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

[Bernard Cleyet <bernardcleyet@redshift.com>]

I am forwarding the nearly ever helpful Denker's answer to my friend.  
(Not the best English.)  I add:   Graphing the processes below is a good 
idea.

bc who bought one thousand dice to illustrate the below.  (On sale at 
Target. I exhausted the supply at three stores!)

p.s. I don't think he's "that" unwashed, just I'm not that washed as a 
teacher, but thinks it would profit him to "take" a course in analytic 
geometry (pre-calculus).



John Denker wrote:

> 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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