554 MHR • Appendix 3
One way to avoid ambiguity is to express
numbers using scientific notation. Thus, in
5 × 106 bacteria, it is clear that there is one
significant digit.Mathematical Operations
You will sometimes need to perform calculations
with measured quantities. It is important to
keep track of which digits in your calculations
are significant. Your results should not express
more certainty than your measured quantities
justify. Be particularly careful when using
calculators — they usually report results with
far more figures than your data warrant.
The following three rules will help you keep
track of significant digits in calculations and
results.Operations and Counted Quantities
If you see four bacteria in a microscope field,
the value “4” is an exact number. If you say
someone has two eyes, you mean exactly two,
not 2.1 or 2.3. Counted quantities have no
uncertainty. In other words, they do not limit
the number of significant digits in a calculation.
When you multiply using counted quantities,
you should treat the operation as an addition.
For example, suppose you have four beakers,
each containing 25.0 mL of solution. What is
the total volume of solution?
4 ×25.0 mL=100.0 mL. Why are there four
significant digits in the answer instead of three?
The operation 4 ×25.0 mLcan be represented
as an addition operation:
25.0 mL+25.0 mL+25.0 mL+25.0 mL
=100.0 mL. Thus, the number of decimal
places in the measured quantity being
multiplied determines the number of decimal
places in the answer. This is only the case
when a measured quantity is multiplied by an
exact, counted quantity.Instant Practice
1.How many significant digits are there in the
following measurements?
(a)2004 mm (b)0.007 V
(c)26.30 cm (d)21.6°C2.Determine the number of significant digits
in the following measurements. If you are
unable to tell, briefly explain why.
(a)8.923× 10 −^7 mg (b)7.000 cm
(c)30 mL
(d)5.000 201× 102 s3.Calculate the following and round to the
correct number of significant digits:
(a)705 mL+0.40 mL (b)13.07 s−2.1 s
(c)206 mm×0.83 mm(d)4.25 g/3 L4.Perform the following calculations. Include
the appropriate number of significant digits
in your answer.
(a)55.682 g+43.3 g (b)
1.732× 102 g
15.5 cm^3
(c)0.0050 mL−0.000 350 mL+1.0 mL
(d) 5 ×37.0 mL, where 5 is a counted
quantityRule 1: Multiplying and Dividing
The value with the fewest number of
significant digits, going into the calculation,
determines the number of significant digits
that you should report in your answer.
For example, 25.00 cm×3.00 cm=75.0 cm^2.Rule 2: Adding and Subtracting
The value with the fewest number of
decimal places, going into the calculation,
determines the number of decimal places
that you should report in your answer.
For example, 34.50 mL+23.1 mL=57.6 mLRule 3: Rounding
To get the appropriate number of significant
digits (rule 1) or decimal places (rule 2), you
may need to round your answer.
If your answer ends in a number that is
greater than 5, increase the preceding digit
by 1. For example, 2.346 can be rounded
to 2.35.
If your answer ends with a number that is
less than 5, leave the preceding number
unchanged. For example, 5.73 can be
rounded to 5.7.
If your answer ends with 5, increase the
preceding number by 1 if it is odd. Leave
the preceding number unchanged if it is
even. For example, 18.35 can be rounded
to 18.4, but 18.25 is rounded to 18.2.