Re: [Phys-l] quadratic uncertainty

[John Denker <jsd@av8n.com>]

On 08/25/2010 08:30 PM, Stefan Jeglinski wrote:
> > 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]

> Just one question for starters - is there intentionally no 
> uncertainty quoted for the RHS of the quadratic equation? 

Any uncertainty in the RHS is indistinguishable from
the uncertainty associated with the zeroth-order term (c)
on the LHS ... which is stated and which is nontrivial.

Writing the 0 on the RHS is an empty formality.  You
could equally well write 

     a x^2 + b x = -c         [2]

if you find that more appealing.  The answer is the
same either way.

===========

This question is entirely on-topic.  The quadratic thing
is interesting unto itself, but also it serves to stir up
a broader discussion of uncertainty.  And this is a good
thing to do this time of year, since some chemistry classes
heavily emphasize "significant figures" in the first week
of the semester.  The prevalence of this has been declining
in recent years, and I am trying to hasten the decline.

FWIW I think that even in the "sig figs" world 0 is exact.  
You could interpret it as 0 ± 10% ... but ten percent of 
zero is still zero.

As a piece of sociology, when *I* write a number, even a
decimal such as 2.54, if I don't state the uncertainty then
(unless I have made a mistake) you can assume that either
  a) the uncertainty doesn't matter, or
  b) the uncertainty is zero, as in (1 inch) / (1 cm) = 2.54

There is no scenario where I will intentionally use the number
of digits to encode the uncertainty (or anything else). For
details on how to do things right, see
  http://www.av8n.com/physics/uncertainty.htm

On 08/25/2010 08:30 PM, ludwik kowalski wrote:
> > One possibility is to use the Monte Carlo method. 

A definite possibility.  Verrry powerful.

> > a) Select a, b, c by using a random number generator. Then solve for x
> > b) Repeat this 1000 times, for example, and plot the resulting
> > distribution of x

a) It's easier than that.  Fifty repetitions is enough
to give you a good start.  You can do more later if you
decide you want to.

b) Item (a) is important because it means you can do
worthwhile Monte Carlos using only a spreadsheet.

c) Item (b) is important because quite a few students 
who would be hopelessly intimidated by any scientific
programming language are reasonably comfortable with
a spreadsheet.

> > But for any given of (a,b,c) we have two solutions for x. I am not sure
> > how to deal with this. /

Calculate them both.  Plot them both.

> > Is selecting + or - randomly acceptable? 

It's easier to just calculate them both.

> > Suppose the distribution of uncertainties (in a, b and c) are gaussian. 

Ah, well, you raise an interesting point, because quite
a few spreadsheets have no built-in Gaussian random number 
generator.  That's perhaps the biggest single barrier for 
a new user to get started doing Monte Carlo on a spreadsheet.
Sometimes a Gaussian RNG is available as an "add in" and 
sometimes not.

Huge hint:  Box-Muller transform.

Even huger hint:  =sqrt(-2*ln(rand()))*sin(2*pi()*rand())+0*rand()