Re: [Phys-l] Help! An explanation for the laity

[John Denker <jsd@av8n.com>]

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