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msakowski
Average Member
USA
18 Posts |
 Posted - 09/11/2007 : 07:07:15
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Euler's number fascinates me. It shows up everywhere. Analytically solve a differential equation, and chances are very high it will be part of the solution.
So how did Euler come up with the definition?
In calculus, "e" is defined as LN e = the integral of 1/t dt from 1 to e such that this integral evaluates to 1. So is this the original definition?
How do we get the definition e = lim (1 + 1/n)^n as n=>inf from this?
Did this definition precede the integral definition?
Once I get to the latter limit definition, I am able to prove the derivative formulas for e, but not until then.
It is easy to demonstrate why "e" occurs "naturally". Simply work a compound interest problem with the number of compounding periods approaching infinity and do a substitution. But here again you have to use the limit definition of e.
Anybody know more about this? |
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tkhunny
Advanced Member
USA
1001 Posts |
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msakowski
Average Member
USA
18 Posts |
Posted - 09/11/2007 : 09:34:08
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Lot's of stuff about e, that's for sure. Thanks for the link.
In the article at http://www-history.mcs.st-and.ac.uk/HistTopics/e.html it states
[Euler] showed that
e = 1 + 1/1! + 1/2! + 1/3! + ...
and that e is the limit of (1 + 1/n)n as n tends to infinity.
My question is, HOW could he show this unless he already had a definition of e? Sure, he could demonstrate that the series matched the reoccurring value to a number of satisfactory places, but that is not proof. So, what was the very first definition of e?
From the article, it sounds like mathematicians all over the place kept encountering this quantity, 2.71821... , and they were aware that this was some sort of universal number. Euler seemed to be the first to provide a solid definition for this quantity. I would like to know what that "first definition" was. |
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Subhotosh Khan
Advanced Member
USA
9114 Posts |
Posted - 09/11/2007 : 15:45:58
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Somewhere I read (I don't quite remember where)
e^x came from the definition:
if df(x)/dx = f(x) then f(x) = e^x |
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msakowski
Average Member
USA
18 Posts |
Posted - 09/12/2007 : 16:26:48
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Subhotosh, your reply
e^x came from the definition:
if df(x)/dx = f(x) then f(x) = e^x
makes sense.
From this you get the series definition and I would imagine the integral property as well.
How is the formula (1 + 1/n)^n for n=>inf derived from this? |
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pka
Advanced Member
USA
2731 Posts |
Posted - 09/12/2007 : 17:38:48
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