This is an audio version of the Wikipedia Article: 00:00:35 1 History
00:01:10 2 Applications
00:01:45 2.1 Compound interest
00:02:21 2.2 Bernoulli trials
00:02:56 2.3 Standard normal distribution
00:03:14 2.4 Derangements
00:03:49 2.5 Optimal planning problems
00:04:24 2.6 Asymptotics
00:04:59 3 In calculus
00:05:35 3.1 Alternative characterizations
00:06:28 4 Properties
00:07:03 4.1 Calculus
00:07:38 4.2 Inequalities
00:08:13 4.3 Exponential-like functions
00:08:49 4.4 Number theory
00:09:24 4.5 Complex numbers
00:11:27 4.6 Differential equations
00:12:03 5 Representations
00:12:20 5.1 Stochastic representations
00:12:56 5.2 Known digits
00:13:31 6 In computer culture
00:14:06 7 Notes
00:14:41 8 Further reading
00:15:17 9 External links
00:15:52 Complex numbers
00:17:03 Differential equations
00:17:38 Representations
00:19:59 Stochastic representations
00:20:34 e.
00:21:10 Known digits
00:21:45 In computer culture
00:22:20 Notes
00:22:38 Further reading
00:23:13 External links
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SUMMARY
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The number e is a mathematical constant that is the base of the natural logarithm: the unique number whose natural logarithm is equal to one. It is approximately equal to 2.71828, and 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
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{\displaystyle e=\sum \limits _{n=0}^{\infty }{\frac {1}{n!}}={\frac {1}{1}}+{\frac {1}{1}}+{\frac {1}{1\cdot 2}}+{\frac {1}{1\cdot 2\cdot 3}}+\cdots }
The constant can be characterized in many different ways. For example, it can be defined as the unique positive number a such that the graph of the function y = ax has unit slope at x = 0. The function f(x) = ex is called the (natural) exponential function, and is the unique exponential function equal to its own derivative. 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 alternative 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 as Napier's constant. However, Euler's choice of the symbol e is said to have been retained in his honor. The constant was discovered by the Swiss mathematician Jacob Bernoulli while studying compound interest.The number e is of eminent importance in mathematics, alongside 0, 1, π, and i. 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. Like the constant π, e is irrational: it is not a ratio of integers. Also like π, e is transcendental: it is not a root of any non-zero polynomial with rational ...
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