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[[File:Numeral Systems of the World.svg|264px|thumb|right|Numbers written in different numeral systems.]]
{{About|expressing numbers with symbols|different kinds of numbers|Number system|expressing numbers with words|Numeral (linguistics)}}
{{more footnotes|date=January 2011}}
{{Numeral systems}}
A '''numeral system''' (or '''system of numeration''') is a [[writing system]] for expressing numbers; that is, a [[mathematical notation]] for representing [[number]]s of a given set, using [[Numerical digit|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 [[computer]]s), and the number ''two'' in the [[unary numeral system]] (e.g. used in [[Tally marks|tallying]] scores).
The number the numeral represents is called its value. Not all number systems can represent all numbers that are considered in the modern days; for example, Roman numerals have no zero.
Ideally, a numeral system will:
*Represent a useful set of numbers (e.g. all [[integer]]s, or [[rational number]]s)
*Give every number represented a unique representation (or at least a standard representation)
*Reflect the [[algebra|algebraic]] and [[arithmetic]] structure of the numbers.
For example, the usual [[decimal representation]] gives every nonzero [[natural number]] a unique representation as a [[finite set|finite]] [[sequence]] of [[numerical digit|digits]], beginning with a non-zero digit.
Numeral systems are sometimes called ''[[number system]]s'', but that name is ambiguous, as it could refer to different systems of numbers, such as the system of [[real number]]s, the system of [[complex number]]s, the system of [[p-adic number|''p''-adic numbers]], etc. Such systems are, however, not the topic of this article.
==Main numeral systems==
{{main|List of numeral systems}}
The most commonly used system of numerals is [[decimal]]. [[Indian mathematicians]] are credited with developing the integer version, the [[Hindu–Arabic numeral system]].<ref>{{cite book |author=David Eugene Smith |author2=Louis Charles Karpinski |title=The Hindu-Arabic numerals |url=https://archive.org/details/hinduarabicnume05karpgoog |year=1911 |publisher=Ginn and Company}}</ref> [[Aryabhata]] of [[Patna|Kusumapura]] developed the [[place-value notation]] in the 5th century and a century later [[Brahmagupta]] introduced the symbol for [[zero]]. The system slowly spread to other surrounding regions like Arabia due to their commercial and military activities with India. [[Middle-East|Middle-Eastern]] mathematicians extended the system to include negative powers of 10 ([[fractions]]), as recorded in a treatise by [[Syrian]] mathematician [[Abu'l-Hasan al-Uqlidisi]] in 952–953, and the [[decimal point]] notation was introduced{{when|date=February 2021}} by [[Sind ibn Ali]], who also wrote the earliest treatise on Arabic numerals. The Hindu-Arabic numeral system then spread to Europe due to merchants trading, and the digits used in Europe are called [[Arabic numerals]], as they learned them from the Arabs.
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 {{mono|/}} is chosen, for example, then the number seven would be represented by {{mono|///////}}. [[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 arbitrary-sized 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 {{mono|+++ ////}} and the number 123 as {{mono|+ − − ///}} without any need for zero. This is called [[sign-value 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 [[language]]s, 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 dix-neuf'' ({{nowrap|60 + 10 + 9}}) and in Welsh is ''pedwar ar bymtheg a thrigain'' ({{nowrap|4 + (5 + 10) + (3 × 20)}}) or (somewhat archaic) ''pedwar ugain namyn un'' ({{nowrap|4 × 20 − 1}}). In English, one could say "four score less one", as in the famous [[Gettysburg Address]] representing "87 years ago" as "four score and seven years ago".
More elegant is a ''[[positional notation|positional system]]'', also known as place-value 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 {{nowrap|304 {{=}} 3×100 + 0×10 + 4×1}} or more precisely {{nowrap|3×10<sup>2</sup> + 0×10<sup>1</sup> + 4×10<sup>0</sup>}}. Zero, which is not needed in the other systems, is of crucial importance here, in order to be able to "skip" a power. The Hindu–Arabic 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).<ref>{{Cite book|last=Chowdhury|first=Arnab|url=https://books.google.com/books?id=WXn-mT3K6dgC&q=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).&pg=PA2|title=Design of an Efficient Multiplier using DBNS|publisher=GIAP Journals|isbn=978-93-83006-18-2|language=en}}</ref>
The positional decimal system is presently universally used in human writing. The base 1000 is also used (albeit not universally), by grouping the digits and considering a sequence of three decimal digits as a single digit. This is the meaning of the common notation 1,000,234,567 used for very large numbers.
In [[computers]], the main numeral systems are based on the positional system in base 2 ([[binary numeral system]]), with two [[binary digit]]s, 0 and 1. Positional systems obtained by grouping binary digits by three ([[octal numeral system]]) or four ([[hexadecimal numeral system]]) are commonly used. For very large integers, bases 2<sup>32</sup> or 2<sup>64</sup> (grouping binary digits by 32 or 64, the length of the [[machine word]]) are used, as, for example, in [[GNU Multiple Precision Arithmetic Library|GMP]].
In certain biological systems, the [[unary coding]] system is employed. Unary numerals used in the [[neural circuit]]s responsible for [[birdsong]] production.<ref> Fiete, I. R.; Seung, H. S. (2007). "Neural network models of birdsong production, learning, and coding". In Squire, L.; Albright, T.; Bloom, F.; Gage, F.; Spitzer, N. New Encyclopedia of Neuroscience.</ref> The nucleus in the brain of the songbirds that plays a part in both the learning and the production of bird song is the HVC ([[high vocal center]]). The command signals for different notes in the birdsong emanate from different points in the HVC. This coding works as space coding which is an efficient strategy for biological circuits due to its inherent simplicity and robustness.
The numerals used when writing numbers with digits or symbols can be divided into two types that might be called the [[Arithmetic sequence|arithmetic]] numerals (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) and the [[Geometric sequence|geometric]] numerals (1, 10, 100, 1000, 10000 ...), respectively. The sign-value systems use only the geometric numerals and the positional systems use only the arithmetic numerals. A sign-value system does not need arithmetic numerals because they are made by repetition (except for the [[Greek numerals|Ionic system]]), and a 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 base ''k'' positional system is used, called [[bijective numeration]], with digits 1, 2, ..., ''k'' ({{nowrap|''k'' ≥ 1}}), and zero being represented by an empty string. This establishes a [[bijection]] between the set of all such digit-strings and the set of non-negative integers, avoiding the non-uniqueness caused by leading zeros. Bijective base-''k'' numeration is also called ''k''-adic notation, not to be confused with [[p-adic number|''p''-adic numbers]]. Bijective base 1 is the same as unary.
==Positional systems in detail==<!--This section is linked from [[Cantor set]]-->
{{See also|Positional notation}}
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 {{math|('''4'''×10<sup>3</sup>) + ('''3'''×10<sup>2</sup>) + ('''2'''×10<sup>1</sup>) + ('''7'''×10<sup>0</sup>)}}, noting that {{math|10<sup>0</sup> {{=}} 1}}.
In general, if ''b'' is the base, one writes a number in the numeral system of base ''b'' by expressing it in the form {{math|''a''<sub>''n''</sub>''b''<sup>''n''</sup> + ''a''<sub>''n'' − 1</sub>''b''<sup>''n'' − 1</sup> + ''a''<sub>''n'' − 2</sub>''b''<sup>''n'' − 2</sup> + ... + ''a''<sub>0</sub>''b''<sup>0</sup>}} and writing the enumerated digits {{math|''a''<sub>''n''</sub>''a''<sub>''n'' − 1</sub>''a''<sub>''n'' − 2</sub> ... ''a''<sub>0</sub>}} in descending order. The digits are natural numbers between 0 and {{math|''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<sub>base</sub>. 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 {{math|1×2<sup>1</sup> + 0×2<sup>0</sup> + 1×2<sup>−1</sup> + 1×2<sup>−2</sup> {{=}} 2.75}}.
In general, numbers in the base ''b'' system are of the form:
:<math>
(a_na_{n-1}\cdots a_1a_0.c_1 c_2 c_3\cdots)_b =
\sum_{k=0}^n a_kb^k + \sum_{k=1}^\infty c_kb^{-k}.
</math>
The numbers ''b''<sup>''k''</sup> and ''b''<sup>−''k''</sup> are the [[weight function|weights]] of the corresponding digits. The position ''k'' is the [[logarithm]] of the corresponding weight ''w'', that is <math>k = \log_{b} w = \log_{b} b^k</math>. 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 <math>k + 1 = \log_{b} w + 1</math>, for ''k'' ≥ 0. For example, to describe the weight 1000 then four digits are needed because <math>\log_{10} 1000 + 1 = 3 + 1</math>. The number of digits required to ''describe the position'' is <math>\log_b k + 1 = \log_b \log_b w + 1</math> (in positions 1, 10, 100,... only for simplicity in the decimal example).
:<math>\begin{array}{l|rrrrrrr}
\text{Position}
& 3
& 2
& 1
& 0
& -1
& -2
& \cdots
\\
\hline
\text{Weight}
& b^3 & b^2 & b^1 & b^0 & b^{-1} & b^{-2} & \cdots
\\
\text{Digit}
& a_3 & a_2 & a_1 & a_0 & c_1 & c_2 & \cdots
\\
\hline
\text{Decimal example weight}
& 1000 & 100 & 10 & 1 & 0.1 & 0.01 & \cdots
\\
\text{Decimal example digit}
& 4 & 3 & 2 & 7 & 0 & 0 & \cdots
\end{array}
</math>
A number has a terminating or repeating expansion [[if and only if]] it is [[rational number|rational]]; this does not depend on the base. A number that terminates in one base may repeat in another (thus {{math|0.3<sub>10</sub> {{=}} 0.0100110011001...<sub>2</sub>}}). An irrational number stays aperiodic (with an infinite number of non-repeating digits) in all integral bases. Thus, for example in base 2, {{math|[[pi|π]] {{=}} 3.1415926...<sub>10</sub>}} can be written as the aperiodic 11.001001000011111...<sub>2</sub>.
Putting [[overline|overscores]], {{overline|''n''}}, or dots, ''ṅ'', above the common digits is a convention used to represent repeating rational expansions. Thus:
:14/11 = 1.272727272727... = 1.{{overline|27}} or 321.3217878787878... = 321.321{{Overline|78}}.
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 number|''p''-adic numbers]].
==Generalized variable-length integers==
{{main|Punycode}}
More general is using a [[mixed radix]] notation (here written [[Endianness|little-endian]]) like <math>a_0 a_1 a_2</math> for <math>a_0 + a_1 b_1 + a_2 b_1 b_2</math>, 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. There are also so-called threshold values (<math>t_0, t_1, ...</math>) which are fixed for every position in the number. A digit <math>a_i</math> (in a given position in the number) that is lower than its corresponding threshold value <math>t_i</math> means that it is the most-significant digit, hence in the string this is the end of the number, and the next symbol (if present) is the least-significant digit of the next 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, first-digit range is only b–9 (i.e. 1–35), therefore the weight ''b''<sub>1</sub> is 35 instead of 36. More generally, if ''t<sub>n</sub>'' is the threshold for the ''n''-th digit, it is easy to show that <math>b_{n+1}=36-t_n</math>.
Suppose the threshold values for the second and third digits are ''c'' (i.e. 2), then the second-digit range is a–b (i.e. 0–1) with the second digit being most significant, while the range is c–9 (i.e. 2–35) in the presence of a third digit. Generally, for any ''n'', the weight of the (''n''+1)-th digit is the weight of the previous one times (36 − threshold of the ''n''-th digit). So the weight of the second symbol is <math>36 - t_0 = 35</math>. And the weight of the third symbol is <math>35 * (36 - t_1) = 35*34 = 1190</math>.
So we have the following sequence of the numbers with at most 3 digits:
''a'' (0), ''ba'' (1), ''ca'' (2), ..., ''9a'' (35), ''bb'' (36), ''cb'' (37), ..., ''9b'' (70), ''bca'' (71), ..., ''99a'' (1260), ''bcb'' (1261), ..., ''99b'' (2450).
Unlike a regular n-based numeral system, there are numbers like ''9b'' where ''9'' and ''b'' each represent 35; yet the representation is unique because ''ac'' and ''aca'' are not allowed – the first ''a'' would terminate each of the se numbers.
The flexibility in choosing threshold values allows optimization for number of digits 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.
==See also==
{{columns-list|colwidth=22em|
*[[List of numeral systems]]
*[[Computer numbering formats]]
*[[Golden ratio base]]
*[[History of ancient numeral systems]]
*[[History of numbers]]
*[[List of numeral system topics]]
*[[N-ary (disambiguation)|''n''-ary]]
*[[Numeral (linguistics)|Number names]]
*[[Quater-imaginary base]]
*[[Quipu]]
*[[Recurring decimal]]
*[[Residue numeral system]]
*[[Long and short scales|Short and long scales]]
*[[Scientific notation]]
*[[-yllion]]
*[[Numerical cognition]]
*[[Number system]]
*[[Unary numeral system]]}}
* [[0.999...]] - every nonzero terminating decimal has two equal representations
==References==
<references/>
==Sources==
*Georges Ifrah. ''The Universal History of Numbers : From Prehistory to the Invention of the Computer'', Wiley, 1999. {{isbn|0-471-37568-3}}.
*[[Donald Knuth|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.
*{{cite book |author=Hans J. Nissen |author2=Peter Damerow |author3=Robert K. Englund |title=Archaic Bookkeeping: Early Writing and Techniques of Economic Administration in the Ancient Near East |year=1993 |publisher=[[University Of Chicago Press]] |isbn=978-0-226-58659-5}}
*{{cite book |last=Schmandt-Besserat |author-link=Denise Schmandt-Besserat |first=Denise |title=How Writing Came About |year=1996 |publisher=[[University of Texas Press]] |isbn=978-0-292-77704-0}}
*{{cite book |last=Zaslavsky |first=Claudia |title=Africa counts: number and pattern in African cultures |year=1999 |publisher=Chicago Review Press |isbn=978-1-55652-350-2}}
==External links==
{{Wiktionary|numeration|numeral}}
*{{Commonscat-inline|Numeral systems}}
{{Authority control}}
{{DEFAULTSORT:Numeral System}}
[[Category:Numeral systems| ]]
[[Category:Graphemes]]
[[Category:Mathematical notation]]
[[Category:Writing systems]]
[[te:తెలుగు]]' |
New page wikitext, after the edit (new_wikitext) | '==Main numeral systems==
{{main|List of numeral systems}}
The most commonly used system of numerals is [[decimal]]. [[Indian mathematicians]] are credited with developing the integer version, the [[Hindu–Arabic numeral system]].<ref>{{cite book |author=David Eugene Smith |author2=Louis Charles Karpinski |title=The Hindu-Arabic numerals |url=https://archive.org/details/hinduarabicnume05karpgoog |year=1911 |publisher=Ginn and Company}}</ref> [[Aryabhata]] of [[Patna|Kusumapura]] developed the [[place-value notation]] in the 5th century and a century later [[Brahmagupta]] introduced the symbol for [[zero]]. The system slowly spread to other surrounding regions like Arabia due to their commercial and military activities with India. [[Middle-East|Middle-Eastern]] mathematicians extended the system to include negative powers of 10 ([[fractions]]), as recorded in a treatise by [[Syrian]] mathematician [[Abu'l-Hasan al-Uqlidisi]] in 952–953, and the [[decimal point]] notation was introduced{{when|date=February 2021}} by [[Sind ibn Ali]], who also wrote the earliest treatise on Arabic numerals. The Hindu-Arabic numeral system then spread to Europe due to merchants trading, and the digits used in Europe are called [[Arabic numerals]], as they learned them from the Arabs.
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 {{mono|/}} is chosen, for example, then the number seven would be represented by {{mono|///////}}. [[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 arbitrary-sized 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 {{mono|+++ ////}} and the number 123 as {{mono|+ − − ///}} without any need for zero. This is called [[sign-value 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 [[language]]s, 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 dix-neuf'' ({{nowrap|60 + 10 + 9}}) and in Welsh is ''pedwar ar bymtheg a thrigain'' ({{nowrap|4 + (5 + 10) + (3 × 20)}}) or (somewhat archaic) ''pedwar ugain namyn un'' ({{nowrap|4 × 20 − 1}}). In English, one could say "four score less one", as in the famous [[Gettysburg Address]] representing "87 years ago" as "four score and seven years ago".
More elegant is a ''[[positional notation|positional system]]'', also known as place-value 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 {{nowrap|304 {{=}} 3×100 + 0×10 + 4×1}} or more precisely {{nowrap|3×10<sup>2</sup> + 0×10<sup>1</sup> + 4×10<sup>0</sup>}}. Zero, which is not needed in the other systems, is of crucial importance here, in order to be able to "skip" a power. The Hindu–Arabic 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).<ref>{{Cite book|last=Chowdhury|first=Arnab|url=https://books.google.com/books?id=WXn-mT3K6dgC&q=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).&pg=PA2|title=Design of an Efficient Multiplier using DBNS|publisher=GIAP Journals|isbn=978-93-83006-18-2|language=en}}</ref>
The positional decimal system is presently universally used in human writing. The base 1000 is also used (albeit not universally), by grouping the digits and considering a sequence of three decimal digits as a single digit. This is the meaning of the common notation 1,000,234,567 used for very large numbers.
In [[computers]], the main numeral systems are based on the positional system in base 2 ([[binary numeral system]]), with two [[binary digit]]s, 0 and 1. Positional systems obtained by grouping binary digits by three ([[octal numeral system]]) or four ([[hexadecimal numeral system]]) are commonly used. For very large integers, bases 2<sup>32</sup> or 2<sup>64</sup> (grouping binary digits by 32 or 64, the length of the [[machine word]]) are used, as, for example, in [[GNU Multiple Precision Arithmetic Library|GMP]].
In certain biological systems, the [[unary coding]] system is employed. Unary numerals used in the [[neural circuit]]s responsible for [[birdsong]] production.<ref> Fiete, I. R.; Seung, H. S. (2007). "Neural network models of birdsong production, learning, and coding". In Squire, L.; Albright, T.; Bloom, F.; Gage, F.; Spitzer, N. New Encyclopedia of Neuroscience.</ref> The nucleus in the brain of the songbirds that plays a part in both the learning and the production of bird song is the HVC ([[high vocal center]]). The command signals for different notes in the birdsong emanate from different points in the HVC. This coding works as space coding which is an efficient strategy for biological circuits due to its inherent simplicity and robustness.
The numerals used when writing numbers with digits or symbols can be divided into two types that might be called the [[Arithmetic sequence|arithmetic]] numerals (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) and the [[Geometric sequence|geometric]] numerals (1, 10, 100, 1000, 10000 ...), respectively. The sign-value systems use only the geometric numerals and the positional systems use only the arithmetic numerals. A sign-value system does not need arithmetic numerals because they are made by repetition (except for the [[Greek numerals|Ionic system]]), and a 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 base ''k'' positional system is used, called [[bijective numeration]], with digits 1, 2, ..., ''k'' ({{nowrap|''k'' ≥ 1}}), and zero being represented by an empty string. This establishes a [[bijection]] between the set of all such digit-strings and the set of non-negative integers, avoiding the non-uniqueness caused by leading zeros. Bijective base-''k'' numeration is also called ''k''-adic notation, not to be confused with [[p-adic number|''p''-adic numbers]]. Bijective base 1 is the same as unary.
==Positional systems in detail==<!--This section is linked from [[Cantor set]]-->
{{See also|Positional notation}}
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 {{math|('''4'''×10<sup>3</sup>) + ('''3'''×10<sup>2</sup>) + ('''2'''×10<sup>1</sup>) + ('''7'''×10<sup>0</sup>)}}, noting that {{math|10<sup>0</sup> {{=}} 1}}.
In general, if ''b'' is the base, one writes a number in the numeral system of base ''b'' by expressing it in the form {{math|''a''<sub>''n''</sub>''b''<sup>''n''</sup> + ''a''<sub>''n'' − 1</sub>''b''<sup>''n'' − 1</sup> + ''a''<sub>''n'' − 2</sub>''b''<sup>''n'' − 2</sup> + ... + ''a''<sub>0</sub>''b''<sup>0</sup>}} and writing the enumerated digits {{math|''a''<sub>''n''</sub>''a''<sub>''n'' − 1</sub>''a''<sub>''n'' − 2</sub> ... ''a''<sub>0</sub>}} in descending order. The digits are natural numbers between 0 and {{math|''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<sub>base</sub>. 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 {{math|1×2<sup>1</sup> + 0×2<sup>0</sup> + 1×2<sup>−1</sup> + 1×2<sup>−2</sup> {{=}} 2.75}}.
In general, numbers in the base ''b'' system are of the form:
:<math>
(a_na_{n-1}\cdots a_1a_0.c_1 c_2 c_3\cdots)_b =
\sum_{k=0}^n a_kb^k + \sum_{k=1}^\infty c_kb^{-k}.
</math>
The numbers ''b''<sup>''k''</sup> and ''b''<sup>−''k''</sup> are the [[weight function|weights]] of the corresponding digits. The position ''k'' is the [[logarithm]] of the corresponding weight ''w'', that is <math>k = \log_{b} w = \log_{b} b^k</math>. 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 <math>k + 1 = \log_{b} w + 1</math>, for ''k'' ≥ 0. For example, to describe the weight 1000 then four digits are needed because <math>\log_{10} 1000 + 1 = 3 + 1</math>. The number of digits required to ''describe the position'' is <math>\log_b k + 1 = \log_b \log_b w + 1</math> (in positions 1, 10, 100,... only for simplicity in the decimal example).
:<math>\begin{array}{l|rrrrrrr}
\text{Position}
& 3
& 2
& 1
& 0
& -1
& -2
& \cdots
\\
\hline
\text{Weight}
& b^3 & b^2 & b^1 & b^0 & b^{-1} & b^{-2} & \cdots
\\
\text{Digit}
& a_3 & a_2 & a_1 & a_0 & c_1 & c_2 & \cdots
\\
\hline
\text{Decimal example weight}
& 1000 & 100 & 10 & 1 & 0.1 & 0.01 & \cdots
\\
\text{Decimal example digit}
& 4 & 3 & 2 & 7 & 0 & 0 & \cdots
\end{array}
</math>
A number has a terminating or repeating expansion [[if and only if]] it is [[rational number|rational]]; this does not depend on the base. A number that terminates in one base may repeat in another (thus {{math|0.3<sub>10</sub> {{=}} 0.0100110011001...<sub>2</sub>}}). An irrational number stays aperiodic (with an infinite number of non-repeating digits) in all integral bases. Thus, for example in base 2, {{math|[[pi|π]] {{=}} 3.1415926...<sub>10</sub>}} can be written as the aperiodic 11.001001000011111...<sub>2</sub>.
Putting [[overline|overscores]], {{overline|''n''}}, or dots, ''ṅ'', above the common digits is a convention used to represent repeating rational expansions. Thus:
:14/11 = 1.272727272727... = 1.{{overline|27}} or 321.3217878787878... = 321.321{{Overline|78}}.
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 number|''p''-adic numbers]].
==Generalized variable-length integers==
{{main|Punycode}}
More general is using a [[mixed radix]] notation (here written [[Endianness|little-endian]]) like <math>a_0 a_1 a_2</math> for <math>a_0 + a_1 b_1 + a_2 b_1 b_2</math>, 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. There are also so-called threshold values (<math>t_0, t_1, ...</math>) which are fixed for every position in the number. A digit <math>a_i</math> (in a given position in the number) that is lower than its corresponding threshold value <math>t_i</math> means that it is the most-significant digit, hence in the string this is the end of the number, and the next symbol (if present) is the least-significant digit of the next 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, first-digit range is only b–9 (i.e. 1–35), therefore the weight ''b''<sub>1</sub> is 35 instead of 36. More generally, if ''t<sub>n</sub>'' is the threshold for the ''n''-th digit, it is easy to show that <math>b_{n+1}=36-t_n</math>.
Suppose the threshold values for the second and third digits are ''c'' (i.e. 2), then the second-digit range is a–b (i.e. 0–1) with the second digit being most significant, while the range is c–9 (i.e. 2–35) in the presence of a third digit. Generally, for any ''n'', the weight of the (''n''+1)-th digit is the weight of the previous one times (36 − threshold of the ''n''-th digit). So the weight of the second symbol is <math>36 - t_0 = 35</math>. And the weight of the third symbol is <math>35 * (36 - t_1) = 35*34 = 1190</math>.
So we have the following sequence of the numbers with at most 3 digits:
''a'' (0), ''ba'' (1), ''ca'' (2), ..., ''9a'' (35), ''bb'' (36), ''cb'' (37), ..., ''9b'' (70), ''bca'' (71), ..., ''99a'' (1260), ''bcb'' (1261), ..., ''99b'' (2450).
Unlike a regular n-based numeral system, there are numbers like ''9b'' where ''9'' and ''b'' each represent 35; yet the representation is unique because ''ac'' and ''aca'' are not allowed – the first ''a'' would terminate each of the se numbers.
The flexibility in choosing threshold values allows optimization for number of digits 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.
==See also==
{{columns-list|colwidth=22em|
*[[List of numeral systems]]
*[[Computer numbering formats]]
*[[Golden ratio base]]
*[[History of ancient numeral systems]]
*[[History of numbers]]
*[[List of numeral system topics]]
*[[N-ary (disambiguation)|''n''-ary]]
*[[Numeral (linguistics)|Number names]]
*[[Quater-imaginary base]]
*[[Quipu]]
*[[Recurring decimal]]
*[[Residue numeral system]]
*[[Long and short scales|Short and long scales]]
*[[Scientific notation]]
*[[-yllion]]
*[[Numerical cognition]]
*[[Number system]]
*[[Unary numeral system]]}}
* [[0.999...]] - every nonzero terminating decimal has two equal representations
==References==
<references/>
==Sources==
*Georges Ifrah. ''The Universal History of Numbers : From Prehistory to the Invention of the Computer'', Wiley, 1999. {{isbn|0-471-37568-3}}.
*[[Donald Knuth|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.
*{{cite book |author=Hans J. Nissen |author2=Peter Damerow |author3=Robert K. Englund |title=Archaic Bookkeeping: Early Writing and Techniques of Economic Administration in the Ancient Near East |year=1993 |publisher=[[University Of Chicago Press]] |isbn=978-0-226-58659-5}}
*{{cite book |last=Schmandt-Besserat |author-link=Denise Schmandt-Besserat |first=Denise |title=How Writing Came About |year=1996 |publisher=[[University of Texas Press]] |isbn=978-0-292-77704-0}}
*{{cite book |last=Zaslavsky |first=Claudia |title=Africa counts: number and pattern in African cultures |year=1999 |publisher=Chicago Review Press |isbn=978-1-55652-350-2}}
==External links==
{{Wiktionary|numeration|numeral}}
*{{Commonscat-inline|Numeral systems}}
{{Authority control}}
{{DEFAULTSORT:Numeral System}}
[[Category:Numeral systems| ]]
[[Category:Graphemes]]
[[Category:Mathematical notation]]
[[Category:Writing systems]]
[[te:తెలుగు]]' |
Unified diff of changes made by edit (edit_diff) | '@@ -1,23 +1,2 @@
-{{short description|Notation for expressing numbers}}
-[[File:Numeral Systems of the World.svg|264px|thumb|right|Numbers written in different numeral systems.]]
-{{About|expressing numbers with symbols|different kinds of numbers|Number system|expressing numbers with words|Numeral (linguistics)}}
-{{more footnotes|date=January 2011}}
-{{Numeral systems}}
-
-A '''numeral system''' (or '''system of numeration''') is a [[writing system]] for expressing numbers; that is, a [[mathematical notation]] for representing [[number]]s of a given set, using [[Numerical digit|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 [[computer]]s), and the number ''two'' in the [[unary numeral system]] (e.g. used in [[Tally marks|tallying]] scores).
-
-The number the numeral represents is called its value. Not all number systems can represent all numbers that are considered in the modern days; for example, Roman numerals have no zero.
-
-Ideally, a numeral system will:
-*Represent a useful set of numbers (e.g. all [[integer]]s, or [[rational number]]s)
-*Give every number represented a unique representation (or at least a standard representation)
-*Reflect the [[algebra|algebraic]] and [[arithmetic]] structure of the numbers.
-
-For example, the usual [[decimal representation]] gives every nonzero [[natural number]] a unique representation as a [[finite set|finite]] [[sequence]] of [[numerical digit|digits]], beginning with a non-zero digit.
-
-Numeral systems are sometimes called ''[[number system]]s'', but that name is ambiguous, as it could refer to different systems of numbers, such as the system of [[real number]]s, the system of [[complex number]]s, the system of [[p-adic number|''p''-adic numbers]], etc. Such systems are, however, not the topic of this article.
-
==Main numeral systems==
{{main|List of numeral systems}}
' |
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6 => 'A '''numeral system''' (or '''system of numeration''') is a [[writing system]] for expressing numbers; that is, a [[mathematical notation]] for representing [[number]]s of a given set, using [[Numerical digit|digits]] or other symbols in a consistent manner.',
7 => '',
8 => '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 [[computer]]s), and the number ''two'' in the [[unary numeral system]] (e.g. used in [[Tally marks|tallying]] scores).',
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18 => '',
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Parsed HTML source of the new revision (new_html) | '<div class="mw-parser-output"><div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none" /><div class="toctitle" lang="en" dir="ltr"><h2 id="mw-toc-heading">Contents</h2><span class="toctogglespan"><label class="toctogglelabel" for="toctogglecheckbox"></label></span></div>
<ul>
<li class="toclevel-1 tocsection-1"><a href="#Main_numeral_systems"><span class="tocnumber">1</span> <span class="toctext">Main numeral systems</span></a></li>
<li class="toclevel-1 tocsection-2"><a href="#Positional_systems_in_detail"><span class="tocnumber">2</span> <span class="toctext">Positional systems in detail</span></a></li>
<li class="toclevel-1 tocsection-3"><a href="#Generalized_variable-length_integers"><span class="tocnumber">3</span> <span class="toctext">Generalized variable-length integers</span></a></li>
<li class="toclevel-1 tocsection-4"><a href="#See_also"><span class="tocnumber">4</span> <span class="toctext">See also</span></a></li>
<li class="toclevel-1 tocsection-5"><a href="#References"><span class="tocnumber">5</span> <span class="toctext">References</span></a></li>
<li class="toclevel-1 tocsection-6"><a href="#Sources"><span class="tocnumber">6</span> <span class="toctext">Sources</span></a></li>
<li class="toclevel-1 tocsection-7"><a href="#External_links"><span class="tocnumber">7</span> <span class="toctext">External links</span></a></li>
</ul>
</div>
<h2><span class="mw-headline" id="Main_numeral_systems">Main numeral systems</span></h2>
<style data-mw-deduplicate="TemplateStyles:r1033289096">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/List_of_numeral_systems" title="List of numeral systems">List of numeral systems</a></div>
<p>The most commonly used system of numerals is <a href="/wiki/Decimal" title="Decimal">decimal</a>. <a href="/wiki/Indian_mathematicians" class="mw-redirect" title="Indian mathematicians">Indian mathematicians</a> are credited with developing the integer version, the <a href="/wiki/Hindu%E2%80%93Arabic_numeral_system" title="Hindu–Arabic numeral system">Hindu–Arabic numeral system</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1">[1]</a></sup> <a href="/wiki/Aryabhata" title="Aryabhata">Aryabhata</a> of <a href="/wiki/Patna" title="Patna">Kusumapura</a> developed the <a href="/wiki/Place-value_notation" class="mw-redirect" title="Place-value notation">place-value notation</a> in the 5th century and a century later <a href="/wiki/Brahmagupta" title="Brahmagupta">Brahmagupta</a> introduced the symbol for <a href="/wiki/Zero" class="mw-redirect" title="Zero">zero</a>. The system slowly spread to other surrounding regions like Arabia due to their commercial and military activities with India. <a href="/wiki/Middle-East" class="mw-redirect" title="Middle-East">Middle-Eastern</a> mathematicians extended the system to include negative powers of 10 (<a href="/wiki/Fractions" class="mw-redirect" title="Fractions">fractions</a>), as recorded in a treatise by <a href="/wiki/Syrian" class="mw-redirect" title="Syrian">Syrian</a> mathematician <a href="/wiki/Abu%27l-Hasan_al-Uqlidisi" title="Abu'l-Hasan al-Uqlidisi">Abu'l-Hasan al-Uqlidisi</a> in 952–953, and the <a href="/wiki/Decimal_point" class="mw-redirect" title="Decimal point">decimal point</a> notation was introduced<sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Manual_of_Style/Dates_and_numbers#Chronological_items" title="Wikipedia:Manual of Style/Dates and numbers"><span title="The time period mentioned near this tag is ambiguous. (February 2021)">when?</span></a></i>]</sup> by <a href="/wiki/Sind_ibn_Ali" class="mw-redirect" title="Sind ibn Ali">Sind ibn Ali</a>, who also wrote the earliest treatise on Arabic numerals. The Hindu-Arabic numeral system then spread to Europe due to merchants trading, and the digits used in Europe are called <a href="/wiki/Arabic_numerals" title="Arabic numerals">Arabic numerals</a>, as they learned them from the Arabs.
</p><p>The simplest numeral system is the <a href="/wiki/Unary_numeral_system" title="Unary numeral system">unary numeral system</a>, in which every <a href="/wiki/Natural_number" title="Natural number">natural number</a> is represented by a corresponding number of symbols. If the symbol <style data-mw-deduplicate="TemplateStyles:r886049734">.mw-parser-output .monospaced{font-family:monospace,monospace}</style><span class="monospaced">/</span> is chosen, for example, then the number seven would be represented by <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734"/><span class="monospaced">///////</span>. <a href="/wiki/Tally_marks" title="Tally marks">Tally marks</a> represent one such system still in common use. The unary system is only useful for small numbers, although it plays an important role in <a href="/wiki/Theoretical_computer_science" title="Theoretical computer science">theoretical computer science</a>. <a href="/wiki/Elias_gamma_coding" title="Elias gamma coding">Elias gamma coding</a>, which is commonly used in <a href="/wiki/Data_compression" title="Data compression">data compression</a>, expresses arbitrary-sized numbers by using unary to indicate the length of a binary numeral.
</p><p>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 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734"/><span class="monospaced">+++ ////</span> and the number 123 as <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734"/><span class="monospaced">+ − − ///</span> without any need for zero. This is called <a href="/wiki/Sign-value_notation" title="Sign-value notation">sign-value notation</a>. The ancient <a href="/wiki/Egyptian_numeral_system" class="mw-redirect" title="Egyptian numeral system">Egyptian numeral system</a> was of this type, and the <a href="/wiki/Roman_numeral_system" class="mw-redirect" title="Roman numeral system">Roman numeral system</a> was a modification of this idea.
</p><p>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 <a href="/wiki/Chinese_numerals" title="Chinese numerals">Chinese numerals</a> and other East Asian numerals based on Chinese. The number system of the <a href="/wiki/English_language" title="English language">English language</a> is of this type ("three hundred [and] four"), as are those of other spoken <a href="/wiki/Language" title="Language">languages</a>, regardless of what written systems they have adopted. However, many languages use mixtures of bases, and other features, for instance 79 in French is <i>soixante dix-neuf</i> (<span class="nowrap">60 + 10 + 9</span>) and in Welsh is <i>pedwar ar bymtheg a thrigain</i> (<span class="nowrap">4 + (5 + 10) + (3 × 20)</span>) or (somewhat archaic) <i>pedwar ugain namyn un</i> (<span class="nowrap">4 × 20 − 1</span>). In English, one could say "four score less one", as in the famous <a href="/wiki/Gettysburg_Address" title="Gettysburg Address">Gettysburg Address</a> representing "87 years ago" as "four score and seven years ago".
</p><p>More elegant is a <i><a href="/wiki/Positional_notation" title="Positional notation">positional system</a></i>, also known as place-value 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 <span class="nowrap">304 = 3×100 + 0×10 + 4×1</span> or more precisely <span class="nowrap">3×10<sup>2</sup> + 0×10<sup>1</sup> + 4×10<sup>0</sup></span>. Zero, which is not needed in the other systems, is of crucial importance here, in order to be able to "skip" a power. The Hindu–Arabic numeral system, which originated in India and is now used throughout the world, is a positional base 10 system.
</p><p>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).<sup id="cite_ref-2" class="reference"><a href="#cite_note-2">[2]</a></sup>
</p><p>The positional decimal system is presently universally used in human writing. The base 1000 is also used (albeit not universally), by grouping the digits and considering a sequence of three decimal digits as a single digit. This is the meaning of the common notation 1,000,234,567 used for very large numbers.
</p><p>In <a href="/wiki/Computers" class="mw-redirect" title="Computers">computers</a>, the main numeral systems are based on the positional system in base 2 (<a href="/wiki/Binary_numeral_system" class="mw-redirect" title="Binary numeral system">binary numeral system</a>), with two <a href="/wiki/Binary_digit" class="mw-redirect" title="Binary digit">binary digits</a>, 0 and 1. Positional systems obtained by grouping binary digits by three (<a href="/wiki/Octal_numeral_system" class="mw-redirect" title="Octal numeral system">octal numeral system</a>) or four (<a href="/wiki/Hexadecimal_numeral_system" class="mw-redirect" title="Hexadecimal numeral system">hexadecimal numeral system</a>) are commonly used. For very large integers, bases 2<sup>32</sup> or 2<sup>64</sup> (grouping binary digits by 32 or 64, the length of the <a href="/wiki/Machine_word" class="mw-redirect" title="Machine word">machine word</a>) are used, as, for example, in <a href="/wiki/GNU_Multiple_Precision_Arithmetic_Library" title="GNU Multiple Precision Arithmetic Library">GMP</a>.
</p><p>In certain biological systems, the <a href="/wiki/Unary_coding" title="Unary coding">unary coding</a> system is employed. Unary numerals used in the <a href="/wiki/Neural_circuit" title="Neural circuit">neural circuits</a> responsible for <a href="/wiki/Birdsong" class="mw-redirect" title="Birdsong">birdsong</a> production.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3">[3]</a></sup> The nucleus in the brain of the songbirds that plays a part in both the learning and the production of bird song is the HVC (<a href="/wiki/High_vocal_center" class="mw-redirect" title="High vocal center">high vocal center</a>). The command signals for different notes in the birdsong emanate from different points in the HVC. This coding works as space coding which is an efficient strategy for biological circuits due to its inherent simplicity and robustness.
</p><p>The numerals used when writing numbers with digits or symbols can be divided into two types that might be called the <a href="/wiki/Arithmetic_sequence" class="mw-redirect" title="Arithmetic sequence">arithmetic</a> numerals (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) and the <a href="/wiki/Geometric_sequence" class="mw-redirect" title="Geometric sequence">geometric</a> numerals (1, 10, 100, 1000, 10000 ...), respectively. The sign-value systems use only the geometric numerals and the positional systems use only the arithmetic numerals. A sign-value system does not need arithmetic numerals because they are made by repetition (except for the <a href="/wiki/Greek_numerals" title="Greek numerals">Ionic system</a>), and a positional system does not need geometric numerals because they are made by position. However, the spoken language uses <i>both</i> arithmetic and geometric numerals.
</p><p>In certain areas of computer science, a modified base <i>k</i> positional system is used, called <a href="/wiki/Bijective_numeration" title="Bijective numeration">bijective numeration</a>, with digits 1, 2, ..., <i>k</i> (<span class="nowrap"><i>k</i> ≥ 1</span>), and zero being represented by an empty string. This establishes a <a href="/wiki/Bijection" title="Bijection">bijection</a> between the set of all such digit-strings and the set of non-negative integers, avoiding the non-uniqueness caused by leading zeros. Bijective base-<i>k</i> numeration is also called <i>k</i>-adic notation, not to be confused with <a href="/wiki/P-adic_number" title="P-adic number"><i>p</i>-adic numbers</a>. Bijective base 1 is the same as unary.
</p>
<h2><span class="mw-headline" id="Positional_systems_in_detail">Positional systems in detail</span></h2>
<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1033289096"/><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Positional_notation" title="Positional notation">Positional notation</a></div>
<p>In a positional base <i>b</i> numeral system (with <i>b</i> a <a href="/wiki/Natural_number" title="Natural number">natural number</a> greater than 1 known as the <a href="/wiki/Radix" title="Radix">radix</a>), <i>b</i> basic symbols (or digits) corresponding to the first <i>b</i> 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 <i>b</i>.
</p><p>For example, in the <a href="/wiki/Decimal" title="Decimal">decimal</a> system (base 10), the numeral 4327 means <span class="texhtml">(<b>4</b>×10<sup>3</sup>) + (<b>3</b>×10<sup>2</sup>) + (<b>2</b>×10<sup>1</sup>) + (<b>7</b>×10<sup>0</sup>)</span>, noting that <span class="texhtml">10<sup>0</sup> = 1</span>.
</p><p>In general, if <i>b</i> is the base, one writes a number in the numeral system of base <i>b</i> by expressing it in the form <span class="texhtml"><i>a</i><sub><i>n</i></sub><i>b</i><sup><i>n</i></sup> + <i>a</i><sub><i>n</i> − 1</sub><i>b</i><sup><i>n</i> − 1</sup> + <i>a</i><sub><i>n</i> − 2</sub><i>b</i><sup><i>n</i> − 2</sup> + ... + <i>a</i><sub>0</sub><i>b</i><sup>0</sup></span> and writing the enumerated digits <span class="texhtml"><i>a</i><sub><i>n</i></sub><i>a</i><sub><i>n</i> − 1</sub><i>a</i><sub><i>n</i> − 2</sub> ... <i>a</i><sub>0</sub></span> in descending order. The digits are natural numbers between 0 and <span class="texhtml"><i>b</i> − 1</span>, inclusive.
</p><p>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<sub>base</sub>. Unless specified by context, numbers without subscript are considered to be decimal.
</p><p>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 <span class="texhtml">1×2<sup>1</sup> + 0×2<sup>0</sup> + 1×2<sup>−1</sup> + 1×2<sup>−2</sup> = 2.75</span>.
</p><p>In general, numbers in the base <i>b</i> system are of the form:
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a_{n}a_{n-1}\cdots a_{1}a_{0}.c_{1}c_{2}c_{3}\cdots )_{b}=\sum _{k=0}^{n}a_{k}b^{k}+\sum _{k=1}^{\infty }c_{k}b^{-k}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msub>
<mo>⋯<!-- ⋯ --></mo>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo>.</mo>
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msub>
<mo>⋯<!-- ⋯ --></mo>
<msub>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>b</mi>
</mrow>
</msub>
<mo>=</mo>
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
<mo>=</mo>
<mn>0</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</munderover>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<msup>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msup>
<mo>+</mo>
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">∞<!-- ∞ --></mi>
</mrow>
</munderover>
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<msup>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mi>k</mi>
</mrow>
</msup>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (a_{n}a_{n-1}\cdots a_{1}a_{0}.c_{1}c_{2}c_{3}\cdots )_{b}=\sum _{k=0}^{n}a_{k}b^{k}+\sum _{k=1}^{\infty }c_{k}b^{-k}.}</annotation>
</semantics>
</math></span><img src="/media/api/rest_v1/media/math/render/svg/ed875ba981decb322a05335f7efdb5490244d67f" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -3.171ex; width:52.072ex; height:7.009ex;" alt="(a_{n}a_{n-1}\cdots a_{1}a_{0}.c_{1}c_{2}c_{3}\cdots )_{b}=\sum _{k=0}^{n}a_{k}b^{k}+\sum _{k=1}^{\infty }c_{k}b^{-k}."/></span></dd></dl>
<p>The numbers <i>b</i><sup><i>k</i></sup> and <i>b</i><sup>−<i>k</i></sup> are the <a href="/wiki/Weight_function" title="Weight function">weights</a> of the corresponding digits. The position <i>k</i> is the <a href="/wiki/Logarithm" title="Logarithm">logarithm</a> of the corresponding weight <i>w</i>, that is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=\log _{b}w=\log _{b}b^{k}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>k</mi>
<mo>=</mo>
<msub>
<mi>log</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>b</mi>
</mrow>
</msub>
<mo>⁡<!-- --></mo>
<mi>w</mi>
<mo>=</mo>
<msub>
<mi>log</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>b</mi>
</mrow>
</msub>
<mo>⁡<!-- --></mo>
<msup>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle k=\log _{b}w=\log _{b}b^{k}}</annotation>
</semantics>
</math></span><img src="/media/api/rest_v1/media/math/render/svg/c2fe6f9d7c4f8a8275a0c41afb2bfd406150d4db" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:19.752ex; height:3.176ex;" alt="k=\log _{b}w=\log _{b}b^{k}"/></span>. The highest used position is close to the <a href="/wiki/Order_of_magnitude" title="Order of magnitude">order of magnitude</a> of the number.
</p><p>The number of <a href="/wiki/Tally_marks" title="Tally marks">tally marks</a> required in the <a href="/wiki/Unary_numeral_system" title="Unary numeral system">unary numeral system</a> for <i>describing the weight</i> would have been <b>w</b>. In the positional system, the number of digits required to describe it is only <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k+1=\log _{b}w+1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
<mo>=</mo>
<msub>
<mi>log</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>b</mi>
</mrow>
</msub>
<mo>⁡<!-- --></mo>
<mi>w</mi>
<mo>+</mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle k+1=\log _{b}w+1}</annotation>
</semantics>
</math></span><img src="/media/api/rest_v1/media/math/render/svg/ffa94ae36727c9da610fe1b54cc4f5ea559d77aa" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:18.276ex; height:2.676ex;" alt="{\displaystyle k+1=\log _{b}w+1}"/></span>, for <i>k</i> ≥ 0. For example, to describe the weight 1000 then four digits are needed because <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log _{10}1000+1=3+1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>log</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>10</mn>
</mrow>
</msub>
<mo>⁡<!-- --></mo>
<mn>1000</mn>
<mo>+</mo>
<mn>1</mn>
<mo>=</mo>
<mn>3</mn>
<mo>+</mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \log _{10}1000+1=3+1}</annotation>
</semantics>
</math></span><img src="/media/api/rest_v1/media/math/render/svg/9276fbacb3b57e34c84dd39e5c725d1ce6e554f5" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:22.152ex; height:2.676ex;" alt="\log _{10}1000+1=3+1"/></span>. The number of digits required to <i>describe the position</i> is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log _{b}k+1=\log _{b}\log _{b}w+1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>log</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>b</mi>
</mrow>
</msub>
<mo>⁡<!-- --></mo>
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
<mo>=</mo>
<msub>
<mi>log</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>b</mi>
</mrow>
</msub>
<mo>⁡<!-- --></mo>
<msub>
<mi>log</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>b</mi>
</mrow>
</msub>
<mo>⁡<!-- --></mo>
<mi>w</mi>
<mo>+</mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \log _{b}k+1=\log _{b}\log _{b}w+1}</annotation>
</semantics>
</math></span><img src="/media/api/rest_v1/media/math/render/svg/e1772ff8f82c45f3dc17e901a92d6c80d5d80985" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:26.869ex; height:2.676ex;" alt="\log _{b}k+1=\log _{b}\log _{b}w+1"/></span> (in positions 1, 10, 100,... only for simplicity in the decimal example).
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{l|rrrrrrr}{\text{Position}}&3&2&1&0&-1&-2&\cdots \\\hline {\text{Weight}}&b^{3}&b^{2}&b^{1}&b^{0}&b^{-1}&b^{-2}&\cdots \\{\text{Digit}}&a_{3}&a_{2}&a_{1}&a_{0}&c_{1}&c_{2}&\cdots \\\hline {\text{Decimal example weight}}&1000&100&10&1&0.1&0.01&\cdots \\{\text{Decimal example digit}}&4&3&2&7&0&0&\cdots \end{array}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mtable columnalign="left right right right right right right right" rowspacing="4pt" columnspacing="1em" rowlines="solid none solid none" columnlines="solid none none none none none none">
<mtr>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mtext>Position</mtext>
</mrow>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mtd>
<mtd>
<mo>−<!-- − --></mo>
<mn>2</mn>
</mtd>
<mtd>
<mo>⋯<!-- ⋯ --></mo>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mtext>Weight</mtext>
</mrow>
</mtd>
<mtd>
<msup>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msup>
</mtd>
<mtd>
<msup>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mtd>
<mtd>
<msup>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msup>
</mtd>
<mtd>
<msup>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msup>
</mtd>
<mtd>
<msup>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
</mtd>
<mtd>
<msup>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>2</mn>
</mrow>
</msup>
</mtd>
<mtd>
<mo>⋯<!-- ⋯ --></mo>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mtext>Digit</mtext>
</mrow>
</mtd>
<mtd>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msub>
</mtd>
<mtd>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mtd>
<mtd>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
</mtd>
<mtd>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mtd>
<mtd>
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
</mtd>
<mtd>
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mtd>
<mtd>
<mo>⋯<!-- ⋯ --></mo>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mtext>Decimal example weight</mtext>
</mrow>
</mtd>
<mtd>
<mn>1000</mn>
</mtd>
<mtd>
<mn>100</mn>
</mtd>
<mtd>
<mn>10</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>0.1</mn>
</mtd>
<mtd>
<mn>0.01</mn>
</mtd>
<mtd>
<mo>⋯<!-- ⋯ --></mo>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mtext>Decimal example digit</mtext>
</mrow>
</mtd>
<mtd>
<mn>4</mn>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>7</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mo>⋯<!-- ⋯ --></mo>
</mtd>
</mtr>
</mtable>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\begin{array}{l|rrrrrrr}{\text{Position}}&3&2&1&0&-1&-2&\cdots \\\hline {\text{Weight}}&b^{3}&b^{2}&b^{1}&b^{0}&b^{-1}&b^{-2}&\cdots \\{\text{Digit}}&a_{3}&a_{2}&a_{1}&a_{0}&c_{1}&c_{2}&\cdots \\\hline {\text{Decimal example weight}}&1000&100&10&1&0.1&0.01&\cdots \\{\text{Decimal example digit}}&4&3&2&7&0&0&\cdots \end{array}}}</annotation>
</semantics>
</math></span><img src="/media/api/rest_v1/media/math/render/svg/6c26c9b64098769a9ffd3549c273a753b6a922cb" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -7.838ex; width:66.242ex; height:16.843ex;" alt="{\displaystyle {\begin{array}{l|rrrrrrr}{\text{Position}}&3&2&1&0&-1&-2&\cdots \\\hline {\text{Weight}}&b^{3}&b^{2}&b^{1}&b^{0}&b^{-1}&b^{-2}&\cdots \\{\text{Digit}}&a_{3}&a_{2}&a_{1}&a_{0}&c_{1}&c_{2}&\cdots \\\hline {\text{Decimal example weight}}&1000&100&10&1&0.1&0.01&\cdots \\{\text{Decimal example digit}}&4&3&2&7&0&0&\cdots \end{array}}}"/></span></dd></dl>
<p>A number has a terminating or repeating expansion <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> it is <a href="/wiki/Rational_number" title="Rational number">rational</a>; this does not depend on the base. A number that terminates in one base may repeat in another (thus <span class="texhtml">0.3<sub>10</sub> = 0.0100110011001...<sub>2</sub></span>). An irrational number stays aperiodic (with an infinite number of non-repeating digits) in all integral bases. Thus, for example in base 2, <span class="texhtml"><a href="/wiki/Pi" title="Pi">π</a> = 3.1415926...<sub>10</sub></span> can be written as the aperiodic 11.001001000011111...<sub>2</sub>.
</p><p>Putting <a href="/wiki/Overline" title="Overline">overscores</a>, <span style="text-decoration:overline;"><i>n</i></span>, or dots, <i>ṅ</i>, above the common digits is a convention used to represent repeating rational expansions. Thus:
</p>
<dl><dd>14/11 = 1.272727272727... = 1.<span style="text-decoration:overline;">27</span>   or   321.3217878787878... = 321.321<span style="text-decoration:overline;">78</span>.</dd></dl>
<p>If <i>b</i> = <i>p</i> is a <a href="/wiki/Prime_number" title="Prime number">prime number</a>, one can define base-<i>p</i> numerals whose expansion to the left never stops; these are called the <a href="/wiki/P-adic_number" title="P-adic number"><i>p</i>-adic numbers</a>.
</p>
<h2><span class="mw-headline" id="Generalized_variable-length_integers">Generalized variable-length integers</span></h2>
<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1033289096"/><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Punycode" title="Punycode">Punycode</a></div>
<p>More general is using a <a href="/wiki/Mixed_radix" title="Mixed radix">mixed radix</a> notation (here written <a href="/wiki/Endianness" title="Endianness">little-endian</a>) like <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{0}a_{1}a_{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a_{0}a_{1}a_{2}}</annotation>
</semantics>
</math></span><img src="/media/api/rest_v1/media/math/render/svg/594f4bccf62e3e1214c2720d00534ba007e5f03a" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:6.852ex; height:2.009ex;" alt="a_{0}a_{1}a_{2}"/></span> for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{0}+a_{1}b_{1}+a_{2}b_{1}b_{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<msub>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<msub>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<msub>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a_{0}+a_{1}b_{1}+a_{2}b_{1}b_{2}}</annotation>
</semantics>
</math></span><img src="/media/api/rest_v1/media/math/render/svg/c5eb27e8a6336e82759411076ccfacfd5bdc040a" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:18.688ex; height:2.509ex;" alt="a_{0}+a_{1}b_{1}+a_{2}b_{1}b_{2}"/></span>, etc.
</p><p>This is used in <a href="/wiki/Punycode" title="Punycode">Punycode</a>, 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. There are also so-called threshold values (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{0},t_{1},...}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>.</mo>
<mo>.</mo>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle t_{0},t_{1},...}</annotation>
</semantics>
</math></span><img src="/media/api/rest_v1/media/math/render/svg/86b1b1e6ff737e8616a99c3953dcfc6314313937" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:8.57ex; height:2.343ex;" alt="{\displaystyle t_{0},t_{1},...}"/></span>) which are fixed for every position in the number. A digit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{i}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a_{i}}</annotation>
</semantics>
</math></span><img src="/media/api/rest_v1/media/math/render/svg/0bc77764b2e74e64a63341054fa90f3e07db275f" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:2.029ex; height:2.009ex;" alt="a_{i}"/></span> (in a given position in the number) that is lower than its corresponding threshold value <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{i}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle t_{i}}</annotation>
</semantics>
</math></span><img src="/media/api/rest_v1/media/math/render/svg/8b61e3d4d909be4a19c9a554a301684232f59e5a" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:1.639ex; height:2.343ex;" alt="t_{i}"/></span> means that it is the most-significant digit, hence in the string this is the end of the number, and the next symbol (if present) is the least-significant digit of the next number.
</p><p>For example, if the threshold value for the first digit is <i>b</i> (i.e. 1) then <i>a</i> (i.e. 0) marks the end of the number (it has just one digit), so in numbers of more than one digit, first-digit range is only b–9 (i.e. 1–35), therefore the weight <i>b</i><sub>1</sub> is 35 instead of 36. More generally, if <i>t<sub>n</sub></i> is the threshold for the <i>n</i>-th digit, it is easy to show that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{n+1}=36-t_{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msub>
<mo>=</mo>
<mn>36</mn>
<mo>−<!-- − --></mo>
<msub>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle b_{n+1}=36-t_{n}}</annotation>
</semantics>
</math></span><img src="/media/api/rest_v1/media/math/render/svg/486a14aabdc64357bf42d01337b677ca5fb5ec76" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:14.638ex; height:2.509ex;" alt="{\displaystyle b_{n+1}=36-t_{n}}"/></span>.
Suppose the threshold values for the second and third digits are <i>c</i> (i.e. 2), then the second-digit range is a–b (i.e. 0–1) with the second digit being most significant, while the range is c–9 (i.e. 2–35) in the presence of a third digit. Generally, for any <i>n</i>, the weight of the (<i>n</i>+1)-th digit is the weight of the previous one times (36 − threshold of the <i>n</i>-th digit). So the weight of the second symbol is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 36-t_{0}=35}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>36</mn>
<mo>−<!-- − --></mo>
<msub>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo>=</mo>
<mn>35</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 36-t_{0}=35}</annotation>
</semantics>
</math></span><img src="/media/api/rest_v1/media/math/render/svg/e2608d79127bafa4f5f85fab420769cceb5e388c" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:12.483ex; height:2.509ex;" alt="{\displaystyle 36-t_{0}=35}"/></span>. And the weight of the third symbol is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 35*(36-t_{1})=35*34=1190}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>35</mn>
<mo>∗<!-- ∗ --></mo>
<mo stretchy="false">(</mo>
<mn>36</mn>
<mo>−<!-- − --></mo>
<msub>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>35</mn>
<mo>∗<!-- ∗ --></mo>
<mn>34</mn>
<mo>=</mo>
<mn>1190</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 35*(36-t_{1})=35*34=1190}</annotation>
</semantics>
</math></span><img src="/media/api/rest_v1/media/math/render/svg/1fa80fba1cdaef757adb16cd8f60548aa94574c3" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:31.079ex; height:2.843ex;" alt="{\displaystyle 35*(36-t_{1})=35*34=1190}"/></span>.
</p><p>So we have the following sequence of the numbers with at most 3 digits:
</p><p><i>a</i> (0), <i>ba</i> (1), <i>ca</i> (2), ..., <i>9a</i> (35), <i>bb</i> (36), <i>cb</i> (37), ..., <i>9b</i> (70), <i>bca</i> (71), ..., <i>99a</i> (1260), <i>bcb</i> (1261), ..., <i>99b</i> (2450).
</p><p>Unlike a regular n-based numeral system, there are numbers like <i>9b</i> where <i>9</i> and <i>b</i> each represent 35; yet the representation is unique because <i>ac</i> and <i>aca</i> are not allowed – the first <i>a</i> would terminate each of the se numbers.
</p><p>The flexibility in choosing threshold values allows optimization for number of digits depending on the frequency of occurrence of numbers of various sizes.
</p><p>The case with all threshold values equal to 1 corresponds to <a href="/wiki/Bijective_numeration" title="Bijective numeration">bijective numeration</a>, where the zeros correspond to separators of numbers with digits which are non-zero.
</p>
<h2><span class="mw-headline" id="See_also">See also</span></h2>
<style data-mw-deduplicate="TemplateStyles:r998391716">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 22em;">
<ul><li><a href="/wiki/List_of_numeral_systems" title="List of numeral systems">List of numeral systems</a></li>
<li><a href="/wiki/Computer_numbering_formats" class="mw-redirect" title="Computer numbering formats">Computer numbering formats</a></li>
<li><a href="/wiki/Golden_ratio_base" title="Golden ratio base">Golden ratio base</a></li>
<li><a href="/wiki/History_of_ancient_numeral_systems" title="History of ancient numeral systems">History of ancient numeral systems</a></li>
<li><a href="/wiki/History_of_numbers" class="mw-redirect" title="History of numbers">History of numbers</a></li>
<li><a href="/wiki/List_of_numeral_system_topics" title="List of numeral system topics">List of numeral system topics</a></li>
<li><a href="/wiki/N-ary_(disambiguation)" class="mw-redirect mw-disambig" title="N-ary (disambiguation)"><i>n</i>-ary</a></li>
<li><a href="/wiki/Numeral_(linguistics)" title="Numeral (linguistics)">Number names</a></li>
<li><a href="/wiki/Quater-imaginary_base" title="Quater-imaginary base">Quater-imaginary base</a></li>
<li><a href="/wiki/Quipu" title="Quipu">Quipu</a></li>
<li><a href="/wiki/Recurring_decimal" class="mw-redirect" title="Recurring decimal">Recurring decimal</a></li>
<li><a href="/wiki/Residue_numeral_system" class="mw-redirect" title="Residue numeral system">Residue numeral system</a></li>
<li><a href="/wiki/Long_and_short_scales" class="mw-redirect" title="Long and short scales">Short and long scales</a></li>
<li><a href="/wiki/Scientific_notation" title="Scientific notation">Scientific notation</a></li>
<li><a href="/wiki/-yllion" title="-yllion">-yllion</a></li>
<li><a href="/wiki/Numerical_cognition" title="Numerical cognition">Numerical cognition</a></li>
<li><a href="/wiki/Number_system" class="mw-redirect" title="Number system">Number system</a></li>
<li><a href="/wiki/Unary_numeral_system" title="Unary numeral system">Unary numeral system</a></li></ul></div>
<li><a href="/wiki/0.999..." title="0.999...">0.999...</a> - every nonzero terminating decimal has two equal representations</li>
<h2><span class="mw-headline" id="References">References</span></h2>
<div class="mw-references-wrap"><ol class="references">
<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1067248974">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free a,.mw-parser-output .citation .cs1-lock-free a{background:linear-gradient(transparent,transparent),url("/media/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited a,.mw-parser-output .id-lock-registration a,.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:linear-gradient(transparent,transparent),url("/media/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription a,.mw-parser-output .citation .cs1-lock-subscription a{background:linear-gradient(transparent,transparent),url("/media/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:linear-gradient(transparent,transparent),url("/media/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:#d33}.mw-parser-output .cs1-visible-error{color:#d33}.mw-parser-output .cs1-maint{display:none;color:#3a3;margin-left:0.3em}.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}</style><cite id="CITEREFDavid_Eugene_SmithLouis_Charles_Karpinski1911" class="citation book cs1">David Eugene Smith; Louis Charles Karpinski (1911). <a rel="nofollow" class="external text" href="https://archive.org/details/hinduarabicnume05karpgoog"><i>The Hindu-Arabic numerals</i></a>. Ginn and Company.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Hindu-Arabic+numerals&rft.pub=Ginn+and+Company&rft.date=1911&rft.au=David+Eugene+Smith&rft.au=Louis+Charles+Karpinski&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fhinduarabicnume05karpgoog&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumeral+system" class="Z3988"></span></span>
</li>
<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1067248974"/><cite id="CITEREFChowdhury" class="citation book cs1">Chowdhury, Arnab. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=WXn-mT3K6dgC&q=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).&pg=PA2"><i>Design of an Efficient Multiplier using DBNS</i></a>. GIAP Journals. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-93-83006-18-2" title="Special:BookSources/978-93-83006-18-2"><bdi>978-93-83006-18-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Design+of+an+Efficient+Multiplier+using+DBNS&rft.pub=GIAP+Journals&rft.isbn=978-93-83006-18-2&rft.aulast=Chowdhury&rft.aufirst=Arnab&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DWXn-mT3K6dgC%26q%3DArithmetic%2Bis%2Bmuch%2Beasier%2Bin%2Bpositional%2Bsystems%2Bthan%2Bin%2Bthe%2Bearlier%2Badditive%2Bones%3B%2Bfurthermore%2C%2Badditive%2Bsystems%2Bneed%2Ba%2Blarge%2Bnumber%2Bof%2Bdifferent%2Bsymbols%2Bfor%2Bthe%2Bdifferent%2Bpowers%2Bof%2B10%3B%2Ba%2Bpositional%2Bsystem%2Bneeds%2Bonly%2Bten%2Bdifferent%2Bsymbols%2B%28assuming%2Bthat%2Bit%2Buses%2Bbase%2B10%29.%26pg%3DPA2&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumeral+system" class="Z3988"></span></span>
</li>
<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"> Fiete, I. R.; Seung, H. S. (2007). "Neural network models of birdsong production, learning, and coding". In Squire, L.; Albright, T.; Bloom, F.; Gage, F.; Spitzer, N. New Encyclopedia of Neuroscience.</span>
</li>
</ol></div>
<h2><span class="mw-headline" id="Sources">Sources</span></h2>
<ul><li>Georges Ifrah. <i>The Universal History of Numbers : From Prehistory to the Invention of the Computer</i>, Wiley, 1999. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1067248974"/><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-37568-3" title="Special:BookSources/0-471-37568-3">0-471-37568-3</a>.</li>
<li><a href="/wiki/Donald_Knuth" title="Donald Knuth">D. Knuth</a>. <i><a href="/wiki/The_Art_of_Computer_Programming" title="The Art of Computer Programming">The Art of Computer Programming</a></i>. Volume 2, 3rd Ed. <a href="/wiki/Addison%E2%80%93Wesley" class="mw-redirect" title="Addison–Wesley">Addison–Wesley</a>. pp. 194–213, "Positional Number Systems".</li>
<li><a href="/wiki/A.L._Kroeber" class="mw-redirect" title="A.L. Kroeber">A.L. Kroeber</a> (Alfred Louis Kroeber) (1876–1960), Handbook of the Indians of California, Bulletin 78 of the Bureau of American Ethnology of the Smithsonian Institution (1919)</li>
<li>J.P. Mallory and D.Q. Adams, <i>Encyclopedia of Indo-European Culture</i>, Fitzroy Dearborn Publishers, London and Chicago, 1997.</li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1067248974"/><cite id="CITEREFHans_J._NissenPeter_DamerowRobert_K._Englund1993" class="citation book cs1">Hans J. Nissen; Peter Damerow; Robert K. Englund (1993). <i>Archaic Bookkeeping: Early Writing and Techniques of Economic Administration in the Ancient Near East</i>. <a href="/wiki/University_Of_Chicago_Press" class="mw-redirect" title="University Of Chicago Press">University Of Chicago Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-226-58659-5" title="Special:BookSources/978-0-226-58659-5"><bdi>978-0-226-58659-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Archaic+Bookkeeping%3A+Early+Writing+and+Techniques+of+Economic+Administration+in+the+Ancient+Near+East&rft.pub=University+Of+Chicago+Press&rft.date=1993&rft.isbn=978-0-226-58659-5&rft.au=Hans+J.+Nissen&rft.au=Peter+Damerow&rft.au=Robert+K.+Englund&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumeral+system" class="Z3988"></span></li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1067248974"/><cite id="CITEREFSchmandt-Besserat1996" class="citation book cs1"><a href="/wiki/Denise_Schmandt-Besserat" title="Denise Schmandt-Besserat">Schmandt-Besserat, Denise</a> (1996). <i>How Writing Came About</i>. <a href="/wiki/University_of_Texas_Press" title="University of Texas Press">University of Texas Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-292-77704-0" title="Special:BookSources/978-0-292-77704-0"><bdi>978-0-292-77704-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=How+Writing+Came+About&rft.pub=University+of+Texas+Press&rft.date=1996&rft.isbn=978-0-292-77704-0&rft.aulast=Schmandt-Besserat&rft.aufirst=Denise&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumeral+system" class="Z3988"></span></li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1067248974"/><cite id="CITEREFZaslavsky1999" class="citation book cs1">Zaslavsky, Claudia (1999). <i>Africa counts: number and pattern in African cultures</i>. Chicago Review Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-55652-350-2" title="Special:BookSources/978-1-55652-350-2"><bdi>978-1-55652-350-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Africa+counts%3A+number+and+pattern+in+African+cultures&rft.pub=Chicago+Review+Press&rft.date=1999&rft.isbn=978-1-55652-350-2&rft.aulast=Zaslavsky&rft.aufirst=Claudia&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumeral+system" class="Z3988"></span></li></ul>
<h2><span class="mw-headline" id="External_links">External links</span></h2>
<table role="presentation" class="mbox-small plainlinks sistersitebox" style="background-color:#f9f9f9;border:1px solid #aaa;color:#000">
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<td class="mbox-image"><img alt="" src="/media/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/40px-Wiktionary-logo-en-v2.svg.png" decoding="async" width="40" height="40" class="noviewer" srcset="/media/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/60px-Wiktionary-logo-en-v2.svg.png 1.5x, /media/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/80px-Wiktionary-logo-en-v2.svg.png 2x" data-file-width="512" data-file-height="512" /></td>
<td class="mbox-text plainlist">Look up <i><b><a href="https://en.wiktionary.org/wiki/numeration" class="extiw" title="wiktionary:numeration">numeration</a></b></i> or <i><b><a href="https://en.wiktionary.org/wiki/numeral" class="extiw" title="wiktionary:numeral">numeral</a></b></i> in Wiktionary, the free dictionary.</td></tr>
</tbody></table>
<ul><li><a href="/wiki/File:Commons-logo.svg" class="image"><img alt="" src="/media/wikipedia/en/thumb/4/4a/Commons-logo.svg/12px-Commons-logo.svg.png" decoding="async" width="12" height="16" class="noviewer" srcset="/media/wikipedia/en/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png 1.5x, /media/wikipedia/en/thumb/4/4a/Commons-logo.svg/24px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a> Media related to <a href="/wiki/Category:Numeral_systems" class="extiw" title="commons:Category:Numeral systems">Numeral systems</a> at Wikimedia Commons</li></ul>
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Whether or not the change was made through a Tor exit node (tor_exit_node) | false |
Unix timestamp of change (timestamp) | 1649820841 |
