Definition and Understanding of e Number

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e number is a number that is the base of natural logarithms. There are several ways of defining it. Probably the most satisfactory is this. First, define natural logarithms (ln) as in approach 2 to the logarithmic function. Then define exponential (exp) as the inverse function of ln in exponential function. Then define e as equal to exp 1. This amounts to saying that e is the number that makes

$\int_{1}^{e}\frac{1}{t}dt$

It is necessary to go on to show that $e^{x}$ and exp x are equal and so are identical as functions, and also that ln and $log_{e}$ are identical functions.

The number e has important properties derived from some of the properties of ln and exp. For example,

$e=\lim_{h\rightarrow 0}(1+h)^\frac{1}{h}=\lim_{n\rightarrow \mathbb{\sim }}\left ( 1+\frac{1}{n} \right )^n$

Also, e is the sum of the series

$1+\frac{1}{1!}+\frac{1}{2!}+...+\frac{1}{n!}+...$

Another approach, but not a recommended one, is to make of one these properties the definition of e. Then exp x would be defined as $e^{x}$ and exp x, ln x would be defined as its inverse function, and the properties of these functions would have to be proved.

The value of e is 2.718 281 83 (to 8 decimal places). The proof that e is irrational is comparatively easy. In 1873, Hermite proved that e is transcendental, and his proof was subsequently simplified by Hilbert.

13 Aug 2015