To a certain extent this whole discussion is silly. There are many good
uses for significant figures, and just as many misuses. The reasons for
teaching it come from several sources. The first is getting students to
understand that measurements are inherently inaccurate at some level. The
other is so that they understand common usage. Common usage is where you
just write down the number without errors. Certainly in science one should
communicate the precision by using errors, but this is usually neglected
when doing problems and in everyday usage.
So if you see the population of a small town ,where I grew up, given as 475,
you know that this might still be accurate, but if you see the population of
a city as 1,670,453 you know this may have only been accurate at one time.
So a more realistic number would be 1,670,000 which might be rounded off to
the inherent uncertainty. Of course you have no way of knowing which zeroes
are rounded and which are precise. So significant figures are telling the
next person how accurate the number might be. But of course it is not an
exact communication. To be exact you need errors, or at least an indicator
of which digits are significant.
The big problem with SFs has already been brought out when they are used in
calculations. Doing rounding as you go can result in a poor or nonexistent
answer. This is where SFs have been abused. Rounding off the final answer
should not cause problems. The only time it may cause problems is when your
answer consists of several numbers which may be used again in calculations.
I once took a course where the instructor insisted you round off as you go.
But we were calculating averages and STDs, which could be done easily using
my scientific calculator. His answers did not agree with mine, because the
calculator did not round off in each step. So I had to do the calculations
one step at a time slowly so as to pass the test. He was abusing both SFs
and students. But in all fairness he didn't understand this. He was very
old fashioned, and due to retire soon. If I were teaching the course I
would allow for small variations in the final answers, but he did not. So
rather than trying to convince him I did it his way and of course got an A.
It is amazing what you have to do to get certification. You have to put up
with taking courses involving subjects that you are over-qualified to teach!
Unfortunately most explanations of SFs ignore the problems and have students
round off numbers too much. If you estimate that you only have 3 SFs you
should round off to 4SFs so that you have a guard digit. This takes cared
of the problem where 0.99 is 2SFs by the rules, but 1.00 is called 3SFs.
They are both the same accuracy which is about 1%.
The intelligent involved students will often recognize how the SF rules are
slightly off kilter when it comes to the previous example. Indeed the nerds
will often understand why SF rules actually work, but the average student
will not. So the conventional teaching of SFs is not very effective. It
just gives students rules that they don't understand. And they ignore
little problems like how absolute error grows during addition, but %error
decreases. Incidentally we should use the term random variation rather than
error, because error has the wrong implication in student minds.
So at some point the explanation of how this works should include examples
of what happens to answers when you vary the numbers. This can be done in
one class where each group experiments with varying numbers and then looks
at how the answer changes. Then they begin to understand why quoting
9.376108 as the answer to a problem is wrong, when the input numbers do not
have similar precision. Unfortunately texts just give the rules and assume
wrongly that students understand why they work. The students who quote far
too many SFs probably have poor numeracy and just forcing them to obey the
rules will not change it!
No journal would generally allow you to quote 9.376108 +-0.1 . You would
have to round off the number to 9.37. Insisting it be rounded to 9.3 is
wrong as that significantly increases the imprecision!
SFs probably are a good way of acquainting students with how precision of
numbers is important. You would not teach lower level or younger students
to do strict error analysis. Instead you introduce the idea of SFs. Later
on, and for advanced students error analysis would be appropriate,
especially if they are taking calculus.
Most of this discussion has been about how SFs are abused, and there is no
argument there. It should be used in conjunction with error analysis to
state the precision of the answer. But SFs are actually common usage and it
is impossible to make them totally consistent scientific usage. That is
where the problem lies. The rules are just rules of thumb, and like any
such rules has points at which they don't work well. So uncommon sense is
needed when using SFs.