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 an object. Ten people are recruited,
and each given an instrument to measure lengths. They report the
following values:
963 mm 958 mm 961 mm 955 mm 959 mm
964 mm 962 mm 958 mm 960 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 object 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 object 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 roundingoff).
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.
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
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 10digit display – if we had a calculator with a
25digit 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 has an accuracy of 5 significant digits
0.000076 has an accuracy of 2 significant digits
27.66 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.
