 Numeral system

This article is about different methods of expressing numbers with symbols. For classifying numbers in mathematics, see number system. For how numbers are expressed using words, see number names.
Numeral systems by culture HinduArabic numerals Western Arabic (Hindu numerals)
Eastern Arabic
Indian family
TamilBurmese
Khmer
Lao
Mongolian
ThaiEast Asian numerals Chinese
Japanese
SuzhouKorean
Vietnamese
Counting rodsAlphabetic numerals Abjad
Armenian
Āryabhaṭa
CyrillicGe'ez
Greek (Ionian)
HebrewOther systems Aegean
Attic
Babylonian
Brahmi
Egyptian
Etruscan
InuitKharosthi
Mayan
Quipu
Roman
Sumerian
UrnfieldList of numeral system topics Positional systems by base Decimal (10) 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 20, 24, 30, 36, 60, 64 List of numeral systems 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 graphemes or symbols in a consistent manner. It can be seen as the context that allows the symbols "11" to be interpreted as the binary symbol for three, the decimal symbol for eleven, or a symbol for other numbers in different bases.
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 whole number a unique representation as a finite sequence of digits. 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 padic numbers, etc. Such systems are not the topic of this article.
Contents
Types of numeral systems
The most commonly used system of numerals is known as Arabic numerals or HinduArabic numerals. Two Indian mathematicians are credited with developing them. Aryabhata of Kusumapura developed the placevalue notation in the 5th century and a century later Brahmagupta introduced the symbol for zero.^{[1]}
The simplest numeral system is the unary numeral system, in which every natural number is represented by a corresponding number of symbols. If the symbol / is chosen, for example, then the number seven would be represented by ///////. Tally marks represent one such system still in common use. The unary system is only useful for small numbers, although it plays an important role in theoretical computer science. Elias gamma coding, which is commonly used in data compression, expresses arbitrarysized numbers by using unary to indicate the length of a binary numeral.
The unary notation can be abbreviated by introducing different symbols for certain new values. Very commonly, these values are powers of 10; so for instance, if / stands for one, − for ten and + for 100, then the number 304 can be compactly represented as +++ //// and the number 123 as + − − /// without any need for zero. This is called signvalue notation. The ancient Egyptian numeral system was of this type, and the Roman numeral system was a modification of this idea.
More useful still are systems which employ special abbreviations for repetitions of symbols; for example, using the first nine letters of the alphabet for these abbreviations, with A standing for "one occurrence", B "two occurrences", and so on, one could then write C+ D/ for the number 304. This system is used when writing Chinese numerals and other East Asian numerals based on Chinese. The number system of the English language is of this type ("three hundred [and] four"), as are those of other spoken languages, regardless of what written systems they have adopted. However, many languages use mixtures of bases, and other features, for instance 79 in French is soixante dixneuf (60+10+9) and in Welsh is pedwar ar bymtheg a thrigain (4+(5+10)+(3 × 20)) or (somewhat archaic) pedwar ugain namyn un (4 × 20 − 1). In English, you could say "four score less one", as in the famous Gettysburg Address representing 87 as "four score and seven years ago".
More elegant is a positional system, also known as placevalue notation. Again working in base 10, ten different digits 0, ..., 9 are used and the position of a digit is used to signify the power of ten that the digit is to be multiplied with, as in 304 = 3×100 + 0×10 + 4×1. Note that zero, which is not needed in the other systems, is of crucial importance here, in order to be able to "skip" a power. The HinduArabic numeral system, which originated in India and is now used throughout the world, is a positional base 10 system.
Arithmetic is much easier in positional systems than in the earlier additive ones; furthermore, additive systems need a large number of different symbols for the different powers of 10; a positional system needs only ten different symbols (assuming that it uses base 10).
The numerals used when writing numbers with digits or symbols can be divided into two types that might be called the arithmetic numerals 0,1,2,3,4,5,6,7,8,9 and the geometric numerals 1,10,100,1000,10000... respectively. The signvalue systems use only the geometric numerals and the positional systems use only the arithmetic numerals. The signvalue system does not need arithmetic numerals because they are made by repetition (except for the Ionic system), and the positional system does not need geometric numerals because they are made by position. However, the spoken language uses both arithmetic and geometric numerals.
In certain areas of computer science, a modified basek positional system is used, called bijective numeration, with digits 1, 2, ..., k (k ≥ 1), and zero being represented by an empty string. This establishes a bijection between the set of all such digitstrings and the set of nonnegative integers, avoiding the nonuniqueness caused by leading zeros. Bijective basek numeration is also called kadic notation, not to be confused with padic numbers. Bijective base1 is the same as unary.
Positional systems in detail
See also: Positional notationIn a positional baseb numeral system (with b a positive natural number 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 (4×10^{3}) + (3×10^{2}) + (2×10^{1}) + (7×10^{0}), noting that 10^{0} = 1.
In general, if b is the base, we write a number in the numeral system of base b by expressing it in the form a_{n}b^{n} + a_{n − 1}b^{n − 1} + a_{n − 2}b^{n − 2} + ... + a_{0}b^{0} and writing the enumerated digits a_{n}a_{n − 1}a_{n − 2} ... a_{0} in descending order. The digits are natural numbers between 0 and b − 1, 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: number_{base}. 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 base2 numeral 10.11 denotes 1×2^{1} + 0×2^{0} + 1×2^{−1} + 1×2^{−2} = 2.75.
In general, numbers in the base b system are of the form:
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 _{b}w = log _{b}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 _{b}w + 1, for . E.g. to describe the weight 1000 then four digits are needed since log _{10}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).
Position 3 2 1 0 −1 −2 . . . Weight b^{3} b^{2} b^{1} b^{0} b ^{− 1} b ^{− 2} Digit a_{3} a_{2} a_{1} a_{0} c_{1} c_{2} Decimal example weight 1000 100 10 1 0.1 0.01 . . . Decimal example digit 4 3 2 7 0 0 . . . Note that 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 0.3_{10} = 0.0100110011001..._{2}). An irrational number stays unperiodic (infinite amount of unrepeating digits) in all integral bases. Thus, for example in base 2, π = 3.1415926..._{10} can be written down as the unperiodic 11.001001000011111..._{2}.
Putting overscores, n, or dots, ṅ, above the common digits is a convention used to represent repeating rational expansions. Thus:
 14/11 = 1.272727272727... = 1.27 or 321.3217878787878... = 321.3217̇8̇.
If b = p is a prime number, one can define basep numerals whose expansion to the left never stops; these are called the padic numbers.
Generalized variablelength integers
More general is using a notation (here written littleendian) 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 nonnegative 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 mostsignificant 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 the 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 nonzero.
See also
 Babylonian numerals, a base60 system
 Computer numbering formats
 Golden ratio base
 List of numeral system topics
 Maya numerals, a base20 system
 nary
 Number names
 Quaterimaginary base
 Quipu
 Recurring decimal
 Residue number system
 Short and long scales
 Subtractive notation
 yllion
References
 ^ Hindu Arabic Numerals by David Eugene Smith Google Books)
 Georges Ifrah. The Universal History of Numbers : From Prehistory to the Invention of the Computer, Wiley, 1999. ISBN 0471375683.
 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 IndoEuropean Culture, Fitzroy Dearborn Publishers, London and Chicago, 1997.
 Hans J. Nissen, P. Damerow, R. Englund, Archaic Bookkeeping, University of Chicago Press, 1993, ISBN 0226586596.
 Denise SchmandtBesserat, How Writing Came About, University of Texas Press, 1992, ISBN 0292777043.
 Claudia Zaslavsky, Africa Counts: Number and Pattern in African Cultures, Lawrence Hill Books, 1999, ISBN 1556523505.
External links
 Numerical Mechanisms and Children's Concept of Numbers
 Software for converting from one numeral system to another
 Online conversion of fractional numbers between numeral systems
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