Explain the process E in the form of an equation.
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The number e, also known as Euler's number, is a mathematical constant approximately equal to 2.71828, and can be characterized in many ways. It is the base of the natural logarithm.[1][2][3] It 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[4][5]
Graph of the equation y = 1/x. Here, e is the unique number larger than 1 that makes the shaded area equal to 1.
{\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 }{\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 also the unique positive number a such that the graph of the function y = ax has a slope of 1 at x = 0.[6]
The (natural) exponential function f(x) = ex is the unique function which is equal to its own derivative, with the initial value f(0) = 1 (and hence one may define e as f(1)). The natural logarithm, or logarithm to base e, is the inverse function to the natural exponential function. The natural logarithm of a number k > 1 can be defined directly as the area under the curve y = 1/x between x = 1 and x = k, in which case e is the value of k for which this area equals one (see image). There are various other characterizations.
e is sometimes called Euler's number, after the Swiss mathematician Leonhard Euler (not to be confused with γ, the Euler–Mascheroni constant, sometimes called simply Euler's constant), or Napier's constant.[5] However, Euler's choice of the symbol e is said to have been retained in his honor.[7] The constant was discovered by the Swiss mathematician Jacob Bernoulli while studying compound interest.[8][9]
The number e has eminent importance in mathematics,[10] alongside 0, 1, π, and i. All five appear in one formulation of Euler's identity, and play important and recurring roles across mathematics.[11][12] Like the constant π, e is irrational (that is, it cannot be represented as a ratio of integers) and transcendental (that is, it is not a root of any non-zero polynomial with rational coefficients).[5] To 50 decimal places the value of e is: