There are two kinds of numbers in scientific work: exact numbers (those whose values are known exactly) and inexact numbers (those whose values have some uncertainty). Most of the exact numbers have defined values. For example, there are exactly 12 eggs in a dozen, exactly 1000 g in a kilogram, and exactly 2.54 cm in an inch. The number 1 in any conversion factor between units, as in 1 m 100 cm or 1 kg 2.2046 lb, is also an exact number. Exact numbers can also result from counting numbers of objects. For example, we can count the exact number of marbles in a jar or the exact number of people in a classroom.

Numbers obtained by measurement are always inexact. There are always inherent limitations in the equipment used to measure quantities (equipment errors), and there are differences in how different people make the same measurement (human errors). Suppose that 10 students with 10 different balances are given the same dime to weigh. The 10 measurements will vary slightly. The balances might be calibrated slightly differently, and there might be differences in how each student reads the mass from the balance. Counting very large numbers of objects usually has some associated error as well. Consider, for example, how difficult it is to obtain accurate census information for a city or vote counts for an election. Remember: Uncertainties always exist in measured quantities.

Precision and Accuracy

The terms precision and accuracy are often used in discussing the uncertainties of measured values. Precision is a measure of how closely individual measurements agree with one another. Accuracy refers to how closely individual measurements agree with the correct, or "true," value. The analogy of darts stuck in a dartboard pictured in Figure 1.25 illustrates the difference between these two concepts.

In the laboratory we often perform several different "trials" of the same experiment. We gain confidence in the accuracy of our measurements if we obtain nearly the same value each time. Figure 1.25 should remind us, however, that precise measurements can be inaccurate. For example, if a very sensitive balance is poorly calibrated, the masses we measure will be consistently either high or low. They will be inaccurate even if they are precise.

Significant Figures

Suppose you weigh a dime on a balance capable of measuring to the nearest 0.0001 g. You could report the mass as The notation (read "plus or minus") expresses the uncertainty of a measurement. In much scientific work we drop the notation with the understanding that an uncertainty of at least one unit exists in the last digit of the measured quantity. That is, measured quantities are generally reported in such a way that only the last digit is uncertain.

Figure 1.26 shows a thermometer with its liquid column between the scale marks. We can read the certain digits from the scale and estimate the uncertain one. From the scale marks, we see that the liquid is between the and marks. We might estimate the temperature to be being somewhat uncertain of the second digit of our measurement.

All digits of a measured quantity, including the uncertain one, are called significant figures. A measured mass reported as 2.2 g has two significant figures, whereas one reported as 2.2405 g has five significant figures. The greater the number of significant figures, the greater is the certainty implied for the measurement.

In any measurement that is properly reported, all nonzero digits are significant. Zeros, however, can be used either as part of the measured value or merely to locate the decimal point. Thus, zeros may or may not be significant, depending on how they appear in the number. The following guidelines describe the different situations involving zeros:

1. Zeros between nonzero digits are always significant—1005 kg (4 significant figures); 1.03 cm (three significant figures).

2. Zeros at the beginning of a number are never significant; they merely indicate the position of the decimal point—0.02 g (one significant figure); 0.0026 cm (two significant figures).

3. Zeros that fall both at the end of a number and after the decimal point are always significant—0.0200 g (3 significant figures); 3.0 cm (2 significant figures).

4. When a number ends in zeros but contains no decimal point, the zeros may or may not be significant—130 cm (two or three significant figures); 10,300 g (three, four, or five significant figures).

The use of exponential notation (Appendix A) eliminates the potential ambiguity of whether the zeros at the end of a number are significant (rule 4). For example, a mass of 10,300 g can be written in exponential notation showing three, four, or five significant figures:

   (three significant figures)

   (four significant figures)

   (five significant figures)

In these numbers all the zeros to the right of the decimal point are significant (rules 1 and 3). (All significant figures come before the exponent; the exponential term does not add to the number of significant figures.)

Exact numbers can be treated as if they have an infinite number of significant figures. This rule applies to many definitions between units. Thus, when we say, "There are 12 inches in 1 foot," the number 12 is exact and we need not worry about the number of significant figures in it.

Significant Figures in Calculations

When carrying measured quantities through calculations, observe these points: (1) The least certain measurement used in a calculation limits the certainty of the calculated quantity. (2) The final answer for any calculation should be reported with only one uncertain digit.

To keep track of significant figures in calculations, we will make frequent use of two rules. The first involves multiplication and division, and the second involves addition and subtraction. In multiplication and division the result must be reported with the same number of significant figures as the measurement with the fewest significant figures. When the result contains more than the correct number of significant figures, it must be rounded off.

For example, the area of a rectangle whose measured edge lengths are 6.221 cm and 5.2 cm should be reported as 32 cm² even though a calculator shows the product of 6.221 and 5.2 to have more digits:

We round off to two significant figures because the least precise number—5.2 cm—has only two significant figures.

In rounding off numbers, look at the leftmost digit to be dropped:

1. If the leftmost digit to be removed is less than 5, the preceding number is left unchanged. Thus, rounding 7.248 to two significant figures gives 7.2.

2. If the leftmost digit to be removed is 5 or greater, the preceding number is increased by 1. Rounding 4.735 to three significant figures gives 4.74, and rounding 2.376 to two significant figures gives 2.4.*

The guidelines used to determine the number of significant figures in addition and subtraction are different from those for multiplication and division. In addition and subtraction the result can have no more decimal places than the measurement with the fewest number of decimal places. In the following example the uncertain digits appear in color:

When a calculation involves two or more steps and you write down answers for intermediate steps, retain at least one additional digit—past the number of significant figures—for the intermediate answers. This procedure ensures that small errors from rounding at each step do not combine to affect the final result. When using a calculator, you may enter the numbers one after another, rounding only the final answer. Accumulated rounding-off errors may account for small differences between results you obtain and answers given in the text for numerical problems.