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1
Rounding Rules
Tactical Style in Mathematics
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Working with Measured Values
With the exception of the measured value of counting which is naturally
exact, measured values bear a certain degree of uncertainty. As an example
of this, consider the problem posed in Figure – below.
Figure
We can immediately tell that the value is greater than 34. Now, in your
mind’s eye break the interval from 34 to 35 into 10 parts and guess at the
number of tenths that fit into the length extending beyond 34. A reasonable
value would be 2 so that our reading now is 34.2 m. The number 34.2 is te
result of making a measurement using a measuring instrument. Based on
the process described above, digits 3 and 4 are certain to be correct while
digit 2 is guessed at.
Note that it would be unreasonable to write a value such as 34.2173 m.
The measuring instrument’s smallest divisions set a limit on which digits
can reasonable be written down. In the case of the problem above, how
could any of us, using the ruler depicted in the diagram, guess at how many
hundredths, thousandths and tenthousandths there are in the length?
One could demand more accuracy but to do so we would need to have a
measuring instument with finer divisions. Such a scenario is shown in Figure
– below.
Figure
We can now tell that the length of the line is about 34.18. Now that we have
a ruler with a smaller subdivision, we can generate a more accurate value.
Here digits 3, 4 and 1 are certain to be correct while digit 8 is guessed at.
The pattern that we see is that all digits up to and including the estimated digit come from us making an act of measurement. Such digits are
examples of certain digits called significant digits. The rightmost significant digit is called the estimated digit. Note that the estimated digit is
3
4
still significant as it results from us making a measurement.1
We also understand how the smallest divisions on the measuring instrument limit the number of digits that can be reasonably put down when making a measurement. We should also understand how to specify the smallest
divisions on the measuring instrument used to make the measurement. The
smalles division corresponds to the place value of the significant digit on the
left side of the estimated digit. As an example, given the measured value
34.2 cm, the smalles divisions on the measuring instrument are the ones (1
cm) whereas in 34.28 cm, the smallest division on the masuring instrument
is the tenths (0.1 cm or 1 mm).
This answers a common question on the difference between numbers such
as 3.1 and 3.10 as measured values. The first number was generated using a
measuring instrument that had, as its smallest subdivisions, that of a unit.
The second number was generated using a measuring instrument that had,
as its smallest subdivisions, that of a tenth.
In addition, when we make a measurement and write 3.1, it means that
the tenths digit was estimated. We are not really making any claims about
whether the digit in the hundredths should be a 0, 1, 2, 3, or 4. Any lengths
of 3.10, 3.11, 3.12, 3.13, and 3.14 would have registered as 3.1 on this measuring instrument. However, when we write 3.10 as the measured value, we
claim that the we did look at how many hundredths fit into the length and
that not even one such hundredth fits in. In effect, we now claim that the
hundredths digits is not a 1 or a 2 or a 3 or a 4 but 0.
This discussion shows that, while in performing calculations the rightmost 0 in 3.10 may not matter, from the point of view of the accuracy of
the measurement, the 0 plays an important role and, it should be written
down (i.e., if it is significant), then it must be written down.
In addition to the above, one must exercise caution when making calculations that involve measured values. It does not make sense to report
digits running into the millionths when performing operations on measured
values whose estimated digits are in the tenths and hundredths. One should
report a reasonable number of digits in such cases but what criteria tells us
exactly how many are not too many?
There are many different approaches to answering this question. Below
we will present three of these approaches: The first is dictated by the findings
of a branch of math called Interval Math. The second is a simplified version
called the Standard Rule. And the third is a further-simplified version called
1
We will discuss significant digits and the estimated digit in detail shortly.
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the Three-Significant Digit Rule.2 These rules for rounding the results of
operations performed on measured values require a knowledge of significant
digits, the estimated digit and exact values. We now turn our attention to
these matters.
Significant Digits
A digit in a measured value is significant if it either comes directly from
making an act of measurement or it can be interpreted as such. When we
measure the length of a table and write 1.54 m, the digits 1 and 5 were seen
on the measuring tape and the 4 was estimated. All three digits result from
making an act of measurement and all three are therefore significant digits.
Significant digits also arise during the intermediate stages in the problemsolving process while we add, subtract, multiply, divide and perform other
mathematical operations on measured values. It is reasonable to say the
number of significan digits reported for such intermediate values must be
in line with those that we started with in the measured values themselves.
If distance covered is 215 km and travel time is 3.25 h, is it reasonable to
report a speed of 66.153 846 m/s? The distance was measured up to the
unit and the time is accurate up to hundredths. Is it reasonable to report
the value of speed as if it is accurate to the millionths?
In such situations only some of the digits should be thought of as being
within the reasonable range. These are called significant digits as well. To
indicate which digits are significant our notation for significant digits must
provide us with the means of labelling them as such.
The conventions and associated notations for the identification of the significant digits in a measured value or intermediate values during calculations
are as follows:
1. Leftend 0s are not significant. In all scenrios below leftend 0s are
excluded from the discussion.
2. If there is a dot under a digit, then all digits on the left up to and
including the digit with the dot under are significant. If there is no
dot under any of the digits, then
a. if the number is an integer, then all digits other then rightend 0s
are significant.
b. if the number is written in decimal notation or scientific notation, all
digits in the decimal notation and the mantissa in scientific notation
are significant.
2
In any particular situation we recommend the reader to consult the practitioners in
the field to determine whether one of these or some other approach is used.
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Example
How many significant digits are there in
1.
2.
3.
4.
35.29
2.50
0021
−2.1 × 103
5.
6.
7.
8.
−3.05
4.10 × 10−4
3.9 × 100
0.0024
9.
10.
11.
12.
9.20
−0.310
920
3.418 32.8 4
5.
6.
7.
8.
3
3
2
2
9.
10.
11.
12.
3
3
2
6
Answers
1.
2.
3.
4.
4
3
2
2
The Estimated Digits
The estimated digit is the rightmost significant digit. Note that this is not
the same as the rightmost digit. In 4.25.79 the estimated digit is digit 5 and
in 29 000 the estimated digit is digit 9. Neither of these is the rightmost
digit but they are the righmost significant digit.
Example
In each case determine which digit is estimated.
1.
2.
3.
4.
35.29
2.50
0021
−2.1 × 103
5.
6.
7.
8.
−3.05
4.10 × 10−4
3.9 × 100
0.0024
9.
10.
11.
12.
9.20
−0.310
920
3.418 32.8 4
5.
6.
7.
8.
5
0
9
4
9.
10.
11.
12.
0
Rightend 0
2
2
Answers
1.
2.
3.
4.
9
0
1
1
Exact Values
Exact values are values that are known to be correct with no level of uncertainty in their expresisons. Exact values have two sources: They either
arise from counting (an act of measurement) or from theory.
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With respect to counting, it is somewhat obvious that the measured
value is exact. When we report that we counted the number of molecules in
a sample and we report 4128 of them, then there are exactly 4128 of them,
not even the slightest bit more or less. Note that for this claim to exactness one must make a direct measurement of the amount present and not
report an indirect measurment based on the calculation of, say, the mass
of the sample. Such indirect calculations are subject to uncertainties in the
measured values used to calculate them, making the measured amount approximate.
With respect to theory, any specific numbers that arise are thought to
be exact. As an example of this, in the formula for kinetic energy
Ek = 12 mv 2
the 1 and both 2s (in the denominator and as exponent of v) arise from
theoretical considerations and all three are, therefore, exact. Such values
are always reported as actual numbers in the formulas of science.
Note the difference between an exact value and a constant. While exact
values are constant, not all constant values are exact. In the formula for
acceleration due to gravity near the surface of the Earth
Ep = mgh
we know that the value of g is approximately 9.8 m/s2 . However, no physics
book worthy of its name would write the formula as Ep = m × 9.8 × h as
this leaves the impression that the value 9.8 is exact. The value is measured
and is subject to approximation.
Example
Identify the exact values.
1.
2.
3.
4.
5.
4.52 kg
12 943 people (measured directly)
d = d0 + v0 t + 21 at2
3526 atoms (measured indirectly)
(G is constant)
F = GmM
r2
Answers
1. Not exact
2. Exact
3. The numbers 1 and the two 2s are exact.
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4. Not exact
5. The number 2 is exact.
One of the characteristics of an exact value is that every digit in it is
known. A mass of 4.2 kg may be written as 4.2??????? kg as the digits
in the hundredths, thousandths and so on are unknown. In a direct measuremnt of 120 as the number of particles, however, we know that we can
write the value as 120.000000 . . . as we are certain that these digits are 0s.
These rightmost digits in exact values are never written down but they are
assumed to be there. This is important because in a sense they, too, come
from the act of making a measurement, that of counting. In this sense they
are significant digits and, therefore, an exact value can be said to have an
infinite number of significant digits. The estimated digit does not exist as
all digits are known to be exact. Some think of it as being the rightmost
digit at infinity.
We now turn our attention to the manner in which measured values are
processed. We will begin with Interval Math, followed by the Standard Rule
and the Three-Significant-Digit Rule. These are rules that specify when and
how an act of rounding should take place.
Interval Math
Interval Math is that branch of math that deals with the types of questions
that we have posed. Its rules are rooted firmly in logic and it is capable of
producing results that can be relied on. However, the rules proposed by Interval Math are complex and in most practical situations the level of rigour
that it provides seems excessive. We will not discuss Interval Math in this
textbook as its results are seldom used at the introductory levels as is the
case in most introductory science courses.
A the introductory level it is often the case that simplified versions of the
rules of Interval Math are used. It should not come as a surprise that the
simplicity afforded to us by the simplified rules is offset by the approximate
natures of their conclusions. Any simplified version must be simple enough
to use by the majority of the people with relative ease and not sacrifice much
in area of accuracy.
The Standard Rule
The standard rule presents algorithms for dealing with chain addition/subtraction,
chain multiplication/division and individual rules for higher levels of operations (such as exponents, roots, logarithms and so on). The rules for chain
addition/subtraction and multiplication/division are presented below followed by a note on working with expressions in general.
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CHAIN ADDITION/SUBTRACTION
When two or more measured values are added and/or subtracted, use the
following algorithm to compute the result.
1. Finish the chain addition/subtraction exactly (i.e., no rounding) and
write the result down.
2. From now on ignore any exact values.3
3. Find the place values of the estimated digits in each measured value
that you started with.
4. Round the result in Step 1 to the largest place value in Step 3.
Example
Evaluate and round according to the Standard Rule.
1.
2.
3.
4.
45.2 + 3.18
1500 − 2110 + 1821
2.100 − 1.51 + 120 120 is exact.
4.3 × 104 + 1.8 × 103
Solution
1. 45.2 + 3.18 = 48.38 Exact
= 48.4 Rounded
In going from 48.38 to 48.4 we reasoned as follows: The estimated
digit in 45.2 is 2 which is in the tenths place. The estimated digit in
3.18 is 8 which is in the hundredths place. Since a tenth is larger than
a hundredth, we round the result to the tenths to get 48.4.
2. 1500 − 2110 + 1821 = 1211
= 1200
Exact
Rounded
In going from 1211 to 1200 we reasoned as follows: The estimated
digit in 1500 is 5 which is in the hundreds place. The estimated digit
in 2110 is the rightmost 1 which is in the tens place. The estimated
digit in 1821 is the rightmost 1 which is in the ones place. Since a
hundred is larger than a ten and a one, we round the result to the
hundreds to get 1200.
3
Exact values do not offer the largest place value of the estimated digits in values
involved as they do not have estimated digits or one can think of the estimated digit being
at infinity. They should, therefore, be ignored when these rules are applied.
10
3. 2.100 − 1.51 + 120 = 120.59
= 120.59
Exact
Rounded
The estimated digit in 2.100 is the rightmost 0 which is in the thousandths place. The estimated digit in 1.51 is the rightmost 1 which is
in the hundredths place. We ignore 120 which has been declared to be
exact. Since a hundredth is larger than a thousandth, we round the
result to the hundredths to get 120.59.
4. 4.3 × 104 + 1.8 × 103 = 4.3 × 104 + 0.18 × 104
= 4.48 × 104 Exact
= 4.5 × 104 Rounded
Before hunting down the place values of the estimated digits, we must
make sure that the exponents of 10 are the same otherwise place values will not line up and they won’t compare. We, therefore, find the
estimated digits in 4.3 × 104 and 0.18 × 104 , not the original numbers.
With these modified numbers, the estimated digit in 4.3×104 is 3 with
an apparent place value of tenths. The estimated digit in 0.18 × 104 is
8 with an apparent place value of hundredths. Since a tenth is larger
than a hundredth, we round to the tenths to get 4.5 × 104 .
Interesting result happen in some calculations that involve additions and
subtractions with the rule above. As an example, consider the addition
12 000 + 0.03. The exact value is 12 000.03 which rounds to 12 000 making
it appear as if nothing was added to 12 000. However, a little reflection shows
that this behaviour is expected. On the one hand, if this was length, would
it really matter to add 1 cm to 12 km? In most situations we would continue
to call it 12 km, dropping the insignificant 1 cm from the conversation. Note
that with the 12 km, we have estimated the 2 so the measuremnt could be
off by 1 km. The tiny 1 cm would get lost in the large estimation for the
measured 12 km. If the addition of the 1 cm would need to be accoutned for,
then the measured value 12 km would have to be measure more accurately
as well, here, down to the centimetre and recorded as 12 000.00. Now the
addition of 0.03 would result in a rounded value of 12 000.00.
CHAIN MULTIPLICATION/DIVISION
When two or more measured values are multiplied and/or divided, use the
following algorithm to compute the result.
1. Finish the chain multiplicaiton/division exactly (i.e., no
rounding) and write the result down.
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2. From now on ignore any exact values.4
3. Find the number of significant digits in each measured
value that you started with.
4. Round the result in Step 1 to the minimum number of
significant digits from Step 3.
Example
Evaluate and round according to the Standard Rule.
1. 2.4 × 8.74
3.45
2. 2.104
× 3210
3. 14001829
× 4.1
4. 5.923 ×
0.0346
5.
3 is exact.
5.18 × 103 × 2.900 × 102
−9.05 × 106 × 3.50 × 10−2
Solution
1. 2.4 × 8.74 = 20.976
= 21
Exact
Rounded
In going from 20.976 to 21 we reasoned as follows: 2.4 has two significant digits. 8.74 has three significant digits. The minimum number of
significant digits is two. We round the result to two significant digits
to get 21.
3.45 = 1.639 733 840 . . .
2. 2.104
= 1.64 Rounded
Exact
In going from 1.639 733 840 . . . to 1.64 we reasoned as follows: 3.45
has three significant digits. 2.104 has four significant digits. The minimum number of significant digits is three. We round the result to
three significant digits to get 1.64.
× 3210 = 2457.080 371 787 . . .
3. 14001829
= 2500 Rounded
Exact
In going from 2457.080 371 787 . . . to 2500 we reasoned as follows: 1400
4
Exact values do not offer the smallest number of significant digits as they have an
infinite number of significant digits.
12
has two significant digits. 3210 has three significant digits. And 1829
has four significant digits. The minimum number of significant digits
is two. We round the result to two significant digits to get 2500.
× 4.1
4. 5.923 ×
0.0346 = 60.049 211 060 . . .
= 60. Rounded
Exactvspace*.2cm
In going from 60.049 211 060 . . . to 20. we reasoned as follows: Since
3 is exact, we will ignore it. 4.1 has two significant digits. 5.92 has
three significant digits. 0.0346 has three significant digits. The minimum number of significant digits is two. We round the result to two
significant digits to get 20.. Note the use of the dot under the 0.
× 3210 = 2457.080 371 787 . . .
5. 14001829
= 2500 Rounded
Exact
In going from 2457.080 371 787 . . . to 2500 we reasoned as follows: 1400
has two significant digits. 3210 has three significant digits. And 1829
has four significant digits. The minimum number of significant digits
is two. We round the result to two significant digits to get 2500.
6.
5.18 × 103 × 2.900 × 102 = 0.474 254 143 646 . . . × 103−2−6+2
−9.05 × 106 × 3.50 × 10−2
= 0.474 254 143 646 . . . × 10−3
= 4.741 254 436 464 . . . × 10−4
= 4.74 × 10−4
Exact
Rounded
In going from the exact value to the rounded value we reasoned as follows: 5.18 has three significant digits. 2.900 has four significant digits.
−9.05 has three significant digits. And 3.50 has three significant digits. The minimum number of significant digits is three. We round the
result to three significant digits to get 4.74 × 10−4 .
EXPRESSIONS
The above cases focused on expressions with chain addition/subtraction
and chain multiplication/division. Expressions that involve a mix of these
operations and others can be evaluated using the following algorithm.
1. Break the expression into terms and the terms into factors.
2. Evaluate the factors and round according to the rule for each factor.
Write the exact result down and continue to work with the exact value
but use a dot to identify the digits that are significant. Exact values
should be processed exactly.
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3. Use the rule for chain multiplication/division to evaluate each term.
Write the exact result down for each term and continue to work with
the exact values but use a dot to identify the digits that are significant.
4. Use the rule for chain addition/subtraction to evaluate the expression.
Example
Evaluate and round according to the Standard Rule.
1. 4.56 × 3.9 + −2.84 × 3.11
2. 4.59 + 3.8 × 2.18
Solution
1. 4.56 × 3.9 + −2.84 × 3.11 = 17..784 − 8.83.24
= 26.6164
Exact
= 27
2. 4.59 + 3.8 × 2.18 = 4.59 + 8.2.84
= 12.874 Exact
= 12.9 Rounded
We will not delve deeper into the Standard Rule. However, we remind the reader that the algorithm is approximate and, beyond chain addition/subtraction and chain multiplication/division, the result is often dependent on the specific steps taken by the problem-solver. An added complexity is that it is difficult to apply the Standard Rule to problem-solving
techniques that employ a step-by-step philosophy in carrying out their solutions. In such cases it is easier to apply the rules after each addition,
subtraction, multiplication, division, etc.
The Three-Significant-Digit Rule
The approximate nature of the Standard Rule and the complexities that arise
in the applicaiton of the rules tend to make the approach less appealing to
the general practitioner. For these reasons, many argue that one might as
well round the results of all calcualtions to three significant digits.
Example
Evaluate and round the following expression according to the Three-SignificanDigit Rule.
4.5 × 3.29 + 1.892 × 3.7 × 2.19
14
Solution
4.5 × 3.29 + 1.892 × 3.7 × 2.19
14.8 + 7.00 × 2.19
14.8 + 15.3
30.1
Application of the Standard Rule to the above expression yields 30. as
the answer.
The reason why three significant digits are used rather than two or four
or some other is that it seems to be the most reasonable number of significant
digits in a measured value. If I measure the length of a table, I would likely
want to account for the fact that it is 2 m long, that it is an extra 4 decimetres
long and also that it has about 3 centimetres more, perhaps venturing into
the fact that there seems to be an additional 1 millimetre, i.e., the number
2.43 m or 2.431 m would be resonable with the former bordering on being
not so accurate and the latter bordering on being too accurate.
Exercises
1. How many significant digits are there in each of the following measured
values?
a.
b.
c.
d.
247
3.24
0056
0.184
e.
f.
g.
h.
0.0029
5002
0.702
24 000
i.
j.
k.
l.
24 0.00
24 00.0
24 000.
24 000.
m.
n.
o.
p.
3.20
0.20100
24.0
200.000
2. Evaluate each of the following. All values are measured except when
noted otherwise.
a. 4.2 × 3.71
b. 3.6 × 2.45 × 9.8
c. 1.6 × 3.62 × 1.50, 1.6 is exact.
d. 0.025 × 300.
e. 1902 × 420̇0
f. 0.00 160 × 240.
g. 0046 × 205
h. 3.6 × 60 × 60, Both 60s are exact.
i. 32.5 × 8.4
k. 62.5
18
j. 10.201 × 42 × 120.0
9.6
l. 4.25
15
9.6 , 9.6 is exact.
m. 4.25
× 4.5
p. 2.681
6.23
9.6 , 4.25 is exact.
n. 4.25
q. 4201200
× 10.0
2400
o. 30
0.00
132 000
r. 4200
. × 10
3.6 × 400 , 400, 3.5, and 3.6 are all exact.
s. 3.5
× 1.82
3. Find the estimated digit in each of the following measured values.
a.
b.
c.
d.
247
3.24
0056
0.184
e.
f.
g.
h.
0.0029
5002
0.702
24 000
i.
j.
k.
l.
24 0.00
24 00.0
24 000.
24 000.
m.
n.
o.
p.
3.20
0.20100
24.0
200.000
4. Evaluate each of the following. All values are measured except when
noted otherwise.
a.
b.
c.
d.
e.
425 + 32 + 15.68
90 + 32.4 − 0.26 + 42.5
4.25 + 0.086 − 200
4.25 + 0.086 − 20.0
4.25 + 0.086 − 200.
f.
g.
h.
i.
j.
4.25
4.25
4.25
4.25
94.6
+
+
+
+
+
0.086
0.086
0.086
0.086
180
−
−
−
−
200.
200.0
200.00
200.000
k. 3000 − 2560 + 1020 − 1800, 3000 is exact.
l. 0.026 − 0.00182 + 0.0513, 0.026 is exact.
m. 50 200 + 32 000 − 1010 + 21 000, 1010 is exact.
5. Evaluate each of the following expressions. All values are measured
unless noted otherwise.
a. 4.2 × 6 + 18 × 9.45, 6 is exact.
b. 3.0 × 425 + 9.68 × 3 × 14.0
c. 1.6 + 124 × 0.1 + 0.094
1
× 15 + 1.8 × 4 −
d. 4.23.61
3.6 × 1.5
18
e. 4.56
3.6 + 100 , 3.6 and 18 are both exact.
f. 21 × 19 × 3.4 + 23 × 4500. × 0.065
1.000
1.00
g. 4.2 × 160. + 0.62
× 51 − 2.4 × 3.68
16
h. 14 200 + 16 × 2000. − 56 000
1
i. 2450 + 3000. × 24 × 650.
j. 4 × 9.4 + 3 × 0.64 − 2 × 14.5, 4, 3, and 2 are exact.
Answers
1. a. 3
b. 3
c. 2
2. a.
b.
c.
d.
e.
d. 3
e. 2
f. 4
16
86
8.69
7.5
8 000 000
3. a. 7
b. 4
j.
i.
k.
l.
m.
n.
o.
p.
the
the
the
the
0
the
0
the
g. 3
h. 2
i. 3
f.
g.
h.
i.
j.
j. 4
k. 5
l. 5
0.384
9400
13 000
270
510 000
c. 6
d. 4
k.
l.
m.
n.
o.
m. 3
n. 5
o. 3
3.5
2.3
2.26
2.3
0.080
e. 9
f. 2
p.
q.
r.
s.
1.9
0.029
3
226
g. 2
h. 4
third 0 from the right
second 0 from the right
rightmost 0
rightmost 0
rightmost 0
rightmost 0
4. a. 473
b. 160
c. −200
d. −20.0
e. −196
f. −196
5. a. 20.0
b. 1700
d. 24
g. 670
e. 1.4
h. −10.000
i. 2600
c. 10
p. 6
f. 200
g. −196.7
h. −196.66
i. −196.66
j. 270
k. −300
l. 0.0755
m. 102 000
j. 11