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A numeral system (or system of numeration) is a writing system for expressing numbers; that is, a mathematical notation for representing numbers of a given set, using digits or other symbols in a consistent manner. The same sequence of symbols may represent different numbers in different numeral systems. For example, "11" represents the number ''eleven'' in the decimal numeral system (used in common life), the number ''three'' in the binary numeral system (used in computers), and the number ''two'' in the unary numeral system (e.g. used in tallying scores). The number the numeral represents is called its value. Ideally, a numeral system will: *Represent a useful set of numbers (e.g. all integers, or rational numbers) *Give every number represented a unique representation (or at least a standard representation) *Reflect the algebraic and arithmetic structure of the numbers. For example, the usual decimal representation of whole numbers gives every nonzero whole number a unique representation as a finite sequence of digits, beginning with a non-zero digit. However, when decimal representation is used for the rational or real numbers, such numbers, in general, have an infinite number of representations, for example 2.31 can also be written as 2.310, 2.3100000, 2.309999999..., etc., all of which have the same meaning except for some scientific and other contexts where greater precision is implied by a larger number of figures shown. Numeral systems are sometimes called ''number systems'', but that name is ambiguous, as it could refer to different systems of numbers, such as the system of real numbers, the system of complex numbers, the system of ''p''-adic numbers, etc. Such systems are, however, not the topic of this article.

Main numeral systems

Positional systems in detail

In a positional base ''b'' numeral system (with ''b'' a natural number greater than 1 known as the radix), ''b'' basic symbols (or digits) corresponding to the first ''b'' natural numbers including zero are used. To generate the rest of the numerals, the position of the symbol in the figure is used. The symbol in the last position has its own value, and as it moves to the left its value is multiplied by ''b''. For example, in the decimal system (base 10), the numeral 4327 means , noting that . In general, if ''b'' is the base, one writes a number in the numeral system of base ''b'' by expressing it in the form and writing the enumerated digits in descending order. The digits are natural numbers between 0 and , inclusive. If a text (such as this one) discusses multiple bases, and if ambiguity exists, the base (itself represented in base 10) is added in subscript to the right of the number, like this: numberbase. Unless specified by context, numbers without subscript are considered to be decimal. By using a dot to divide the digits into two groups, one can also write fractions in the positional system. For example, the base 2 numeral 10.11 denotes . In general, numbers in the base ''b'' system are of the form: :$\left(a_na_\cdots a_1a_0.c_1 c_2 c_3\cdots\right)_b = \sum_^n a_kb^k + \sum_^\infty c_kb^.$ The numbers ''b''''k'' and ''b''−''k'' are the weights of the corresponding digits. The position ''k'' is the logarithm of the corresponding weight ''w'', that is $k = \log_ w = \log_ b^k$. The highest used position is close to the order of magnitude of the number. The number of tally marks required in the unary numeral system for ''describing the weight'' would have been w. In the positional system, the number of digits required to describe it is only $k + 1 = \log_ w + 1$, for ''k'' ≥ 0. For example, to describe the weight 1000 then four digits are needed because $\log_ 1000 + 1 = 3 + 1$. The number of digits required to ''describe the position'' is $\log_b k + 1 = \log_b \log_b w + 1$ (in positions 1, 10, 100,... only for simplicity in the decimal example). :$\begin \text & 3 & 2 & 1 & 0 & -1 & -2 & \cdots \\ \hline \text & b^3 & b^2 & b^1 & b^0 & b^ & b^ & \cdots \\ \text & a_3 & a_2 & a_1 & a_0 & c_1 & c_2 & \cdots \\ \hline \text & 1000 & 100 & 10 & 1 & 0.1 & 0.01 & \cdots \\ \text & 4 & 3 & 2 & 7 & 0 & 0 & \cdots \end$ A number has a terminating or repeating expansion if and only if it is rational; this does not depend on the base. A number that terminates in one base may repeat in another (thus ). An irrational number stays aperiodic (with an infinite number of non-repeating digits) in all integral bases. Thus, for example in base 2, can be written as the aperiodic 11.001001000011111...2. Putting overscores, , or dots, ''ṅ'', above the common digits is a convention used to represent repeating rational expansions. Thus: :14/11 = 1.272727272727... = 1.   or   321.3217878787878... = 321.321. If ''b'' = ''p'' is a prime number, one can define base-''p'' numerals whose expansion to the left never stops; these are called the ''p''-adic numbers.

Generalized variable-length integers

More general is using a mixed radix notation (here written little-endian) like $a_0 a_1 a_2$ for $a_0 + a_1 b_1 + a_2 b_1 b_2$, etc. This is used in punycode, one aspect of which is the representation of a sequence of non-negative integers of arbitrary size in the form of a sequence without delimiters, of "digits" from a collection of 36: a–z and 0–9, representing 0–25 and 26–35 respectively. A digit lower than a threshold value marks that it is the most-significant digit, hence the end of the number. The threshold value depends on the position in the number. For example, if the threshold value for the first digit is b (i.e. 1) then a (i.e. 0) marks the end of the number (it has just one digit), so in numbers of more than one digit, range is only b–9 (1–35), therefore the weight ''b''1 is 35 instead of 36. Suppose the threshold values for the second and third digits are c (2), then the third digit has a weight 34 × 35 = 1190 and we have the following sequence: a (0), ba (1), ca (2), .., 9a (35), bb (36), cb (37), .., 9b (70), bca (71), .., 99a (1260), bcb (1261), etc. Unlike a regular based numeral system, there are numbers like 9b where 9 and b each represents 35; yet the representation is unique because ac and aca are not allowed – the a would terminate the number. The flexibility in choosing threshold values allows optimization depending on the frequency of occurrence of numbers of various sizes. The case with all threshold values equal to 1 corresponds to bijective numeration, where the zeros correspond to separators of numbers with digits which are non-zero.

* 0.999... - every nonzero terminating decimal has two equal representations

References

Sources

*Georges Ifrah. ''The Universal History of Numbers : From Prehistory to the Invention of the Computer'', Wiley, 1999. . *D. Knuth. ''The Art of Computer Programming''. Volume 2, 3rd Ed. Addison–Wesley. pp. 194–213, "Positional Number Systems". *A.L. Kroeber (Alfred Louis Kroeber) (1876–1960), Handbook of the Indians of California, Bulletin 78 of the Bureau of American Ethnology of the Smithsonian Institution (1919) *J.P. Mallory and D.Q. Adams, ''Encyclopedia of Indo-European Culture'', Fitzroy Dearborn Publishers, London and Chicago, 1997. * * *