TheInfoList

A numeral system (or system of numeration) is a writing system for expressing numbers; that is, a mathematical notation for representing
number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deduct ...

s of a given set, using digits or other symbols in a consistent manner. The same sequence of symbols may represent different numbers in different numeral systems. For example, "11" represents the number ''eleven'' in the
decimal numeral system The decimal numeral system (also called the base-ten positional numeral system, and occasionally called denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hi ...
(used in common life), the number ''three'' in the
binary numeral system In mathematics and digital electronics Digital electronics is a field of electronics Electronics comprises the physics, engineering, technology and applications that deal with the emission, flow and control of electrons in vacuum and matter ...
(used in
computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern computers can perform generic sets of operations known as Computer program, programs. These programs enable compu ...

s), and the number ''two'' in the
unary numeral system The unary numeral system is the simplest numeral system to represent natural numbers: to represent a number ''N'', a symbol representing 1 is repeated ''N'' times. In the unary system, the number 0 (zero) is represented by the empty string, that i ...
(e.g. used in 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 An integer (from the Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of t ...
s, or
rational number In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
s) *Give every number represented a unique representation (or at least a standard representation) *Reflect the and
arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne', 'art' or 'cra ...
structure of the numbers. For example, the usual
decimal representation A decimal representation of a non-negative In mathematics, the sign of a real number is its property of being either positive, negative number, negative, or zero. Depending on local conventions, zero may be considered as being neither positive n ...
gives every nonzero
natural number File:Three Baskets.svg, Natural numbers can be used for counting (one apple, two apples, three apples, ...) In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, o ...
a unique representation as a
finite Finite is the opposite of Infinity, infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected ...
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order theory, order matters. Like a Set (mathematics), set, it contains Element (mathematics), members (also called ''elements'', or ''terms''). ...

of digits, beginning with a non-zero digit. Numeral systems are sometimes called ''
number system A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deduc ...
s'', but that name is ambiguous, as it could refer to different systems of numbers, such as the system of
real number In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
s, the system of
complex number In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

s, the system of ''p''-adic numbers, etc. Such systems are, however, not the topic of this article.

# Main numeral systems

The most commonly used system of numerals is
decimal The decimal numeral system A numeral system (or system of numeration) is a writing system A writing system is a method of visually representing verbal communication Communication (from Latin ''communicare'', meaning "to share") is t ...
.
Indian mathematicians The chronology of Indian mathematicians spans from the Indus Valley Civilization oxen for pulling a cart and the presence of the chicken The chicken (''Gallus gallus domesticus''), a subspecies of the red junglefowl, is a type of d ...
are credited with developing the integer version, the
Hindu–Arabic numeral system The Hindu–Arabic numeral system or Indo-Arabic numeral system Audun HolmeGeometry: Our Cultural Heritage 2000 (also called the Arabic numeral system or Hindu numeral system) is a positional notation, positional decimal numeral system, and is t ...
.
Aryabhata Aryabhata (, ISO: ) or Aryabhata I (476–550 CE) was the first of the major mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such ...
of developed the
place-value notation Positional notation (or place-value notation, or positional numeral system) denotes usually the extension to any base of the Hindu–Arabic numeral system (or decimal system). More generally, a positional system is a numeral system in which the ...
in the 5th century and a century later
Brahmagupta Brahmagupta ( – ) was an Indian mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematic ...

introduced the symbol for
zero 0 (zero) is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in languag ...

. The system slowly spread to other surrounding regions like Arabia due to their commercial and military activities with India.
Middle-Eastern The Middle East is a transcontinental region in Afro-Eurasia which generally includes Western Asia Western Asia, also West Asia, is the westernmost subregion of Asia. It is entirely a part of the Greater Middle East. It includes Anatoli ...
mathematicians extended the system to include negative powers of 10 (
fractions A fraction (from Latin ', "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, ...
), as recorded in a treatise by
Syrian Syrians ( ar, سوريون, ''Sūriyyūn''), also known as the Syrian people ( ar, الشعب السوري, ALA-LC: ''al-sha‘ab al-Sūrī''; syr, ܣܘܪܝܝܢ), are the majority inhabitants of Syria and share common Levantine Semitic roots. ...

mathematician
Abu'l-Hasan al-UqlidisiAbu'l Hasan Ahmad ibn Ibrahim Al-Uqlidisi ( ar, أبو الحسن أحمد بن ابراهيم الإقليدسي) was a Muslim Muslims () are people who follow or practice Islam, a Monotheism, monotheistic Abrahamic religions, Abrahamic religi ...
in 952–953, and the
decimal point A decimal separator is a symbol used to separate the integer An integer (from the Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spok ...
notation was introduced by
Sind ibn AliAbu al-Tayyib Sanad ibn Ali al-Yahudi (died c. 864 C.E.), was an eighth-century Iraqi Jewish astronomer An astronomer is a scientist in the field of astronomy who focuses their studies on a specific question or field outside the scope of Earth. T ...
, 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 Arabic numerals are the ten numerical digit A numerical digit (often shortened to just digit) is a single symbol used alone (such as "2") or in combinations (such as "25"), to represent numbers in a Positional notation, positional numeral sy ...

, as they learned them from the Arabs. The simplest numeral system is the
unary numeral system The unary numeral system is the simplest numeral system to represent natural numbers: to represent a number ''N'', a symbol representing 1 is repeated ''N'' times. In the unary system, the number 0 (zero) is represented by the empty string, that i ...
, in which every
natural number File:Three Baskets.svg, Natural numbers can be used for counting (one apple, two apples, three apples, ...) In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, o ...
is represented by a corresponding number of symbols. If the symbol is chosen, for example, then the number seven would be represented by .
Tally marks frame, Brahmi numerals (lower row) in India in the 1st century CE. Note the similarity of the first three numerals to the Chinese characters for one through three (一 二 三), plus the resemblance of both sets of numerals to horizontal tally ma ...

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 Image:Maquina.png, An artistic representation of a Turing machine. Turing machines are used to model general computing devices. Theoretical computer science (TCS) is a subset of general computer science and mathematics that focuses on mathematical ...

. Elias gamma coding, which is commonly used in
data compression In signal processing Signal processing is an electrical engineering subfield that focuses on analysing, modifying, and synthesizing signals such as audio signal processing, sound, image processing, images, and scientific measurements. Signal ...
, 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 and the number 123 as without any need for zero. This is called
sign-value notation A sign-value notation represents numbers by a series of numeric signs that added together equal the number represented. In Roman numerals Roman numerals are a numeral system that originated in ancient Rome and remained the usual way of writin ...
. The ancient Egyptian numeral system was of this type, and the
Roman numeral system Roman numerals are a numeral system A numeral system (or system of numeration) is a writing system A writing system is a method of visually representing verbal communication Communication (from Latin ''communicare'', meaning "to s ...
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 Chinese numerals are words and characters used to denote number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has bee ...
and other East Asian numerals based on Chinese. The number system of the
English language English is a West Germanic languages, West Germanic language first spoken in History of Anglo-Saxon England, early medieval England, which has eventually become the World language, leading language of international discourse in the 21st centu ...

is of this type ("three hundred four"), as are those of other spoken
language A language is a structured system of communication used by humans, including speech (spoken language), gestures (Signed language, sign language) and writing. Most languages have a writing system composed of glyphs to inscribe the original soun ...

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'' () and in Welsh is ''pedwar ar bymtheg a thrigain'' () or (somewhat archaic) ''pedwar ugain namyn un'' (). In English, one could say "four score less one", as in the famous
Gettysburg Address The Gettysburg Address is a speech Speech is human vocal communication using language. Each language uses Phonetics, phonetic combinations of vowel and consonant sounds that form the sound of its words (that is, all English words sound diff ...

representing "87 years ago" as "four score and seven years ago". More elegant is a ''
positional system Positional notation (or place-value notation, or positional numeral system) denotes usually the extension to any radix, base of the Hindu–Arabic numeral system (or decimal, decimal system). More generally, a positional system is a numeral 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 or more precisely . 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). 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 A computer is a machine that can be programmed to Execution (computing), carry out sequences of arithmetic or logical operations automatically. Modern computers can perform generic sets of operations known as Computer program, programs. These p ...

, the main numeral systems are based on the positional system in base 2 (
binary numeral system In mathematics and digital electronics Digital electronics is a field of electronics Electronics comprises the physics, engineering, technology and applications that deal with the emission, flow and control of electrons in vacuum and matter ...
), with two
binary digit Binary may refer to: Science and technology Mathematics * Binary number In mathematics and digital electronics, a binary number is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal ...
s, 0 and 1. Positional systems obtained by grouping binary digits by three ( octal numeral system) or four (
hexadecimal numeral system In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
) are commonly used. For very large integers, bases 232 or 264 (grouping binary digits by 32 or 64, the length of the
machine word In computing Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithmic processes and development of both computer hardware , hardware and softwa ...
) are used, as, for example, in GMP. In certain biological systems, the
unary codingUnary coding, or the unary numeral system The unary numeral system is the simplest numeral system to represent natural numbers: to represent a number ''N'', a symbol representing 1 is repeated ''N'' times. In the unary system, the number 0 (zero) ...
system is employed. Unary numerals used in the
neural circuit A neural circuit is a population of neuron A neuron or nerve cell is an membrane potential#Cell excitability, electrically excitable cell (biology), cell that communicates with other cells via specialized connections called synapses. It is the ...
s responsible for
birdsong (''Turdus merula'') singing, Bogense havn, Funen, Denmark. Bird vocalization includes both bird calls and bird songs. In non-technical use, bird Birds are a group of warm-blooded vertebrates constituting the class (biology), class Aves ...
production. 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. 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 HVC in the context of the song-learning pathway in birds. HVC (formerly, hyperstriatum ventrale, pars caudalis (HVc), and high vocal center) is a nucleus in the brain of the songbirds (order passeriformes) necessary for both the learning and th ...
). 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 Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne', 'art' or 'cra ...

numerals (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) and the
geometric Geometry (from the grc, γεωμετρία; '' geo-'' "earth", '' -metron'' "measurement") is, with arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt ...

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 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 Bijective numeration is any numeral system A numeral system (or system of numeration) is a writing system A writing system is a method of visually representing verbal communication Communication (from Latin ''communicare'', meaning " ...
, with digits 1, 2, ..., ''k'' (), and zero being represented by an empty string. This establishes a
bijection In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

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 numbers. Bijective base 1 is the same as unary.

# Positional systems in detail

In a positional base ''b'' numeral system (with ''b'' a
natural number File:Three Baskets.svg, Natural numbers can be used for counting (one apple, two apples, three apples, ...) In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, o ...
greater than 1 known as the
radix In a positional numeral system, the radix or base is the number of unique digits, including the digit zero, used to represent numbers. For example, for the decimal/denary system (the most common system in use today) the radix (base number) is t ...

), ''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 The decimal numeral system A numeral system (or system of numeration) is a writing system A writing system is a method of visually representing verbal communication Communication (from Latin ''communicare'', meaning "to share") is t ...
system (base 10), the numeral 4327 means , noting that . In general, if ''b'' is the base, one writes a number in the numeral system of base ''b'' by expressing it in the form and writing the enumerated digits in descending order. The digits are natural numbers between 0 and , inclusive. If a text (such as this one) discusses multiple bases, and if ambiguity exists, the base (itself represented in base 10) is added in subscript to the right of the number, like this: numberbase. Unless specified by context, numbers without subscript are considered to be decimal. By using a dot to divide the digits into two groups, one can also write fractions in the positional system. For example, the base 2 numeral 10.11 denotes . In general, numbers in the base ''b'' system are of the form: :$\left(a_na_\cdots a_1a_0.c_1 c_2 c_3\cdots\right)_b = \sum_^n a_kb^k + \sum_^\infty c_kb^.$ The numbers ''b''''k'' and ''b''−''k'' are the
weights Weight is a measurement of the gravitational force acting on an object. In non-scientific contexts it may refer to an object's mass (quantity of matter). Figuratively, it refers to the seriousness or depth of an idea or thought, or the danger and u ...
of the corresponding digits. The position ''k'' is the
logarithm In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a given number  is the exponent to which another fixed number, the ''base (exponentiation), base'' , must be raised, to produce that ...

of the corresponding weight ''w'', that is $k = \log_ w = \log_ b^k$. The highest used position is close to the
order of magnitude An order of magnitude is an approximation of the logarithm In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry ...
of the number. The number of
tally marks frame, Brahmi numerals (lower row) in India in the 1st century CE. Note the similarity of the first three numerals to the Chinese characters for one through three (一 二 三), plus the resemblance of both sets of numerals to horizontal tally ma ...

required in the
unary numeral system The unary numeral system is the simplest numeral system to represent natural numbers: to represent a number ''N'', a symbol representing 1 is repeated ''N'' times. In the unary system, the number 0 (zero) is represented by the empty string, that i ...
for ''describing the weight'' would have been w. In the positional system, the number of digits required to describe it is only $k + 1 = \log_ w + 1$, for ''k'' ≥ 0. For example, to describe the weight 1000 then four digits are needed because $\log_ 1000 + 1 = 3 + 1$. The number of digits required to ''describe the position'' is $\log_b k + 1 = \log_b \log_b w + 1$ (in positions 1, 10, 100,... only for simplicity in the decimal example). :$\begin \text & 3 & 2 & 1 & 0 & -1 & -2 & \cdots \\ \hline \text & b^3 & b^2 & b^1 & b^0 & b^ & b^ & \cdots \\ \text & a_3 & a_2 & a_1 & a_0 & c_1 & c_2 & \cdots \\ \hline \text & 1000 & 100 & 10 & 1 & 0.1 & 0.01 & \cdots \\ \text & 4 & 3 & 2 & 7 & 0 & 0 & \cdots \end$ A number has a terminating or repeating expansion
if and only if In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, l ...
it is
rational Rationality is the quality or state of being rational – that is, being based on or agreeable to reason Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογι ...
; this does not depend on the base. A number that terminates in one base may repeat in another (thus ). An irrational number stays aperiodic (with an infinite number of non-repeating digits) in all integral bases. Thus, for example in base 2, can be written as the aperiodic 11.001001000011111...2. Putting , , or dots, ''ṅ'', above the common digits is a convention used to represent repeating rational expansions. Thus: :14/11 = 1.272727272727... = 1.   or   321.3217878787878... = 321.321. If ''b'' = ''p'' is a
prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
, one can define base-''p'' numerals whose expansion to the left never stops; these are called the ''p''-adic numbers.

# Generalized variable-length integers

More general is using a
mixed radix Mixed radix numeral system A numeral system (or system of numeration) is a writing system A writing system is a method of visually representing verbal communication Communication (from Latin ''communicare'', meaning "to share") is th ...
notation (here written
little-endian In computing Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithmic processes and development of both computer hardware , hardware and softw ...
) like $a_0 a_1 a_2$ for $a_0 + a_1 b_1 + a_2 b_1 b_2$, etc. This is used in
punycodePunycode is a representation of Unicode Unicode is an information technology Technical standard, standard for the consistent character encoding, encoding, representation, and handling of Character (computing), text expressed in most of the world ...
, one aspect of which is the representation of a sequence of non-negative integers of arbitrary size in the form of a sequence without delimiters, of "digits" from a collection of 36: a–z and 0–9, representing 0–25 and 26–35 respectively. A digit lower than a threshold value marks that it is the most-significant digit, hence the end of the number. The threshold value depends on the position in the number. For example, if the threshold value for the first digit is b (i.e. 1) then a (i.e. 0) marks the end of the number (it has just one digit), so in numbers of more than one digit, range is only b–9 (1–35), therefore the weight ''b''1 is 35 instead of 36. Suppose the threshold values for the second and third digits are c (2), then the third digit has a weight 35''b''2, determined from $\left(99b\right)_+1=35+35b_1+b_1b_2+1=\left(bcca\right)_=1+2b_1+2b_1b_2\therefore b_2=1+35-2=34,\ b_1b_2=1190 \text$ with the subscript ''pc'' referring to the code described, and we have the following sequence: a (0), ba (1), ca (2), .., 9a (35), bb (36), cb (37), .., 9b (70), bca (71), .., 99a (1260), bcb (1261), .., 99b (2450). Unlike a regular based numeral system, there are numbers like 9b where 9 and b each represents 35; yet the representation is unique because ac and aca are not allowed – the first a would terminate the number. More generally, if ''tn'' is the threshold for the ''n''-th digit, it is easy to show that $b_=36-t_n$. The flexibility in choosing threshold values allows optimization depending on the frequency of occurrence of numbers of various sizes. The case with all threshold values equal to 1 corresponds to
bijective numeration Bijective numeration is any numeral system A numeral system (or system of numeration) is a writing system A writing system is a method of visually representing verbal communication Communication (from Latin ''communicare'', meaning " ...
, where the zeros correspond to separators of numbers with digits which are non-zero.

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

# Sources

*Georges Ifrah. ''The Universal History of Numbers : From Prehistory to the Invention of the Computer'', Wiley, 1999. . * D. Knuth. ''
The Art of Computer Programming ''The Art of Computer Programming'' (''TAOCP'') is a comprehensive monograph written by computer scientist Donald Knuth that covers many kinds of Computer programming, programming algorithms and analysis of algorithms, their analysis. Knuth began ...
''. Volume 2, 3rd Ed.
Addison–Wesley Addison-Wesley is a publisher of textbooks and computer literature. It is an imprint of Pearson PLC, a global publishing and education company. In addition to publishing books, Addison-Wesley also distributes its technical titles through the Safa ...
. 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. * * *