Significant Figures in Long Sequences of Calculation

[Lightfuzz]

After losing marks in an exam due to significant figures, I have decided to clear up all my doubts about this concept. But since my teacher hasn't been very helpful, I've decided to post my question here.

I understand the rules for significant figures in both single-step multiplication/division and addition/subtraction calculations, but I am uncertain about what to do with them in long sequences of calculations, especially ones involving both types of arithmetic calculations. Based on my brief research on the Internet, I have found that intermediate answers should not be rounded as this would lead to accumulated error. Instead one should take a few more digits in the calculation process than required in the final answer and then round off at the end. But what I don't understand is, how do we decide how many significant figures to round off to at the end? Do I use the rule for multiplication/division (as many significant digits as the data with the least significant digits) or the rule for addition/subtraction (as many decimal places as the data with the least decimal places)?

Any help will be appreciated. Thank you.

 

[tiny-tim]

Hi Lightfuzz!

… But what I don't understand is, how do we decide how many significant figures to round off to at the end? Do I use the rule for multiplication/division (as many significant digits as the data with the least significant digits) or the rule for addition/subtraction (as many decimal places as the data with the least decimal places)?

Not really following you …

what is wrong with using the multiplication/division rule for multiplication/division and the addition/subtraction rule for addition/subtraction?

(If it's eg (A+B)C, then use the addition rule for A+B to find the number of decimal places, convert that into significant figures, then use that result in the multiplication rule for the final result)

 

[Lightfuzz]

Thanks for the response.

So would the following example be correct?

(1.5 + 2.03) x 1.1
= 3.53 x 1.1
= 3.883

But according to the addition rule, the addition intermediate step yields a result with 1 decimal place (3.5), which contains 2 significant figures. Therefore when this intermediate result is multiplied to a number with 2 significant figures, the answer should contain 2 significant figures (3.9).

 

[tiny-tim]

Hi Lightfuzz!

So would the following example be correct?

(1.5 + 2.03) x 1.1
= 3.53 x 1.1
= 3.883

But according to the addition rule, the addition intermediate step yields a result with 1 decimal place (3.5), which contains 2 significant figures. Therefore when this intermediate result is multiplied to a number with 2 significant figures, the answer should contain 2 significant figures (3.9).

Yes, that's exactly correct

Though it would be 2 significant figures anyway, since that is the smallest number of sig figs in the original

a better example would be
(11.5 - 2.72) x 2.22
= 8.78 x 2.22
= 19.4916

which should be written as 19 (to 2 sig figs), even though all the original terms had 3 sig figs

(and which, if you round off too early, gives you 8.8 x 2.22 = 19.536, which would be 20 )

 

[HallsofIvy]

You don't really need separate "rules" for addition or multiplication. The basic rule is "the result of a calculation has the same number of significant figures as the number, in the original data, with the least number of significant figures.

Here, both 1.1 and 1.5 have two significant figures while 2.03 has three. The smaller of those numbers is "2" so the result of ANY calculation with those numbers would have two significant figures.

 

[tiny-tim]

The basic rule is "the result of a calculation has the same number of significant figures as the number, in the original data, with the least number of significant figures.

no, i think for addition or subtraction it has to be based on decimal places

eg in 3.16 - 3.15 + 0.111, all three terms have 3 significant figures,

but the correct answer is 0.12 (2 decimal places, and only 2 sig figs)