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{{short description|Notation for expressing numbers}}
{{About|expressing numbers with symbols|different kinds of numbers|Number system|expressing numbers with words|Numeral (linguistics)}}
{{more footnotes|date=January 2011}}
{{Numeral systems}}
[[File:Numeral Systems of the World.svg|264px|thumb|right|Numbers written in different numeral systems]]

A '''numeral system''' is a writing system for expressing [[Number|numbers]]; that is, a [[mathematical notation]] for representing numbers 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|decimal or base-10]] numeral system (today, the most common system globally), the number ''three'' in the [[binary number|binary or base-2]] numeral system (used in modern computers), and the number ''two'' in the [[unary numeral system]] (used in [[Tally marks|tallying]] scores).

The number the numeral represents is called its value. Not all number systems can represent the same set of numbers; for example, [[Roman numerals]] cannot represent the number 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 [[sequence]] of 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, various [[hypercomplex number]] systems, the system of [[p-adic number|''p''-adic numbers]], etc. Such systems are, however, not the topic of this article.

==History==
{{see also|Numerical digit#History}}
{{Expand section|date=July 2024}}
<div style="float:right;">
{| class="wikitable zebra"
|-
!Western Arabic
|0 ||1 ||2 ||3 ||4 ||5 ||6 ||7 ||8 ||9
|-
!Eastern Arabic
|٠ ||١ ||٢ ||٣ ||٤ ||٥ ||٦ ||٧ ||٨ ||٩
|-
!Persian
|۰ ||۱ ||۲ ||۳ ||۴ ||۵ ||۶ ||۷ ||۸ ||۹
|-
!Devanagari
|० ||१ ||२ ||३ ||४ ||५ ||६ ||७ ||८ ||९
|}
</div>
The first true written [[positional numeral system]] is considered to be the [[Hindu–Arabic numeral system]]. This system was established by the 7th&nbsp;century in India,<ref name="O'Connor and Robertson">O'Connor, J. J. and Robertson, E. F. [http://www-history.mcs.st-andrews.ac.uk/HistTopics/Arabic_numerals.html Arabic Numerals]. January 2001. Retrieved on 2007-02-20.</ref> but was not yet in its modern form because the use of the digit [[zero]] had not yet been widely accepted. Instead of a zero sometimes the digits were marked with dots to indicate their significance, or a space was used as a placeholder. The first widely acknowledged use of zero was in 876.<ref>{{cite web|url=https://www.ams.org/featurecolumn/archive/india-zero.html |title=All for Nought |work=Feature Column |author=Bill Casselman |author-link=Bill Casselman (mathematician) |publisher=AMS |date=February 2007}}</ref> The original numerals were very similar to the modern ones, even down to the [[glyph]]s used to represent digits.<ref name="O'Connor and Robertson"/>

[[Image:Maya.svg|thumb|left|150px|The digits of the Maya numeral system]]
By the 13th century, [[Western Arabic numerals]] were accepted in European mathematical circles ([[Fibonacci]] used them in his {{Lang|la|[[Liber Abaci]]}}). They began to enter common use in the 15th&nbsp;century.<ref>{{Cite web|last=Bradley|first=Jeremy|title=How Arabic Numbers Were Invented|url=https://www.theclassroom.com/how-to-identify-numbers-on-brass-from-india-12082499.html|access-date=2020-07-22|website=www.theclassroom.com}}</ref> By the end of the 20th&nbsp;century virtually all non-computerized calculations in the world were done with Arabic numerals, which have replaced native numeral systems in most cultures.

===Other historical numeral systems using digits===
The exact age of the [[Maya numerals]] is unclear, but it is possible that it is older than the Hindu–Arabic system. The system was [[vigesimal]] (base&nbsp;20), so it has twenty digits. The Mayas used a shell symbol to represent zero. Numerals were written vertically, with the ones place at the bottom. The [[Mayas]] had no equivalent of the modern [[decimal separator]], so their system could not represent fractions.{{Citation needed|date=November 2024}}

The [[Thai numerals|Thai numeral system]] is identical to the [[Hindu–Arabic numeral system]] except for the symbols used to represent digits. The use of these digits is less common in [[Thailand]] than it once was, but they are still used alongside Arabic numerals.<ref>{{cite news |title=Wissanu rejects dumping Thai numerals |url=https://www.bangkokpost.com/thailand/general/2318442/wissanu-rejects-dumping-thai-numerals |access-date=27 November 2024 |work=Bangkok Post |date=31 May 2022}}</ref>

The rod numerals, the written forms of [[counting rods]] once used by [[China|Chinese]] and [[Japan]]ese mathematicians, are a decimal positional system used for performing decimal calculations. Rods were placed on a counting board and slid forwards or backwards to change the decimal place. The ''[[Sunzi Suanjing|Sūnzĭ Suànjīng]]'', a mathematical treatise dated to between the 3rd and 5th centuries AD, provides detailed instructions for the system, which is thought to have been in use since at least the 4th century BC.<ref>{{MacTutor|class=HistTopics|id=Chinese_numerals|title=Chinese numerals|date=January 2004}}</ref> Zero was not initially treated as a number, but as a vacant position.<ref name="Crossley">{{Cite book |last1=Shen |first1=Kangshen |url=https://books.google.com/books?id=eiTJHRGTG6YC |title=The Nine Chapters on the Mathematical Art: Companion and Commentary |last2=Crossley |first2=John N. |last3=Lun |first3=Anthony W.-C. |publisher=Oxford University Press |year=1999 |isbn=978-0-19-853936-0 |page=35 |quote=zero was regarded as a number in India ... whereas the Chinese employed a vacant position |author1-mask=Shen Kanshen}}</ref> Later sources introduced conventions for the expression of zero and negative numbers. The use of a round symbol {{Lang|zh|〇}} for zero is first attested in the ''[[Mathematical Treatise in Nine Sections]]'' of 1247 AD.<ref name="Qin">{{Cite web |title=Mathematics in the Near and Far East |url=http://grmath4.phpnet.us/istoria/the_history_of%20math_greece/the_history_of%20math_greece_3-5.pdf |url-status=live |archive-url=https://web.archive.org/web/20131104120005/http://grmath4.phpnet.us/istoria/the_history_of%20math_greece/the_history_of%20math_greece_3-5.pdf |archive-date=4 November 2013 |access-date=7 June 2012 |website=grmath4.phpnet.us |page=262}}</ref> The origin of this symbol is unknown; it may have been produced by modifying a square symbol.<ref>{{cite book |last=Martzloff |first=Jean-Claude |title=A History of Chinese Mathematics |publisher=Springer |year=2007 |isbn=978-3-540-33783-6 |page=208 |translator-last1=Wilson |translator-first1=Stephen S.}}</ref> The [[Suzhou numerals]], a descendant of rod numerals, are still used today for some commercial purposes.{{Citation needed|date=November 2024}}

{| class="wikitable" style="text-align:center"
|+ Rod numerals (vertical)
|-
! style="width:50px" | 0
! style="width:50px" | 1
! style="width:50px" | 2
! style="width:50px" | 3
! style="width:50px" | 4
! style="width:50px" | 5
! style="width:50px" | 6
! style="width:50px" | 7
! style="width:50px" | 8
! style="width:50px" | 9
|-
| [[Image:Counting rod 0.png]]
| [[Image:Counting rod v1.png]]
| [[Image:Counting rod v2.png]]
| [[Image:Counting rod v3.png]]
| [[Image:Counting rod v4.png]]
| [[Image:Counting rod v5.png]]
| [[Image:Counting rod v6.png]]
| [[Image:Counting rod v7.png]]
| [[Image:Counting rod v8.png]]
| [[Image:Counting rod v9.png]]
|-
! style="width:50px" | –0
! style="width:50px" | –1
! style="width:50px" | –2
! style="width:50px" | –3
! style="width:50px" | –4
! style="width:50px" | –5
! style="width:50px" | –6
! style="width:50px" | –7
! style="width:50px" | –8
! style="width:50px" | –9
|-
| [[Image:Counting rod -0.png]]
| [[Image:Counting rod v-1.png]]
| [[Image:Counting rod v-2.png]]
| [[Image:Counting rod v-3.png]]
| [[Image:Counting rod v-4.png]]
| [[Image:Counting rod v-5.png]]
| [[Image:Counting rod v-6.png]]
| [[Image:Counting rod v-7.png]]
| [[Image:Counting rod v-8.png]]
| [[Image:Counting rod v-9.png]]
|}

==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&nbsp;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-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 (the number of these abbreviations is sometimes called the ''base'' of the system). 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 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. The positional systems are classified by their ''base'' or ''[[radix]]'', which is the number of symbols called ''[[numerical digit|digit]]s'' used by the system. In base&nbsp;10, ten different digits 0,&nbsp;..., 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&nbsp;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&nbsp;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&nbsp;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&nbsp;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&nbsp;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 some areas of computer science, a modified base ''k'' positional system is used, called [[bijective numeration]], with digits 1, 2,&nbsp;..., ''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&nbsp;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]] or ''base'' of the system), ''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&nbsp;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&nbsp;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&nbsp;2, {{math|[[pi|{{pi}}]] {{=}} 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}} &nbsp; or &nbsp; 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]].

It is also possible to define a variation of base ''b'' in which digits may be positive or negative; this is called a [[signed-digit representation]].

==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, \ldots</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''&nbsp;+&nbsp;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\cdot34 = 1190</math>.

So we have the following sequence of the numbers with at most 3 digits:

''a'' (0), ''ba'' (1), ''ca'' (2), ..., 9''a'' (35), ''bb'' (36), ''cb'' (37), ..., 9''b'' (70), ''bca'' (71), ..., 99''a'' (1260), ''bcb'' (1261), ..., 99''b'' (2450).

Unlike a regular ''n''-based numeral system, there are numbers like 9''b'' 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 these 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 number format]]s
*[[Non-standard positional numeral systems]]
*[[History of ancient numeral systems]]
*[[History of numbers]]
*[[List of numeral system topics]]
*[[Numeral (linguistics)|Number names]]
*[[Repeating decimal]]
*[[Residue numeral system]]
*[[Long and short scales]]
*[[Scientific notation]]
*[[-yllion]]
*[[Numerical cognition]]
*[[Number system]]
*[[Hebrew numerals]]
}}

==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.&nbsp;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; 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 |author-link=Claudia Zaslavsky |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}}

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{{DEFAULTSORT:Numeral System}}
[[Category:Numeral systems| ]]
[[Category:Numbers]]
[[Category:Mathematical notation]]

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