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topic 003

SIGNIFICANT FIGURES

RULE #1 - All digits 1 through 9 are significant
          If the mass of an object is measured as 15.8 g, this means that the mass is known to lie between 15.7 and 15.9g. There are 3 significant figures in 15.8.  THE INSTRUMENT USED TO MAKE THIS  MASS MEASUREMENT CAN ONLY DETECT DIFFERENCES OF 0.1G OR 100 MG.
          If the mass of an object is measured as 12.3456 g, this means that the  mass is known to lie between 12.3455 and 12.3457 g. There are 6 significant figures in 12.3456. THE INSTRUMENT USED TO MAKE THIS MASS MEASUREMENT CAN ONLY DETECT DIFFERENCES OF 0.1 MG OR 100 MICROGRAMS.
Overall, you are counting all of the "certain" digits and a final digit that is uncertain, but nevertheless significant.


RULE #2 - Zero is significant when it is between two non-zero digits
          The quantities 306, 30.6, 3.06 and 0.306 all contain 3 significant figures since the 0 between the 3 and 6 is significant. The number 306 means that the true value rests somewhere between 305 and 307, thus, the zero is known with certainty and is significant.  That is, zeros within  a number are always significant.

RULE #3 - Trailing zeros that are not used to hold the zero are not significant
          The quantities 279.0, 27.90 and 2.790 all contain 4 significant figures.Again, the first three numbers are known with certainty and the final number is always taken as significant.
The quantities 0.2790 and 0.27900 have 4 and 5 significant figures, respectively. In the number 0.27900, it does NOT matter that there are two consecutive zeros.  The first zero is known with certainty and the final zero while not known with certainty is still significant. Thus, 4.000 has 4 significant figures.

RULE #4 - A zero used to fix a decimal point  is never significant.
         The quantities 0.456, 0.0456 and 0.00456 all contain 3 significant figures. In this case, you need to think in terms of exponential numbers.  0.0456 is 4.56 x 10-2 (only 3 significant figures) and 0.00456 is 4.56 x 10-3 (again, only three significant numbers).  Thus, 470,000 has only 2 significant figures.  However, 470,000 with a line drawn above the final zero or the presence of a decimal point indicate that the measurement had six significant figures.  0.000000004 has only one significant figure; the remaining zeros were used to fix the decimal point.


Questions:
1.How many significant figures are in the number 0.01020?
    Ask yourself: how many figures are known with certainty? Ans. = 5

2. Which of the following has the smallest number of significant figures?0.00030, 123, 0.4005, 2.04, 2.004, 123 and 2.04 each has 3 significant figures but 0.00030 is the same as 3.0 x 10-4, so it has only 2 significant figures.

3. How many significant figures are in the number 20.010?
Once again, four digits are known with certainty and there are five significant figures.

4.How many significant figures are in the number 0.01000?
This is tricky and requires some thought.  As an exponential number it can be represented as 1.000 x 10-2 and then the presence of FOUR significant figures becomes evident.

5.How many of the following numbers have 4 significant figures? 3.0156,  18.00,  0.007000,  3.45 x 104  and  0.0021


Timothy C.K. Su, Professor
Chemistry Department
UMass Dartmouth
Suppose a ruler is used to measure the length of an object as shown in the figure below.
 
 
 

Some of you may say its length is 6.75 cm. Some may say it is 6.74 cm or 6.76 cm. We are quite
certain that the length is somewhere between 6.7 cm and 6.8 cm. The third (last) digit is a reasonable
guess. In other words, we can only estimate to the nearest hundredth of a centimeter. There is an
uncertainty of at least 0.01cm.
    It is unreasonable to report a value like 6.75342183 cm since we are not even sure about the third digit. The last six digits are meaningless. Suppose we take the value 6.75 cm. In this reported measurement, the first two digits are definitely significant. The third digit is also significant but has some uncertainty associate with it. It is our best estimate of where it is between 6.7 and 6.8 cm. Therefore, there are three significant figures in the measured quantity reported above.

Similarly, all measured quantities are generally reported in such a way that the last digit is uncertain.
All digits known with certainty including the uncertain one in a measurement are called significant figures.

      Value                     # of S. F.
      2.456                         4
1003.2                             5
      1.03000                     6
      0.0000402                 3
         230000                   2 - 6
    In the last example,  it is not clear how many significant figures there are. Suppose there are
three significant figures, the number represents 230,000±1,000. If there are two significant figures,
the number represents 230,000±10,000. To overcome this ambiguity as well as for ease of
manipulation, such numbers should always be written in exponential (scientific) notation: 2.30 x 105
(3 significant figures). Very large and very small numbers are usually expressed in exponential
notation:
0.00000001230 = 1.230 x 10-8 (four significant figures)
3000000000. = 3.0 x 109 (two significant figures)

N.B. Only those digits before the exponent are used to express the number of significant figures. Do
not add the exponential term to the number of significant figures.


Exact numbers are considered to have an infinite number of significant figures. For example, if you
said, "A is twice (or two times) as large as B", the number 2 would be exact. Or, if you said "There
are 4 quarts in a gallon", the number 4 would be exact. Exact numbers usually involve counted values
or definitions.
IF YOU WANT MORE PRACTICE OR THE CONCEPT IS STILL NOT CLEAR TO YOU, THEN POINT YOUR BROWSER TO:
http://www.chem.tamu.edu/class/fyp/mathrev/mr-sigfg.html