uniqueness of digital representation

uniqueness of digital representation

Theorem.

Let the

positive integer

b

be the base of a

http://planetmath.org/node/3313

positional digital system.  Every positive integer

a

may be represented uniquely in the form

a

=

s

n

⁢

b

n

+

s

n

-

1

⁢

b

n

-

1

+

…

+

s

1

⁢

b

+

s

0

,

(1)

where the

integers

s

i

satisfy

0

≦

s

i

≦

b

-

1

and

s

n

≠

0

.

The theorem means that the integer

a

may be represented e.g. in the

http://planetmath.org/node/9839

decimal system in the form

s

n

⁢

s

n

-

1

⁢

…

⁢

s

1

⁢

s

0

in one and only one way.

Proof.

Let

b

n

be the highest integer power of

b

not exceeding

a

.  By the

division algorithm for integers

, we obtain in succession

a

=

s

n

⁢

b

n

+

r

1

(

0

<

s

n

<

b

,

0

≦

r

1

<

b

n

)

,

r

1

=

s

n

-

1

⁢

b

n

-

1

+

r

2

(

0

≦

s

n

-

1

<

b

,

0

≦

r

2

<

b

n

-

1

)

,

r

2

=

s

n

-

2

⁢

b

n

-

2

+

r

3

(

0

≦

s

n

-

2

<

b

,

0

≦

r

3

<

b

n

-

2

)

,

⁢

⋮

r

n

-

2

=

s

2

⁢

b

2

+

r

n

-

1

(

0

≦

s

2

<

b

,

0

≦

r

n

-

1

<

b

2

)

,

r

n

-

1

=

s

1

⁢

b

+

s

0

(

0

≦

s

1

<

b

,

0

≦

s

0

<

b

)

.

Adding these equations yields the equation (1) with

0

≦

s

i

≦

b

-

1

,

s

n

≠

0

.

For showing the uniqueness of (1) we suppose also another

a

=

t

m

⁢

b

m

+

t

m

-

1

⁢

b

m

-

1

+

…

+

t

1

⁢

b

+

t

0

with

0

≦

t

i

≦

s

-

1

,

b

m

≠

0

.  The equality

s

n

⁢

b

n

+

s

n

-

1

⁢

b

n

-

1

+

…

+

s

1

⁢

b

+

s

0

=

t

m

⁢

b

m

+

t

m

-

1

⁢

b

m

-

1

+

…

+

t

1

⁢

b

+

t

0

(2)

immediately implies

s

0

≡

t

0

(

mod

b

)

,

i.e.

b

∣

s

0

-

t

0

.

Since now

|

s

0

-

t

0

|

≦

b

-

1

,  we infer that

s

0

-

t

0

=

0

and thus

t

0

=

s

0

.  Consequently, we can then infer from (2) that

s

1

⁢

b

≡

t

1

⁢

b

(

mod

b

2

)

, whence

s

1

≡

t

1

(

mod

b

)

,  and as before,

t

1

=

s

1

.  We may continue in manner and see that always

t

i

=

s

i

, whence also

m

=

n

.  Accordingly, the both are identical. Q.E.D.

Remark.

There is the following generalisation of the theorem.  — If we have an infinite

sequence

b

1

,

b

2

,

…

of integers greater than 1, then

a

may be represented uniquely in the form

a

=

∑

i

=

0

n

s

i

⁢

b

1

⁢

b

2

⁢

⋯

⁢

b

i

where the integers

s

i

satisfy

0

≦

s

i

<

b

i

+

1

and

s

n

≠

0

.  Cf. the

factorial base

.

Title

uniqueness of digital representation

Canonical name

UniquenessOfDigitalRepresentation

Date of creation

2013-03-22 18:52:16

Last modified on

2013-03-22 18:52:16

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

15

Author

pahio (2872)

Entry type

Theorem

Classification

msc 11A63

Classification

msc 11A05

Synonym

uniqueness of decimal representation

Synonym

digital representation of integer

Related topic

UnambiguityOfFactorialBaseRepresentation

Related topic

UniquenessOfFourierExpansion

Related topic

UniquenessOfLaurentExpansion

Related topic

ZeckendorfsTheorem

Related topic

RepresentationOfRealNumbers

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