Never forget that significant figures conventions were invented to prevent misunderstanding and wasted effort, not to establish rigid rules. There are circumstances in which common sense dictates violation of the conventions. For instance, suppose a runner finishes a 10 km race in 36 minutes, 17.31 seconds; it is reasonable to assume that the distance has actually been measured to the nearest few meters, so if it were expressed in the ``sig fig" convention it would be 10,000 m. But it usually isn't. Would you then feel obliged to ignore all but the first two significant figures of the mean speed calculated by dividing the distance by the time? Of course not. You are expected to use your judgement in applying significant figures.
Another situation in which significant figures can be misleading is the lengthy calculation involving many combinations of measured numbers with various uncertainties. In such cases there is always the danger that ``round-off errors" can accumulate as one proceeds through the steps of the calculation, yielding a final result that differs by more than the last digit from the one made ignoring the significant-figures convention. Therefore it is not considered bad form to retain one extra significant figure in all intermediate calculations and save the strict convention for the final result. If you are making calculations in an electronic calculator, of course, there is no reason to truncate any of the intermediate results unless you are transcribing them to paper to show your work.
Finally, the significant figures convention (like ``52.11")
is just a very compact (and slightly crude) means of expressing
uncertainty; if you want to say exactly what you mean, just use
the explicit notation ``
" or the shorthand
version ``52.113(6)" where the number in parentheses is the
uncertainty in the last digit. Such explicit notation
is always used in describing the results of lengthy and/or
expensive experiments like those at high energy particle
accelerators.
[ In fact, most of the hard work in analyzing the data from elementary particle physics experiments goes into meticulous estimation of uncertainties. You will often see results expressed in the form 14#14 where even the uncertainty is expressed to several significant figures! ]
The general rule is, use common sense and make sure your notation conveys clearly what you know and what you don't know.