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So here is my question: Say we have measured something to be 15,67 mm and the significant error is $\pm 0,01$mm. then we convert the measurement to meter to be 0,01567m would the significant error then be $\pm 0.00001$m?

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  • $\begingroup$ yes for @Qmechanic 's edited version, no for OP's version $\endgroup$ – DavePhD May 10 '14 at 12:17
  • $\begingroup$ Of Course. Once you change the unit of the measurement then you also have to change the unit of error into the same. Hence you had guessed it right! [For the recently edited one] $\endgroup$ – Saharsh May 10 '14 at 18:47
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In some sense this question is silly. You have made a measurement. The value you measured was $L \pm \Delta L$. You found $L=15.67\textrm{ mm}$ and $\Delta L = .01 \textrm{ mm}$. But really you should be thinking of $L$ and $\Delta L$ as lengths, independent of any representation in terms of a specific choice of units.

Now thinking of it this way, since $\Delta L = 0.01 \textrm{ mm}$, it necessarily follows that $\Delta L = 0.00001 \textrm{ m}$. This is just two different ways of representing the same length using different units.

So I say that this question is a little "silly" because you don't need to know anything about uncertainties to know that $0.01 \textrm{ mm} = 0.00001 \textrm{ m}$.

The one caveat is that it is preferable to make a consistent choice of units when giving the value of $L\pm \Delta L$. So for example it is ok to say $15.67 \pm .01 \textrm{ mm}$ or $0.01567 \pm 0.00001 \textrm{ m}$, but writing $ 15.67\textrm{ mm} \pm 0.00001 \textrm{ m}$ or $0.01567 \textrm{ m} \pm 0.01 \textrm{ mm}$ is considered a sub-optimal way of giving the value. I think if you read it, you will consider it to be more confusing as well. However, it is still logically consistent.

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You are correct that an uncertainty of 0.01 mm is the same as an uncertainty of 0.000 01 m. The unedited version of your question contained a factor-of-ten error.

For numbers with many significant digits, or numbers which are for whatever reason written in cumbersome units, there are a couple of tricks people use to avoid getting confused and making factor-of-ten errors. One is to write the digits in groups separated by spaces. (And if you're fussy about such things and your software supports it, you can use non-breaking thin spaces). Another is to write the uncertainty in the last digits of a measurement in parentheses next to the measurement — in your case, $$15.67(1)\text{ mm} \quad\text{or}\quad 0.015\,67(1)\text{ m}.$$ This format is common among precision measurement folks.

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