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Check the figure:

pencil measurement using ruler

As you can see from the figure, the pencil has 5.65 cm length with 0.05 cm uncertainty. We simply could say the length has 3 significant figures considering 5 and 6 are known digits and 5 is estimated digit. But we also have 0.05 cm uncertainty, meaning that length could be 5.60 cm or 5.70 cm, right? So how can we say 5.65 ± 0.05 cm has 3 significant figures when it could be 5.70 cm? The digit "6" is also seems estimated or suspicious in first place (it is not well-known and could be "7")!

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  • $\begingroup$ A trailing zero, as in 5.70, can be a significant figure just like any other digit. Measuring 5.7 and measuring 5.70 isn’t the same thing because the latter is a more precise measurement. $\endgroup$
    – G. Smith
    Dec 15, 2019 at 18:33
  • $\begingroup$ But we end up with two suspicious digits in 5.65 cm becuse both digits after decimal point could be changed. I would say 5.65 ± 0.05 cm has 2 significant figures because of uncertainity. One is for known digit and the second is for not well-known digit. $\endgroup$
    – linx
    Dec 15, 2019 at 18:46
  • $\begingroup$ When the test comes you can find out whether your teacher agrees with you. I think the convention is to say it has 3. Part of learning physics is learning what is important and what is just convention. $\endgroup$
    – G. Smith
    Dec 15, 2019 at 18:51

1 Answer 1

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You are trying to apply significant figure rules to numbers with uncertainty, but you actually don't need to think about significant figures when the uncertainty is explicitly reported.

The purpose of significant figures is to loosely indicate the uncertainty in the measurement. In this case, like you are suggesting in your question, if we just said the length is $5.65\,\mathrm{cm}$ then this means we are sure about the $5.6$ part, but we aren't exactly sure about the $0.05$ part. In relation to the measuring device, with the given tick marks we are essentially saying, "My ruler has ticks every $0.1\,\mathrm{cm}$, so I'm sure this pencil is longer than $5.6\,\mathrm{cm}$. However, I don't have tick marks more precise than this, so I'm going to use my best guess based on my vision and ability to line things up to say it is half way between the $5.6$ and $5.7$ tick marks at $5.65\,\mathrm{cm}$." Based on this, it is obvious that the final significant digit tells others that we are not sure about that last $0.05\,\mathrm{cm}$. Even the math we do with keeping track of significant figures and the rules for addition vs. multiplication with using the correct amount of significant figures is to make sure we keep consistent with the uncertain digits.

The issue though is that significant figures don't tell us how uncertain we are. Do we think we have amazing sight and eatimation so we are within $0.01\,\mathrm{cm}$ of the $5.65\,\mathrm{cm}$? Or are we less confident in the $5.65\,\mathrm{cm}$ and want to report a larger uncertainty? Significant figures don't allow us to do this. Therefore, we can instead explicitly report the uncertainty as $5.65\pm0.05\,\mathrm{cm}$. At this point we don't need to worry about what that extra $0.05\,\mathrm{cm}$ in the measurement represents. The reported uncertainty takes care of this for us. And there are rules for doing math with numbers with uncertainty that keep our uncertainties consistent with the operations we choose to use, just like we have for significant figures.

Therefore, you don't need to worry about significant figures if uncertainty is explicitly reported. The uncertainty already tells us what is actually certain and what is not, thus it replaces (and improves) what significant figures are supposed to do.

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