numeration system

numeration system

A

numeration system

is a triple

N

=

(

b

,

Σ

,

d

)

, where

b

>

1

is a

positive integer

,

Σ

is a non-empty

alphabet

, and

d

is a one-to-one function from

Σ

to the set of non-negative

integers

ℕ

∪

{

0

}

. Order elements of

Σ

so that their values are in increasing order:

Σ

=

{

a

1

,

…

,

a

k

}

, where

d

⁢

(

a

i

)

<

d

⁢

(

a

i

+

1

)

for

i

=

1

,

…

,

k

-

1

.

b

is called the

base

of numeration system

N

, and the

elements

a

1

,

…

,

a

k

the

digits

of

N

. Words over

Σ

are called

numeral words

.

Given a numeral word

u

=

c

1

⁢

⋯

⁢

c

m

with

c

j

∈

Σ

, the integer non-negative

n

is said to be

represented

by

u

if

n

=

c

1

⁢

b

m

-

1

+

⋯

+

c

j

⁢

b

m

-

j

+

⋯

+

c

m

.

An integer

n

is said to be representable in

N

if there is a numeral word

u

representing

n

.

Examples

.

•

The most common numeration system is the decimal system:

D

=

(

10

,

{

0

,

1

,

2

,

3

,

4

,

5

,

6

,

7

,

8

,

9

}

,

d

)

where

d

⁢

(

i

)

=

i

is the

identity function

.

•

Just as common is the binary digital system:

B

=

(

2

,

{

0

,

1

}

,

d

)

where

d

again is the identity function.

•

In fact, any digital system is a numeration system

(

n

,

Σ

,

d

)

, where

Σ

=

{

a

0

,

…

,

a

n

-

1

}

and

d

⁢

(

a

i

)

=

i

.

•

Consider the system

B

1

=

(

2

,

{

1

}

,

d

)

, where

d

⁢

(

1

)

=

1

. Since any word over

{

1

}

is just a string of

1

’s,

n

consecutive strings of

1

represent

1

+

2

+

⋯

+

2

n

-

1

=

2

n

-

1

. We conclude that the integers representable by

N

have the form

2

n

-

1

for any positive integer

n

.

•

Consider the system

D

1

=

(

10

,

{

[

0

]

,

[

1

]

,

…

,

[

9

]

,

[

10

]

,

[

100

]

,

[

1000

]

,

[

10000

]

,

[

10000000

]

,

[

10000000000

]

}

,

d

)

where

d

⁢

(

[

i

]

)

=

i

. It is easy to see that every integer is representable by

D

1

. However, some integers may be represented by more than one numeral words. For example,

[

1000

]

=

[

100

]

⁢

[

0

]

=

[

10

]

⁢

[

0

]

⁢

[

0

]

=

[

1

]

⁢

[

0

]

⁢

[

0

]

⁢

[

0

]

.

The numeration system

D

1

is used by the Chinese.

•

Consider the system

N

=

(

3

,

{

[

1

]

,

[

2

]

,

[

4

]

}

,

d

)

where

d

⁢

(

[

i

]

)

=

i

. Then

[

2

]

⁢

[

1

]

⁢

[

4

]

⁢

[

1

]

=

2

×

3

3

+

1

×

3

2

+

4

×

3

+

1

=

184

. Notice that

0

can not be represented

N

. Also, note that

[

4

]

=

4

=

1

×

3

+

1

=

[

1

]

⁢

[

1

]

.

A numeration system

N

is said to be

complete

if every non-negative integer has at least one representation in

N

; and

unambiguous

if every non-negative integer has at most one representation in

N

.

N

is

ambiguous

if

N

is not unambiguous. Every digital system is complete and unambiguous. In the examples above,

D

1

is complete but ambiguous;

B

1

is unambiguous but not complete;

N

is neither complete nor unambiguous.

Remark

. Representation non-negative integers by a numeration system can be extended to

rational numbers

. The corresponding

concepts

of completeness and ambiguity may be defined similarly.

References

1

A. Salomaa

Computation

and

Automata

, Encyclopedia of Mathematics and Its Applications, Vol. 25

. Cambridge (1985).

Title

numeration system

Canonical name

NumerationSystem

Date of creation

2013-03-22 18:57:46

Last modified on

2013-03-22 18:57:46

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

8

Author

CWoo (3771)

Entry type

Definition

Classification

msc 11A67

Synonym

numeral system

Synonym

number system

Related topic

Base3

Defines

base

Defines

digit

Defines

complete numeration system

Defines

unambiguous numeration system

Defines

ambiguous numeration system

Generated on Fri Feb 9 15:24:49 2018 by

LaTeXML