Page 29 - Theory and Problems of BEGINNING CHEMISTRY

18 MATHEMATICAL METHODS IN CHEMISTRY [CHAP. 2

but you cannot estimate the number of micrometers or even tenths of a millimeter, no matter how much you
try. You should report the length of the block as 5.4 cm. Using the ruler at the bottom of the block, which
has divisions in tenths of centimeters (millimeters), allows you to see for certain that the block is more than
5.4 cm but less than 5.5 cm. You can estimate it as 5.43 cm. Using the extra digit when you report the value
allows the person who reads the result to determine that you used the more accurate ruler to make this latter
measurement.

Centimeter ruler
0 1 2 3 4 5 6 cm

0 1 2 3 4 5 6 cm
Millimeter ruler
Fig. 2-1. Accuracy of measurement

EXAMPLE 2.20. Which of the two rulers shown in Fig. 2-1 was used to make each of the following measurements?
(a) 2.75 cm, (b) 1.3 cm, (c) 5.11 cm, (d ) 4.2 cm, and (e) 0.90 cm.
Ans. The measurements reported in (a), (c), and (e) can easily be seen to have two decimal places. Since they are reported
to the nearest hundredth of a centimeter, they must have been made by the more accurate ruler, the millimeter ruler.
The measurements reported in (b) and (d) were made with the centimeter ruler at the top. In part (e), the 0 at the end
shows that this measurement was made with the more accurate ruler. Here the distance was measured as more nearly
0.90 cm than 0.89 or 0.91 cm. Thus, the results are estimated to the nearest hundredth of a centimeter, but that value
just happens to havea0asthe estimated digit.

Zeros as Significant Digits
Suppose that we want to report the measurement 4.95 cm in terms of meters. Is our measurement any more or
less precise? No, changing to another set of units does not increase or decrease the precision of the measurement.
Therefore, we must use the same number of significant digits to report the result. How do we change a number
of centimeters to meters?
1m

4.95 cm = 0.0495 m
100 cm
The zeros in 0.0495 m do not indicate anything about the precision with which the measurement was made;
they are not significant. (They are important, however.) In a properly reported number, all nonzero digits are
significant. Zeros are significant only when they help to indicate the precision of the measurement. The following
rules are used to determine when zeros are significant in a properly reported number:
1. All zeros to the left of the first nonzero digit are nonsignificant. The zeros in 0.018 and 007 are not significant
(except perhaps to James Bond).
2. All zeros between significant digits are significant. The 0 in 4.03 is significant.
3. All zeros to the right of the decimal point and to the right of the last nonzero digit are significant. The zeros
in 7.000 and 6.0 are significant.
4. Zeros to the right of the last nonzero digit in a number with no decimal places are uncertain; they may or
may not be significant. The zeros in 500 and 8 000 000 are uncertain. They may be present merely to indicate
the magnitude of the number (i.e., to locate the decimal point), or they may also indicate something about
the precision of measurement. [Note: Some elementary texts use an overbar to denote the last significant 0
¯
in such numbers (100). Other texts use a decimal point at the end of an integer, as in 100., to signify that the
zeros are significant. However, these practices are not carried into most regular general chemistry texts or
into the chemical literature.] A way to avoid the ambiguity is given in Example 2.23.