Suppose you use a meterstick to measure a pen, and you find that the end of the pen is just past 14.3 cm. This measurement has three valid digits: two you are sure of, and one you estimated. The valid digits in a measurement are called significant digits. The last digit given for any measurement is the uncertain digit. All nonzero digits in a measurement are significant.

No. For example, in the measurement 0.0860 m, the first two zeros serve only to locate the decimal point and are not significant. The last zero, however, is the estimated digit and is significant. The measurement 172,000 m could have 3, 4, 5, or 6 significant digits. This ambiguity is one reason to use scientific notation: it is clear that the measurement $1.7200 \times 10^5$ m has five significant digits.

When you perform any arithmetic operation, it is important to remember that the result never can be more precise than the least-precise measurement.

To add or subtract measurements, first perform the operation, then round off the result to correspond to the least-precise value involved. For example:

$3.86\ \text{m} + 2.4\ \text{m} = 6.3\ \text{m}$

because the least-precise measure is to one-tenth of a meter.

To multiply or divide measurements, perform the calculation and then round to the same number of significant digits as the least-precise measurement. For example:

$\frac{409.2\ \text{km}}{11.4\ \text{L}} = 35.9\ \text{km/L}$

because the least-precise measure has three significant digits.

Some calculators display several additional digits, as shown in Figure 1-4, while others round at different points. Be sure to record your answers with the correct number of digits. Note that significant digits are considered only when calculating with measurements. There is no uncertainty associated with counting (4 washers) or exact conversion factors (24 hours in 1 day).
 

Solve the following problems.

1. 9
   a. 6.201 cm + 7.4 cm + 0.68 cm + 12.0 cm
   b. 1.6 km + 1.62 m + 1200 cm
2. 10
   a. 10.8 g − 8.264 g
   b. 4.75 m − 0.4168 m
3. 11
   a. 139 cm × 2.3 cm
   b. 3.2145 km × 4.23 km
4. 12
   a. 13.78 g ÷ 11.3 mL
   b. 18.21 g ÷ 4.4 cm³