The Mathematics 11
Competency Test
Rules for Arithmetic With
Approximate Numbers
The reason we have to be concerned with concepts such as exact and approximate numbers,
and significant digits, precision and accuracy of approximate numbers, is to make sure that when
we work with numbers representing actual measurements of physical quantities, we understand
specifically what these numerical digits are telling us about the size of the physical quantity. In
particular, it is important to avoid writing down digits in such a number which really convey no
meaningful or justifiable information. This is really a topic that requires more detailed discussion
than is possible in these notes. However, perhaps one hypothetical example will give you an
insight into this issue.
Suppose the goal was to get an “accurate” measurement of the length of a piece of pipe. Ten
people are recruited, and each given an instrument to measure lengths. They report the following
values:
963 mm
964 mm
958 mm
962 mm
961 mm
958 mm
955 mm
960 mm
959 mm
961 mm
Now, although a couple of pairs of these measurements are the same, there is also quite a
variation in results, ranging from a low value of 955 mm to a high value of 964 mm. In this
situation, we would have to conclude that we really don’t know the length of the piece of pipe
precise to 1 mm, since at that level of precision, there is no general agreement between the ten
measurements. To be honest then, it appears we should state the result as 960 mm, meaning
that the true length could be as much as 5 mm shorter or 5 mm longer than this value. So, to say
that this pipe is 958 mm long might be misleading, since the data we have doesn’t narrow the
length down more precisely than “some value between 955 mm and 965 mm.”
This example illustrates in a superficial way that it is important to be wary of stating results with
an unwarranted number of significant digits when they are based on either measurements or
calculations involving approximate numbers. What we will state below are the simplest of rules
for deciding when digits in the result of a calculation are not really warranted and so should be
discarded (through rounding-off).
For simple arithmetic calculations, there are two rules for rounding results obtained from
approximate numbers:
Rule 1: When two or more approximate numbers are added and/or subtracted, the result is
rounded to the precision of the least precise approximate number involved.
Rule 2: When two or more approximate numbers are multiplied and/or divided, the result is
rounded off to the accuracy of the least accurate approximate number involved.
In practice, one does the entire calculation first, getting an overall result. Then the rules are
applied as appropriate to round this final result. We do not do any rounding before the final result
is obtained.
Example 1: Compute the result of
528.63 + 816.4 – 921.072
Assume that each of these numbers are approximate numbers.
David W. Sabo (2003)
Rules for Arithmetic With Approximate Numbers
Page 1 of 3
solution:
Just entering these numbers into a calculator gives the apparent result 423.958. Since this
calculation involves only addition and subtraction, we need to use the precision of the three
numbers involved to decide rounding for the final result (Rule 1). Now
528.63 has a precision of 2 decimal places
816.4 has a precision of 1 decimal place
921.072 has a precision of 3 decimal places.
The smallest precision here is one decimal place, for the second number, 816.4. Therefore the
final result must be rounded to just one decimal place, giving 424.0.
Example 2: Compute the result of
( 45683 )( 0.000076 )
27.66
assuming each number is an approximate number.
solution:
When we enter these numbers into a calculator, the result is something like 0.125520896 (in this
case, with a calculator having a 10-digit display – if we had a calculator with a 25-digit display,
we’d get 0.1255208966015907447577730, so obviously there is no avoiding the question of
having to round off the result of this calculation in some way!)
This calculation involves only multiplications and divisions, so it is the accuracy of the individual
numbers which determines the accuracy of the result (Rule 2). Here
45683
0.000076
27.66
has an accuracy of 5 significant digits
has an accuracy of 2 significant digits
has an accuracy of 4 significant digits
We see that the least accurate number is the second one, 0.000076, with an accuracy of two
significant digits. Therefore, by Rule 2, the final result should be rounded to two significant digits,
giving 0.13 .
Example 3: Four packages are to be loaded on an airplane. The labels state their weights
(masses) as 4.32 kg, 55.4 kg, 8.791 kg, and 453 kg. Compute the total weight of the four
packages and round your result appropriately.
solution:
Obviously the total weight will just be the sum of the four weights of the individual packages.
Since these numbers would have been obtained by weighing the packages, they are approximate
numbers and so our arithmetic is governed by the rules of rounding given above. First, we sum
the four values to get an initial sum, which must then be rounded.
4.32 kg
55.4
kg
8.791 kg
David W. Sabo (2003)
precision
two decimal places
one decimal place
three decimal places
Rules for Arithmetic With Approximate Numbers
Page 2 of 3
453.
kg
units place
521.511 kg
Since we’re adding here, the rounding of the final result is governed by the precision of the
numbers involved. From the list of precisions on the right above, we see that the least precise of
these four values is the last one, 453 kg, with a precision of units. Thus, our final result should
be rounded to units, giving 522 kg. Thus, the total weight of the four packages, rounded
appropriately is 522 kg.
Example 4: In a test run, a computer is able to scan a sample of 34000 client records in 18.56
minutes. How many seconds does it spend on each record?
solution:
We will present a formal method for handling unit conversion problems later in these notes.
However, we can sort out this problem fairly easily without those methods. We should get the
required answer by dividing the total number of seconds taken by the total number of records
scanned, to get the number of seconds spent to scan each record.
Now
18.56 minutes = 18.56 x 60 seconds
and so
number of seconds/record =
(18.56 )( 60 ) ≅ 0.032752941
34000
on a calculator with a ten-digit display.
This calculation involves just multiplications and divisions, so it is the accuracy of the numbers
involved that determine the accuracy to which the final result should be rounded. Checking each
number in turn, we see that
18.56 is an approximate number (the result of a measurement of a time interval) and has
an accuracy of four significant digits.
60 is an exact number, because it is the definition of how many seconds there are in a
minute. You can consider an exact number to have an infinite number of
significant digits for the purposes of applying the rounding rules here. This is the
same thing as saying we need not take exact numbers into account at all in
applying the rules.
34000 is also an exact number, because it is the result of counting objects. Thus, it too
has no bearing on how we should round off our final result.
So, the least accurate number here is the only approximate number actually involved in the
calculation, the 18.56 minutes, which has an accuracy of four significant digits. So, our final
result should be rounded to four significant digits, giving the conclusion: the computer requires
0.03275 seconds to scan one client record.
David W. Sabo (2003)
Rules for Arithmetic With Approximate Numbers
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