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An electronic balance was used to determine that a sample has a mass of 9.00060 g. If the balance's precision is ± 1 mg, what is the correct number of significant figures for this measurement?
actually the answer is 4 but I cant figure out from where has it come from? Isn't it 5?

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The significant digits you use are the values of which you are confident, plus one more value which may be uncertain.

In this case, the balance is accurate to $\pm0.001$ mg, so you're confident about the first 2 decimal places, but the 3rd place is uncertain.

The measurement given is $9.00060$, but the decimal part is smaller than our uncertainty, so we can't count on it. The most accurate we could record is $9.001$, because the first decimal we're uncertain about is the 3rd place, which leaves us 4 significant figures.

If the measurement had been $19.00060$ instead, then the same process would result in 5 significant figures.

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