Definition of Significant Figures

Question

In my textbooks, significant figures are defined as:

1. “Significant figures by definition are the reliable digits in a number that are known with certainty.”

2. “A significant figure is the one which is known to be reasonably reliable.”

Reliable means giving the same result on successive trials or reliable information can be trusted to be accurate. However, at the end of each of the two definitions it is also written that ‘the last digit of a number is generally considered uncertain in the absence of qualifying information.' For example if the mass of an object is 12.248 gm, the last digit which is ‘8’ is uncertain by plus or minus 0.001 gm. This uncertainty is unreliability in the information. So, the digit ‘8’ is uncertain, and thus according to the definition 1 it is not a significant figure. However, the rules for significant figures say that ‘non-zero digits are all significant’. Due to the last digit ‘8’, if the object was weighed with careful handling minimizing the chances of error, the figures preceding ‘8’ that are (1, 2, 2, 4) are certain and reliable. If the mass of another object is 2 gm, then it is uncertain by plus or minus 1 gm i.e. its mass could be 3 gm or could be 1 gm. ‘2’ is therefore not certain!
What is the reason that significant figures are defined to be the reliable digits in a number? In what sense they are said to be reliable?

Answer 1

I encounter significant digits much more often in chemistry than in physics. So basically:

Say you have a ruler with centimeter and millimeter markings. You measure the length of a pencil, and it comes out to somewhere in between 8.6 cm and 8.7 cm. It seems a touch closer to 8.6 than to 8.7. So, you say that the pencil is 8.63 cm long. The last digit implies that it is $\pm.01$. This way, the value could be 8.62, 8.63, 8.64, or anywhere in between. The most that you know is that it is definitely closer to 8.6 than 8.7, and the range from 8.62-8.64 just about covers your uncertainty about the measurement.

If you wanted to be absolutely precise, every single measurement you make and quantity you calculate would have a tolerance based on the limitations of your measuring apparatus. Of course, it would be cumbersome to keep writing $\pm.01$ every time, so it is simply assumed that the value is known exactly except for the last digit, which is uncertain.

Now when you do calculations, you can't use the value that you found, because it has some uncertainty associated with it. To be correct, you would have to carry out multiple calculations, first on the lower bound, and then on the upper bound, to figure out what the uncertainties of your new quantity are. This doubles the number of calculations you need to make, and is just cumbersome and tedious. That is why the rules for significant digits arose. They are a guideline for figuring out what kind of uncertainty your new quantity has without having to make any extraneous calculations. Thus, when you multiply 2 numbers, one with 3 sig figs and the other with 2, you will know for sure that the product will have 2 sig figs, one of which is absolutely certain, the other slightly uncertain.

To reiterate, in the example above, our value of 8.63 cm for the length of the pencil has 3 significant digits; two of which are absolutely certain (8 and 6), and the last one certain to $\pm1$.

Answer 2

I think you are correct, the definition quoted from your textbook seems inadequate - as you pointed out there is a contradiction around the use of the word "reliable." Also the reference you cite no where mentions anything about "reliability."

That is a problem with textbooks, authors sometimes don't explain the material very well they consider to be preliminary.

If you want a mathematically rigorous explanation of this, you might have to find a book about Significance Arithmetic.

Here is a link to a Decimal Arithmetic FAQ from IBM that explains the difference between decimal arithmetic and significance arithmetic.

If you are interested in computer science, you might have to find a book about Floating Point variables.

Experimental physicists need to 1) read instruments and 2) communicate the precision of measured or computed values correctly. The rules for reporting measured values in print vary with field and editorial convention. Measured values, and values calculated from measured values, have significant figures. Constants do not have significant figures.

Here is an example of significance arithmetic applied to experimental chemistry.

Put another way, the significant figures in a measurement value are not guaranteed to be accurate (i.e. reliable.) The number of significant figures can be defined for a very poor measurement.