decimal fraction

decimal fraction

A

rational number

d

is called a

decimal fraction

if

10

k

⁢

d

is an

integer

for some non-negative integer

k

. For example, any integer, as well as rationals such as

0.23123

,

3

4

,

236

125

are all decimal fractions. Rational numbers such as

1

3

,

-

227

12

,

2

.

312

¯

are not.

There are two other ways of characterizing a decimal fraction: for a rational number

d

,

1.

d

is as in the above definition;

2.

d

can be written as a

fraction

p

q

, where

p

and

q

are integers, and

q

=

2

m

⁢

5

n

for some non-negative integers

m

and

n

;

3.

d

has a terminating decimal expansion, meaning that it has a decimal representation

a

.

d

1

⁢

d

2

⁢

⋯

⁢

d

n

⁢

000

⁢

⋯

where

a

is a

nonnegative integer

and

d

i

is any one of the digits

0

,

…

,

9

.

A decimal fraction is sometimes called a

decimal number

, although a decimal number in the most general sense may have non-terminating

decimal expansions

.

Remarks

. Let

D

⊂

ℚ

be the set of all decimal fractions.

•

If

a

,

b

∈

D

, then

a

⋅

b

and

a

+

b

∈

D

as well. Also,

-

a

∈

D

whenever

a

∈

D

. In other words,

D

is a subring of

ℚ

. Furthermore, as an abelian group,

D

is

2

-

divisible

and

5

-divisible. However, unlike

ℚ

,

D

is not divisible (

http://

planetmath

.org/DivisibleGroup

).

•

As inherited from

ℚ

,

D

has a

total order

structure

. It is easy to see that

D

is dense (

http://planetmath.org/DenseTotalOrder

): for any

a

,

b

∈

D

with

a

<

b

, there is

c

∈

D

such that

a

<

c

<

b

. Simply take

c

=

a

+

b

2

.

•

From a topological point of view,

D

, as a subset of

ℝ

, is

dense in

ℝ

. This is essentially the fact that every real number has a decimal expansion, so that every real number can be “approximated” by a decimal fraction to any degree of accuracy.

•

We can

associate

each decimal fraction

d

with the least non-negative integer

k

⁢

(

d

)

such that

10

k

⁢

(

d

)

⁢

d

is an integer. This integer is uniquely determined by

d

. In fact,

k

⁢

(

d

)

is the last decimal place where its corresponding digit is non-zero in its decimal representation. For example,

k

⁢

(

1.41243

)

=

5

and

k

⁢

(

7

/

25

)

=

2

. It is not hard to see that if we write

d

=

p

2

m

⁢

5

n

, where

p

and

2

m

⁢

5

n

are

coprime

, then

k

⁢

(

d

)

=

max

⁡

(

m

,

n

)

.

•

For each non-negative integer

i

, let

D

⁢

(

i

)

be the set of all

d

∈

D

such that

k

⁢

(

d

)

=

i

. Then

D

can be partitioned into sets

D

=

D

⁢

(

0

)

∪

D

⁢

(

1

)

∪

⋯

∪

D

⁢

(

n

)

∪

⋯

.

Note that

D

⁢

(

0

)

=

ℤ

. Another basic property is that if

a

∈

D

⁢

(

i

)

and

b

∈

D

⁢

(

j

)

with

i

<

j

, then

a

+

b

∈

D

⁢

(

j

)

.

Title

decimal fraction

Canonical name

DecimalFraction

Date of creation

2013-03-22 17:27:15

Last modified on

2013-03-22 17:27:15

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

10

Author

CWoo (3771)

Entry type

Definition

Classification

msc 11-01

Related topic

RationalNumber

Defines

decimal number

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