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I had thought that the non significant figures determined whether the least significant figure was to be rounded up. For example, given the number $9.1145$, if the number is to be expressed as three significant figures, then, I had thought that, it would be expressed as $9.12$.

However, I noticed that in one of my chemistry professor's calculations in which the quantities were expressed as significant to three figures, she expressed $8.125 \times 10^2$ as $813$. In another calculation she expressed $9.1145 \times 10^{-3}$ as $0.00911$. However, it seems to me that $0.00912$ is the correct expression of $9.1145 \times 10^{-3}$, as the $5$ requires rounding up the $4$, which requires rounding up the least significant $1$ to $2$.

Is there a rule for rounding that my professor is following that I don't know?

(Note: this isn't a homework question, but the homework tag seemed to be the most appropriate one.)

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    $\begingroup$ Never round twice. If that was the rule, then 0.46 could be rounded up to 1, which is clearly not the case. $\endgroup$ – HDE 226868 Oct 4 '15 at 19:04
  • $\begingroup$ Relevant meta discussion $\endgroup$ – M.A.R. Oct 4 '15 at 19:55
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The international standard ISO 80000 Quantities and units – Part 1: General includes the normative Annex B Rounding of numbers, which stipulates:

Rounding in more than one stage (…) may lead to errors. Therefore, the rounding shall always be carried out in only one step.

Example: 12.254 should be rounded to 12.3 and not first to 12.25 and then to 12.2.

(Note that, in the given example, the ISO standard follows the round half to even rule in which the even multiple is selected as the rounded number. This is the default rounding mode for floating-point computation. Anyway, the rule “rounding shall always be carried out in only one step” is independent of the used tie-breaking rule.)

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