsto e^{x},} one has also e = e 1. {\displaystyle e=e^{1}.}
  
  The logarithm of base b can be defined as the inverse function of the function x ↦ b x. {\displaystyle x\mapsto b^{x}.} Since b = b 1, {\displaystyle b=b^{1},} one has log b b = 1. {\displaystyle \log _{b}b=1.} The equation e = e 1 {\displaystyle e=e^{1}} implies therefore that e is the base of the natural logarithm.
  
  The number e can also be characterized in terms of an integral: ∫ 1 e d x x = 1. {\displaystyle \int _{1}^{e}{\frac {dx}{x}}=1.}
  
  For other characterizations, see § Representations.