The Foundations of Chemistry

EXAMPLE 1-2 Significant Figures (Multiplication)

What is the area of a rectangle 1.23 cm wide and 12.34 cm long?

Plan

The area of a rectangle is its length times its width. We must first check to see that the width
and length are expressed in the same units. (They are, but if they were not, we must first con-
vert one to the units of the other.) Then we multiply the width by the length. We then follow
Rule 5 for significant figures to find the correct number of significant figures. The units for
the result are equal to the product of the units for the individual terms in the multiplication.

Solution

A

w
(12.34 cm)(1.23 cm)
 15.2 cm^2

(calculator result
15.1782)

Because three is the smallest number of significant figures used, the answer should contain only
three significant figures. The number generated by an electronic calculator (15.1782) implies
more accuracy than is justified; the result cannot be more accurate than the information that
led to it. Calculators have no judgment, so you must exercise yours.

You should now work Exercise 27.

The step-by-step calculation in the margin demonstrates why the area is reported as

15.2 cm^2 rather than 15.1782 cm^2. The length, 12.34 cm, contains four significant fig-
ures, whereas the width, 1.23 cm, contains only three. If we underline each uncertain fig-
ure, as well as each figure obtained from an uncertain figure, the step-by-step multipli-
cation gives the result reported in Example 1-2. We see that there are only two certain
figures (15) in the result. We report the first doubtful figure (.2), but no more. Division
is just the reverse of multiplication, and the same rules apply.
In the three simple arithmetic operations we have performed, the number combina-
tion generated by an electronic calculator is not the “answer” in a single case! The cor-
rect result of each calculation, however, can be obtained by “rounding off.” The rules of
significant figures tell us where to round off.
In rounding off, certain conventions have been adopted. When the number to be
dropped is less than 5, the preceding number is left unchanged (e.g., 7.34 rounds off to
7.3). When it is more than 5, the preceding number is increased by 1 (e.g., 7.37 rounds
off to 7.4). When the number to be dropped is 5, the preceding number is set to the near-
est evennumber (e.g., 7.45 rounds off to 7.4, and 7.35 rounds off to 7.4).

With many examples we suggest

selected exercises from the end of the

chapter. These exercises use the skills

or concepts from that example. Now

you should work Exercise 27 from the

end of this chapter.

12.34cm

 1.23cm

3702

2468

1234

15.1 782 cm^2 
15.2 cm^2

Rounding off to an even number is

intended to reduce the accumulation of

errors in chains of calculations.

1-8 Use of Numbers 25

Problem-Solving Tip:When Do We Round?

When a calculation involves several steps, we often show the answer to each step to the

correct number of significant figures. We carry all digits in the calculator to the end of

the calculation, however. Then we round the final answer to the appropriate number of

significant figures. When carrying out such a calculation, it is safest to carry extra fig-

ures through all steps and then to round the final answer appropriately.