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Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base of the
Hindu–Arabic numeral system The Hindu–Arabic numeral system or Indo-Arabic numeral system Audun HolmeGeometry: Our Cultural Heritage 2000 (also called the Hindu numeral system or Arabic numeral system) is a positional decimal numeral system, and is the most common syste ...
(or decimal system). More generally, a positional system is a numeral system in which the contribution of a digit to the value of a number is the value of the digit multiplied by a factor determined by the position of the digit. In early
numeral system A numeral system (or system of numeration) is a writing system for expressing numbers; that is, a mathematical notation for representing numbers of a given set, using Numerical digit, digits or other symbols in a consistent manner. The same s ...
s, such as
Roman numerals Roman numerals are a numeral system that originated in ancient Rome and remained the usual way of writing numbers throughout Europe well into the Late Middle Ages. Numbers are written with combinations of letters from the Latin alphabet, eac ...
, a digit has only one value: I means one, X means ten and C a hundred (however, the value may be negated if placed before another digit). In modern positional systems, such as the decimal system, the position of the digit means that its value must be multiplied by some value: in 555, the three identical symbols represent five hundreds, five tens, and five units, respectively, due to their different positions in the digit string. The Babylonian numeral system, base 60, was the first positional system to be developed, and its influence is present today in the way time and angles are counted in tallies related to 60, such as 60 minutes in an hour and 360 degrees in a circle. Today, the Hindu–Arabic numeral system (
base ten The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral ...
) is the most commonly used system globally. However, the
binary numeral system A binary number is a number expressed in the base-2 numeral system or binary numeral system, a method of mathematical expression which uses only two symbols: typically "0" (zero) and "1" ( one). The base-2 numeral system is a positional notatio ...
(base two) is used in almost all
computer A computer is a machine that can be programmed to Execution (computing), carry out sequences of arithmetic or logical operations (computation) automatically. Modern digital electronic computers can perform generic sets of operations known as C ...
s and
electronic device The field of electronics is a branch of physics and electrical engineering that deals with the emission, behaviour and effects of electrons using electronic devices. Electronics uses active devices to control electron flow by amplification ...
s because it is easier to implement efficiently in
electronic circuit An electronic circuit is composed of individual electronic components, such as resistors, transistors, capacitors, inductors and diodes, connected by conductive wires or traces through which electric current can flow. It is a type of electrical ...
s. Systems with negative base,
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
base or negative digits have been described. Most of them do not require a minus sign for designating negative numbers. The use of a
radix point A decimal separator is a symbol used to separate the integer part from the fractional part of a number written in decimal form (e.g., "." in 12.45). Different countries officially designate different symbols for use as the separator. The choi ...
(decimal point in base ten), extends to include
fractions A fraction (from la, fractus, "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 ...
and allows representing any
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
with arbitrary accuracy. With positional notation, arithmetical computations are much simpler than with any older numeral system; this led to the rapid spread of the notation when it was introduced in western Europe.


History

Today, the base-10 (
decimal The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral ...
) system, which is presumably motivated by counting with the ten
finger A finger is a limb of the body and a type of digit, an organ of manipulation and sensation found in the hands of most of the Tetrapods, so also with humans and other primates. Most land vertebrates have five fingers ( Pentadactyly). Chambers ...
s, is ubiquitous. Other bases have been used in the past, and some continue to be used today. For example, the Babylonian numeral system, credited as the first positional numeral system, was
base-60 Sexagesimal, also known as base 60 or sexagenary, is a numeral system with sixty as its base. It originated with the ancient Sumerians in the 3rd millennium BC, was passed down to the ancient Babylonians, and is still used—in a modified form ...
. However, it lacked a real
zero 0 (zero) is a number representing an empty quantity. In place-value notation Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base of the Hindu–Arabic numeral system (or ...
. Initially inferred only from context, later, by about 700 BC, zero came to be indicated by a "space" or a "punctuation symbol" (such as two slanted wedges) between numerals. It was a
placeholder Placeholder may refer to: Language * Placeholder name, a term or terms referring to something or somebody whose name is not known or, in that particular context, is not significant or relevant. * Filler text, text generated to fill space or provi ...
rather than a true zero because it was not used alone or at the end of a number. Numbers like 2 and 120 (2×60) looked the same because the larger number lacked a final placeholder. Only context could differentiate them. The polymath
Archimedes Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientists ...
(ca. 287–212 BC) invented a decimal positional system in his
Sand Reckoner ''The Sand Reckoner'' ( el, Ψαμμίτης, ''Psammites'') is a work by Archimedes, an Ancient Greek mathematician of the 3rd century BC, in which he set out to determine an upper bound for the number of grains of sand that fit into the unive ...
which was based on 108 and later led the German mathematician
Carl Friedrich Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
to lament what heights science would have already reached in his days if Archimedes had fully realized the potential of his ingenious discovery. Before positional notation became standard, simple additive systems (
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 for example, X means ten and L means fifty. Hence LXXX means eighty (50 + 10 + 10  ...
) such as
Roman numerals Roman numerals are a numeral system that originated in ancient Rome and remained the usual way of writing numbers throughout Europe well into the Late Middle Ages. Numbers are written with combinations of letters from the Latin alphabet, eac ...
were used, and accountants in ancient Rome and during the Middle Ages used the
abacus The abacus (''plural'' abaci or abacuses), also called a counting frame, is a calculating tool which has been used since ancient times. It was used in the ancient Near East, Europe, China, and Russia, centuries before the adoption of the Hin ...
or stone counters to do arithmetic.
Counting rods Counting rods () are small bars, typically 3–14 cm long, that were used by mathematicians for calculation in ancient East Asia. They are placed either horizontally or vertically to represent any integer or rational number. The written fo ...
and most abacuses have been used to represent numbers in a positional numeral system. With counting rods or
abacus The abacus (''plural'' abaci or abacuses), also called a counting frame, is a calculating tool which has been used since ancient times. It was used in the ancient Near East, Europe, China, and Russia, centuries before the adoption of the Hin ...
to perform arithmetic operations, the writing of the starting, intermediate and final values of a calculation could easily be done with a simple additive system in each position or column. This approach required no memorization of tables (as does positional notation) and could produce practical results quickly. The oldest extant positional notation system is either that of Chinese
rod numerals Counting rods () are small bars, typically 3–14 cm long, that were used by mathematicians for calculation in ancient East Asia. They are placed either horizontally or vertically to represent any integer or rational number. The written fo ...
, used from at least the early 8th century, or perhaps
Khmer numerals Khmer numerals are the numerals used in the Khmer language. They have been in use since at least the early 7th century, with the earliest known use being on a stele dated to AD 604 found in Prasat Bayang, near Angkor Borei, Cambodia. Numera ...
, showing possible usages of positional-numbers in the 7th century. Khmer numerals and other
Indian numerals Indian or Indians may refer to: Peoples South Asia * Indian people, people of Indian nationality, or people who have an Indian ancestor ** Non-resident Indian, a citizen of India who has temporarily emigrated to another country * South Asia ...
originate with the
Brahmi numerals The Brahmi numerals are a numeral system attested from the 3rd century BCE (somewhat later in the case of most of the tens). They are a non positional decimal system. They are the direct graphic ancestors of the modern Hindu–Arabic numeral s ...
of about the 3rd century BC, which symbols were, at the time, not used positionally. Medieval Indian numerals are positional, as are the derived
Arabic numerals Arabic numerals are the ten numerical digits: , , , , , , , , and . They are the most commonly used symbols to write Decimal, decimal numbers. They are also used for writing numbers in other systems such as octal, and for writing identifiers ...
, recorded from the 10th century. After the
French Revolution The French Revolution ( ) was a period of radical political and societal change in France that began with the Estates General of 1789 and ended with the formation of the French Consulate in November 1799. Many of its ideas are considere ...
(1789–1799), the new French government promoted the extension of the decimal system. Some of those pro-decimal efforts—such as
decimal time Decimal time is the representation of the time of day using units which are decimally related. This term is often used specifically to refer to the time system used in France for a few years beginning in 1792 during the French Revolution, whic ...
and the
decimal calendar A decimal calendar is a calendar which includes units of time based on the decimal system. For example a "decimal month" would consist of a year with 10 months and 36.52422 days per month. History Egyptian calendar The ancient Egyptian calenda ...
—were unsuccessful. Other French pro-decimal efforts—currency
decimalisation Decimalisation or decimalization (see spelling differences) is the conversion of a system of currency or of weights and measures to units related by powers of 10. Most countries have decimalised their currencies, converting them from non-decimal ...
and the
metrication Metrication or metrification is the act or process of converting to the metric system of measurement. All over the world, countries have transitioned from local and traditional units of measurement to the metric system. This process began in F ...
of weights and measures—spread widely out of France to almost the whole world.


History of positional fractions

J. Lennart Berggren notes that positional decimal fractions were used for the first time by Arab mathematician
Abu'l-Hasan al-Uqlidisi Abu'l Hasan Ahmad ibn Ibrahim Al-Uqlidisi ( ar, أبو الحسن أحمد بن ابراهيم الإقليدسي) was a Muslim Arab mathematician, who was active in Damascus and Baghdad. He wrote the earliest surviving book on the positional use ...
as early as the 10th century. The Jewish mathematician
Immanuel Bonfils Immanuel ben Jacob Bonfils (c. 1300 – 1377) was a French-Jewish mathematician and astronomer in medieval times who flourished from 1340 to 1377, a rabbi who was a pioneer of exponential calculus and is credited with inventing the system of decimal ...
used decimal fractions around 1350, but did not develop any notation to represent them. The Persian mathematician
Jamshīd al-Kāshī Ghiyāth al-Dīn Jamshīd Masʿūd al-Kāshī (or al-Kāshānī) ( fa, غیاث الدین جمشید کاشانی ''Ghiyās-ud-dīn Jamshīd Kāshānī'') (c. 1380 Kashan, Iran – 22 June 1429 Samarkand, Transoxania) was a Persian astronomer a ...
made the same discovery of decimal fractions in the 15th century.
Al Khwarizmi Muḥammad ibn Mūsā al-Khwārizmī ( ar, محمد بن موسى الخوارزمي, Muḥammad ibn Musā al-Khwārazmi; ), or al-Khwarizmi, was a Persian polymath from Khwarazm, who produced vastly influential works in mathematics, astronomy ...
introduced fractions to Islamic countries in the early 9th century; his fraction presentation was similar to the traditional Chinese mathematical fractions from
Sunzi Suanjing ''Sunzi Suanjing'' () was a mathematical treatise written during 3rd to 5th centuries AD which was listed as one of the Ten Computational Canons during the Tang dynasty. The specific identity of its author Sunzi (lit. "Master Sun") is still ...
.
Lam Lay Yong Lam Lay Yong (maiden name Oon Lay Yong, ; born 1936) is a retired Professor of Mathematics. Academic career From 1988 to 1996 she was Professor at the Department of Mathematics, National University of Singapore (NUS). She graduated from the Un ...
, "The Development of Hindu-Arabic and Traditional Chinese Arithmetic", ''Chinese Science'', 1996 p38, Kurt Vogel notation
This form of fraction with numerator on top and denominator at bottom without a horizontal bar was also used by 10th century
Abu'l-Hasan al-Uqlidisi Abu'l Hasan Ahmad ibn Ibrahim Al-Uqlidisi ( ar, أبو الحسن أحمد بن ابراهيم الإقليدسي) was a Muslim Arab mathematician, who was active in Damascus and Baghdad. He wrote the earliest surviving book on the positional use ...
and 15th century
Jamshīd al-Kāshī Ghiyāth al-Dīn Jamshīd Masʿūd al-Kāshī (or al-Kāshānī) ( fa, غیاث الدین جمشید کاشانی ''Ghiyās-ud-dīn Jamshīd Kāshānī'') (c. 1380 Kashan, Iran – 22 June 1429 Samarkand, Transoxania) was a Persian astronomer a ...
's work "Arithmetic Key".
The adoption of the
decimal representation A decimal representation of a non-negative real number is its expression as a sequence of symbols consisting of decimal digits traditionally written with a single separator: r = b_k b_\ldots b_0.a_1a_2\ldots Here is the decimal separator, is ...
of numbers less than one, a
fraction A fraction (from la, fractus, "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 ...
, is often credited to
Simon Stevin Simon Stevin (; 1548–1620), sometimes called Stevinus, was a Flemish mathematician, scientist and music theorist. He made various contributions in many areas of science and engineering, both theoretical and practical. He also translated vario ...
through his textbook
De Thiende ''De Thiende'', published in 1585 in the Dutch language by Simon Stevin, is remembered for extending positional notation to the use of decimals to represent fractions. A French version, ''La Disme'', was issued the same year by Stevin. Stevin i ...
; but both Stevin and
E. J. Dijksterhuis Eduard Jan Dijksterhuis (28 October 1892, in Tilburg – 18 May 1965, in De Bilt) was a Dutch historian of science. Career Dijksterhuis studied mathematics at the University of Groningen from 1911 to 1918. His Ph.d. thesis was entitled "A Contrib ...
indicate that
Regiomontanus Johannes Müller von Königsberg (6 June 1436 – 6 July 1476), better known as Regiomontanus (), was a mathematician, astrologer and astronomer of the German Renaissance, active in Vienna, Buda and Nuremberg. His contributions were instrumental ...
contributed to the European adoption of general
decimal The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral ...
s:
E. J. Dijksterhuis Eduard Jan Dijksterhuis (28 October 1892, in Tilburg – 18 May 1965, in De Bilt) was a Dutch historian of science. Career Dijksterhuis studied mathematics at the University of Groningen from 1911 to 1918. His Ph.d. thesis was entitled "A Contrib ...
(1970) ''Simon Stevin: Science in the Netherlands around 1600'',
Martinus Nijhoff Publishers Brill Academic Publishers (known as E. J. Brill, Koninklijke Brill, Brill ()) is a Dutch international academic publisher founded in 1683 in Leiden, Netherlands. With offices in Leiden, Boston, Paderborn and Singapore, Brill today publishes 27 ...
, Dutch original 1943
:European mathematicians, when taking over from the Hindus, ''via'' the Arabs, the idea of positional value for integers, neglected to extend this idea to fractions. For some centuries they confined themselves to using common and
sexagesimal Sexagesimal, also known as base 60 or sexagenary, is a numeral system with sixty as its base. It originated with the ancient Sumerians in the 3rd millennium BC, was passed down to the ancient Babylonians, and is still used—in a modified form ...
fractions... This half-heartedness has never been completely overcome, and sexagesimal fractions still form the basis of our trigonometry, astronomy and measurement of time. ¶ ... Mathematicians sought to avoid fractions by taking the radius ''R'' equal to a number of units of length of the form 10n and then assuming for ''n'' so great an integral value that all occurring quantities could be expressed with sufficient accuracy by integers. ¶ The first to apply this method was the German astronomer Regiomontanus. To the extent that he expressed goniometrical line-segments in a unit ''R''/10n, Regiomontanus may be called an anticipator of the doctrine of decimal positional fractions. In the estimation of Dijksterhuis, "after the publication of
De Thiende ''De Thiende'', published in 1585 in the Dutch language by Simon Stevin, is remembered for extending positional notation to the use of decimals to represent fractions. A French version, ''La Disme'', was issued the same year by Stevin. Stevin i ...
only a small advance was required to establish the complete system of decimal positional fractions, and this step was taken promptly by a number of writers ... next to Stevin the most important figure in this development was Regiomontanus." Dijksterhuis noted that
tevin Tevin may refer to: *Tevin (given name) *Tevin, Kandovan, Iran *Tovin, Kaghazkonan, Iran See also *Teven (disambiguation) Teven may refer to: *Teven, New South Wales, Australia *Teven Jenkins (born 1998), American football player See also *Tevin ...
"gives full credit to Regiomontanus for his prior contribution, saying that the trigonometric tables of the German astronomer actually contain the whole theory of 'numbers of the tenth progress'."


Mathematics


Base of the numeral system

In mathematical numeral systems the radix is usually the number of unique digits, including zero, that a positional numeral system uses to represent numbers. In some cases, such as with a
negative base A negative base (or negative radix) may be used to construct a non-standard positional numeral system. Like other place-value systems, each position holds multiples of the appropriate power of the system's base; but that base is negative—that i ...
, the radix is the
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
r=, b, of the base . For example, for the decimal system the radix (and base) is ten, because it uses the ten digits from 0 through 9. When a number "hits" 9, the next number will not be another different symbol, but a "1" followed by a "0". In binary, the radix is two, since after it hits "1", instead of "2" or another written symbol, it jumps straight to "10", followed by "11" and "100". The highest symbol of a positional numeral system usually has the value one less than the value of the radix of that numeral system. The standard positional numeral systems differ from one another only in the base they use. The radix is an integer that is greater than 1, since a radix of zero would not have any digits, and a radix of 1 would only have the zero digit. Negative bases are rarely used. In a system with more than , b, unique digits, numbers may have many different possible representations. It is important that the radix is finite, from which follows that the number of digits is quite low. Otherwise, the length of a numeral would not necessarily be
logarithm In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number  to the base  is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 o ...
ic in its size. (In certain
non-standard positional numeral systems Non-standard positional numeral systems here designates numeral systems that may loosely be described as positional systems, but that do not entirely comply with the following description of standard positional systems: :In a standard positional ...
, including
bijective numeration Bijective numeration is any numeral system in which every non-negative integer can be represented in exactly one way using a finite string of digits. The name refers to the bijection (i.e. one-to-one correspondence) that exists in this case betw ...
, the definition of the base or the allowed digits deviates from the above.) In standard base-ten (
decimal The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral ...
) positional notation, there are ten
decimal 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 numeral system. The name "digit" comes from the fact that the ten digits (Latin ...
s and the number :5305_ = (5 \times 10^3) + (3 \times 10^2) + (0 \times 10^1) + (5 \times 10^0). In standard base-sixteen (
hexadecimal In mathematics and computing, the hexadecimal (also base-16 or simply hex) numeral system is a positional numeral system that represents numbers using a radix (base) of 16. Unlike the decimal system representing numbers using 10 symbols, hexa ...
), there are the sixteen hexadecimal digits (0–9 and A–F) and the number :14\mathrm9_ = (1 \times 16^3) + (4 \times 16^2) + (\mathrm \times 16^1) + (9 \times 16^0) \qquad (= 5305_) , where B represents the number eleven as a single symbol. In general, in base-''b'', there are ''b'' digits \ =:D and the number :(a_3 a_2 a_1 a_0)_b = (a_3 \times b^3) + (a_2 \times b^2) + (a_1 \times b^1) + (a_0 \times b^0) has \forall k \colon a_k \in D . Note that a_3 a_2 a_1 a_0 represents a sequence of digits, not
multiplication Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being additi ...
.


Notation

When describing base in
mathematical notation Mathematical notation consists of using symbols for representing operations, unspecified numbers, relations and any other mathematical objects, and assembling them into expressions and formulas. Mathematical notation is widely used in mathematic ...
, the letter ''b'' is generally used as a
symbol A symbol is a mark, sign, or word that indicates, signifies, or is understood as representing an idea, object, or relationship. Symbols allow people to go beyond what is known or seen by creating linkages between otherwise very different conc ...
for this concept, so, for a
binary Binary may refer to: Science and technology Mathematics * Binary number, a representation of numbers using only two digits (0 and 1) * Binary function, a function that takes two arguments * Binary operation, a mathematical operation that t ...
system, ''b'' equals 2. Another common way of expressing the base is writing it as a decimal subscript after the number that is being represented (this notation is used in this article). 11110112 implies that the number 1111011 is a base-2 number, equal to 12310 (a
decimal notation The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numer ...
representation), 1738 (
octal The octal numeral system, or oct for short, is the radix, base-8 number system, and uses the Numerical digit, digits 0 to 7. This is to say that 10octal represents eight and 100octal represents sixty-four. However, English, like most languages, ...
) and 7B16 (
hexadecimal In mathematics and computing, the hexadecimal (also base-16 or simply hex) numeral system is a positional numeral system that represents numbers using a radix (base) of 16. Unlike the decimal system representing numbers using 10 symbols, hexa ...
). In books and articles, when using initially the written abbreviations of number bases, the base is not subsequently printed: it is assumed that binary 1111011 is the same as 11110112. The base ''b'' may also be indicated by the phrase "base-''b''". So binary numbers are "base-2"; octal numbers are "base-8"; decimal numbers are "base-10"; and so on. To a given radix ''b'' the set of digits is called the standard set of digits. Thus, binary numbers have digits ; decimal numbers have digits and so on. Therefore, the following are notational errors: 522, 22, 1A9. (In all cases, one or more digits is not in the set of allowed digits for the given base.)


Exponentiation

Positional numeral systems work using
exponentiation Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to re ...
of the base. A digit's value is the digit multiplied by the value of its place. Place values are the number of the base raised to the ''n''th power, where ''n'' is the number of other digits between a given digit and the
radix point A decimal separator is a symbol used to separate the integer part from the fractional part of a number written in decimal form (e.g., "." in 12.45). Different countries officially designate different symbols for use as the separator. The choi ...
. If a given digit is on the left hand side of the radix point (i.e. its value is an
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
) then ''n'' is positive or zero; if the digit is on the right hand side of the radix point (i.e., its value is fractional) then ''n'' is negative. As an example of usage, the number 465 in its respective base ''b'' (which must be at least base 7 because the highest digit in it is 6) is equal to: :4\times b^2 + 6\times b^1 + 5\times b^0 If the number 465 was in base-10, then it would equal: :4\times 10^2 + 6\times 10^1 + 5\times 10^0 = 4\times 100 + 6\times 10 + 5\times 1 = 465 (46510 = 46510) If however, the number were in base 7, then it would equal: :4\times 7^2 + 6\times 7^1 + 5\times 7^0 = 4\times 49 + 6\times 7 + 5\times 1 = 243 (4657 = 24310) 10''b'' = ''b'' for any base ''b'', since 10''b'' = 1×''b''1 + 0×''b''0. For example, 102 = 2; 103 = 3; 1016 = 1610. Note that the last "16" is indicated to be in base 10. The base makes no difference for one-digit numerals. This concept can be demonstrated using a diagram. One object represents one unit. When the number of objects is equal to or greater than the base ''b'', then a group of objects is created with ''b'' objects. When the number of these groups exceeds ''b'', then a group of these groups of objects is created with ''b'' groups of ''b'' objects; and so on. Thus the same number in different bases will have different values: 241 in base 5: 2 groups of 52 (25) 4 groups of 5 1 group of 1 ooooo ooooo ooooo ooooo ooooo ooooo ooooo ooooo + + o ooooo ooooo ooooo ooooo ooooo ooooo 241 in base 8: 2 groups of 82 (64) 4 groups of 8 1 group of 1 oooooooo oooooooo oooooooo oooooooo oooooooo oooooooo oooooooo oooooooo oooooooo oooooooo + + o oooooooo oooooooo oooooooo oooooooo oooooooo oooooooo oooooooo oooooooo oooooooo oooooooo The notation can be further augmented by allowing a leading minus sign. This allows the representation of negative numbers. For a given base, every representation corresponds to exactly one
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
and every real number has at least one representation. The representations of rational numbers are those representations that are finite, use the bar notation, or end with an infinitely repeating cycle of digits.


Digits and numerals

A ''digit'' is a symbol that is used for positional notation, and a ''numeral'' consists of one or more digits used for representing a
number A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers c ...
with positional notation. Today's most common digits are the decimal digits "0", "1", "2", "3", "4", "5", "6", "7", "8", and "9". The distinction between a digit and a numeral is most pronounced in the context of a number base. A non-zero ''numeral'' with more than one digit position will mean a different number in a different number base, but in general, the ''digits'' will mean the same. For example, the base-8 numeral 238 contains two digits, "2" and "3", and with a base number (subscripted) "8". When converted to base-10, the 238 is equivalent to 1910, i.e. 238 = 1910. In our notation here, the subscript "8" of the numeral 238 is part of the numeral, but this may not always be the case. Imagine the numeral "23" as having an ambiguous base number. Then "23" could likely be any base, from base-4 up. In base-4, the "23" means 1110, i.e. 234 = 1110. In base-60, the "23" means the number 12310, i.e. 2360 = 12310. The numeral "23" then, in this case, corresponds to the set of base-10 numbers while its digits "2" and "3" always retain their original meaning: the "2" means "two of", and the "3" means "three of". In certain applications when a numeral with a fixed number of positions needs to represent a greater number, a higher number-base with more digits per position can be used. A three-digit, decimal numeral can represent only up to 999. But if the number-base is increased to 11, say, by adding the digit "A", then the same three positions, maximized to "AAA", can represent a number as great as 1330. We could increase the number base again and assign "B" to 11, and so on (but there is also a possible encryption between number and digit in the number-digit-numeral hierarchy). A three-digit numeral "ZZZ" in base-60 could mean . If we use the entire collection of our
alphanumerics Alphanumericals or alphanumeric characters are a combination of alphabetical and numbers, numerical Character (symbol), characters. More specifically, they are the collection of Latin alphabet, Latin letters and Arabic numerals, Arabic digits. A ...
we could ultimately serve a base-''62'' numeral system, but we remove two digits, uppercase "I" and uppercase "O", to reduce confusion with digits "1" and "0". We are left with a base-60, or sexagesimal numeral system utilizing 60 of the 62 standard alphanumerics. (But see ''
Sexagesimal system Sexagesimal, also known as base 60 or sexagenary, is a numeral system with sixty as its base. It originated with the ancient Sumerians in the 3rd millennium BC, was passed down to the ancient Babylonians, and is still used—in a modified form ...
'' below.) In general, the number of possible values that can be represented by a d digit number in base r is r^d. The common numeral systems in computer science are binary (radix 2), octal (radix 8), and hexadecimal (radix 16). In
binary Binary may refer to: Science and technology Mathematics * Binary number, a representation of numbers using only two digits (0 and 1) * Binary function, a function that takes two arguments * Binary operation, a mathematical operation that t ...
only digits "0" and "1" are in the numerals. In the
octal The octal numeral system, or oct for short, is the radix, base-8 number system, and uses the Numerical digit, digits 0 to 7. This is to say that 10octal represents eight and 100octal represents sixty-four. However, English, like most languages, ...
numerals, are the eight digits 0–7. Hex is 0–9 A–F, where the ten numerics retain their usual meaning, and the alphabetics correspond to values 10–15, for a total of sixteen digits. The numeral "10" is binary numeral "2", octal numeral "8", or hexadecimal numeral "16".


Radix point

The notation can be extended into the negative exponents of the base ''b''. Thereby the so-called radix point, mostly ».«, is used as separator of the positions with non-negative from those with negative exponent. Numbers that are not
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s use places beyond the
radix point A decimal separator is a symbol used to separate the integer part from the fractional part of a number written in decimal form (e.g., "." in 12.45). Different countries officially designate different symbols for use as the separator. The choi ...
. For every position behind this point (and thus after the units digit), the exponent ''n'' of the power ''b''''n'' decreases by 1 and the power approaches 0. For example, the number 2.35 is equal to: :2\times 10^0 + 3\times 10^ + 5\times 10^


Sign

If the base and all the digits in the set of digits are non-negative, negative numbers cannot be expressed. To overcome this, a
minus sign The plus and minus signs, and , are mathematical symbols used to represent the notions of positive and negative, respectively. In addition, represents the operation of addition, which results in a sum, while represents subtraction, resulti ...
, here »-«, is added to the numeral system. In the usual notation it is prepended to the string of digits representing the otherwise non-negative number.


Base conversion

The conversion to a base b_2 of an integer represented in base b_1 can be done by a succession of
Euclidean division In arithmetic, Euclidean division – or division with remainder – is the process of dividing one integer (the dividend) by another (the divisor), in a way that produces an integer quotient and a natural number remainder strictly smaller than ...
s by b_2: the right-most digit in base b_2 is the remainder of the division of by b_2; the second right-most digit is the remainder of the division of the quotient by b_2, and so on. The left-most digit is the last quotient. In general, the th digit from the right is the remainder of the division by b_2 of the th quotient. For example: converting A10BHex to decimal (41227): 0xA10B/10 = 0x101A R: 7 (ones place) 0x101A/10 = 0x19C R: 2 (tens place) 0x19C/10 = 0x29 R: 2 (hundreds place) 0x29/10 = 0x4 R: 1 ... 4 When converting to a larger base (such as from binary to decimal), the remainder represents b_2 as a single digit, using digits from b_1. For example: converting 0b11111001 (binary) to 249 (decimal): 0b11111001/10 = 0b11000 R: 0b1001 (0b1001 = "9" for ones place) 0b11000/10 = 0b10 R: 0b100 (0b100 = "4" for tens) 0b10/10 = 0b0 R: 0b10 (0b10 = "2" for hundreds) For the fractional part, conversion can be done by taking digits after the radix point (the numerator), and dividing it by the implied denominator in the target radix. Approximation may be needed due to a possibility of non-terminating digits if the reduced fraction's denominator has a prime factor other than any of the base's prime factor(s) to convert to. For example, 0.1 in decimal (1/10) is 0b1/0b1010 in binary, by dividing this in that radix, the result is 0b0.00011 (because one of the prime factors of 10 is 5). For more general fractions and bases see the algorithm for positive bases. In practice,
Horner's method In mathematics and computer science, Horner's method (or Horner's scheme) is an algorithm for polynomial evaluation. Although named after William George Horner, this method is much older, as it has been attributed to Joseph-Louis Lagrange by Hor ...
is more efficient than the repeated division required above. A number in positional notation can be thought of as a polynomial, where each digit is a coefficient. Coefficients can be larger than one digit, so an efficient way to convert bases is to convert each digit, then evaluate the polynomial via Horner's method within the target base. Converting each digit is a simple
lookup table In computer science, a lookup table (LUT) is an array that replaces runtime computation with a simpler array indexing operation. The process is termed as "direct addressing" and LUTs differ from hash tables in a way that, to retrieve a value v wi ...
, removing the need for expensive division or modulus operations; and multiplication by x becomes right-shifting. However, other polynomial evaluation algorithms would work as well, like
repeated squaring A rerun or repeat is a rebroadcast of an episode of a radio or television Broadcast programming, program. There are two types of reruns – those that occur during a Hiatus (television), hiatus, and those that occur when a program is Broadcast sy ...
for single or sparse digits. Example: Convert 0xA10B to 41227 A10B = (10*16^3) + (1*16^2) + (0*16^1) + (11*16^0) Lookup table: 0x0 = 0 0x1 = 1 ... 0x9 = 9 0xA = 10 0xB = 11 0xC = 12 0xD = 13 0xE = 14 0xF = 15 Therefore 0xA10B's decimal digits are 10, 1, 0, and 11. Lay out the digits out like this. The most significant digit (10) is "dropped": 10 1 0 11 <- Digits of 0xA10B --------------- 10 Then we multiply the bottom number from the source base (16), the product is placed under the next digit of the source value, and then add: 10 1 0 11 160 --------------- 10 161 Repeat until the final addition is performed: 10 1 0 11 160 2576 41216 --------------- 10 161 2576 41227 and that is 41227 in decimal. Convert 0b11111001 to 249 Lookup table: 0b0 = 0 0b1 = 1 Result: 1 1 1 1 1 0 0 1 <- Digits of 0b11111001 2 6 14 30 62 124 248 ------------------------- 1 3 7 15 31 62 124 249


Terminating fractions

The numbers which have a finite representation form the
semiring In abstract algebra, a semiring is an algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse. The term rig is also used occasionally—this originated as a joke, suggesting that rigs ar ...
:\frac := \left\ . More explicitly, if p_1^ \cdot \ldots \cdot p_n^ := b is a
factorization In mathematics, factorization (or factorisation, see American and British English spelling differences#-ise, -ize (-isation, -ization), English spelling differences) or factoring consists of writing a number or another mathematical object as a p ...
of b into the primes p_1, \ldots ,p_n \in \mathbb P with exponents then with the non-empty set of denominators S := \ we have : \Z_S := \left\ = b^ \, \Z = ^\Z where \langle S\rangle is the group generated by the p\in S and ^\Z is the so-called
localization Localization or localisation may refer to: Biology * Localization of function, locating psychological functions in the brain or nervous system; see Linguistic intelligence * Localization of sensation, ability to tell what part of the body is a ...
of \Z with respect to The
denominator A fraction (from la, fractus, "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 ...
of an element of \Z_S contains if reduced to lowest terms only prime factors out of S. This
ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
of all terminating fractions to base b is
dense Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ...
in the field of
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...
s \Q. Its completion for the usual (Archimedean) metric is the same as for \Q, namely the real numbers \R. So, if S = \ then \Z_ has not to be confused with \Z_ , the
discrete valuation ring In abstract algebra, a discrete valuation ring (DVR) is a principal ideal domain (PID) with exactly one non-zero maximal ideal. This means a DVR is an integral domain ''R'' which satisfies any one of the following equivalent conditions: # ''R'' ...
for the
prime A prime number (or a prime) is a natural number greater than 1 that is not a 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 because the only ways ...
p, which is equal to \Z_ with T = \mathbb P \setminus \ . If b divides c, we have b^ \, \Z \subseteq c^ \, \Z.


Infinite representations


Rational numbers

The representation of non-integers can be extended to allow an infinite string of digits beyond the point. For example, 1.12112111211112 ... base-3 represents the sum of the infinite
series Series may refer to: People with the name * Caroline Series (born 1951), English mathematician, daughter of George Series * George Series (1920–1995), English physicist Arts, entertainment, and media Music * Series, the ordered sets used i ...
: :\begin 1\times 3^ + \\ 1\times 3^ + 2\times 3^ + \\ 1\times 3^ + 1\times 3^ + 2\times 3^ + \\ 1\times 3^ + 1\times 3^ + 1\times 3^ + 2\times 3^ + \\ 1\times 3^ + 1\times 3^ + 1\times 3^ + 1\times 3^ + 2\times 3^ + \cdots \end Since a complete infinite string of digits cannot be explicitly written, the trailing ellipsis (...) designates the omitted digits, which may or may not follow a pattern of some kind. One common pattern is when a finite sequence of digits repeats infinitely. This is designated by drawing a vinculum across the repeating block: :2.42\overline_5 = 2.42314314314314314\dots_5 This is the repeating decimal notation (to which there does not exist a single universally accepted notation or phrasing). For base 10 it is called a repeating decimal or recurring decimal. An
irrational number In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integ ...
has an infinite non-repeating representation in all integer bases. Whether a
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...
has a finite representation or requires an infinite repeating representation depends on the base. For example, one third can be represented by: :0.1_3 :0.\overline3_ = 0.3333333\dots_ ::or, with the base implied: ::0.\overline3 = 0.3333333\dots (see also 0.999...) :0.\overline_2 = 0.010101\dots_2 :0.2_6 For integers ''p'' and ''q'' with ''gcd'' (''p'', ''q'') = 1, the
fraction A fraction (from la, fractus, "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 ...
''p''/''q'' has a finite representation in base ''b'' if and only if each
prime factor A prime number (or a prime) is a natural number greater than 1 that is not a 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 because the only ways ...
of ''q'' is also a prime factor of ''b''. For a given base, any number that can be represented by a finite number of digits (without using the bar notation) will have multiple representations, including one or two infinite representations: :1. A finite or infinite number of zeroes can be appended: ::3.46_7 = 3.460_7 = 3.460000_7 = 3.46\overline0_7 :2. The last non-zero digit can be reduced by one and an infinite string of digits, each corresponding to one less than the base, are appended (or replace any following zero digits): ::3.46_7 = 3.45\overline6_7 ::1_ = 0.\overline9_\qquad (see also 0.999...) ::220_5 = 214.\overline4_5


Irrational numbers

A (real) irrational number has an infinite non-repeating representation in all integer bases. Examples are the non-solvable ''n''th roots :y = \sqrt with y^n = x and , numbers which are called algebraic, or numbers like :\pi,e which are transcendental. The number of transcendentals is
uncountable In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal numb ...
and the sole way to write them down with a finite number of symbols is to give them a symbol or a finite sequence of symbols.


Applications


Decimal system

In the
decimal The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral ...
(base-10)
Hindu–Arabic numeral system The Hindu–Arabic numeral system or Indo-Arabic numeral system Audun HolmeGeometry: Our Cultural Heritage 2000 (also called the Hindu numeral system or Arabic numeral system) is a positional decimal numeral system, and is the most common syste ...
, each position starting from the right is a higher power of 10. The first position represents 100 (1), the second position 101 (10), the third position 102 ( or 100), the fourth position 103 ( or 1000), and so on.
Fraction A fraction (from la, fractus, "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 ...
al values are indicated by a separator, which can vary in different locations. Usually this separator is a period or
full stop The full stop (Commonwealth English), period (North American English), or full point , is a punctuation mark. It is used for several purposes, most often to mark the end of a declarative sentence (as distinguished from a question or exclamation ...
, or a
comma The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline ...
. Digits to the right of it are multiplied by 10 raised to a negative power or exponent. The first position to the right of the separator indicates 10−1 (0.1), the second position 10−2 (0.01), and so on for each successive position. As an example, the number 2674 in a base-10 numeral system is: :(2 × 103) + (6 × 102) + (7 × 101) + (4 × 100) or :(2 × 1000) + (6 × 100) + (7 × 10) + (4 × 1).


Sexagesimal system

The
sexagesimal Sexagesimal, also known as base 60 or sexagenary, is a numeral system with sixty as its base. It originated with the ancient Sumerians in the 3rd millennium BC, was passed down to the ancient Babylonians, and is still used—in a modified form ...
or base-60 system was used for the integral and fractional portions of
Babylonian numerals Assyro-Chaldean Babylonian cuneiform numerals were written in cuneiform (script), cuneiform, using a wedge-tipped Phragmites, reed stylus to make a mark on a soft clay tablet which would be exposed in the sun to harden to create a permanent record ...
and other Mesopotamian systems, by
Hellenistic In Classical antiquity, the Hellenistic period covers the time in Mediterranean history after Classical Greece, between the death of Alexander the Great in 323 BC and the emergence of the Roman Empire, as signified by the Battle of Actium in ...
astronomers using
Greek numerals Greek numerals, also known as Ionic, Ionian, Milesian, or Alexandrian numerals, are a system of writing numbers using the letters of the Greek alphabet. In modern Greece, they are still used for ordinal numbers and in contexts similar to tho ...
for the fractional portion only, and is still used for modern time and angles, but only for minutes and seconds. However, not all of these uses were positional. Modern time separates each position by a colon or a
prime symbol The prime symbol , double prime symbol , triple prime symbol , and quadruple prime symbol are used to designate units and for other purposes in mathematics, science, linguistics and music. Although the characters differ little in appearance fr ...
. For example, the time might be 10:25:59 (10 hours 25 minutes 59 seconds). Angles use similar notation. For example, an angle might be (10
degree Degree may refer to: As a unit of measurement * Degree (angle), a unit of angle measurement ** Degree of geographical latitude ** Degree of geographical longitude * Degree symbol (°), a notation used in science, engineering, and mathematics ...
s 25
minute The minute is a unit of time usually equal to (the first sexagesimal fraction) of an hour, or 60 seconds. In the UTC time standard, a minute on rare occasions has 61 seconds, a consequence of leap seconds (there is a provision to insert a nega ...
s 59
second The second (symbol: s) is the unit of time in the International System of Units (SI), historically defined as of a day – this factor derived from the division of the day first into 24 hours, then to 60 minutes and finally to 60 seconds ...
s). In both cases, only minutes and seconds use sexagesimal notation—angular degrees can be larger than 59 (one rotation around a circle is 360°, two rotations are 720°, etc.), and both time and angles use decimal fractions of a second. This contrasts with the numbers used by Hellenistic and
Renaissance The Renaissance ( , ) , from , with the same meanings. is a period in European history marking the transition from the Middle Ages to modernity and covering the 15th and 16th centuries, characterized by an effort to revive and surpass ideas ...
astronomers, who used
third Third or 3rd may refer to: Numbers * 3rd, the ordinal form of the cardinal number 3 * , a fraction of one third * Second#Sexagesimal divisions of calendar time and day, 1⁄60 of a ''second'', or 1⁄3600 of a ''minute'' Places * 3rd Street (d ...
s, fourths, etc. for finer increments. Where we might write , they would have written or 10°2559233112. Using a digit set of digits with upper and lowercase letters allows short notation for sexagesimal numbers, e.g. 10:25:59 becomes 'ARz' (by omitting I and O, but not i and o), which is useful for use in URLs, etc., but it is not very intelligible to humans. In the 1930s,
Otto Neugebauer Otto Eduard Neugebauer (May 26, 1899 – February 19, 1990) was an Austrian-American mathematician and historian of science who became known for his research on the history of astronomy and the other exact sciences as they were practiced in anti ...
introduced a modern notational system for Babylonian and Hellenistic numbers that substitutes modern decimal notation from 0 to 59 in each position, while using a semicolon (;) to separate the integral and fractional portions of the number and using a comma (,) to separate the positions within each portion. For example, the mean
synodic month In lunar calendars, a lunar month is the time between two successive syzygies of the same type: new moons or full moons. The precise definition varies, especially for the beginning of the month. Variations In Shona, Middle Eastern, and Europ ...
used by both Babylonian and Hellenistic astronomers and still used in the
Hebrew calendar The Hebrew calendar ( he, הַלּוּחַ הָעִבְרִי, translit=HaLuah HaIvri), also called the Jewish calendar, is a lunisolar calendar used today for Jewish religious observance, and as an official calendar of the state of Israel. I ...
is 29;31,50,8,20 days, and the angle used in the example above would be written 10;25,59,23,31,12 degrees.


Computing

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 hardware and software. Computing has scientific, e ...
, the
binary Binary may refer to: Science and technology Mathematics * Binary number, a representation of numbers using only two digits (0 and 1) * Binary function, a function that takes two arguments * Binary operation, a mathematical operation that t ...
(base-2), octal (base-8) and
hexadecimal In mathematics and computing, the hexadecimal (also base-16 or simply hex) numeral system is a positional numeral system that represents numbers using a radix (base) of 16. Unlike the decimal system representing numbers using 10 symbols, hexa ...
(base-16) bases are most commonly used. Computers, at the most basic level, deal only with sequences of conventional zeroes and ones, thus it is easier in this sense to deal with powers of two. The hexadecimal system is used as "shorthand" for binary—every 4 binary digits (bits) relate to one and only one hexadecimal digit. In hexadecimal, the six digits after 9 are denoted by A, B, C, D, E, and F (and sometimes a, b, c, d, e, and f). The
octal The octal numeral system, or oct for short, is the radix, base-8 number system, and uses the Numerical digit, digits 0 to 7. This is to say that 10octal represents eight and 100octal represents sixty-four. However, English, like most languages, ...
numbering system is also used as another way to represent binary numbers. In this case the base is 8 and therefore only digits 0, 1, 2, 3, 4, 5, 6, and 7 are used. When converting from binary to octal every 3 bits relate to one and only one octal digit. Hexadecimal, decimal, octal, and a wide variety of other bases have been used for
binary-to-text encoding A binary-to-text encoding is encoding of data in plain text. More precisely, it is an encoding of binary data in a sequence of printable characters. These encodings are necessary for transmission of data when the channel does not allow binary dat ...
, implementations of
arbitrary-precision arithmetic In computer science, arbitrary-precision arithmetic, also called bignum arithmetic, multiple-precision arithmetic, or sometimes infinite-precision arithmetic, indicates that calculations are performed on numbers whose digits of precision are li ...
, and other applications. ''For a list of bases and their applications, see
list of numeral systems There are many different numeral systems, that is, writing systems for expressing numbers. By culture / time period By type of notation Numeral systems are classified here as to whether they use positional notation (also known as place-value ...
.''


Other bases in human language

Base-12 systems (
duodecimal The duodecimal system (also known as base 12, dozenal, or, rarely, uncial) is a positional notation numeral system using twelve as its base. The number twelve (that is, the number written as "12" in the decimal numerical system) is instead wri ...
or dozenal) have been popular because multiplication and division are easier than in base-10, with addition and subtraction being just as easy. Twelve is a useful base because it has many
factors Factor, a Latin word meaning "who/which acts", may refer to: Commerce * Factor (agent), a person who acts for, notably a mercantile and colonial agent * Factor (Scotland), a person or firm managing a Scottish estate * Factors of production, suc ...
. It is the smallest common multiple of one, two, three, four and six. There is still a special word for "dozen" in English, and by analogy with the word for 102, ''hundred'', commerce developed a word for 122, ''gross''. The standard 12-hour clock and common use of 12 in English units emphasize the utility of the base. In addition, prior to its conversion to decimal, the old British currency
Pound Sterling Sterling (abbreviation: stg; Other spelling styles, such as STG and Stg, are also seen. ISO code: GBP) is the currency of the United Kingdom and nine of its associated territories. The pound ( sign: £) is the main unit of sterling, and t ...
(GBP) ''partially'' used base-12; there were 12 pence (d) in a shilling (s), 20 shillings in a pound (£), and therefore 240 pence in a pound. Hence the term LSD or, more properly,
£sd £sd (occasionally written Lsd, spoken as "pounds, shillings and pence" or pronounced ) is the popular name for the pre-decimal currencies once common throughout Europe, especially in the British Isles and hence in several countries of the B ...
. The
Maya civilization The Maya civilization () of the Mesoamerican people is known by its ancient temples and glyphs. Its Maya script is the most sophisticated and highly developed writing system in the pre-Columbian Americas. It is also noted for its art, archit ...
and other civilizations of
pre-Columbian In the history of the Americas, the pre-Columbian era spans from the original settlement of North and South America in the Upper Paleolithic period through European colonization, which began with Christopher Columbus's voyage of 1492. Usually, th ...
Mesoamerica Mesoamerica is a historical region and cultural area in southern North America and most of Central America. It extends from approximately central Mexico through Belize, Guatemala, El Salvador, Honduras, Nicaragua, and northern Costa Rica. W ...
used base-20 (
vigesimal vigesimal () or base-20 (base-score) numeral system is based on twenty (in the same way in which the decimal numeral system is based on ten). '' Vigesimal'' is derived from the Latin adjective '' vicesimus'', meaning 'twentieth'. Places In ...
), as did several North American tribes (two being in southern California). Evidence of base-20 counting systems is also found in the languages of central and western
Africa Africa is the world's second-largest and second-most populous continent, after Asia in both cases. At about 30.3 million km2 (11.7 million square miles) including adjacent islands, it covers 6% of Earth's total surface area ...
. Remnants of a
Gaulish Gaulish was an ancient Celtic languages, Celtic language spoken in parts of Continental Europe before and during the period of the Roman Empire. In the narrow sense, Gaulish was the language of the Celts of Gaul (now France, Luxembourg, Belgium ...
base-20 system also exist in French, as seen today in the names of the numbers from 60 through 99. For example, sixty-five is ''soixante-cinq'' (literally, "sixty ndfive"), while seventy-five is ''soixante-quinze'' (literally, "sixty ndfifteen"). Furthermore, for any number between 80 and 99, the "tens-column" number is expressed as a multiple of twenty. For example, eighty-two is ''quatre-vingt-deux'' (literally, four twenty ndtwo), while ninety-two is ''quatre-vingt-douze'' (literally, four twenty ndtwelve). In Old French, forty was expressed as two twenties and sixty was three twenties, so that fifty-three was expressed as two twenties ndthirteen, and so on. In English the same base-20 counting appears in the use of " scores". Although mostly historical, it is occasionally used colloquially. Verse 10 of Psalm 90 in the King James Version of the Bible starts: "The days of our years are threescore years and ten; and if by reason of strength they be fourscore years, yet is their strength labour and sorrow". The Gettysburg Address starts: "Four score and seven years ago". The
Irish language Irish ( Standard Irish: ), also known as Gaelic, is a Goidelic language of the Insular Celtic branch of the Celtic language family, which is a part of the Indo-European language family. Irish is indigenous to the island of Ireland and was ...
also used base-20 in the past, twenty being ''fichid'', forty ''dhá fhichid'', sixty ''trí fhichid'' and eighty ''ceithre fhichid''. A remnant of this system may be seen in the modern word for 40, ''daoichead''. The
Welsh language Welsh ( or ) is a Celtic language family, Celtic language of the Brittonic languages, Brittonic subgroup that is native to the Welsh people. Welsh is spoken natively in Wales, by some in England, and in Y Wladfa (the Welsh colony in Chubut P ...
continues to use a
base-20 vigesimal () or base-20 (base-score) numeral system is based on twenty (in the same way in which the decimal numeral system is based on ten). '' Vigesimal'' is derived from the Latin adjective '' vicesimus'', meaning 'twentieth'. Places In ...
counting system In linguistics, a numeral (or number word) in the broadest sense is a word or phrase that describes a numerical quantity. Some theories of grammar use the word "numeral" to refer to cardinal numbers that act as a determiner that specify the quant ...
, particularly for the age of people, dates and in common phrases. 15 is also important, with 16–19 being "one on 15", "two on 15" etc. 18 is normally "two nines". A decimal system is commonly used. The
Inuit languages The Inuit languages are a closely related group of indigenous American languages traditionally spoken across the North American Arctic and adjacent subarctic, reaching farthest south in Labrador. The related Yupik languages (spoken in western ...
use a
base-20 vigesimal () or base-20 (base-score) numeral system is based on twenty (in the same way in which the decimal numeral system is based on ten). '' Vigesimal'' is derived from the Latin adjective '' vicesimus'', meaning 'twentieth'. Places In ...
counting system. Students from
Kaktovik, Alaska Kaktovik (; ik, Qaaktuġvik, ) is a City (Alaska), city in North Slope Borough, Alaska, North Slope Borough, Alaska, United States. The population was 283 at the 2020 United States census, 2020 census. History Until the late nineteenth century ...
invented a base-20 numeral system in 1994 Danish numerals display a similar
base-20 vigesimal () or base-20 (base-score) numeral system is based on twenty (in the same way in which the decimal numeral system is based on ten). '' Vigesimal'' is derived from the Latin adjective '' vicesimus'', meaning 'twentieth'. Places In ...
structure. The
Māori language Māori (), or ('the Māori language'), also known as ('the language'), is an Eastern Polynesian language spoken by the Māori people, the indigenous population of mainland New Zealand. Closely related to Cook Islands Māori, Tuamotuan, and ...
of New Zealand also has evidence of an underlying base-20 system as seen in the terms ''Te Hokowhitu a Tu'' referring to a war party (literally "the seven 20s of Tu") and ''Tama-hokotahi'', referring to a great warrior ("the one man equal to 20"). The binary system was used in the Egyptian Old Kingdom, 3000 BC to 2050 BC. It was cursive by rounding off rational numbers smaller than 1 to , with a 1/64 term thrown away (the system was called the
Eye of Horus The Eye of Horus, ''wedjat'' eye or ''udjat'' eye is a concept and symbol in ancient Egyptian religion that represents well-being, healing, and protection. It derives from the mythical conflict between the god Horus with his rival Set, in wh ...
). A number of
Australian Aboriginal languages The Indigenous languages of Australia number in the hundreds, the precise number being quite uncertain, although there is a range of estimates from a minimum of around 250 (using the technical definition of 'language' as non-mutually intellig ...
employ binary or binary-like counting systems. For example, in
Kala Lagaw Ya ''Kalau Lagau Ya'', ''Kalaw Lagaw Ya'', ''Kala Lagaw Ya'' (), or the ''Western Torres Strait language'' (also several other names, see below), is the language indigenous to the central and western Torres Strait Islands, Queensland, Australia. O ...
, the numbers one through six are ''urapon'', ''ukasar'', ''ukasar-urapon'', ''ukasar-ukasar'', ''ukasar-ukasar-urapon'', ''ukasar-ukasar-ukasar''. North and Central American natives used base-4 (
quaternary The Quaternary ( ) is the current and most recent of the three periods of the Cenozoic Era in the geologic time scale of the International Commission on Stratigraphy (ICS). It follows the Neogene Period and spans from 2.58 million years ...
) to represent the four cardinal directions. Mesoamericans tended to add a second base-5 system to create a modified base-20 system. A base-5 system (
quinary Quinary (base-5 or pental) is a numeral system with five as the base. A possible origination of a quinary system is that there are five digits on either hand. In the quinary place system, five numerals, from 0 to 4, are used to represent an ...
) has been used in many cultures for counting. Plainly it is based on the number of digits on a human hand. It may also be regarded as a sub-base of other bases, such as base-10, base-20, and base-60. A base-8 system (
octal The octal numeral system, or oct for short, is the radix, base-8 number system, and uses the Numerical digit, digits 0 to 7. This is to say that 10octal represents eight and 100octal represents sixty-four. However, English, like most languages, ...
) was devised by the
Yuki tribe The Yuki (also known as Yukiah) are an indigenous people of California, whose traditional territory is around Round Valley, Mendocino County. Today they are enrolled members of the Round Valley Indian Tribes of the Round Valley Reservation. Before ...
of Northern California, who used the spaces between the fingers to count, corresponding to the digits one through eight. There is also linguistic evidence which suggests that the Bronze Age
Proto-Indo European Proto-Indo-European (PIE) is the reconstructed common ancestor of the Indo-European language family. Its proposed features have been derived by linguistic reconstruction from documented Indo-European languages. No direct record of Proto-Indo-E ...
s (from whom most European and Indic languages descend) might have replaced a base-8 system (or a system which could only count up to 8) with a base-10 system. The evidence is that the word for 9, ''newm'', is suggested by some to derive from the word for "new", ''newo-'', suggesting that the number 9 had been recently invented and called the "new number". Many ancient counting systems use five as a primary base, almost surely coming from the number of fingers on a person's hand. Often these systems are supplemented with a secondary base, sometimes ten, sometimes twenty. In some
African languages The languages of Africa are divided into several major language families: * Niger–Congo or perhaps Atlantic–Congo languages (includes Bantu and non-Bantu, and possibly Mande and others) are spoken in West, Central, Southeast and Southern A ...
the word for five is the same as "hand" or "fist" (
Dyola language Jola (french: Diola; Jola: ''Joola''), also called Jola-Fonyi (french: Diola-Fogny) and Kujamataak, is a language spoken by half a million people in the Casamance region of Senegal, and neighboring countries. Jola-Fonyi is one of several closely ...
of
Guinea-Bissau Guinea-Bissau ( ; pt, Guiné-Bissau; ff, italic=no, 𞤘𞤭𞤲𞤫 𞤄𞤭𞤧𞤢𞥄𞤱𞤮, Gine-Bisaawo, script=Adlm; Mandinka: ''Gine-Bisawo''), officially the Republic of Guinea-Bissau ( pt, República da Guiné-Bissau, links=no ) ...
, Banda language of
Central Africa Central Africa is a subregion of the African continent comprising various countries according to different definitions. Angola, Burundi, the Central African Republic, Chad, the Democratic Republic of the Congo, the Republic of the Congo, ...
). Counting continues by adding 1, 2, 3, or 4 to combinations of 5, until the secondary base is reached. In the case of twenty, this word often means "man complete". This system is referred to as ''quinquavigesimal''. It is found in many languages of the
Sudan Sudan ( or ; ar, السودان, as-Sūdān, officially the Republic of the Sudan ( ar, جمهورية السودان, link=no, Jumhūriyyat as-Sūdān), is a country in Northeast Africa. It shares borders with the Central African Republic t ...
region. The
Telefol language Telefol is a language spoken by the Telefol people in Papua New Guinea, notable for possessing a base-27 numeral system. History The Iligimin people also spoke Telefol, but they were defeated by the Telefol proper. Orthography Single and r ...
, spoken in
Papua New Guinea Papua New Guinea (abbreviated PNG; , ; tpi, Papua Niugini; ho, Papua Niu Gini), officially the Independent State of Papua New Guinea ( tpi, Independen Stet bilong Papua Niugini; ho, Independen Stet bilong Papua Niu Gini), is a country i ...
, is notable for possessing a base-27 numeral system.


Non-standard positional numeral systems

Interesting properties exist when the base is not fixed or positive and when the digit symbol sets denote negative values. There are many more variations. These systems are of practical and theoretic value to computer scientists.
Balanced ternary Balanced ternary is a ternary numeral system (i.e. base 3 with three digits) that uses a balanced signed-digit representation of the integers in which the digits have the values −1, 0, and 1. This stands in contrast to the standard (unbalance ...
uses a base of 3 but the digit set is instead of . The "" has an equivalent value of −1. The negation of a number is easily formed by switching the on the 1s. This system can be used to solve the
balance problem Balance or balancing may refer to: Common meanings * Balance (ability) in biomechanics * Balance (accounting) * Balance or weighing scale * Equality (mathematics), Balance as in equality or equilibrium Arts and entertainment Film * Balance (1983 ...
, which requires finding a minimal set of known counter-weights to determine an unknown weight. Weights of 1, 3, 9, ... 3''n'' known units can be used to determine any unknown weight up to 1 + 3 + ... + 3''n'' units. A weight can be used on either side of the balance or not at all. Weights used on the balance pan with the unknown weight are designated with , with 1 if used on the empty pan, and with 0 if not used. If an unknown weight ''W'' is balanced with 3 (31) on its pan and 1 and 27 (30 and 33) on the other, then its weight in decimal is 25 or 101 in balanced base-3. : The
factorial number system In combinatorics, the factorial number system, also called factoradic, is a mixed radix numeral system adapted to numbering permutations. It is also called factorial base, although factorials do not function as base, but as place value of digit ...
uses a varying radix, giving
factorial In mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times (n-1) \times (n-2) \t ...
s as place values; they are related to
Chinese remainder theorem In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer ''n'' by several integers, then one can determine uniquely the remainder of the division of ''n'' by the product of thes ...
and
residue number system A residue numeral system (RNS) is a numeral system representing integers by their values modulo several pairwise coprime integers called the moduli. This representation is allowed by the Chinese remainder theorem, which asserts that, if is the pro ...
enumerations. This system effectively enumerates permutations. A derivative of this uses the
Towers of Hanoi The Tower of Hanoi (also called The problem of Benares Temple or Tower of Brahma or Lucas' Tower and sometimes pluralized as Towers, or simply pyramid puzzle) is a mathematical game or puzzle consisting of three rods and a number of disks of va ...
puzzle configuration as a counting system. The configuration of the towers can be put into 1-to-1 correspondence with the decimal count of the step at which the configuration occurs and vice versa.


Non-positional positions

Each position does not need to be positional itself. Babylonian sexagesimal numerals were positional, but in each position were groups of two kinds of wedges representing ones and tens (a narrow vertical wedge , for the one and an open left pointing wedge ⟨ for the ten) — up to 5+9=14 symbols per position (i.e. 5 tens ⟨⟨⟨⟨⟨ and 9 ones , , , , , , , , , grouped into one or two near squares containing up to three tiers of symbols, or a place holder (\\) for the lack of a position). Hellenistic astronomers used one or two alphabetic Greek numerals for each position (one chosen from 5 letters representing 10–50 and/or one chosen from 9 letters representing 1–9, or a zero symbol).Ifrah, pages 261–264


See also

Examples: *
List of numeral systems There are many different numeral systems, that is, writing systems for expressing numbers. By culture / time period By type of notation Numeral systems are classified here as to whether they use positional notation (also known as place-value ...
* : Positional numeral systems Related topics: *
Algorism Algorism is the technique of performing basic arithmetic by writing numbers in place value form and applying a set of memorized rules and mathematical table, facts to the digits. One who practices algorism is known as an algorist. This positiona ...
*
Hindu–Arabic numeral system The Hindu–Arabic numeral system or Indo-Arabic numeral system Audun HolmeGeometry: Our Cultural Heritage 2000 (also called the Hindu numeral system or Arabic numeral system) is a positional decimal numeral system, and is the most common syste ...
*
Mixed radix Mixed radix numeral systems are non-standard positional numeral systems in which the numerical radix, base varies from position to position. Such numerical representation applies when a quantity is expressed using a sequence of units that are eac ...
*
Non-standard positional numeral systems Non-standard positional numeral systems here designates numeral systems that may loosely be described as positional systems, but that do not entirely comply with the following description of standard positional systems: :In a standard positional ...
*
Numeral system A numeral system (or system of numeration) is a writing system for expressing numbers; that is, a mathematical notation for representing numbers of a given set, using Numerical digit, digits or other symbols in a consistent manner. The same s ...
*
Scientific notation Scientific notation is a way of expressing numbers that are too large or too small (usually would result in a long string of digits) to be conveniently written in decimal form. It may be referred to as scientific form or standard index form, o ...
Other: *
Significant figures Significant figures (also known as the significant digits, ''precision'' or ''resolution'') of a number in positional notation are digits in the number that are reliable and necessary to indicate the quantity of something. If a number expre ...


Notes


References

* * * * *


External links


Accurate Base ConversionThe Development of Hindu Arabic and Traditional Chinese ArithmeticsImplementation of Base Conversion
at
cut-the-knot Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet-born Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow Institute of Electronics and Math ...

Learn to count other bases on your fingersOnline Arbitrary Precision Base Converter
{{DEFAULTSORT:Positional Notation Mathematical notation Articles containing proofs