Need help conceptualizing the constant "e"

[Enrique]

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Mathematicians were looking for a function that is equal to its own derivative. They narrowed down the search to functions of the form f(x) = a^x, where a is real.

For a fixed x,

f'(x) = lim (1/h)( f(x+h) - f(x) ) where h--> infinity

a^x = lim (1/h)( a^(x+h) - a^x ) where h---> infinity

factoring out a^x from the right hand side

a^x = a^x * lim (1/h)( a^x - 1) where h--> infinity

1 = lim (1/h) (a^x - 1 ) where h--->infinity

e is defined to be the unique value of a such that the equation above is true. You can massage the equation above and substitute h = 1/n to get the definition provided by previous posters.

[/ QUOTE ]

I don't think this is true. I think Euler was the first one to talk about the constant and he was trying to sum power series. Working out properties of summing power series, he found "e" although of course he didn't call it e and he noticed it was an important constant for summing stuff.

The property that DS mentions about everyone getting a new seat, is a cool probability that Euler discovered while working on what is called the hat problem: If you have n people entering a party and every one leaves his hat at the door to dance. If you give them their hats back randomly, what is the probability that no one got his hat back? The answer is 1/e.