uniqueness of digital representation
Theorem. Let the positive integer be the base of a http://planetmath.org/node/3313positional digital system. Every positive integer may be represented uniquely in the form
where the integers satisfy and .
The theorem means that the integer may be represented e.g. in the http://planetmath.org/node/9839decimal system in the form
in one and only one way.
Proof. Let be the highest integer power of not exceeding . By the division algorithm for integers, we obtain in succession
Adding these equations yields the equation (1) with , .
For showing the uniqueness of (1) we suppose also another
with , . The equality
Since now , we infer that and thus
. Consequently, we can then infer from (2) that , whence
, and as before, . We may continue in manner and see that always
, whence also . Accordingly, the both are identical. Q.E.D.
Remark. There is the following generalisation of the theorem. — If we have an infinite sequence of integers greater than 1, then may be represented uniquely in the form
where the integers satisfy and . Cf. the factorial base.
|Title||uniqueness of digital representation|
|Date of creation||2013-03-22 18:52:16|
|Last modified on||2013-03-22 18:52:16|
|Last modified by||pahio (2872)|
|Synonym||uniqueness of decimal representation|
|Synonym||digital representation of integer|