Say we divide 1858.54 cm by 5 — where 5 is an exact number. We get the quotient 371.708
According to the rules of significant figures, we would answer write 371.708 cm, since "1858.54" is 6 significant figures, and 5, being an exact number doesn't count.
My question is, how do you make sense of 371.708 cm, which is more precise than the original? That is, the quotient is precise to the nearest thousandth while the original is precise to nearest hundredth. To me that doesn't make sense.
Best Answer
Significant figures are a heuristic for keeping track of accuracy in a calculation. If you want to be careful, you need to assign an uncertainty to each number and keep track of how they build up. You also need to decide whether you want the error band on your result to be a worst case or some sort of "$3 \sigma$" based on assumptions of the distribution of errors in your input. Often we assume normal errors because they are easy to compute with, but real world errors tend to have longer tails than that.
For your specific case, absent other information we assume $1858.54$ is $\pm 0.005$. If we divide by an exact $5$ we get $371.708 \pm 0.001$. This is not quite as good as a $6$ significant figure $371.708$, which should be $\pm 0.0005$ but it is better than $371.71\pm 0.005$. For many purposes we do not care to this level of detail and we just track significant figures. If we do care, give the result as $371.708\pm 0.001$ and indicate whether this is a hard limit (like if the original $1858.54$ was measured with a graduated scale) or has some chance of being outside the range.