Re: [Phys-l] quadratic uncertainty

[ludwik kowalski <kowalskil@mail.montclair.edu>]

On Aug 25, 2010, at 11:09 PM, John Denker wrote:

> Here's a little puzzle with some seasonal relevance:
>
> We need to find a good value for x
> /and for the uncertainty associated with x/
> given that:
>     a x^2 + b x + c = 0         [1]
>     a =  1 ± .0001
>     b = -2.08 ± .01
>     c =  1.08 ± .01
>
> This was mentioned in connection with the annual "sig figs"
> donnybrook on the chemistry list.  There are a thousand people
> on that list, and so far nobody has come up with a solution.
> One person came kinda close, but no cigar.
>
> I think it's safe to say that the problem is more interesting
> than it might at first appear.  The interest-to-difficulty
> ratio is pretty good IMHO.  After all, it's just a quadratic,
> so there's a limit to how hard it can be.
>
> The point of the exercise is to propagate the uncertainty
> from the inputs (a,b,c) to the result (x).  A lot of people
> talk the talk about propagation of uncertainty but have not
> much experience actually walking the walk, especially when
> it comes to calculations that require more than two or three
> steps.
>
> Any method of solution you can think of is fair game.

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
Was this question addressed to a mathematician or to a physicist? A  
mathematician would say that each (a,b,c) produces one x, a complex  
number. S/he would be interested in distributions or the real and  
imaginary parts of x, as clarified yesterday.

A physicist or engineer, on the other hand, dealing with a specific  
case, would have to decide whether or not the imaginary (or negative- 
real) solutions make sense.  Suppose the x stands for the mass of an  
object. In that case negative solutions would be ignored.

Ludwik

http://csam.montclair.edu/~kowalski/life/intro.html