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.

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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.

[Fly]

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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


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I've never heard it called the "hat problem". Either "matching problem", "problem of recontre", or for the simplest case, counting derangements.

[borisp]

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...The answer is 1/e...

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I think you mean that the answer tends to 1/e as n approaches infinity. Lol mathaments. Btw, wiki thinks that Bernoulli was the first to "discover" e, and apparently he did it by considering continuously compounded interest. Of course, this could be wrong.

Here is a cooler problem, imo: show that the expected value of the # of people who get their hat back is 1, independent of n.

[David Sklansky]

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By definition e is the limit of (1+1/n)^n as n approaches infinity .

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I think this definition is arbitrary. e shows up in all sorts of places and you could use any one of them as the starting point to define it. My first introduction to e was by way of the area under the curve 1/x. e is that number such that the area from 1 to e under the curve 1/x is 1. The natural log function, ln, is defined as the area under the curve 1/x so from this definition e is that number such that ln(e)=1. From this definition properties like the one above can be derived.

PairTheBoard

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Somebody else must have written this post for you.