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On 12/05/2007 02:19 PM, Bernard Cleyet wrote:
Anything that decreases proportional to its initial value is described by exp[- anything * t]. e.g. nuclear decay and lin. damped SHM. (charges in a resistive medium?) Why?
Why is almost never the right question. Physics needs
to say what happens; it need not and usually does not
say why it happens.
Rather than asking why, it is usually better to ask
"how do we know?"
http://www.av8n.com/physics/causation.htm#sec-1638
Now, to answer the question that should have been asked:
For those who are algebraically unwashed, sometimes an example will get the point across. You can do the
following using just a hand calculator ... or use a spreadsheet if you like:
Start with 1.0
Multiply it by 0.9; what have you got?
Multiply it by 0.9 again; what have you got?
Multiply it by 0.9 again; what have you got?
After ten iterations of a 10% decline, what have you got?
How does the ten-step process compare to just taking 0.9^10 in one step?
How does that compare to 1/exp(1)?
And how about 0.99^100?
And how about 0,999^1000?
When I do it, I conclude that not only do iterated percentage-
wise decreases result in strictly exponential behavior, the base of the exponent is very nearly e, and converges to e in the limit of a nice continuous process, if we take the
"natural" unit of time and divide it into many steps.
======================
Also note that choosing e for the base of exponentials (and
logarithms) is related to choosing to measure angles in
radians.
ln(1.001) = 0.001
sin(0.001) = 0.001
And this is *not* a coincidence. Some guy named Euler had
something to say about this.
http://en.wikipedia.org/wiki/Euler's_formula
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