Rounding Off
You are probably familiar with the notion of rounding numbers.
It is a process of discarding one or more digits from the right
of a number (or replacing them by zeros) in a reasonable way.
The rules are:
(i.) when the first discarded digit is a 4 or less, the last
retained digit is not changed.
(ii.) when the first discarded digit is 5 or greater, the last
retained digit is increased by 1.
(iii.) discarded digits to the left of the decimal point are
replaced by zeros. Discarded digits to the right of the decimal
point are simply dropped.
Example 1:
Round 45.6793 to two decimal places.
solution:
This means that the second digit to the right of the decimal
point, the 7, is the last retained digit, in the terminology of
the rules stated above. The first discarded digit is then the one
following this 7, namely the 9. Since 9 is bigger than 5, we
increment the 7 to 8, and drop the last two digits, the 9 and the
3 (since they are to the right of the decimal point). Thus, the
rounded answer requested is 45.68.
Example 2:
Round 156.857262 to three decimal places.
solution:
In this case, the last retained digit is the 7 in the third
decimal place, and the first discarded digit is the 2 in the
fourth decimal place. Since the first discarded digit is a 2,
which is less than or equal to 4, the last retained digit is
retained unchanged, and all digits to its right (being to the
right of the decimal point) are simply dropped. So, the answer
required here is 156.857.
Example 3:
Round 42865 to hundreds (or, to the nearest one hundred).
solution:
The discarded digits are the 6 and the 5 in the first and
second positions from the right, because the last retained digit
is to be in the position of hundreds (the third digit from the
right). Since the first discarded digit, the 6, is greater than
5, the last retained digit, the 8, is incremented to a 9. Then
the discarded digits are to the left of the (implied) decimal
point, and so are replaced by zeros. The final answer is 42900.
Obviously, the goal of rounding is to obtain the nearest
numerical value of a specified precision to a given number of
apparently greater precision.
Remark
This goal of rounding leads some people to concern over
applying the simple rounding rules above when the only discarded
digit is a 5, or a 5 following just by zeros. Both leaving the
last retained digit unchanged and incrementing the last retained
digit by 1 seem to give equally reasonable (but different)
rounded results. However, applying the given rules strictly means
that the last retained digit is always incremented in these
situations.
The most common “fix” for rules (i) – (iii) to
handle this situation is to adopt either of the two additional
rules. Whenever the only discarded digit is a 5, or the discarded
digits are a 5 following only by zeros, either
(iv.a) round the last retained digit to the nearest even
number, or,
(iv.b) round the last retained digit to the nearest odd
number.
Of the two, version (iv.a) is the most commonly used.
Since this type of situation arises quite rarely, it is
probably safe to ignore rules (iv) unless someone instructs you
to use them.
Example 4:
To round 46.73500 to two decimal places, we would get 46.74
using rule (iv.a) and we would get 46.73 using rule (iv.b).
