The question which arose my confusion:
Two resistors with resistance $R_1 = 100\pm 3 \ \Omega$, and $R_2 = 200\pm 4 \ \Omega$ are connected in (a) series, (b) in parallel. Find the equivalent resistance of both combinations.
The confusion:
In the end, my answer came out to be (a) $300\pm 7 \ \Omega$ and (b) $66.7\pm 2 \ \Omega$. But according to my book, my response for (b) is incorrect.
It said that the equivalent resistance should have an error of $\pm 1.8$ not my rounded off answer $\pm 2$. And it (book) explicitly states that the error in (b) should be "expressed as $\pm 1.8$ to keep in conformity with the rules of significant figures." I don't understand how this makes any sense (especially how it conforms to the rules of significant figures). Help anyone?
Best Answer
The easy part is that $66.7 \pm 2$ is wrong. It is wrong because unless we know the tenths digit of the error, expressing the main value to the tenth's digit doesn't make sense: we'd be force to drop the result in that digit as soon as we add or subtract. So we should write $67 \pm 2$ or write both the main figure and the uncertainty to the tenths column (that is, $66.7 \pm 1.8$).
The harder part (and indeed the part with a little wiggle room) is recognizing that both of the inputs are accurate to better than three percent, so they should be treated as having about three digits of precision. However, if you are old enough to recall the slide-rule convention for leading 1s (which requires that $1.00 \times 10^2$ is a figure with only two digits of precision), you might feel that that fractional errors of a few percent should imply two digits not three.
Part of the problem is that there is no completely internally consistent way to deal with uncertainty using the crude tool that is significant figures. Working scientists don't follow a checklist on significant figures they just always remember to not write figures that have no meaning. And in that frame of mind I would prefer $66.7 \pm 1.8$.