# How do mathematical operations involving exact numbers affect significant figures? [duplicate]

I'm asking for assurance that I'm not making any mistakes because I can't find these rules explicitly stated in my textbook or elsewhere, but it seems to me that:

When multiplying/dividing a measured number by an exact number, the product/quotient has the same number of significant figures as the measured factor.

and

When adding an exact number to a measured number, the sum may have a different number of significant figures as compared to the measured addend.

So for instance, if I were to subtract exactly 32 from a measured -7.0 x 10^1 degrees F, the difference would be -102. and contain 3 significant figures, unlike the original measured number which contained 2 significant figures. And if i then divide the answer by exactly 1.8, then the quotient -56.7 degrees C should still contain 3 significant figures.

Are my rules correct?

• In practical work, nobody really uses or bothers about these significant number rules the way they are taught in general chemistry courses. Do not worry too much about them. They will have no significance after your course. Focus on better ideas such a statistical treatment of data. These significant figures rules originated when people used mechanical devices to make measurements. May 7, 2022 at 0:16
• There is a nice book, which is available on Google (hosted by Berkeley Uni) for physics. It is titled Data Reduction and Error Analysis for the Physical Sciences. May 7, 2022 at 0:20
• As far as rules for sig figs go, the idea is that with mult. & divison, the result has a no. of s.f. equal to the input with the least no. of s.f., and with add.&subtr., the result has a no. of s.f. equal to the input with the least absolute precision (so if one num. goes to the tenths place, and other to the 100ths, the result goes to the tenths). But exact nos. have effectively infinite precision, so don't affect the s.f. of the answer. Hence when mult. or div. by an exact no., you retain the no. of s.f., and when adding or subtr. you retain the no. of digits before or after the decimal. May 7, 2022 at 4:51
• So, yes, your calculations are correct. But your 2nd rule needs to be edited, since it only says what the answer isn't, it doesn't say what it is. Instead, it should be: "When adding an exact number to a measured number, the sum will be expressed with the same absolute precision (i.e., be expressed to the same number of digits before or after the decimal place) as that of the measured addend." May 7, 2022 at 4:59
• The sig figures rules are just simplified handy applications of more general and more accurate rule of error propagation and additivity of variances: For addition/subtraction of 2 uncorrelated variables, squares of absolute standard deviations are additive. The same for multiplication/division and squares of relative standard deviations. May 7, 2022 at 17:23