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 {0}. Order elements of Σ so that their values are in increasing order:

Σ={a1,,ak}, where d(ai)<d(ai+1) for i=1,,k-1.

b is called the base of numeration system N, and the elements a1,,ak the digits of N. Words over Σ are called numeral words.

Given a numeral word u=c1cm with cjΣ, the integer non-negative n is said to be represented by u if


An integer n is said to be representable in N if there is a numeral word u representing n.


  • The most common numeration system is the decimal system:


    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={a0,,an-1} and d(ai)=i.

  • Consider the system B1=(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++2n-1=2n-1. We conclude that the integers representable by N have the form 2n-1 for any positive integer n.

  • Consider the system


    where d([i])=i. It is easy to see that every integer is representable by D1. However, some integers may be represented by more than one numeral words. For example,


    The numeration system D1 is used by the Chinese.

  • Consider the system N=(3,{[1],[2],[4]},d) where d([i])=i. Then [2][1][4][1]=2×33+1×32+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 completePlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath 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, D1 is complete but ambiguous; B1 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 conceptsMathworldPlanetmath of completeness and ambiguity may be defined similarly.


  • 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