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The result of a calculation is rounded to have the same
least number of significant figures as the least number of a EXAMPLE A.2
measurement involved in the calculation. When rounding In a problem it is necessary to multiply 0.0039 km by 15.0 km. The
2
numbers, the last significant figure is increased by 1 if the num- result from a calculator is 0.0585 km . The least number of significant
ber after it is 5 or larger. If the number after the last significant figures involved in this calculation is two (0.0039 has two significant
figures; 15.0 has three—read the rules again to see why). The calculator
figure is 4 or less, the nonsignificant figures are simply dropped.
result is therefore rounded off to have only two significant figures, and
Thus, if two significant figures are called for in the answer of the 2
the answer is recorded as 0.059 km .
previous example, 216.8 is rounded up to 220 because the last
number after the two significant figures is 6 (a number larger
than 5). If the calculation result had been 214.8, the rounded EXAMPLE A.3
number would be 210 miles. The quantities of 10.3 calories, 10.15 calories, and 16.234 calories are
Note that measurement fi gures are the only figures involved added. The result from a calculator is 36.684 calories. The smallest
in the number of significant figures in the answer. Numbers number of decimal points is one digit to the right of the decimal, so the

that are counted or defined are not included in the determi- answer is rounded to 36.7 calories.
nation of significant figures in an answer. For example, when
dividing by 2 to find an average, the 2 is ignored when consid-
ering the number of significant figures. Defined numbers are
defined exactly and are not used in significant figures. Since A.4 CONVERSION OF UNITS
1 kil ogram is defi ned to be exactly 1,000 grams, such a conver-
sion is not a measurement. The measurement of most properties results in both a numerical
3
value and a unit. The statement that a glass contains 50 cm of a
liquid conveys two important concepts—the numerical value of
ADDITION AND SUBTRACTION
50 and the referent unit of cubic centimeters. Both the numeri-
Addition and subtraction operations involving measurements, cal value and the unit are necessary to communicate correctly
as with multiplication and division, cannot result in an answer the volume of the liquid.
that implies greater accuracy than the measurements had be- When you are working with calculations involving mea-
fore the calculation. Recall that the last digit to the right in a surement units, both the numerical value and the units are treated
measure ment is uncertain; that is, it is the result of an estimate. mathematically. As in other mathematical operations, there are
The answer to an addition or subtraction calculation can have general rules to follow.
this uncertain number no farther from the decimal place than
1. Only properties with like units may be added or
it was in the weakest number involved in the calculation. Thus,
subtracted. It should be obvious that adding quantities
when 8.4 is added to 4.926, the weakest number is 8.4, and the
such as 5 dollars and 10 dimes is meaningless. You must
uncertain number is .4, one place to the right of the decimal.
first convert to like units before adding or subtracting.
The sum of 13.326 is therefore rounded to 13.3, reflecting the
2. Like or unlike units may be multiplied or divided and
placement of this weakest doubtful figure.
treated in the same manner as numbers. You have used this
The rules for counting zeros tell us that the numbers 203 2
rule when dealing with area (length × length = length ,
and 0.200 both have three significant figures. Likewise, the 2
for example, cm × cm = cm ) and when dealing with
numbers 230 and 0.23 only have two significant figures. Once 3
volume (length × length × length = length , for example,
you remember the rules, the counting of significant figures 3
cm × cm × cm = cm ).
is straightforward. On the other hand, sometimes you find a
number that seems to make it impossible to follow the rules. You can use these two rules to create a conversion ratio that
For example, how would you write 3,000 with two significant will help you change one unit to another. Suppose you need to
figures? There are several special systems in use for taking convert 2.3 kg to grams. First, write the relationship between
care of problems such as this, including the placing of a lit- kilograms and grams:
tle bar over the last significant digit. One of the convenient
1,000 g = 1 kg
ways of showing significant figures for difficult numbers is
to use scientific notation, which is discussed in the section Next, divide both sides by what you wish to convert from (kilo-
on scientific notation in this appendix. The convention for grams in this example):
writing significant figures is to display one digit to the left of 1,000 g 1 kg
the decimal. The exponents are not considered when showing _ _

=

the number of significant figures in scientific notation. Thus, 1 1 kg
if you want to write three thousand showing one significant One kilogram divided by 1 kg equals 1, just as 10 divided by 10
3
figure, you write 3 × 10 . To show two significant figures, equals 1. Therefore, the right side of the relationship becomes 1
3
it is 3.0 × 10 , and for three significant figures, it becomes and the equation is
3
3.00 × 10 . As you can see, the correct use of scientific nota-
1,000 g
tion leaves little room for doubt about how many significant _

= 1

figures are intended. 1 kg
626 APPENDIX A Mathematical Review A-4