Why does 1101 cm - 1091 cm = 10 cm with 2 significant figures?

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 × 104 kg - 1.01 × 104 kg 0.20 × 104 kg

After you convert the numbers to scientific notation, you see exactly where the uncertain place is (it's lit up in red). You have 2 significant figures in the answer. The last digit just happens to be a zero. (It would be best to write the answer as 2.0 × 10³ kg).

Reader Comments

(I disagree)... 1101-1091 yields 10, and if this number is written as 10, it has only one significant digit and the 0 is a place keeper. (a reader, 01/16/00

1101 is uncertain in the ones place, and 1091 is uncertain in the ones place, so 1101-1091 is uncertain in the ones place. If you are given a 10 without knowing where it came from, then yes, you say "1 significant figure". But if you have other information showing that the uncertainty is in the ones place, not the tens place, you have to say that it has two significant figures. Following your reasoning, if you take 1102 - 1091, you end up with 2 significant digits and if you take 1101-1091 you end up with only one. Why should such a small difference in the calculation result in such a large increase in the uncertainty?

Think about this example. What if you have an equation that looks like this. 1101.0001 - 1101.0000 divided by 0.00425 Here we have 8 sig figs in the numerator and 3 sig figs in the denominator. If we are dealing measurements, I am SURE that you would agree that we must have been using a very detailed measuring instruments (say for mass in grams for instance). If we follow your method and subtract the numbers in the numerator AND round to correct sig figs at that time, the numerator will have only 1 sig fig, as opposed to the 8 it had before the operation. Now your answer is limited to only 1 sig fig and your answer will be 0.02. If you round AFTER completing the math, your answer will have 3 sig figs and will be 0.0235 (clearly a major difference in the answers using different rounding approaches). I would think that you have to agree that it is not the goal to expressing answers in sig figs to lose any numbers that are relevant. From your perspective, looking at the example I have given, how could you possibly justify having only one sig fig in your answer? (a reader, 01/16/00)

Your error is in assuming that no precision is lost in the subtraction part of the calculation. In fact a subtracting two numbers that are almost exactly equal results in a serious loss of significant digits. This loss doesn't arise from rounding off intermediate results (which would obviously be wrong). It arises from the fact that the difference and the uncertainty in the original measurements are nearly the same size.

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!

Author: Fred Senese senese@antoine.frostburg.edu