rounding (original) (raw)

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 r be a real number. Then

rounding up:

rounding up of r is taking the smallest integer that is greater than or equal to r. This integer is denoted by the ceiling function

Examples: ⌈2.1⌉=3, and ⌈62.672⌉=63.

rounding down:

rounding down of r is taking the largest integer that is less than or equal to r. This integer is denoted by the floor function

Examples: ⌊1.24⌋=1, and ⌊-2.63⌋=-3.

truncation:

Examples: [2.354]=2, and [-81.67]=-81.

ordinary rounding:

this is the most commonly used of the rounding methods described so far. (Ordinary) rounding of r is finding the closest integer to r, and if r is exactly half way between two integers, use the larger of the two as the result. Let R⁢(r) represents the ordinary rounding of r. It is easy to see that

Examples: R⁢(-3.37)=-3, while R⁢(7.5)=8.

There is an easy algorithm of rounding r to the nearest integer.

1. (a)
   write r as a decimal number using decimal expansion
2. (b)
3. (c)
   if the tenths decimal place value is at least 5, then R⁢(r)=[r]+1.

banker’s rounding:

a variant of the ordinary rounding is the banker’s rounding: if r is exactly half way between two integers, and the integer portion of r is even, round down r. Otherwise, use ordinary rounding on r. If B⁢(r) denotes the banker’s rounding of r, then it can be defined as

For example, B⁢(3.5)=4, while B⁢(2.5)=2.

stochastic rounding:

this rounding method requires the aid of a random number generator. Rounding of r may be done using any of the above methods when r is not exactly half way between two consecutive integers. Otherwise, r 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 [0,1] 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 r 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:

1. (a)
   the first number in the sequence is rounded using any of the above methods;
2. (b)
   when the n-th number is rounded, the (n+1)-th number is rounded as follows: if the number is exactly half way between two consecutive integers, then it is rounded down if the n-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 r to a decimal fraction. The general procedure is as follows:

1. 1. First, specify how accurately we want to round r. This can be accomplished by specifying to what decimal place we want to approximate r. Let this place be n (note that n>0 if it is to the right of the decimal point and n<0 otherwise).
2. 1. Write r as a decimal number using decimal expansion.
3. 1. Multiply r by 10n. By doing this, we are basically moving the decimal point so it is positioned between the n-th decimal place and the (n+1)-th decimal place.
4. 1. Use any of the four methods above to round 10n⁢r.
5. 1. Divide the rounded number by 10n to get the result.

In practice, steps 3 through 5 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 π=3.14159⁢… to the nearest thousandths place is 3.142, the thousandths place value 1 is increased to 2 because the ten thousandths place is 5.

Remark. In general, rounding to the n-th decimal place can be thought of as a function f from ℝ to D, the set of all decimal fractions, such that

• •
  |f⁢(r)-r|≤10n, and
• •
  f⁢(r)=r if 10n⁢r∈ℤ.

If g:ℝ→ℤ denotes any of the four rounding methods described in the previous section, and gn corresponds to rounding to the n-th decimal place using method g in step 4 above, then the entire rounding process can be summarized by a single formula: