numeration system
A numeration system is a triple $N=(b,\mathrm{\Sigma},d)$, where $b>1$ is a positive integer, $\mathrm{\Sigma}$ is a nonempty alphabet, and $d$ is a onetoone function from $\mathrm{\Sigma}$ to the set of nonnegative integers $\mathbb{N}\cup \{0\}$. Order elements of $\mathrm{\Sigma}$ so that their values are in increasing order:
$\mathrm{\Sigma}=\{{a}_{1},\mathrm{\dots},{a}_{k}\}$, where $$ for $i=1,\mathrm{\dots},k1$.
$b$ is called the base of numeration system $N$, and the elements ${a}_{1},\mathrm{\dots},{a}_{k}$ the digits of $N$. Words over $\mathrm{\Sigma}$ are called numeral words.
Given a numeral word $u={c}_{1}\mathrm{\cdots}{c}_{m}$ with ${c}_{j}\in \mathrm{\Sigma}$, the integer nonnegative $n$ is said to be represented by $u$ if
$$n={c}_{1}{b}^{m1}+\mathrm{\cdots}+{c}_{j}{b}^{mj}+\mathrm{\cdots}+{c}_{m}.$$ 
An integer $n$ is said to be representable in $N$ if there is a numeral word $u$ representing $n$.
Examples.

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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.

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Just as common is the binary digital system: $B=(2,\{0,1\},d)$ where $d$ again is the identity function.

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In fact, any digital system is a numeration system $(n,\mathrm{\Sigma},d)$, where $\mathrm{\Sigma}=\{{a}_{0},\mathrm{\dots},{a}_{n1}\}$ and $d({a}_{i})=i$.

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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 $1+2+\mathrm{\cdots}+{2}^{n1}={2}^{n1}$. We conclude that the integers representable by $N$ have the form ${2}^{n}1$ for any positive integer $n$.

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Consider the system
$${D}_{1}=(10,\{[0],[1],\mathrm{\dots},[9],[10],[100],[1000],[10000],[10000000],[10000000000]\},d)$$ 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.

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Consider the system $N=(3,\{[1],[2],[4]\},d)$ where $d([i])=i$. Then $[2][1][4][1]=2\times {3}^{3}+1\times {3}^{2}+4\times 3+1=184$. Notice that $0$ can not be represented $N$. Also, note that $[4]=4=1\times 3+1=[1][1]$.
A numeration system $N$ is said to be complete^{} if every nonnegative integer has at least one representation in $N$; and unambiguous if every nonnegative 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 nonnegative 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  20130322 18:57:46 
Last modified on  20130322 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 