Significant error conversion

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?

• yes for @Qmechanic 's edited version, no for OP's version – DavePhD May 10 '14 at 12:17
• 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] – Saharsh May 10 '14 at 18:47

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.
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.