## Uncertainties

A quick question about uncertainties and significant figures:

Say, we have some numbers with a particular uncertainty 0.1 of each of them. What happens if the sum of these numbers has a greater amount of significant figures than each of the numbers alone? For instance,

1.01 + 9.99 = 11.0 or 11.00? (I believe it is the former)

Then, suppose I keep the uncertainty of the sum 11.0, +/-(2 x 0.1), as percentage, say x per cent, and make some calculations. The sum 11.0 will then be changed.

Would I have to take x per cent of the result? For instance,

11.0 x 2 = 22.0 +/- x%?

Or should I not convert the uncertainty to percentage? If not, that if the units are different?
 Recognitions: Gold Member Science Advisor Staff Emeritus If your measurements have an uncertainty of 0.1 then you should write them as 1.01 or 9.99 since those imply they are correct to +/- 0.005, not +/- 0.1. Assuming that you really mean 0.005, then the lowest possible value of the first is 1.01- 0.005= 1.005 and the largest possible value is 1.015. Similarly, the lowest possible value of the second is 9.99- 0.005= 9.985 and its highest value is 9.995. That means that the lowest possible value of their sum is 1.005+ 9.985= 10.990 and the highest value is 1.015+ 9.995= 11.010. The midpoint of those is 11.000 and they are 11.000+/- 0.005 again so that should be written as 11.00 to indicate that the uncetainty is the last digit. There is an engineer's "rule of thumb" that if two measurements are added or subtracted, their "errors" are added (NOT subtracted!) and if two measurements are multiplied or divided, their percentage "errors" are added.
 I was taught that when adding or subracting you round to the least precice decimal place. so 9.99 +1.01 would be 11.00 but when multiplying or dividing you round to the least number of significant digits, so 1.4*456.2 = 640 because 1.4 only has 2 sig digs.