section_path: Real exponents/Exponential function
section_name: Exponential function
paragraph_text: Exponential function
  
  The exponential function is often defined as x ↦ e x, {\displaystyle x\mapsto e^{x},} where e ≈ 2.718 {\displaystyle e\approx 2.718} is Euler's number. To avoid circular reasoning, this definition cannot be used here. So, a definition of the exponential function, denoted exp (x), {\displaystyle \exp(x),} and of Euler's number are given, which rely only on exponentiation with positive integer exponents. Then a proof is sketched that, if one uses the definition of exponentiation given in preceding sections, one has
  
  There are many equivalent ways to define the exponential function, one of them being
  
  One has exp (0) = 1, {\displaystyle \exp(0)=1,} and the exponential identity exp (x + y) = exp (x) exp (y) {\displaystyle \exp(x+y)=\exp(x)\exp(y)} holds as well, since
  
  and the second-order term x y n 2 {\displaystyle {\frac {xy}{n^{2}}}} does not affect the limit, yielding exp (x) exp (y) = exp (x + y) {\displaystyle \exp(x)\exp(y)=\exp(x+y)}.
  
  Euler's number can be defined as e = exp (1) {\displaystyle e=\exp(1)}. It follows from the preceding equations that exp (x) = e x {\displaystyle \exp(x)=e^{x}} when x is an integer (this results from the repeated-multiplication definition of the exponentiation). If x is real, exp (x) = e x {\displaystyle \exp(x)=e^{x}} results from the definitions given in preceding sections, by using the exponential identity if x is rational, and the continuity of the exponential function otherwise.
  
  The limit that defines the exponential function converges for every complex value of x, and therefore it can be used to extend the definition of exp (z) {\displaystyle \exp(z)}, and thus e z, {\displaystyle e^{z},} from the real numbers to any complex argument z. This extended exponential function still satisfies the exponential identity, and is commonly used for defining exponentiation for complex base and exponent.