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 Pre-Calculus and Calculus
 The Original Definition of e?
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msakowski
Average Member

USA
18 Posts

Posted - 09/11/2007 :  07:07:15  Show Profile
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

Posted - 09/11/2007 :  08:28:14  Show Profile
Did you ask Dr. Math?

http://mathforum.org/dr.math/faq/faq.e.html

You can read for hours!
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msakowski
Average Member

USA
18 Posts

Posted - 09/11/2007 :  09:34:08  Show Profile
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
9117 Posts

Posted - 09/11/2007 :  15:45:58  Show Profile
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  Show Profile
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  Show Profile
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