
There are more sophisticated ways to handle uncertainties in quantities than significant figures. [For an extensive discussion, see the NIST website on Uncertainty of Measurement Results.] However, using this relatively simple formalism should create a sensitivity and appreciation for the connections between measuring devices, their capabilities, and their limitations. Many exercises involve the determination and reporting of one or more quantitative properties. Both the value of a property and its uncertainty will be important. It is essential that each measurement that leads to the value has been made and recorded with the full capability of the device or apparatus used in its determination. SUPL001 contains a brief section on the concept of significant figures. As noted in that supplement:
Two features play a role in the determination of significant figures  measurement, and computation using measured values. 

The buret permits a volume to be read to ± 0.02 mL (onefifth of the smallest division on its scale). How many significant figures does such a quantity have?
Next, consider the analytical balance. The analytical balance is an interesing case. It is capable of displaying weights to ± 0.0001 g and its digital display reports weights with four decimal places, no matter how large or small they may be. (But, see intrinsic precision below.) The number of significant figures represented by weights varies with the total weight of an object. Nevertheless, weights determined on an analytical balance must always be reported to four decimal places, regardless of the weight. Again, the number of signficant figures in the weight depends on the weight. E.g. For weights:


The number of significant figures in a computed quantity may be the same or different from those of the quantities from which it is computed. Consider a container which together with its contents weighs 35.2749 g. A sample of the contents is transferred out, producing a final weight of 35.2235 g. Both the initial and final weights are known with a precision of 6 significant figures. The weight of the sample is: 

35.2749
g


35.2235
g 

0.0514 g 

The 6 significant figures in the initial and final weights produce a number (their difference) with only 3 significant figures (but still with four decimal digits. The zero after the decimal point, while important, is not a significant figure). Suppose the sample weight is now used in the calculation of a concentration by dividing it by a volume of 20.32 mL. Despite the four significant figures in the volume, the resulting concentration should be reported to at most 3 significant figures, namely .00253 g/mL. In general, the number of significant figures in a multiplication or division of quantities will be that of the quantity with the least number of significant figures. 

Perhaps the most confusing aspect of significant figures is how to deal with the digit zero (0). That number is often used as a place holder to indicate the location of a decimal point. When used in this manner, zero is not a significant figure. The only times that 0 is significant is when one or more of them occur between nonzero digits, or when one or more of them in a row is at the end of a string of nonzero digits.
Now consider a
number which is the result of an experimental measurement. Suppose
the invesigator reports the number as
3.16 Indeed. with the usual rules for rounding, we can assume that the result was less than 3.165 and more than 3.155 Suppose instead,
she reports the number as
3.160 The
fact that the number is written as it is, must indicate that
the investigator was certain that the result was less than 3.161 and more
than 3.159. (If we include rounding  less than 3.1605 and
more than 3.1595.) 
a substantial decrease in the uncertainty of the result and an increase
in the precision of the measurement over
the previous representation.
If the number is reported as 3.1600 This must be an indication that the investigator was certain that the number was less than 3.1601 and more than 3.1599. (You should be able to provide the relevant limits assuming the usual rounding rules.) An additional decrease in the uncertainty. These zeros are certainly significant! If you understand the above, you should be able to determine which zeros are significant in the following number: 0.0010600 

For quantitites that are restricted to occur in discrete, indivisible units (e.g., people (excluding halfwits), atoms, etc.), The number of significant figures is infinite. We always assume that integers always have as many significant figures as are necessary when used in a calculation. E.g., in the calculation of the perimeter of a square (4 times the length of a side), we can assume that the number 4 used in the calculation has as many significant figures as the length measurement. One possible exception to this rule is Avogadro's number which, while one would like to think is an integer, has neither been defined, nor is known, to the requisite 23 significant figures. Quantities that are calculated from integers (rational numbers) similarly have infinite precision. E.g., the numbers represented by the quotient and product of the integers 2 and 7, 2/7 and 2*7,can be written to as many places as are required. The astute student will recognize that numbers such as pi, e (2.71828.....), log 2, etc., while not rational, can be represented to as many significant figures as may be required in any given computation (E.g., the calculation of the volume of a sphere from its radius, using V = (4/3) x pi x r^{3 }). [Indeed, in this formula, the 4, the two 3's and the pi are all known to an infinite number of significant figures. All uncertainty in the volume, V is due only to the uncertainty in the radius r. The number of significant figures in V is fixed by those in r^{3} (which in turn is fixed by the precision in r)] 

Scientific notation (i.e, writing numbers in the form y.yyy X 10^{n}) simplifies the use of significant figures considerably. The issue of using zeros as place holders for decimal points can be made to disappear. The multiplier (y.yyy) should always have exactly the number of significant figures as are justified. This may involve a number of trailing zeros. E.g.:
Consider for example the following example of two buret readings whose difference is the net volume. In scientific notation 

Final Volume  3.105 x 10^{1 }mL  
Initial Volume  3.2 x 10^{1 }mL  
Net Volume  3.073 X 10^{1} mL  
In normal numeric representation, that subtraction is represented as follows:  
Final Volume  31.05 mL  
Initial Volume   0.32 mL  
Net Volume  30.73 mL  
This example demonstrates why, in the laboratory course, scientific notation will not always be the best way to handle simple computations  particularly in the addition and subtraction of numbers.  
A discussion of significant figures and uncertainties in logarithmic and exponential quantities is given in the web page on calculations for SUSB010 which includes a link to a more exact treatment. 

The intrinsic precision of a device refers to the precision determined by its design and construction  including its interface with the investigator. E.g., the four decimal places in the digital display of the balance notwithstanding, the manufacturer's specification of the intrinsic precision of the analytical balance is +/ 0.0002 g. This means that independent weighings of the same object will produce results that do not differ by more than 0.0002 g. Nevertheless, weights are always recorded with all four decimals. 
Robert F. Schneider (rschneider at notes.cc.sunysb.edu)  
Last Update: 20120712 