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

[John Mallinckrodt <ajm@CSUPOMONA.EDU>]

I don't disagree with the thrust of what Michael Edmiston says, but I
would offer my opinion that "reporting" is different from
"calculating."   Indeed, I'm quite explicit with students about the
need to maintain guard digits (preferably by learning about storage
registers) to use if one plans to do further calculations with a
reported value, but that doesn't and, IMO, shouldn't mean that one
reports them.

Michael commends John Denker's admonition:

> The proper roundoff rules are:
> -- keep many enough digits to avoid unintended loss of precision.
> -- keep few enough digits to be reasonably convenient.

but it seems to me that one should rarely if ever be in a situation
that requires "rounding" if it doesn't also IMMEDIATELY precede
"reporting."  There's really no particular need EVER to perform
rounding in any other situation. Let the calculator do the rounding
for you.  Even if it merely truncates rather than rounding, it'll be
out in the tenth digit or farther.  If you need more accuracy than
that, you probably shouldn't be using a calculator in the first place.

Michael points out, correctly IMO, that some of the most egregious
cases involve reporting too few digits in a result, provides the
following example (edited, perhaps unfairly, to maintain only the
"data"):

> ... students are trying to see how well the velocity of a glider on
> a level air track stays constant, and/or to see if air friction is
> observable, and their measurements show a velocity of 34.88 cm/s
> near the start and 34.64 cm/s near the end of the track ... they do
> this a bunch of times and if the track is truly level the later
> velocities are always a bit slower.  Clearly the equipment produces
> data good enough to demonstrate air friction. ... The data
> consistently show a slight slow down and rounding to the nearest
> cm/s doesn't consistently show this. ... I would be happy if the
> student reported 34.88 and 34.64 cm/s.  I would certainly tolerate
> 34.879 and 34.638 cm/s and probably would tolerate 34.9 and 34.6
> cm/s.

Again, perhaps I may be fairly accused of nitpicking, but I don't
think we have enough information to know what would constitute proper
reporting here.

In situations that involve reporting the results of primary
measurements rather than the results of calculations based on GIVEN
values, I'd need to see the uncertainties in the measurements to have
ANY idea how many sig figs to report.

If the measurements showed, for instance, that v1 = (34.88 +/- .04)
cm/s and v2 = (34.64 +/- .03) cm/s, then I'd prefer that the  student
simply say so.  But if they were going to report the values without
uncertainties for some reason, I'd pretty much want them to say 34.88
cm/s and 34.64 cm/s.  I might have a cringing tolerance for 34.9 cm/s
and 34.6 cm/s, but I would reject 34.879 cm/s and 34.638 cm/s.

It seems to me that the better approach in the situation Michael
describes is to  measure the two velocities many times and report the
results as, for instance,

v1 = (35 +/- 2) cm/s  and  v1 - v2 = (0.24 +/- .05) cm/s

in which case, the "appropriate sig figs" versions would be 35 cm/s
and 0.24 cm/s.

--
John "Slo" Mallinckrodt

Professor of Physics, Cal Poly Pomona
<http://www.csupomona.edu/~ajm>

and

Lead Guitarist, Out-Laws of Physics
<http://www.csupomona.edu/~hsleff/OoPs.html>
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