How to Round Answers with Significant Figures for Weights in Physics

[fojkehelky]

I have two weights. One is 50 grams and one is 100 grams. I don't know how exact the given weight is, i.e. I don't know if the 50 grams is 50.00 grams or 50.0 grams.

I use these numbers (50 and 100) in a calculation involving several other numbers that are not weights but other things. I get an answer that looks something like 9,381562307564 and I have to round it. How do I round it?

I normally take all the numbers that I used in the calculation and look at each number to see how many significant figures it has and then I round the answer to the smallest number of significant figures I found among all the numbers I used in the calculation.

BUT

what I'm thinking about is this: let's say we know the weights are exactly 50.00 grams and 100.00 grams. Wouldn't that mean that 50.00 has 4 significant figures and 100.00 has 5 significant figures? Should I do what I usually do and of these say that 4 is the lowest and use that if it is the lowest of all the numbers?

OR is some other method better? Because isn't the whole point of using significant numbers that the answer reflects the accuracy (or precision, don't know which...) of the numbers I used when I did the calculation? And 50.00 grams and 100.00 grams are both accurate to within 0.01 grams, right? So the accuracy of these two weights should be/is the same even though they have a different amount of digits, right?

I'm confused...

Another thing: when you have the measurement/number (whatever you call it, English is my second language) 100 grams, all you know for certain is that it has at least 1 significant figure, right? But in this case it probably has more than 1 significant figure, right? So I should not use 1 because it's probably accurate to within at least 1 gram and not only to within 100 grams. Is the significant number method not always the best method? Is there a decimal places method or something?

I only learned about this briefly in school and now I have to use it in physics.

 

[Borek]

I would give 9.4 as the answer, assuming 50 has 2 sigfig.

50.00+100.00=150.00 (5 sig figs), rules for addition and subtraction are slightly different than rules for multiplication and division. 100 + 0.12567532673245 = 100, even if the second number has a lot of sig figs.

Finally, sigfigs are only an approximation, better than nothing, but hardly reliable when it comes to calculating real precision of your final result.

 

[phinds]

... let's say we know the weights are exactly 50.00 grams and 100.00 grams

Just FYI, you should be careful with your terminology. "Exactly" generally means "to infinite precision", so "EXACTLY 50.0 mean 50.0000000000000000000000000 ... "

 

[fojkehelky]

I would give 9.4 as the answer, assuming 50 has 2 sigfig.

50.00+100.00=150.00 (5 sig figs), rules for addition and subtraction are slightly different than rules for multiplication and division. 100 + 0.12567532673245 = 100, even if the second number has a lot of sig figs.

Finally, sigfigs are only an approximation, better than nothing, but hardly reliable when it comes to calculating real precision of your final result.

What if you use addition, subtraction, multiplication and division in a calculation and get an answer with a lot of digits and you want to round it?

My main point with this thread was: if you have 100.00 and 50.00 should you really use 5 and 4 significant numbers respectively or should you use some method involving decimal places or some other method instead? Because to me it seems weird that these two weights have different sig figs yet they are of the same accuracy. Both are accurate to within 0.01. I have heard that there are other, better, methods for rounding than using significant figures but I have not yet found what these methods are.

edit: What are those rules for addition/subtraction and multiplication/division? It seems like least amount of sig figs is the rule for addition subtraction, right? What's the rule for multiplication/division?

 

[fojkehelky]

I have been researching this on the internet. I found this:

Multiplication/Division combined with Addition/Subtraction

Use as many digits as possible in intermediate calculations, but round to the appropriate number of sig figs for the final answer.

When measurements are added or subtracted, the answer can contain no more decimal places than the least accurate measurement.

When measurements are multiplied or divided, the answer can contain no more significant figures than the least accurate measurement.

Use the appropriate sig fig rules, as stated above, depending on which operation you are performing at that time. (Example: 1. multiply/divide functions; or 2. add/subtract functions) At the end of each step, you must ask yourself,
"What is the next operation that I will perform on the number that I just calculated?"
If the next operation is in the same group of operations that you just used, (Example:
1. multiply/divide; or 2. add/subtract) then do NOT round off yet.
If the next operation is from the other group, then you must round off that number before
moving on to the next operation.

I copied and pasted segments from these 3 pages and edited some of it slightly:

http://www.edu.pe.ca/gray/class_pages/krcutcliffe/physics521/sigfigs/sigfigRULES.htm

http://chemed.chem.purdue.edu/genchem/topicreview/bp/ch1/sigfigs.html

http://www.spy-hill.net/~myers/notes/SigFigs.html

I hope this will be all I need for the moment. It will have to make do for now. I have to focus on the homework. I've spent hours on this...

 

[fojkehelky]

Just FYI, you should be careful with your terminology. "Exactly" generally means "to infinite precision", so "EXACTLY 50.0 mean 50.0000000000000000000000000 ... "

ok thanks

 

[CWatters]

I remember doing this 40 years ago at school but not all the detail. There is a rigorous way to do this but best I can remember is that..

50 can be any number between 49.5 and 50.49999
so 50 really means 50 +/-1%

So answer can't usually be quoted to better than 1%.

If I remember correctly to squeeze every last decimal point out you need to do a "sensitivity analysis" to find out how sensitive the answer is to errors/tollerance in each value that contributes to it. For example if the problem involves subtracting two similar numbers the answer can be very sensitive. If it involves adding one very large number to one very small number then errors in the small number maybe less important then those of the larger number.

 

[Borek]

I have heard that there are other, better, methods for rounding than using significant figures but I have not yet found what these methods are.

Google "propagation of uncertainty".