Experts of "significant digits" please enter...

[cwes99_03]

I have always used the first method, despite the extra bit of time you have to put into it.

 

The second method could possibly get a bit off, for instance if you were to subtract two numbers and the answer was 9.4 and you left it that way, but with method one you would recognize that your answer should only be 9. This would occur if you said you should have two sig figs, and used 9.4 when really your sig fig rules would only allow you to have a number in the 1's position and no decimal places.

 

Alternatively if you had to subtract before multiplying in the following sense.

 

(9.6-9.3)*8.2=0.3*8.2 which only allows for one sig fig, namely your answer would be 2.

[Garak]

"(100/4.1) ~ 24.39 keeping an extra decimal place.

(100/6.8) ~ 14.71 keeping an extra decimal place.

 

If this were straight division, there would be only two significant figures remaining due to the denominators having the fewest significant figures. However, subtraction gets involved.

 

Round to significant figures before the subtraction, leaving you with 24 and 14 respectively. Then 24-14 is 10, with two signficant figures, which I would express as 1.0*10^1."

 

Hello,

 

Wouldn't that be 24 and 15 respectively?

 

You are right, but the post below yours explains it. While I numerically rounded the wrong way (gack!), the idea is the rounding is done before the subtraction or the addition.

[Garak]

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) ] + f, if I would have to do it the above way, it would take like 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...

 

Alternatively, is it acceptable to do it this way?

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, and don't care about the subtraction problem? *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)

 

This follows only 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 like this and this ignores the addition/subtraction problem and only uses the multiplication rules)

 

 

One major problem. With the addition and subtraction, the rounding has to occur before. Yes, it is rounding before a computation and may appear to be a bad idea. The chemistry people in my experience insisted upon intermediate rounding before addition and subtraction. Such problems involving addition and subtraction were common in chemistry, but seem less common in physics. In physics, the addition/subtraction problem never seemed to be considered as too serious. I can't even think of a problem, offhand, where it was encountered in any significant way.

 

So, to recap, the way I was taught was to round to significant figures after multiplication and before addition or subtraction, then round after addition or subtraction to the least number of decimal places in the addition and subtraction operations. I might suggest consulting Uncle Al. He seems to have a grasp of mathematics, despite being a chemist. :naughty:

[CraigD]

Rather than focusing on numbers of significant digits, one can carefully and precisely account for error in calculations due to limited precision by including error terms in one’s equations, eg:

(a +Ea)*(b +Eb) = a*b +(b*Ea+a*Eb+Ea*Eb)

(a +Ea)+(b +Eb) = a+b +(Ea+Eb)

 

Such attention to detail will usually produce about the same results as the simple rule “calculate everything with as much precision as possible, then use no more significant digits than the least precise term has.”

 

An important additional rule has to do with magnitude under addition. For example for the sum of 2 values

10^6 +-10^2 (4 significant digits)

+ 10^1 +-10^-1 (2 significant digits)

The correct result is about

10^6 +-10^2 (4 significant digits),

not

10^6 +-10^4 (2 significant digits), as the simplistic “use no more significant digits than the least precise term has” rule implies.

 

When a lot of info about the error terms - precise probability distributions, ideally – are available, error accounting can become very complicated, and the results very precisely stated as a probability distribution. Practically, this is valuable only in very critical calculations (Eg: describing a multi-step that depends critically on values within the error range)

[Garak]

Rather than focusing on numbers of significant digits, one can carefully and precisely account for error in calculations due to limited precision by including error terms in one’s equations, eg:

(a +Ea)*(b +Eb) = a*b +(b*Ea+a*Eb+Ea*Eb)

(a +Ea)+(b +Eb) = a+b +(Ea+Eb)

 

Such attention to detail will usually produce about the same results as the simple rule “calculate everything with as much precision as possible, then use no more significant digits than the least precise term has.”

 

An important additional rule has to do with magnitude under addition. For example for the sum of 2 values

10^6 +-10^2 (4 significant digits)

+ 10^1 +-10^-1 (2 significant digits)

The correct result is about

10^6 +-10^2 (4 significant digits),

not

10^6 +-10^4 (2 significant digits), as the simplistic “use no more significant digits than the least precise term has” rule implies.

 

When a lot of info about the error terms - precise probability distributions, ideally – are available, error accounting can become very complicated, and the results very precisely stated as a probability distribution. Practically, this is valuable only in very critical calculations (Eg: describing a multi-step that depends critically on values within the error range)

 

 

That is quite true. When one really wants to calculate errors when it results from actual measured quantities, of course, then it can really get ugly. I had a physics lab instructor who could actually go through ten pages of calculations in my tiny handwriting and find any place that I missed even a decimal in my error calculations, or dind't have a factor where it should be. I'm surprised he's not insane by now. :naughty: