Rounding is a general technique for approximating a real number by a decimal fraction. There are several ways of rounding a real number, five of which are the most common: rounding up, rounding down, truncation, ordinary rounding (or rounding for short), and banker’s rounding.
Rounding to an Integer
The simplest kind of rounding is that of rounding a real number to an integer. Let be a real number. Then
- rounding up:
- rounding down:
rounding down of is taking the largest integer that is less than or equal to . This integer is denoted by the floor function
Examples: , and .
rounding by truncation is done by ignoring all decimals to the right of the decimal point, which is equivalent to taking only the integer part of . The truncation of is denoted by . In terms of rounding up and rounding down: we have
Examples: , and .
- ordinary rounding:
this is the most commonly used of the rounding methods described so far. (Ordinary) rounding of is finding the closest integer to , and if is exactly half way between two integers, use the larger of the two as the result. Let represents the ordinary rounding of . It is easy to see that
Examples: , while .
- banker’s rounding:
a variant of the ordinary rounding is the banker’s rounding: if is exactly half way between two integers, and the integer portion of is even, round down . Otherwise, use ordinary rounding on . If denotes the banker’s rounding of , then it can be defined as
For example, , while .
- stochastic rounding:
this rounding method requires the aid of a random number generator. Rounding of may be done using any of the above methods when is not exactly half way between two consecutive integers. Otherwise, is randomly rounded up or down based on the outcome of randomly selecting a number between 0 and 1 using a random number generator. The choice of rounding up (and thus down) depends on how numbers are in are allocated for rounding up (or down).
- alternate rounding:
this rounding method, like the last one, uses other available methods except when the number in question is exactly half way between two consecutive integers. However, this method is used in a situation where a sequence of numbers needs to be rounded:
the first number in the sequence is rounded using any of the above methods;
when the -th number is rounded, the -th number is rounded as follows: if the number is exactly half way between two consecutive integers, then it is rounded down if the -th number is rounded up, and vice versa. Otherwise, use the rounding method used to round the first number in the sequence.
Rounding to a Decimal Fraction
More generally, the three methods described can be applied to rounding of to a decimal fraction. The general procedure is as follows:
First, specify how accurately we want to round . This can be accomplished by specifying to what decimal place we want to approximate . Let this place be (note that if it is to the right of the decimal point and otherwise).
Write as a decimal number using decimal expansion.
Multiply by . By doing this, we are basically moving the decimal point so it is positioned between the -th decimal place and the -th decimal place.
Use any of the four methods above to round .
Divide the rounded number by to get the result.
In practice, steps through can be combined into one step, simply by performing the rounding operation at the specified decimal place as if it were the ones place. For example, rounding to the nearest thousandths place is , the thousandths place value is increased to because the ten thousandths place is .
Remark. In general, rounding to the -th decimal place can be thought of as a function from to , the set of all decimal fractions, such that
If denotes any of the four rounding methods described in the previous section, and corresponds to rounding to the -th decimal place using method in step above, then the entire rounding process can be summarized by a single formula:
|Date of creation||2013-03-22 17:27:27|
|Last modified on||2013-03-22 17:27:27|
|Last modified by||CWoo (3771)|
|Defines||symmetric arithmetic rounding|