But you know that the 10 cm is the difference between 1101 cm and 1091 cm, which are uncertain in the 1's place. That means the difference between these numbers will be uncertain in the 1's place.
"Significant figures" means "all digits up to and including the first uncertain digit". While you're carrying significant figure counts through a calculation with a lot of steps, keep an eye on where that first uncertain digit is. Converting the numbers to scientific notation will help you keep things straight by discarding the nonsignificant, placeholding zeros from the beginning. For example:
12100 kg -10100 kg 2000 kg | is better written | 1.21 × 10^{4} kg - 1.01 × 10^{4} kg 0.20 × 10^{4} kg |
Look at it from the perspective of uncertainties. There's an uncertainty in the ten-thousandths place in 1101.0001 and in 1101.0000. So the first number is 1101.0001 +/- 0.0001. It could be as low as 1101.0000, or as high as 1101.0002, right? The second number 1101.0000 +/- 0.0001 could be anywhere from 1100.9999 to 1101.0001. The difference between the numbers can be anywhere from 1101.0002 - 1100.9999 = 0.0003 to 1101.0000 - 1101.0001 = -0.0001. The ten-thousandths place is uncertain in the difference. Clearly the difference has only one significant figure.
Now do the whole calculation this way, to show that the range of results reflects one significant digit *without any intermediate rounding*:
The highest possible result is (1101.0002 - 1100.9999)/ 0.00424 = 0.07(0755)
The lowest possible result is (1101.0000 - 1101.0001) / 0.00426 = -0.02(3474)
So the uncertainty in the final result is in the hundredths place, and there is only one significant figure. The lesson is high precision for measurements doesn't guarantee high precision for results calculated from them!
Copyright © 1997-2010 by Fred Senese
Comments & questions to fsenese@frostburg.edu
Last Revised 08/17/15.URL: http://antoine.frostburg.edu/chem/senese/101/measurement/faq/print-difference-sigfig.shtml