Questions about Significant Figures

[nlcsa22]

Hi. Sorry that I abandoned the provided template, but it didn't really apply to the questions that I had..these all deal with significant figures:

1. For the number 100, I understand that there can be one, two, or three
significant figures. If there are three, then I understand that the assumed
uncertainty in the measurement (if no uncertainty is explicity stated) is 1 (perhaps even 2 or 3, but usually 1). So
we can then say 100 +/- 1. So the actual measurement could be as low as 99
or as high as 101. What would the uncertainty be if we were to say that only
two of the figures in 100 were significant (ie. the 1 and the middle 0)?
Would it be plus or minus 10 (meaning that the actual measurement would be
between 90 and 110)? And what would the uncertainty be if we were to say
that there was only one significant figure in 100 (ie. the 1)? Would it be plus or minus 100 (meaning that the actual measurement would be between 0
and 200)?

2. My textbook says that "it is good practice to keep an extra significant
figure or two throughout a calculation, and round off only in the final
result." Can you please explain this? An example would be helpful, too.

3. My textbook says that 1 in. = 2.54 cm. It says that this is exact, so the
2.54 has an infinite number of significant figures, not just 3. In an
example in the book, the author uses this conversion factor to convert 1
square inch to square centimeters. He writes 1sq.in. = (2.54cm)^2 =
6.45sq.cm. Now, 2.54*2.54 = 6.4516. Since 2.54 here has an infinite number
of significant figures, the 6.4516 also must have an infinite number of
significant figures. But the author rounded 6.4516 to 6.45. In doing so, was
he "clipping" the significant figures from infinity to 3, or do we still
consider the 6.45 to have an infinite number of significant figures? If the
former is true (ie. he reduced the significant figures from infinity to 3),
why did he do that?

4. One problem in my textbook involves a posted speed limit of 55 mph. Does
this have an infinite number of significant figures, since it's an exact
number?

Thanks very much

 

[hikaru1221]

1/ The standard format is: A x 10^B, where A and B are numbers. The digits of A are significant numbers. Here when you write "100", we understand that it is "100 x 10^0". If the value is 100 (unofficially written) and the number of significant figures is 2, then you write "10 x 10^1".
Normally if you write "100" without stating the uncertainty, people will understand that the uncertainty is 1. And for "10 x 10^1", it is "1 x 10^1".
And yep, when you write "1 x 10^2" without saying anything about the uncertainty, people understand it's either that you forget to mention the uncertainty or that this value is very unreliable.

2/ For example: A = B + C. Uncertainty: dA = dB + dC. A is the one we need. Say: dB = 1.5, dC = 1.6.
If we round dB and dC up, what do we have for dA? Otherwise, what is dA? See the difference in the 2 cases?

3/ I would say the rounding up might be an approximation, which serves for some particular purpose of the author.

4/ I think besides values which are defined (such as c = 29..., or 1 inch = 2.54cm), the rest should be occupied by uncertainties. Then see my answer for question 1

 

[CharliH]

1. I would put the error in your first case as +/-0.5, then +/-5 and +/-50. I don't think notes are always consistent about that, though. Otherwise, you're correct.

2. This is most applicable when you have varying numbers of sig figs. eg:

5.3*2.13546723435 + 6*7.93548
= 11.3179763 + 47.61288
=58.9308563
=59 to sig figs

If we round of to sig figs immediately, we get:
5.3*2.13546723435 + 6*7.93548
=11 + 50
=61

3. Technically, yes, he did clip the number of sig figs. I would guess that he just didn't think it was worth the hassle of carrying four decimal places. If your school's anything like mine, there's one almighty fuss about sig figs up front and then most of the time you don't bother with them as long as you have a reasonable number of places, at least in theory problems.

4. Yes, it has an infinite number of sig figs, but not because it's an exact number (100 in your first question is an exact number). Only measured quantities need sig figs and the speed limit is not a measured quantity. Any other quantity is taken to have an infinite number of sig figs.

 

[Borek]

we can then say 100 +/- 1. So the actual measurement could be as low as 99
or as high as 101. What would the uncertainty be if we were to say that only
two of the figures in 100 were significant (ie. the 1 and the middle 0)?
Would it be plus or minus 10 (meaning that the actual measurement would be
between 90 and 110)? And what would the uncertainty be if we were to say
that there was only one significant figure in 100 (ie. the 1)? Would it be plus or minus 100 (meaning that the actual measurement would be between 0
and 200)?

I would say it is 100±0.5, 100±5, 100±50. But honestly, you should not treat sig figs too seriously. There are much better ways of explicitly expressing uncertainty.

2. My textbook says that "it is good practice to keep an extra significant
figure or two throughout a calculation, and round off only in the final
result." Can you please explain this? An example would be helpful, too.

These are often called "guard digits".

 

[nlcsa22]

Thanks to everyone who replied to my questions...all the answers were helpful and I feel like I finally understand the whole concept of SigFigs.