Page 31 - Theory and Problems of BEGINNING CHEMISTRY

20 MATHEMATICAL METHODS IN CHEMISTRY [CHAP. 2

Addition and Subtraction
We must report the results of our calculations to the proper number of significant digits. We almost always
use our measurements to calculate other quantities, and the results of the calculations must indicate to the reader
the limit of precision with which the actual measurements were made. The rules for significant digits as the result
of additions or subtractions with measured quantities are as follows.
We may keep digits only as far to the right as the uncertain digit in the least accurate measurement. For
example, suppose you measured a block with the millimeter ruler (Fig. 2-1) as 5.71 cm and another block with
the centimeter ruler as 3.2 cm. What is the length of the two blocks together?
3.2cm
+5.71 cm
8.91 cm → 8.9cm
Since the 2 in the 3.2-cm measurement is uncertain, the 9 in the result is also uncertain. To report 8.91 cm
would indicate that we knew the 8.9 for sure and that the 1 was uncertain. Since this is more precise than our
measurements justify, we must round our reported result to 8.9 cm. That result says that we are unsure of the 9
and certain of the 8.
The rule for addition or subtraction can be stated as follows: Keep digits in the answer only as far to the
right as the measurement in which there are digits least far to the right.

Digit farthest to the right in
least accurate measurement

3.2 cm 16.351 cm
+5.71 cm −0.21 cm
8.91 cm 8.9 cm 16.141 cm 16.14 cm

Last digit retainable
It is not the number of significant digits, but their positions which determine the number of digits in the answer in
addition or subtraction. For example, in the following problems the numbers of significant digits change despite
the final digit being retained in each case:
98.7cm 24.3cm
+42.1cm −21.5cm
140.8cm 2.8cm
Rounding
We have seen that we must sometimes reduce the number of digits in our calculated result to indicate the
precision of the measurements that were made. To reduce the number, we round digits other than integer digits,
using the following rules.
If the first digit which we are to drop is less than 5, we drop the digits without changing the last digit retained.
7.437 → 7.4
If the first digit to be dropped is equal to or greater than 5, the last digit retained is increased by 1:
7.46 → 7.5 7.96 → 8.0 (Increasing the last digit retained caused a carry.)
7.56 → 7.6 7.5501 → 7.6
A slightly more sophisticated method may be used if the first digit to be dropped is a 5 and there are no digits or
only zeros after the 5. Change the last digit remaining only if it is odd to the next higher even digit. The following
numbers are rounded to one decimal place:
7.550 → 7.6 7.55 → 7.6
7.450 → 7.4 7.45 → 7.4
Use this method only if your instructor or text does.