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Application
3-2 Significant Digits in Calculations
date
Objectives
Introduction
If you completed Application 3-1, you learned that scientists indicate
the precision of their measurements by using significant digits (SDs).
Recall that precision refers to the repeatability of a measurement, which
is related to how finely divided the scale of the instrument is. Accuracy
refers to how close a measurement is to the actual value, which is associated with the quality of construction of the instrument and its calibration against a known standard. Scientists regularly have to make calculations using the measurements they have taken in the laboratory. In
order to report their results in a way that correctly shows the precision of
their measurements, they have developed rules for rounding the results
of these calculations. Before you learn these rules, however, you need to
know how to determine the number of significant digits in a measurement and review the rules for rounding off numbers.
Significant Digits
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section
We must first determine the number of SDs in each measurement using
the following rules:
SD Rule 1: Significant digits apply only to measured data. They do not
apply to
a. counted or pure numbers, or
b. fractions that are exactly 1 by definition.
SD Rule 2: All nonzero digits in measured data are significant.
375.42 cm (5 SDs)
SD Rule 3: All zeros between nonzero digits are significant.
208.5 m (4 SDs)
SD Rule 4: Decimal points define significant zeros.
a. If a decimal point is present, all zeros to the right of the last nonzero
digit are significant.
65.10 g (4 SDs)
b. If a decimal point is not present, no trailing zeros (zeros to the right
of the last nonzero digit) are significant.
5300 cm (2 SDs)
Note: Use scientific notation to clearly indicate the number of SDs
in a measurement having trailing zeros. Only the SDs in the number are shown in the decimal portion of scientific notation.
c. If a decimal point is present, none of the leading zeros (zeros to the
left of the first nonzero digit) are significant.
0.0651 g (3 SDs)
After completing this exercise, I will be
able to
✓✓identify the number of significant
digits in a measurement.
✓✓round numbers correctly.
✓✓perform calculations using significant digits.
Introduction
This exercise is the second designed to help your
students understand the determination and
use of significant digits in scientific calculations.
Review this exercise before assigning it. You
should have covered at least Sections 3A–3C of
the textbook beforehand.
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Application 3-2
SD Rule 5: Significant zeros in the one’s place are followed by a decimal
point.
Examples
Number
of Digits
Number of
Significant
Digits
Rule
213 m
3
3
2
210 cm
3
2
4b
3.01 g
3
3
3
5.200 L
4
4
4a
0.031 km
4
2
4c
1.7 × 104 mm
2
2
2, 4b
Measurement
We will use a convention for equalities as
follows:
“=” means “exactly equal to with no rounding”
“@” means “an approximate result rounded
to the correct number of SDs or a truncated
calculator answer”
Practice Counting Significant Digits
State the correct number of SDs or write NA if SDs are not applicable. In
the equations, look at the underlined portions.
g. 2.01 cm 3 a. 123 cm 3 b. 0.012 cm 2 h. 0.0102 cm 3 c. 2100 cm 2 d. 1 gross NA (counted)
e. 0.0101 cm
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f. 1 km/h ≅ 0.621 mi/h
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i. 1020. cm
3
j. 1.020 cm 4
k. 2020 NA l. 1 gal. ≅ 3.786 L
o. 1.03 cm
p. 0.0010 cm
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Note: If the rounded place value is more than one decimal place to
the left of the (assumed) position of the decimal point, fill in the
empty places to the left of the decimal point with zeroes. Do not include a decimal point unless the zero in the one’s place is significant.
38,500 m
Round 38,462 m to the 1000 m place value:
38,000 m
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3. If the first digit to the right of the rounded place value is 5 or greater,
add one to the number in the rounded place value, drop all remaining digits to the right, and substitute with zeroes, if required, as in
Rule 2 above.
r. 1 min. = 60 s
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2
NA
There are different ways to round numbers. The
given way is not necessarily the best, but it is
the simplest and provides consistent results.
This should be a review for most students.
Quiz them to check their understanding before
going on to the next section. Note that we use
the phrasing “round to the nth place value” for
precision in lieu of “round to the nearest n,”
which may be less clear.
Once you cover this material, you must be extra
careful to include a decimal point if trailing
zeros are significant (or use the − notation if
that is your preference). You will find that many
problems that you create will end up with
trivial answers if you do not pay attention to
this notation.
© 2008 BJU Press. Unauthorized reproduction prohibited.
2. If the first digit to the right of the rounded place value is 4 or less,
drop it and all remaining digits to the right.
Round 38,462 m to the 100 m place value:
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q. 0.0012 cm
We round numbers all the time. Certain rules have been established
to ensure numbers are correctly rounded every time, especially when
doing calculations using scientific data. We will use the following familiar rounding rules for our work in this course:
1. Determine the required numerical place value to round to in the
measurement according to the significant digit rules. The rounded
place value is that of the least significant digit.
0.6 m
n. 2000 cm
Rounding Rules
Round 0.561 m to the 0.1 m place value:
m. 2001 cm
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Significant Digits in Calculations
4. For whole numbers, decimal points should not be included after digits in the one’s place unless the zero in the one’s place is significant.
Round 36.5 m to the 1 m place value:
37 m
Round 79.8 m to the 1 m place value:
80. m
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Practice Rounding Measurements
Round the following measurements to the 0.01 place value:
c. 0.0999 km 0.10 km
a. 10.0944 m 10.09 m
d. 1.0149 L 1.01 L
b. 0.1234 s 0.12 s
Round the following measurements to the 10 place value:
c. 294.1 m 290 m
a. 123 cm 120 cm
b. 2354 mm 2350 mm
d. 89 s 90 s
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Rules for Calculations Using Significant Digits
Rules for calculations have been developed to ensure that results reported correctly show the precision of the measurements made in the
laboratory. Let’s say that you measured the edge of one side of a rectangle to be 2.38 cm and the other to be 5.79 cm using a metric ruler
marked in millimeters (0.1 cm). What area would you report for this
rectangle? The calculator answer for this product is 2.38 cm × 5.79 cm =
13.7802 cm2. Let us analyze this answer. In the expanded multiplication
problem shown, all colored digits are either the result from multiplication by an estimated digit or include an estimated digit from the factor
being multiplied. The colored digits in the final product indicate the
digits that are the result of summing estimated digits.
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The results of calculations must follow the same rules as the original
measurements—they may have only one estimated digit. Allowing more
than this would indicate your original measurements were more precise
than they actually were. This would not be honest. Christians must be
honest when reporting their results. In this example, the answer must
be rounded to the 0.1 cm place value, or 13.8 cm2. Rather than doing
such an analysis for each math operation, we have provided some simple
rules that accomplish the same thing. Study the math rules below and
do the practice problems.
Rules for Math Operations with Measurements
Adding and Subtracting Data
Math Rule 1: The data must have the same units. For example, you
cannot add the lengths 3.1 m and 45 cm together without converting
one of the measurements to the other unit.
3.1 m + 45 cm = cannot be done
e. 1.0846 km 1.08 km
f. 23.095 MHz 23.10 MHz
e. 544.9 mL 540 mL
f. 765.23 m/s 770 m/s
The opening example is one way to show
the need for controlling the number of
digits in the answer to a calculation. It is
more difficult if you try to demonstrate this
with division or subtraction. Recall that the
basic arithmetic operations can be related
to addition as follows: Multiplication is
repetitive addition. Subtraction is the
inverse operation of addition, and division
is the inverse operation of multiplication.
To rigorously determine the SD rule for
each type of math operation is beyond the
scope of this course.
It can be truly said that there are as many
ways to determine SDs in calculations as
there are scientists and teachers. Each
branch of science and engineering has
its own set of rules and expectations for
determining precision in calculations. Each
instrument has associated conventions
for determining the precision of its
measurements. A person learns these rules
through working in specific areas. For our
purposes, the accompanying discussion
is a good introduction to the rules for
performing scientific calculations. The
key to success is to enforce the rules and
practice.
While the emphasis in this section seems
to be determining how many digits are
significant, the real intent is to teach
students how to determine which digits in a
measurement are significant. Both skills are
needed to perform scientific calculations.
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Application 3-2
Math Rule 2: The precision of a sum or difference cannot be greater
than that of the least precise data given. In the following example, the
least significant digit of each measurement is underlined. Note the place
value of each.
30 g + 22.5 g = 52.5 g, rounded to 50 g
The least precise measurement of the two addends is 30 g, which is precise only to the nearest 10 g. Therefore, the answer is rounded to 50 g.
Multiplying and Dividing Data
Math Rule 3: A product or quotient cannot have more SDs than the
measurement with the fewest SDs.
2 cm × 4.55 cm = 9.10 cm2, rounded to 9 cm2
Since 2 cm has only one SD and 4.55 cm has three, only one SD is allowed, and the answer is rounded to 9 cm2.
Math Rule 4: The product or quotient of a measurement and a pure
number has the same precision as the original measurement.
For example, compute the radius of a circle whose diameter is 1.35
cm. The radius is found by dividing the radius by two.
r
1.35 cm
=
2
2
d = 0.675 cm ≅ 0.68 cm (0.01 cm precision)
d =
Math Rule 5 (optional): When performing a connected series of operations or when using data obtained from an earlier calculation, be sure
to apply the appropriate SD rules for each type of operation. Always
include one extra digit from the previous result to reduce rounding errors. Rounding after each step of a multiple-step problem introduces
unnecessary errors.
Practice Calculating with Significant Digits
Report the results of the following calculations in a raw answer and
using the appropriate number of SDs. Give the math rule number or
numbers that apply to the calculation.
1.
6.3 cm × 8.05 cm
2.
40.2 mL + 27 mL
3.
6 mL – 2.5 mL
4.
17.0 mm × 5.4 mm
5.
19.80 g – 7.3 g
6.
(See word problem on page 9.)
7.
(See word problem on page 9.)
Raw Answer
Rounded
Answer
Rule
50.715 cm2
67.2 mL
3.5 mL
91.8 mm2
12.50 g
5.00 cm3
122.1 mL
51 cm2
67 mL
4 mL
92 mm2
12.5 g
5.0 cm3
122.1 mL
3
1, 2
1, 2
3
1, 2
3
1, 2
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For another example, consider computing the surface area (SA) of a
cube if the area of one face (A) is 2.25 cm2.
SA = 6A = 6(2.25 cm2)
SA = 13.5 cm2 = 13.50 cm2 (0.01 cm3 precision)
Because the original face area was precise to 0.01 unit, the final result must reflect this precision.
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Significant Digits in Calculations
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8.
16.5 mm ÷ 2
9.
5.5 mL × 3
10.
7.2 cm × 5 cm × 1.25 cm
11.
(See word problem below.)
12.
(See word problem below.) (optional)
8.25 mm
16.5 mL
45 cm3
2.725 km
104.38 m2
Word Problems
6. Mr. Rodriguez, the science teacher, needs to determine the volume
of a block of wood. Its dimensions are length, 2.5 cm; width, 2.00
cm; height, 1.0 cm. What is the volume of the cube?
V = 2.5 cm × 2.00 cm × 1.0 cm
7. Dr. Dickson added 63.7 mL of water to 58.4 mL of sodium hydroxide solution. What is the volume of the new solution?
V = 63.7 mL + 58.4 mL
11. The route that Shane jogs every day is 6.9 km long. Shane’s little
brother wants to run a distance one-fourth as long. What distance
will Shane have to measure for his brother?
d = 6.9 km ÷ 4
12. (Optional) A missionary is going to carpet three classrooms in his
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school. The three rooms measure 3.3 m × 3.95 m, 5.4 m × 6 m, and
7.35 m × 8.08 m. What is the total area of carpet he must buy?
A = (3.3 m × 3.95 m) + (5.4 m × 6 m) + (7.35 m × 8.08 m)
≅ 13.0 m2 + 32 m2 + 59.38 m2
Note: each intermediate product should be truncated (not
rounded) at the place value of the allowed SDs plus 1 digit
before adding (see Rule 5). In practice, the missionary would
need at least 105 m2 of carpet, but this problem is adequate for
illustrating Rule 5.
8.3 mm
16.5 mL
50 cm3
2.7 km
1.0 × 102 m2
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4
3
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5 (optional)