decimal expansion

decimal expansion

Every

rational number

m

n

, where

m

and

n

are

positive integers

, has an endless

decimal expansion

(or

decadic expansion

— according to Greek).  The decimal expansion of

m

n

means the series

ν

.

ν

1

⁢

ν

2

⁢

ν

3

⁢

…

=

ν

+

10

-

1

⁢

ν

1

+

10

-

2

⁢

ν

2

+

10

-

3

⁢

ν

3

+

…

(1)

where

ν

=

⌊

m

n

⌋

is the

integer part

(

http://

planetmath

.org/Floor

) of

m

n

and the

integers

ν

j

are the

remainders

of

⌊

10

j

⋅

m

n

⌋

when divided by 10; thus

0

≦

ν

j

<

10

.

We may suppose that

m

and

n

are

coprime

(if necessary, reduce the

fraction

).  Then the depends only on the denominator

n

.  In the case that

gcd

⁡

(

n

,

10

)

=

1

,  the is the least positive integer

l

such that

10

l

≡

1

(

mod

n

)

(the does not change if we multiply the fraction by a suitable power of 10 and then reduce all

prime factors

of 10 from the denominator).  In every case, the is a factor of the number

φ

⁢

(

n

)

, where

φ

is Euler’s

totient function

.

Examples

1

8

=

0.125000

⁢

…

=

0.124999

⁢

…

(one-digit ; N.B. two possibilities),

1

12

=

0.08333

⁢

…

(one-digit per.),

1

37

=

0

.

′

027

′

027

′

027

′

…

(three-digit per.),

1

82

=

0.0

′

⁢

12195

′

⁢

12195

′

⁢

12195

′

⁢

…

(five-digit per.),

1

25351

=

0.000039446

⁢

…

(hundred-digit per.)

The tail of infinitely many 0’s (as in 0.125000…) is of course usually not written out.  Such a tail is possible only when

n

has no other prime factors except prime factors of the base of the digit system in question.

If the tails of 0’s are not accepted, then the digital expansion of every

positive

rational is unique (then e.g. 0.124999…  is the only for

1

8

in the decimal system).

Completely

similar

results concern the digital expansions in any other positional digit system.  Let the fraction

1

31

be an example (

φ

⁢

(

31

)

=

30

); its is

in the decadic (decimal) digit system

1

31

=

0

.

′

032258064516129

′

…

ten

(15-digit per.),

in the hexadic (senary) digit system

1

51

=

0

.

′

010545

′

010545

′

010545

′

…

six

(6-digit per.),

in the dyadic () digit system

1

11111

=

0.000010000100001

⁢

…

two

(5-digit per.).

Note.

Also any

irrational number

has a unique decimal expansion, but it is non-periodic; for example Liouville’s number (

http://planetmath.org/ExampleOfTranscendentalNumber

)

0.110001 000000 000000 000001 000000

⁢

…

which is

transcendental

over

ℚ

.

Title

decimal expansion

Canonical name

DecimalExpansion

Date of creation

2013-03-22 15:04:01

Last modified on

2013-03-22 15:04:01

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

28

Author

pahio (2872)

Entry type

Result

Classification

msc 11A99

Synonym

decadic expansion

Synonym

digital expansion

Related topic

Base3

Related topic

ExistenceAndUniquenessOfDecimalExpansion

Related topic

0ne1AsRealNumbers

Related topic

Factoradic

Related topic

FactorialBase

Related topic

MatheRealism

Related topic

JosephLiouville

Defines

decadic

Defines

hexadic

Defines

dyadic

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