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Topic: "e"
Replies: 6   Last Post: Dec 19, 2004 3:09 PM

 Messages: [ Previous | Next ]
 Brian Harvey Posts: 26 Registered: 12/6/04
Re: "e"
Posted: Dec 10, 2004 5:34 PM

"Jerry Beeler" &lt;jerrybeeler@att.net&gt; writes:
&gt;Can anyone give me (1) a "non calculus" oriented derivation or explanation
&gt;for "e", and (2) an example or two of where "it often appears in physics and
&gt;math".

The conceptually simplest definition of e is that it's the number such that
the area bounded by y=1/x, y=0, x=1, and x=e is 1. But that isn't a really
satisfying definition -- it doesn't explain why e is important.

That's why you usually see an explanation that uses, if not calculus, at least
limits: most often, the one about continuously compounded interest.

Personally I like to define e^x by its Taylor series, because if you also
show (and verify by examples) the Taylor series for sin x and cos x, you can
get to one of (imho) the most beautiful things in mathematics, Euler's formula
e^{i\theta} = \cos \theta + i \sin \theta
and its special case
e^{i\pi} + 1 = 0
which is what made me a math lover back in high school.

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Date Subject Author
12/10/04 Me
12/10/04 katy
12/10/04 Brian Harvey
12/10/04 Nat Silver
12/11/04 Lisa Belec
12/19/04 GoElf@aol.com
12/19/04 Rob Morewood