Subject: Calculus

# E

e is a mathematical constant that has a value of 2.71828... (the tail numbers goes on forever). It is one of mathematic's most famous constant due to its wide array of uses and some cool stuff. Some of which are used in describing a nonlinear values such as growth or decay and the bell curve(statistics). It is also even used in some probability, counting and prime number distribution problems and even used to describe certain physical and chemical phenomena.

## e as a Formula

The constant e can also be expressed by the following formula;

e = \lim_{x \to \infty} \left (1 + \frac{1}{x}\right )^x

e is sometimes called as Euler's number, Eulerian number or Napier's constant and it can also be represented by this formula;

e = \sum_{i=1}^{\infty}\frac{1}{i!}

## e as an Area

e is also defined as the value that makes the shaded area in the graph of y=\frac{1}{x} below equal to 1.

If we compute the shaded area above from x=1 to some point e, we get

A = \int_{1}^{e} \frac{1}{x}dx = 1
= \ln{x}|_{1}^{e} = \ln{e} - \ln{1} =\ln{e} + 0

which leads us to some basic property of e which is

\ln{e} = 1

## Some Cool Stuffs

Mathematicians found many cool stuffs about e. Some of which are numbered below.

1. e has the value of 2.718 281 828 459 045. Notice the successive re-occurrence of the number 1828.

2. Upon taking the derivative of e^x, you get e^x back again. It is the only number who could do that.

3. Upon taking the integral of e^x, you also get back e^x again. (Of course since integration and differentiation are inverse operations).

### Example #1

Give other names of the mathematical constant e.

1. Euler's number
2. Napier's constant

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