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*e* (mathematical constant)

**e** is a number, about 2.71828. It is a mathematical constant. *e* also has other names, like *Euler's number* (because of the Swiss mathematician Leonhard Euler), or *Napier's constant* (because of the Scottish mathematician John Napier). It is an important number in mathematics, like π and *i.* It is an irrational number, which means it is impossible to write as a fraction with two integers; but some numbers, like 2.71828182845904523536, come close to the true value. The true value of *e* is a number that never ends. Euler himself gave the first 23 digits of e.^{[1]}

The number *e* is very important for exponential functions. For example, the exponential function applied to the number one, has a value of *e*.

## Definitions

There are many different ways to define *e*. Jacob Bernoulli who discovered e, was trying to solve the problem:

- [math]\lim_{n\to\infty} \left(1+\frac{1}{n}\right)^n.[/math]

In other words, there is a number that the expression [math]\left(1+\frac{1}{n}\right)^n[/math] approaches if n is made very big. This number is *e*.

Another definition is to find the solution of the following formula:

- [math]2+\cfrac{2}{2+\cfrac{3}{3+\cfrac{4}{4+\cfrac{5}{5+\cfrac{6}{\ddots\,}}}}}[/math]

## The first 200 places of the number [math]e[/math]

The first 200 digits after the decimal point are:

- [math]e=2{.}71828\;18284\;59045\;23536\;02874\;71352\;66249\;77572\;47093\;69995[/math]
- [math]\;95749\;66967\;62772\;40766\;30353\;54759\;45713\;82178\;52516\;64274[/math]
- [math]\;27466\;39193\;20030\;59921\;81741\;35966\;29043\;57290\;03342\;95260[/math]
- [math]\;59563\;07381\;32328\;62794\;34907\;63233\;82988\;07531\;95251\;01901\,\ldots[/math].

## References

- ↑ Leonhardo Eulero:
*Introductio in analysin infinitorum*Vol. 1, Marcus-Michaelis Bousquet & socii, Lausannæ 1748, (latin; „2,71828182845904523536028“ on p. 90)

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