Re: [Phys-l] Significant figures -- again

[John Denker <jsd@av8n.com>]

On 04/13/2012 01:29 PM, Robert Cohen wrote:
> I don't have a problem with your main message.  It just seems to me
> that an example like Bob and Carol illustrates it better.

We agree, the matrix example is pretty convincing.  I would 
like it even better if it were accessible to high-school 
students (and others) who have never heard of matrices.  
I'm still looking for additional examples, and for ways of 
simplifying the existing examples.

> Or, even simpler...
>
> I worked a year at company A, took some time off, then worked a year
> at company B.  How many days did I work at each company?
>
> Is writing 400 days at each company any better or worse than writing
> 365.2425 days at each company?

That's an interesting and useful example, although it makes
a somewhat different point.

> I would say it would be neither better nor worse without some way of
> conveying the uncertainty. 

We agree that finding a way to clearly express the uncertainty
would be a huge improvement.

As soon as there is a decent expression of the uncertainty, all
the problems with this example go away.  It seems to me that
   t = 1 year ± 10%                       [1a]
and
   t = 365 days ± 10%                     [1b]

are both perfectly reasonable ... and both are preferable to 
   ☠  X=400   ☠                           [2a]
and 
   ☠  X=365.2425  ☠                       [2b]

As usual, I reckon it is a bad idea to spend time discussing
misconceptions in front of students.  My advice is to show them
the right procedure (equation [1]) and not waste time fussing
with equation [2] or anything like that.

Note that the perfectly reasonable expression [1b] departs
strongly from the usual sig-figs rules.  I suppose I could
round off from 365 to 360, but why would I want to?  I can
write 365±10% just as easily as 360±10%.  

  In contrast, converting to 3.6e-2 would expend more work 
  to produce a less-accurate and less-convenient result.
  This strikes me as insane.

===================

Furthermore, we should not let any of these examples distract
us.  We need to see the whole forest, not just this-or-that
tree.

Keep in mind that the whole concept of sig figs has some serious
downsides.  For one thing, it essentially forces students to 
misunderstand the distinction between roundoff error and 
uncertainty.  This is a big deal, because in a well-designed 
experiment, roundoff error is almost never the dominant 
contribution to the overall uncertainty.  Confusing the two 
is wrong in principle, costly in practice, and foolish in 
pedagogical terms.

An even larger point is that teaching sig figs is a grotesque
waste of time.  Other methods must be learned.  More-or-less
everybody concedes that other methods must be used (at least!) 
most of the time, if you want anything resembling accurate 
answers.  Sure, there are isolated cases where you can get 
away with sig figs, but why bother learning a technique that 
only works in isolated, low-priority cases, when other techniques 
are no worse and usually better?  Surely class time is already 
overcommitted.  Surely it would be better to spend time on 
things that have more upside and less downside.
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