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).