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  #11  
Old 11-17-2007, 01:13 PM
Enrique Enrique is offline
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Default Re: Need help conceptualizing the constant \"e\"

[ QUOTE ]
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.
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  #12  
Old 11-17-2007, 01:40 PM
Fly Fly is offline
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Default Re: Need help conceptualizing the constant \"e\"

[ QUOTE ]

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


[/ QUOTE ]

I've never heard it called the "hat problem". Either "matching problem", "problem of recontre", or for the simplest case, counting derangements.
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  #13  
Old 11-17-2007, 06:46 PM
borisp borisp is offline
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Default Re: Need help conceptualizing the constant \"e\"

[ QUOTE ]
The more important thing about e concerns making prop bets when poker tournaments redraw

[/ QUOTE ]
lololololololololol...etc

Oh, and

[ QUOTE ]
Somebody else must have written this post for you

[/ QUOTE ]

lololololololololol...etc
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  #14  
Old 11-17-2007, 10:22 PM
David Sklansky David Sklansky is offline
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Default Re: Need help conceptualizing the constant \"e\"

[ QUOTE ]
[ QUOTE ]

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


[/ QUOTE ]

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

[/ QUOTE ]

Would make a small bet that its known mainly by the words "Euler's problem of the misaddressed letters"
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  #15  
Old 11-17-2007, 10:49 PM
Fly Fly is offline
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Default Re: Need help conceptualizing the constant \"e\"

[ QUOTE ]
[ QUOTE ]
[ QUOTE ]

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


[/ QUOTE ]

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

[/ QUOTE ]

Would make a small bet that its known mainly by the words "Euler's problem of the misaddressed letters"

[/ QUOTE ]

Sure, so long as "mainly" corresponds to what has more (relevant) google hits and/or a wiki entry.

Edit - Spelling Correction

Should be problem of rencontres not recontre
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  #16  
Old 11-17-2007, 11:59 PM
borisp borisp is offline
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Default Re: Need help conceptualizing the constant \"e\"

[ QUOTE ]
...The answer is 1/e...

[/ QUOTE ]
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.
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  #17  
Old 11-18-2007, 12:08 AM
blah_blah blah_blah is offline
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Default Re: Need help conceptualizing the constant \"e\"

I don't want to spoil it, but in fact, if \sigma is an element of the symmetric group on n letters, p is a polynomial, and \fix\sigma denote the number of fixed points of \sigma, then there are methods to evaluate

\sum_{\sigma\in S_n} p(\fix\sigma)

here is a simple proof of the for the case p = id which generalizes to higher degree polynomials. let \fix_i \sigma = 1 if \sigma fixes the ith place and 0 otherwise.

\sum_{\sigma\in S_n} \fix\sigma =
\sum_{\sigma\in S_n} \sum_i \fix_i\sigma =
\sum_i \sum_{\sigma\in S_n} \fix_i\sigma =
\sum_i (n-1)! =
n(n-1)! =
n!

which is the desired result.
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  #18  
Old 11-18-2007, 12:19 AM
Fly Fly is offline
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Default Re: Need help conceptualizing the constant \"e\"

[ QUOTE ]

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.

[/ QUOTE ]

How is this cooler? This is way easier to solve than the original problem, just use <font color="white"> indicator functions </font> &lt;---- answer in white.
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  #19  
Old 11-18-2007, 12:23 AM
borisp borisp is offline
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Default Re: Need help conceptualizing the constant \"e\"

[ QUOTE ]
I don't want to spoil it

[/ QUOTE ]
Then why did you post the answer? [img]/images/graemlins/smile.gif[/img]

Say \sigma is a permutation of n letters, and V is a vector space of dimension n, with basis e_i. Define a linear map e_i \mapsto e_{\sigma(i)}. The trace of this linear map is equal to the number of fixed points of \sigma. This observation, together with the fact that trace is linear, is basically your argument.
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  #20  
Old 11-18-2007, 12:25 AM
borisp borisp is offline
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Default Re: Need help conceptualizing the constant \"e\"

[ QUOTE ]
[ QUOTE ]

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.

[/ QUOTE ]

How is this cooler? This is way easier to solve than the original problem, just use <font color="white"> indicator functions </font> &lt;---- answer in white.

[/ QUOTE ]
Cooler in that it admits several elegant and simple solutions. To me, easier problems are cooler.
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