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When I converted 968 torr to atm, I got 1.27368421053. Should my final sig fig answer be 1.27 ATM or 1.30 atm?
1.27 atm. Your second option implies only two , which should then be written as 1.3 atm, if it were correct.
That is, I mean that if you round ##1.27368421053## to ##1.27##, you treat the ##7## as your last significant digit, but if you round it to ##1.30##, you treat the first ##2## as your last significant digit.
For publication science your number of significant figures should never exceed the lowest accuracy of the data.
For practical use, I recommend that you retain ONLY ONE more significant figure to ensure later calculation accuracy (you can always reduce the number accurately, you cannot increase it once it's been lost).
Assuming that your conversion was 760 torr/atm, the definition assumes any level of significant figure for “atm” that is comparable to the original torr value. Therefore, the conversion is 760 torr/1.00 atm with three significant figures. Thus, your “long value” truncated (or rounded) to three significant figures is 1.27.