[Phys-L] Re: infinite sig. figs.

[John Denker <jsd@AV8N.COM>]

Robert Cohen wrote:

> For example, a car travels 2 m in 7 s.  What is the car's average
> speed?  Can the student give an answer of 0.285714286 m/s and leave
> it at that?
>
> As I wrote before, I wouldn't take off but I would remind the
> student that people typically interpret (correctly or incorrectly)
> the uncertainty of such a number as being around +/- 0.0000000005
> m/s.

I'm an idealist.  Some would call me eccentric, or perhaps something
less flattering.  Remember Feynman's motto, "What do you care what
other people think".  I'm not responsible for what other people
"typically" do.  I'm only responsible for what I do.  If there are
two ways of doing something, i.e. "correctly or incorrectly", I
choose to do it correctly.  There is IMHO overwhelming evidence
that the sooner people stop trying to express uncertainty and/or
significance by counting digits, the better off everyone will be.

> uncertainty in speed as small as
> 0.0000000005 m/s.  And, rightly or wrongly, that is what
> 0.285714286 m/s (without any further information) seems to imply.

That's not what it implies to me.

I think that excess digits are a crime, and we should make the
punishment fit the crime.  Let's do the math:
  -- Black&white printing costs about a penny a page;
  -- There are about 500 words to a page;
  -- Each over-long numeral such as 0.285714286 means there
   will be an entire word less that fits on the page.
  -- Therefore each student who perpetrates such a crime will
   be fined 20 microbucks for each offense.

I figure if a student does that twice a day, every day, all term,
at the end of the term he will have to give me one gummi-bear, to
pay off the accumulated fines.

> Maybe I'm still missing the point between significance and uncertainty.

I see no evidence of that.  Mostly I see the main problem as being
attributing too much meaning to the digit-count.

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

To analyze it another way:  The cost of too many digits is on the
order of 20 microbucks, or maybe less, because the number can always
be rounded later.  The cost of too few digits is the cost of redoing
the measurement or calculation that produced the number, and all
calculations that depend on that number, to rid them of the roundoff
error that was introduced by the unwise rounding.  If you just spent
all day, or all week, or all year setting up an experiment, and if
you round off the observation unwisely, that could be expensive, far
more expensive than keeping extra digits.

When in doubt, write down plenty of guard digits.

If you are worried that some unenlightened person will misintepret
your long number as underestimating the uncertainty, express the
uncertainty explicitly, which you should be doing *anyway*!
For example: 0.285714286 +- 5%.
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