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

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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.

(*Also, in my class, we assume 100km has 3 significant digits, so no worry there...)

Thank you again for helping!

 

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Some simple rules. Whenever multiplying, the answer should have a number of significant digits equal to the least number of significant digits displayed in the multiplied numbers. Whenever adding, the answer should have the same number of decimal places as the least number of decimal places appearing in the added numbers. Finally, when adding and multiplying, you should apply the rules of significant digits before and after each addition .

 

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How do significant digits work for operations such as polynomial, trigonometric, or exponential functions? I.E. You have a measurement of an angle theta to 3 significant digits (degrees). How many significant digits do you have for sin(theta)?

 

When evaluating functions, the function result should have the same number of significant digits as the least number of significant digits among the inputs. So the sine of an angle known to three significant digit has three significant digits. The exponential of a product of two numbers, one with 3 sig figs and one with 2 sig figs should have 2 sig figs. And again, if you have addition anywhere in the process, always apply the rules of sig figs before and after each addition.

 

I am okay with "The rule of number of significant digits in multiplication & division" and "The rule of number of significant digits in addition & subtraction" but the problem is when they are mixed together in a formula, e.g. divisions within subtraction, I searched the web sites and looked in text books but NONE of them say anything about this "combination" cases...terrible...

Back to your post, what actually do you mean apply the rules of sig figs before and after each addition, I don't quite get your meaning...

For exmaple Q2,
Lag time = (100/4.1) - (100/6.8)
=24-15 (the 2 divisions should each has 2 significant digits, but how can I show this work? By writing 24-15? That would be like doing roundings every step...

OR should I do it like this?

=24.3902 - 14.7059 (however, this would be a 4 deciminal places subtraction)
=9.6843 (therefore, the final answer to Q2 should have 4 deciminal places--the same number of significant digits as the one with the least number of significant digits of the subtracting terms...still this doesn't seem to be correct because this 24.3902 and 14.7059 actually has an infinite number of deciminal places when they are not rounded)

HELP!

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Wouldn't you round it up to 10?

This brings back bad memories - I always came unstuck tracking errors.

 

You are not suppose to round until you get to the final answer, but keep track of significant figures which I have color coded.

a = 100/4.1 = 24.390243902439024

b = 100/6.8 = 14.705882352941177

a - b = 9.684361549497847

Correct me if I am wrong, but when you roung 9.7 up, you get 10 and 2 significant figures. Also, you can round the figures as long as they are rounded out of the significant figures range but I just copy and pasted the whole lot. For example, 24.390243902439024 could be rounded to 24.390

 

Generally 0's to the left of the decimal are not considered significant, e.g. 120 has 2 significant figures.

-Dale

 

Kingwinner, although the rules given here are correct, *most* professors will not be terribly picky about you doing it exactly right. Generally it is enough for you to do your calculations to full precision and then simply report the final answer to the least number of significant figures in the supplied data. This effectively ignores the addition and subtraction rules and only uses the multiplication rules.

Only a really anal professor will require more significant figure tracking. I would find out what kind of professor you have before worrying; just ask his policy.

-Dale

 

Yes, the technical way to write 10 with 2 significant figures would be 1.0x10

 

I think it is a given that he has anal professor(s):

Personally, I never really had professors that cared a whole lot about significant figures. Concepts were much more important to them, and I believe this too. Only chemistry nazi's got on my case about significant figures.

 

Agreed. But personally I find scientific notation very cumbersome and don't like to use it unless I really have to. I tend to use engineering notation with the SI prefixes whenever possible. Much more consice.

Could it be that engineers are generally in more of a hurry than scientists?

-Dale

 

It's probably more of the "culture". A scientist's "design" is often his numbers. An engineer's design is more material and numbers are just a tool.

 

Same with my professors, and I think that is the way it should be. I mean, significant digits are much less important to a physics class than learning the concepts of motion, energy, etc. Putting that kind of emphasis on minutia really detracts from important learning.

-Dale

 

In my class, we assume 100km has 3 significant digits, so no worry there...

How about doing it this way?
Q2: Lag time=t(S waves) - t(P waves)
=(100/4.1) - (100/6.8)
=24.39 (only 2 sig fig so no decimal places) -14.71 (only 2 sig fig so no decimal places)
=9.68 (0 decimal places)
=10 (0 decimal places so round up to 10) ?

This way avoid rounding intermediate answers, but for this way I have to keep track of significant digits & sometimes deciminal places (when doing subtraction/addition) for every step, this is driving me crazy and sometimes the formula is just really complicated, like a= d * (b+c) - d * (abc+e), if I would have to do it the above way, it would take forever...

Sometimes I do have strict teachers caring about significant digits, like a question that worth 2 marks, they take off 0.5 marks off even if you got everything right, crazy....this is what makes me to make this long post because I don't want the same thing to happen again in future physics and chemistry or other science courses...

"Kingwinner, although the rules given here are correct, *most* professors will not be terribly picky about you doing it exactly right. Generally it is enough for you to do your calculations to full precision and then simply report the final answer to the least number of significant figures in the supplied data. This effectively ignores the addition and subtraction rules and only uses the multiplication rules."
Lag time=(100/4.1) - (100/6.8)
=9.68446155
=10 (round to 2 significant digits because the least number of significant digits in the measured values is 2 significant digits? *Although this answer "10" is still the same as my above method, conceptially they are different, with the above being rounded to 0 deciminal places and this one being rounded to 2 significant digits, and sometimes doing one way would make a difference compared to the other)

Your way of following the "number of significant digits of multiplication & division" when doing a question involving a combination, say, division & subtraction, really saves a lot of time, but is this the accepted way in high schools and universities? (By the way, I remember I have one teacher in grade 10 saying this same rounding rule as you)

 

High Schools and Universities do not generally set standards on what should be "accepted". It is up to the teacher/professor. In some cases it can be very important to keep track of significant figures. I suspect you are in a class that is just introducing the concept. In which case it is important that you do it correctly and not use the shortcuts presented here.

 

I agree completely with Aer. If you are just learning the concept then you have to go the long and cumbersome route described by others. Otherwise it is entirely up to the the whim of the professor if the shortcut method is acceptable or not.

If you are concerned, you should ask the professor directly about his policy.

-Dale