rounding

rounding

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

⌈

r

⌉

:=

min

⁡

{

n

∈

ℤ

∣

n

≥

r

}

.

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

⌊

r

⌋

:=

max

⁡

{

n

∈

ℤ

∣

n

≤

r

}

=

{

⌈

r

⌉

if

r

is an integer

⌈

r

⌉

-

1

otherwise.

Examples:

⌊

1.24

⌋

=

1

, and

⌊

-

2.63

⌋

=

-

3

.

truncation:

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

r

. The truncation of

r

is denoted by

[

r

]

. In terms of rounding up and rounding down: we have

[

r

]

=

{

⌊

r

⌋

if

r

≥

0

⌈

r

⌉

if

r

<

0

.

If we write

r

as a decimal number using

decimal expansion

, then

[

r

]

is the

integral

portion of

r

.

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

R

⁢

(

r

)

=

⌊

r

+

0.5

⌋

.

Examples:

R

⁢

(

-

3.37

)

=

-

3

, while

R

⁢

(

7.5

)

=

8

.

There is an easy

algorithm

of rounding

r

to the nearest integer.

(a)

write

r

as a decimal number using decimal expansion

(b)

if the tenths

decimal place value

is less than

5

, then

R

⁢

(

r

)

=

[

r

]

(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

B

⁢

(

r

)

=

{

⌊

r

⌋

if

[

r

]

is even, and

2

⁢

r

∈

ℤ

R

⁢

(

r

)

otherwise.

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:

(a)

the first number in the sequence is rounded using any of the above methods;

(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.

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.

Write

r

as a decimal number using decimal expansion.

3.

Multiply

r

by

10

n

. 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.

Use any of the four methods above to round

10

n

⁢

r

.

5.

Divide the rounded number by

10

n

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

|

≤

10

n

, and

•

f

⁢

(

r

)

=

r

if

10

n

⁢

r

∈

ℤ

.

If

g

:

ℝ

→

ℤ

denotes any of the four rounding methods described in the previous

section

, and

g

n

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

:

g

n

⁢

(

r

)

=

g

⁢

(

10

n

⁢

r

)

10

n

.

Title

rounding

Canonical

name

Rounding

Date of creation

2013-03-22 17:27:27

Last modified on

2013-03-22 17:27:27

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

10

Author

CWoo (3771)

Entry type

Definition

Classification

msc 65G99

Classification

msc 65D99

Classification

msc 00A69

Classification

msc 65G50

Synonym

round up

Synonym

round down

Synonym

round to

Defines

rounding up

Defines

rounding down

Defines

symmetric arithmetic rounding

Defines

rounding error

Defines

truncation

Defines

rounded to

Defines

banker’s rounding

Generated on Sat Feb 10 12:36:24 2018 by

LaTeXML