Understanding Significant Figures: Rules & Examples

[babysnatcher]

I do not fully understand significant figures. It seems the physics textbooks ignore their own significant figure rules, eventually. For this problem when I divide 2.3 m by 1.0 s, I get 2.3 m/s, but the answer in the book is 2.30 m/s. In another situation I am divding 48.3 m by 3.0 s to get 16.1 m/s, then I round to get 16 m/s, but the text shows 16.1 m/s as the final answer. Also, what is this thing I have heard of where the 0 in the ones place of the decimal can be significant - I heard this applies when the device can measure that 'ones' place but it happened to be 0( i.e. 0.1 meter).

I have a much easier solution to solve this problem. Can someone just list all the special-case significant figure rules(including the rules derived from the fundamental ones)?

 

[haruspex]

I divide 2.3 m by 1.0 s, I get 2.3 m/s, but the answer in the book is 2.30 m/s.

I agree with you on that one.

In another situation I am divding 48.3 m by 3.0 s to get 16.1 m/s, then I round to get 16 m/s, but the text shows 16.1 m/s as the final answer.

The book answer seems ok to me. What's your logic for dropping the decimal?

Also, what is this thing I have heard of where the 0 in the ones place of the decimal can be significant - I heard this applies when the device can measure that 'ones' place but it happened to be 0( i.e. 0.1 meter).

Not sure I understood that. Are you saying there's an implied difference between .1 and 0.1?

 

[babysnatcher]

I agree with you on that one.

The book answer seems ok to me. What's your logic for dropping the decimal?

The multiplication/division rules for significant figures, and I'm ignoring that 3.0 s is indefinitely accurate because the values are from a table - but it does say to find the average velocity of the car for the last 3 seconds so maybe 3.0 is indefinitely accurate.

Not sure I understood that. Are you saying there's an implied difference between .1 and 0.1?

I mean like when you measure 0.1 m off a meter stick. The stick is capable of measuring that 'ones' place so the zero is significant; also, like when the measured value 3.2 m is subtracted by 3.1 m and becomes 0.1 m, and then is divided by 1.0 s. I'm saying that in this case, the 0 is significant.

 

[Simon Bridge]

I think that last one is that if 1.0 is 2 sig fig, then 0.1 may also be 2 sig fig.
The second one, 3.0 is 2 sig fig while 16.1 is three... so drop the decimal.

It is not unusual for sig fig rules to be indifferently enforced in a text.
The whole thing is just a place-holder until you learn about statistical uncertainties.

A way of checking is to estimate the uncertainty on a quoted measurement at half the lowest order place value ... so 1.0units is (1.0±0.05)units (measurements always have units) while 1.00units is (1.00±0.005)units.

When you add or subtract two, independent[1], measurements - you use pythagoras on the uncertainties... then round the uncertainties to 1sig fig (or two - judgement call here).

So (1.0±0.05)units + (1.00±0.005)units = 2.0±√(0.002525) = 2.0±0.050249 = (2.0±0.5) units ... so, as a shortcut, I'd keep the smallest number of decimal places.

If I multiply or divide two measurements, then the pythagoras thing is applied to the relative uncertainty ... which is the ± part divided by the number.

(2.0±0.05)units the relative uncertainty would be 0.05/2=0.025 (no units this time) or 2.5%.

(2.0±0.05)units x (1.00±0.005)units = 2 ± 2√(0.025^2 + 0.005^2) = (2.0±0.05)units
... so for a shortcut: just keep the smallest sig fig.

48.3/3.0 = (48.3±0.05)/(3.0±0.05) = 16.1 ± 16.1√[(0.05/48.3)^2 + (0.05/3.0)^2] =16.1±0.26885 = (16.1±0.27)units

here it's a judgement call whether to keep the extra decimal place.
usually you'd drop it because the difference of 0.1 is less than the uncertainty of 0.27 ... and we'd actually write the answer as (16.0±0.3)units.

From this example it is a small jump to see that there are situations where it is better to keep the extra figure even though the sig-fig rule-of-thumb says you shouldn't.

It is possible your textbook author is doing this when it's borderline.

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[1] this is where the result of one measurement does not depend on the reult of the other one. When they do, you have to add the uncertainties instead.

 

[haruspex]

I mean like when you measure 0.1 m off a meter stick. The stick is capable of measuring that 'ones' place so the zero is significant; also, like when the measured value 3.2 m is subtracted by 3.1 m and becomes 0.1 m, and then is divided by 1.0 s. I'm saying that in this case, the 0 is significant.

Ah - you mean 'significant' in the sense that it counts towards the number of significant figures. I don't buy that argument. If I measure a distance as 5m with a tape that goes to 99m, does that mean I should write it as 05m, and that counts as two significant figures? Surely not.
I would say .1 and 0.1 each have one significant digit.

 

[Simon Bridge]

Ah - you mean 'significant' in the sense that it counts towards the number of significant figures. I don't buy that argument. If I measure a distance as 5m with a tape that goes to 99m, does that mean I should write it as 05m, and that counts as two significant figures? Surely not.
I would say .1 and 0.1 each have one significant digit.

I concur - signifying something about the measurement does not make the digit "significant" in this sense. These are two different uses of the word.

The sig-fig/dp rules are just an approximation to the full statistical methods outlined in the last post. When you use the stats - the uncertainties or "errors" - then the reasoning becomes more obvious. It makes no difference if it is 1x10^-1 or 01x10^-1.