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Re: "e"
Posted:
Dec 10, 2004 5:34 PM
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"Jerry Beeler" <jerrybeeler@att.net> writes:
>Can anyone give me (1) a "non calculus" oriented derivation or explanation
>for "e", and (2) an example or two of where "it often appears in physics and
>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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