All duly noted and understood. And yet I'd still be flabbergasted.
Let me try another line of explanation.
1) Obviously, I'm not going to lift a finger to write down digits
that I know to be insignificant. It's not worth my time.
2) On the other side of the same coin, I'm not going to lift a
finger to round off something that doesn't need to be rounded
off.
a) It's not worth my time.
b) Roundoff error is an error. Roundoff error never makes anything
better.
3) None of that has anything to do with the central topic of this
thread, which is "significant figures -- again". The crucial point
remains that the number of digits must not be interpreted as saying
anything about the significance, uncertainty, tolerance, or anything
like that.
Conversely, if the number of digits did imply anything about the
uncertainty, I would find time to worry about it ... but it doesn't
so I won't.
4) As I have said before, there are perfectly reasonable situations
where I simply DO NOT KNOW the uncertainty. In such situations, I
will write down a huge number of digits, because the cost of having
extra digits is small, and the cost of having too few digits is
often very very large.
Let's be clear: I am not going to lift a finger to write down
digits I know to be insignificant, but I need to write /something/
even when I DO NOT KNOW the level of significance.
This is important, because it shines a light on one of the most
pernicious things about sig figs doctrine, namely the notion that
you are not allowed to write down a number without "implying"
something about the uncertainty.
The rule should be, and must be: If you have a number, write down
the number. Just write it down already! (Worrying about the
uncertainty can come later, if at all.)
In particular, a student who doesn't know the uncertainty should
be *encouraged* to write down more-than-enough digits. The idea
of teachers docking points from students who are doing the right
thing really makes me cringe.
5) Nowadays people use electronic things called "computers". That
allows keeping lots of digits at virtually zero cost. I can sorta
maybe imagine that back in the days BC (before computers) it was
important to minimize the number of digits at every step of the
calculation ... but those days have been over for a while now.
Maybe in another 100 years of so, textbooks will take into account
the fact that people take data using computers.
By the way, the principle of "don't round off unless you have a
good reason" applies even to computers. In 1970, Forman Acton wrote
that all numerical methods should be done using double precision,
unless there was a compelling reason to optimize for a smaller
word-size. That was more than 40 years ago. That's a lot of dog
years. That was back when it might actually cost you something
to use double precision.
I guarantee you that I know how to optimize things like this. I've
done time-critical arithmetic on eight-bit (and smaller) processors
for real-time applications. I also know that one of the principles
is: Never optimize things that don't need to be optimized. A
related principle is: Don't guess. Typically 1% of the code (or
less) ends up being time-critical, and it is very hard to guess in
advance which 1% that will be. The wise approach is to build a
prototype and /measure/ it so that you know for sure what bits need
to be optimized.
6) I believe in getting the right answer. If students or colleagues
are doing something that interferes with getting the right answer, I
will hassle them about it. Conversely, if it is something that cannot
possibly interfere with getting the right answer, such as carrying
extra guard digits .... Phooey! I've got better things to do than
hassle people about that.
Let's be clear:
-- Following the sig figs rules interferes with getting the right
answer.
-- Carrying lots of guard digits does not.
7) Jeff Bigler raised the issue of "politeness" and cosmetics.
Style is important, but I say getting the right answer comes first.
Later, after we are assured of getting the right answer, *then* we
start worrying about cosmetics and artistic "style points".
I would add that rounding things off to the just-right number of
digits is very far down on the list of cosmetic issues. There are
other things (such as getting all the decimal points to line up) that
are more important, for practical as well as cosmetic reasons.
I would be pleasantly astonished to see an introductory text that
talked about sig figs as a purely cosmetic issue ... or indeed
that said anything about how to do physics with a sense of style,
artistry, or panache.