Thread: Experts of "significant digits" please enter...

I have long been confused with significant digits for questions that involve a combination of addition / subtraction & multiplication / division, in science classes I have lost marks frequently on this, I know the calculations but I lost marks on roundings...so frustrated...I really need an expert on this to help me out...I would really appreciate because I am going to take Physics in which significant digits is particularly important!

Q1: The speed of P waves is 6.8 km/s and the speed of S waves is 4.1 km/s. How long would it take P waves and S waves to travel 100 km respectively?

This is simple!
t(P waves)=d/v
=100/6.8 (3 significant digits divided by 2 significant digits)
=14.7 sec (unround answer)
=15 sec (2 significant digits as a final answer to this question)

Similarily,
t(S waves)=d/v
=100/4.1 (3 significant digits divided by 2 significant digits)
=24.4 sec (unround answer)
=24 sec (2 significant digits as a final answer to this question)

Q2: What is the lag time between the arrival of P waves and S waves over a distance of 100km? (lag time is the time difference in arrival times of P waves and S waves)

I am starting to get confused...
Lag time=t(S waves) - t(P waves)
=(100/4.1) - (100/6.8)
=24.4 - 14.7 (use unrounded intermediate answers from above)
=9.7 or 9 or...? (problem starts to arise here...this calculation of lag time involve a combination of division and subtraction, some possible rounding methods popping off my head:
(i) (100/4.1) - (100/6.8) involve a combination of division and subtraction so it just follow the "rule of number of significant Digits in multiplication & division", i.e. the number of significant digits in an answer should equal the least number of significant digits in any one of the numbers of measurement, in this case, 2 significant digits, thus the final answer is 9.7 sec
(ii) 24.4 - 16.4 both have 1 deciminal place so the final answer should have 1 deciminal place as well, ie 9.7 sec
(iii) Follow "rule of number of significant Digits in multiplication & division" when doing the when doing multiplication/division and also follow "rule of number of significant Digits in addition & subtraction" when doing addition/subtraction, that is, rounding off every step:
=(100/4.1) - (100/6.8)
=24 (2 significant digits) - 15 (2 significant digits)
Now, doing subtraction and both have 0 deciminal place
Thus the final answer is 9 sec (0 deciminal place)
(iv) other...

Can someone point out the correct number of significant digits or deciminal places that should be in the final answer of Q2 and explain why?

Q3: Given the lag time for Austin is 150 sec, the distance from Austin to the location of the earthquake can be found using this formula:
distance = [lag time for Austin (s) x 100 km ] / [lag time for 100 km (s)]
The unrounded answer is 1546.3918 km, again I don't know how many significant digits I should keep in the final answer because I don't know the number of significant digits to the answer of Q2.

Thank you again for helping!

Your solution to Q1 is correct. For Q2 9.7 is correct. The rule for significant figures is that you only round off on your final answer so using the rounded answers in Q1 to get your answer in Q2 is a mistake. The combination of arithmetic operations is irrelevant.

Originally Posted by kingwinner

Q3: Given the lag time for Austin is 150 sec, the distance from Austin to the location of the earthquake can be found using this formula:
distance = [lag time for Austin (s) x 100 km ] / [lag time for 100 km (s)]
The unrounded answer is 1546.3918 km, again I don't know how many significant digits I should keep in the final answer because I don't know the number of significant digits to the answer of Q2.

Significant digits is really about the honest reporting of results. If you report 1546.3918 km without any estimate of the error then you are implying an error of .00005 and you are implying that you measured the velocities of the two types of waves to 8 significant figures. That can cause trouble for people using your results if they rely on this implied accuracy. But there is never any reason to round off results in the midst of a calculation and in fact it is will intoduce additional errors in your calculation.

Originally Posted by mitchellmckain

Your solution to Q1 is correct. For Q2 9.7 is correct. The rule for significant figures is that you only round off on your final answer so using the rounded answers in Q1 to get your answer in Q2 is a mistake. The combination of arithmetic operations is irrelevant.

Originally Posted by kingwinner

Q3: Given the lag time for Austin is 150 sec, the distance from Austin to the location of the earthquake can be found using this formula:
distance = [lag time for Austin (s) x 100 km ] / [lag time for 100 km (s)]
The unrounded answer is 1546.3918 km, again I don't know how many significant digits I should keep in the final answer because I don't know the number of significant digits to the answer of Q2.

Significant digits is really about the honest reporting of results. If you report 1546.3918 km without any estimate of the error then you are implying an error of .00005 and you are implying that you measured the velocities of the two types of waves to 8 significant figures. That can cause trouble for people using your results if they rely on this implied accuracy. But there is never any reason to round off results in the midst of a calculation and in fact it is will intoduce additional errors in your calculation.

But for Q2, it also involve a subtraction, and according to "rule of number of significant Digits in addition & subtraction", when doing subtraction, the final answer should have the same number of DECIMINAL PLACES as the given measurement with the least number of deciminal places. In this question, it also involve a division, so I don't know which rule of rounding to follow, deciminal places? significant digits?

Another question too! I was told that whole numbers have an infinite number of significant digits, so like Q1, "How long would it take P waves and S waves to travel 100 km respectively?", the 100km, would you consider this as having 3 significant digits or an infinite number of that? If the answer is an infinite number of significant digits, that doesn't make sense to me beucase, still, the "100km" is a measurement!

Originally Posted by kingwinner

Another question too! I was told that whole numbers have an infinite number of significant digits, so like Q1, "How long would it take P waves and S waves to travel 100 km respectively?", the 100km, would you consider this as having 3 significant digits or an infinite number of that? If the answer is an infinite number of significant digits, that doesn't make sense to me beucase, still, the "100km" is a measurement!

The only kind of whole numbers which would have an infinite degree of precision are constants in an equation (like diameter = 2 * radius) and an exact count of discrete objects like people or marbles. So you are right 100km does not qualify. In fact, technically the number of significant digits in 100 km is unknown ( it could be 1, 2 or 3). That is why we have scientific notation, in which there is no ambiguity about how many significant figures there are: 1x10^2 has 1 sig fig, 1.0x10^2 has 2 sig figs, and 1.00x10^2 has three sig figs. In a class a teacher may have his own convention, where he asks you to assume that something like 100km has 3 significant figures, if not, I would ask the teacher to clarify if this kind of thing is on an exam.

Originally Posted by kingwinner

But for Q2, it also involve a subtraction, and according to "rule of number of significant Digits in addition & subtraction", when doing subtraction, the final answer should have the same number of DECIMINAL PLACES as the given measurement with the least number of deciminal places. In this question, it also involve a division, so I don't know which rule of rounding to follow, deciminal places? significant digits?

I had not heard this rule but it makes sense, for deals with the problem of the annihilation of significant digits when subtracting numbers which are very close together. You do pose and interesting question and for the answer we must turn to error analysis. The result when applied to your problem is 9.684 +/- .605. This is an even larger error than is suggested by the answer 9 with only 1 significant digit (an implied error of +/- .5). I would not have thought so, but this suggest that your second method is a better one. Applying my rule that you only round off at the end would give you the even better anwer of 10 with 2 significant digits.
24.39 (only 2 sig fig so no decimal places)
-14.71 (only 2 sig fig so no decimal places)
---------
9.68 (no decimal places) so round up to 10 (no decimal places)
which is the same as 10 (2 significant digits)

Another thing that you need to understand is that significant digits is a very rough method of handling errors. There is nothing exact about it. And I think when you are getting down to only one or two significant digits it is really useless and you should start doing a real error analysis.