Computations on significant figures

During multiplication and division, the answer should be written with the same number of significant figures as the operand with least significant figures.

However, while adding and subtracting, the answer should be written with same number of decimal places as the operand with least decimal places.

What is the reason for this difference? The problem is during adding and subtracting, sometimes the answer is more significant than the the operand having least significant numbers. Isn't that logically wrong?

(All the numbers I speak of above are physical measurements only.)

Imagine actually doing some measurements. Say you are measuring the length of a train. You measure all of the cars together and get 1234.5 meters (plus or minus 5 cm). Then you measure the locomotive and get 12.34 meters (plus or minus 5 mm). How precisely do you know the length of the train? Does it matter how short one of the parts you measure is relative to the total length?

If the above example doesn't make it clear, imagine measuring across your room, with the help of your roommate. You notice while taking the measurement that he has his finger stuck between the end of the tape measure and the wall. So the width of the room is equal to the length you measure, plus the thickness of your roommate's finger. I assert that you can know the width of the room to 3 significant digits from this measurement, even though you only know the thickness of your roommate's finger to one significant digit (pun intended).

One tends to forget these 'rules', and so the way to go about is to check that the relative error of the final answer is not less than the relative errors of the quantities involved when multiplying or dividing. When adding or subtracting, the absolute error of the final answer should not be less than the absolute errors of the contributing quantities.

• This approach is to error management is useless for any calculation more complex than unit conversion. (It does at least work for unit conversion, though...) The quantities in an equation contribute differently to error depending on how the final answer is related to them. These effects can and do often result in the relative error of the final answer being less than that of one or more measured quantities. There are also countless calculations in which the relative error of the final answer is several times greater than that of any measured quantity. Apr 23, 2016 at 20:02