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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
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 can ...
s in a
positional Positional notation (or place-value notation, or positional numeral system) usually denotes 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 ...
numeral system. The name "digit" comes from the fact that the ten digits (
Latin Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power of the ...
''digiti'' meaning fingers) of the hands correspond to the ten symbols of the common base 10
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 digits or other symbols in a consistent manner. The same sequence of symbo ...
, i.e. the decimal (ancient Latin adjective ''decem'' meaning ten) digits. For a given numeral system with an integer base, the number of different digits required is given by 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 ...
of the base. For example, the decimal system (base 10) requires ten digits (0 through to 9), whereas the
binary system A binary system is a system of two astronomical bodies which are close enough that their gravitational attraction causes them to orbit each other around a barycenter ''(also see animated examples)''. More restrictive definitions require that thi ...
(base 2) requires two digits (0 and 1).


Overview

In a basic digital system, a numeral is a sequence of digits, which may be of arbitrary length. Each position in the sequence has a
place value Positional notation (or place-value notation, or positional numeral system) usually denotes 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 ...
, and each digit has a value. The value of the numeral is computed by multiplying each digit in the sequence by its place value, and summing the results.


Digital values

Each digit in a number system represents an integer. For example, in
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 ...
the digit "1" represents the integer
one 1 (one, unit, unity) is a number representing a single or the only entity. 1 is also a numerical digit and represents a single unit of counting or measurement. For example, a line segment of ''unit length'' is a line segment of length 1. I ...
, and in the
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 ...
system, the letter "A" represents the number
ten Ten, TEN or 10 may refer to: * 10, an even natural number following 9 and preceding 11 * one of the years 10 BC, AD 10, 1910 and 2010 * October, the tenth month of the year Places * Mount Ten, in Vietnam * Tongren Fenghuang Airport (IATA cod ...
. A
positional number system Positional notation (or place-value notation, or positional numeral system) usually denotes 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 ...
has one unique digit for each integer from
zero 0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usuall ...
up to, but not including, 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 ...
of the number system. Thus in the positional decimal system, the numbers 0 to 9 can be expressed using their respective numerals "0" to "9" in the rightmost "units" position. The number 12 can be expressed with the numeral "2" in the units position, and with the numeral "1" in the "tens" position, to the left of the "2" while the number 312 can be expressed by three numerals: "3" in the "hundreds" position, "1" in the "tens" position, and "2" in the "units" position.


Computation of place values

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 ...
numeral system uses a
decimal separator 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 cho ...
, commonly a
period Period may refer to: Common uses * Era, a length or span of time * Full stop (or period), a punctuation mark Arts, entertainment, and media * Period (music), a concept in musical composition * Periodic sentence (or rhetorical period), a concept ...
in English, 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 o ...
in other European languages, to denote the "ones place" or "units place", which has a place value one. Each successive place to the left of this has a place value equal to the place value of the previous digit times the base. Similarly, each successive place to the right of the separator has a place value equal to the place value of the previous digit divided by the base. For example, in the numeral 10.34 (written in base 10), :the 0 is immediately to the left of the separator, so it is in the ones or units place, and is called the ''units digit'' or ''ones digit''; :the 1 to the left of the ones place is in the tens place, and is called the ''tens digit''; :the 3 is to the right of the ones place, so it is in the tenths place, and is called the ''tenths digit''; :the 4 to the right of the tenths place is in the hundredths place, and is called the ''hundredths digit''. The total value of the number is 1 ten, 0 ones, 3 tenths, and 4 hundredths. Note that the zero, which contributes no value to the number, indicates that the 1 is in the tens place rather than the ones place. The place value of any given digit in a numeral can be given by a simple calculation, which in itself is a complement to the logic behind numeral systems. The calculation involves the multiplication of the given digit by the base raised by the exponent , where ''n'' represents the position of the digit from the separator; the value of ''n'' is positive (+), but this is only if the digit is to the left of the separator. And to the right, the digit is multiplied by the base raised by a negative (−) ''n''. For example, in the number 10.34 (written in base 10), :the 1 is second to the left of the separator, so based on calculation, its value is, :n - 1 = 2 - 1 = 1 :1 \times 10^1 = 10 :the 4 is second to the right of the separator, so based on calculation its value is, :n = -2 :4 \times 10^ = \frac


History

The first true written
positional numeral system Positional notation (or place-value notation, or positional numeral system) usually denotes 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 ...
is considered to be 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 system ...
. This system was established by the 7th century in India,O'Connor, J. J. and Robertson, E. F
Arabic Numerals
January 2001. Retrieved on 2007-02-20.
but was not yet in its modern form because the use of the digit
zero 0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usuall ...
had not yet been widely accepted. Instead of a zero sometimes the digits were marked with dots to indicate their significance, or a space was used as a placeholder. The first widely acknowledged use of zero was in 876. The original numerals were very similar to the modern ones, even down to the
glyph A glyph () is any kind of purposeful mark. In typography, a glyph is "the specific shape, design, or representation of a character". It is a particular graphical representation, in a particular typeface, of an element of written language. A g ...
s used to represent digits. By the 13th century,
Western Arabic numerals Arabic numerals are the ten numerical digits: , , , , , , , , and . They are the most commonly used symbols to write decimal numbers. They are also used for writing numbers in other systems such as octal, and for writing identifiers such as ...
were accepted in European mathematical circles (
Fibonacci Fibonacci (; also , ; – ), also known as Leonardo Bonacci, Leonardo of Pisa, or Leonardo Bigollo Pisano ('Leonardo the Traveller from Pisa'), was an Italian mathematician from the Republic of Pisa, considered to be "the most talented Western ...
used them in his ''
Liber Abaci ''Liber Abaci'' (also spelled as ''Liber Abbaci''; "The Book of Calculation") is a historic 1202 Latin manuscript on arithmetic by Leonardo of Pisa, posthumously known as Fibonacci. ''Liber Abaci'' was among the first Western books to describe ...
''). They began to enter common use in the 15th century. By the end of the 20th century virtually all non-computerized calculations in the world were done with Arabic numerals, which have replaced native numeral systems in most cultures.


Other historical numeral systems using digits

The exact age of the
Maya numerals The Mayan numeral system was the system to represent numbers and calendar dates in the Maya civilization. It was a vigesimal (base-20) positional numeral system. The numerals are made up of three symbols; zero (a shell), one (a dot) and fi ...
is unclear, but it is possible that it is older than the Hindu–Arabic system. The system was
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 ...
(base 20), so it has twenty digits. The Mayas used a shell symbol to represent zero. Numerals were written vertically, with the ones place at the bottom. The
Mayas The Maya peoples () are an ethnolinguistic group of indigenous peoples of Mesoamerica. The ancient Maya civilization was formed by members of this group, and today's Maya are generally descended from people who lived within that historical reg ...
had no equivalent of the modern
decimal separator 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 cho ...
, so their system could not represent fractions. The Thai numeral system is identical to 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 system ...
except for the symbols used to represent digits. The use of these digits is less common in
Thailand Thailand ( ), historically known as Siam () and officially the Kingdom of Thailand, is a country in Southeast Asia, located at the centre of the Indochinese Peninsula, spanning , with a population of almost 70 million. The country is b ...
than it once was, but they are still used alongside Arabic numerals. The rod numerals, the written forms of counting rods once used by
Chinese Chinese can refer to: * Something related to China * Chinese people, people of Chinese nationality, citizenship, and/or ethnicity **''Zhonghua minzu'', the supra-ethnic concept of the Chinese nation ** List of ethnic groups in China, people of v ...
and
Japan Japan ( ja, 日本, or , and formally , ''Nihonkoku'') is an island country in East Asia. It is situated in the northwest Pacific Ocean, and is bordered on the west by the Sea of Japan, while extending from the Sea of Okhotsk in the north ...
ese mathematicians, are a decimal positional system able to represent not only zero but also negative numbers. Counting rods themselves predate the Hindu–Arabic numeral system. The
Suzhou numerals The Suzhou numerals, also known as ' (), is a numeral system used in China before the introduction of Arabic numerals. The Suzhou numerals are also known as ' (), ' (), ' (), ' () and ' (). History The Suzhou numeral system is the only survivin ...
are variants of rod numerals.


Modern digital systems


In computer science

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 ta ...
(base 2),
octal The octal numeral system, or oct for short, is the base-8 number system, and uses the digits 0 to 7. This is to say that 10octal represents eight and 100octal represents sixty-four. However, English, like most languages, uses a base-10 number ...
(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) systems, extensively used in computer science, all follow the conventions 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 system ...
. The binary system uses only the digits "0" and "1", while the octal system uses the digits from "0" through "7". The hexadecimal system uses all the digits from the decimal system, plus the letters "A" through "F", which represent the numbers 10 to 15 respectively. When the binary system is used, the term "bit(s)" is typically used as a alternative for "digit(s)", being a portmanteau of the term "binary digit". Similar terms exist for other number systems, such as "trit(s)" for a ternary system and "dit(s) for the decimal system, although less frequently used.


Unusual systems

The ternary and
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 (unbala ...
systems have sometimes been used. They are both base 3 systems. Balanced ternary is unusual in having the digit values 1, 0 and –1. Balanced ternary turns out to have some useful properties and the system has been used in the experimental Russian Setun computers. Several authors in the last 300 years have noted a facility of
positional 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 decimal system). More generally, a positional system is a numeral system in which the ...
that amounts to a ''modified''
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, i ...
. Some advantages are cited for use of numerical digits that represent negative values. In 1840
Augustin-Louis Cauchy Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He w ...
advocated use of
signed-digit representation In mathematical notation for numbers, a signed-digit representation is a positional numeral system with a set of signed digits used to encode the integers. Signed-digit representation can be used to accomplish fast addition of integers because ...
of numbers, and in 1928
Florian Cajori Florian Cajori (February 28, 1859 – August 14 or 15, 1930) was a Swiss-American historian of mathematics. Biography Florian Cajori was born in Zillis, Switzerland, as the son of Georg Cajori and Catherine Camenisch. He attended schools first ...
presented his collection of references for negative numerals. The concept of signed-digit representation has also been taken up in
computer design In computer engineering, computer architecture is a description of the structure of a computer system made from component parts. It can sometimes be a high-level description that ignores details of the implementation. At a more detailed level, the ...
.


Digits in mathematics

Despite the essential role of digits in describing numbers, they are relatively unimportant to modern mathematics. Nevertheless, there are a few important mathematical concepts that make use of the representation of a number as a sequence of digits.


Digital roots

The digital root is the single-digit number obtained by summing the digits of a given number, then summing the digits of the result, and so on until a single-digit number is obtained.


Casting out nines

Casting out nines Casting out nines is any of three arithmetical procedures: *Adding the decimal digits of a positive whole number, while optionally ignoring any 9s or digits which sum to a multiple of 9. The result of this procedure is a number which is smaller th ...
is a procedure for checking arithmetic done by hand. To describe it, let f(x) represent the
digital root The digital root (also repeated digital sum) of a natural number in a given radix is the (single digit) value obtained by an iterative process of summing digits, on each iteration using the result from the previous iteration to compute a digit s ...
of x, as described above. Casting out nines makes use of the fact that if A + B = C, then f(f(A) + f(B)) = f(C). In the process of casting out nines, both sides of the latter
equation In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for example, in ...
are computed, and if they are not equal, the original addition must have been faulty.


Repunits and repdigits

Repunits are integers that are represented with only the digit 1. For example, 1111 (one thousand, one hundred and eleven) is a repunit.
Repdigit In recreational mathematics, a repdigit or sometimes monodigit is a natural number composed of repeated instances of the same digit in a positional number system (often implicitly decimal). The word is a portmanteau of repeated and digit. Example ...
s are a generalization of repunits; they are integers represented by repeated instances of the same digit. For example, 333 is a repdigit. The
primality 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 repunits is of interest to mathematicians.


Palindromic numbers and Lychrel numbers

Palindromic numbers are numbers that read the same when their digits are reversed. A
Lychrel number A Lychrel number is a natural number that cannot form a palindrome through the iterative process of repeatedly reversing its digits and adding the resulting numbers. This process is sometimes called the ''196-algorithm'', after the most famous numb ...
is a positive integer that never yields a palindromic number when subjected to the iterative process of being added to itself with digits reversed. The question of whether there are any Lychrel numbers in base 10 is an open problem in
recreational mathematics Recreational mathematics is mathematics carried out for recreation (entertainment) rather than as a strictly research and application-based professional activity or as a part of a student's formal education. Although it is not necessarily limited ...
; the smallest candidate is
196 Year 196 ( CXCVI) was a leap year starting on Thursday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Dexter and Messalla (or, less frequently, year 949 ''Ab urbe condita ...
.


History of ancient numbers

Counting aids, especially the use of body parts (counting on fingers), were certainly used in prehistoric times as today. There are many variations. Besides counting ten fingers, some cultures have counted knuckles, the space between fingers, and toes as well as fingers. The Oksapmin culture of New Guinea uses a system of 27 upper body locations to represent numbers. To preserve numerical information, tallies carved in wood, bone, and stone have been used since prehistoric times. Stone age cultures, including ancient indigenous American groups, used tallies for gambling, personal services, and trade-goods. A method of preserving numeric information in clay was invented by the
Sumer Sumer () is the earliest known civilization in the historical region of southern Mesopotamia (south-central Iraq), emerging during the Chalcolithic and early Bronze Ages between the sixth and fifth millennium BC. It is one of the cradles of ...
ians between 8000 and 3500 BC. This was done with small clay tokens of various shapes that were strung like beads on a string. Beginning about 3500 BC, clay tokens were gradually replaced by number signs impressed with a round stylus at different angles in clay tablets (originally containers for tokens) which were then baked. About 3100  BC, written numbers were dissociated from the things being counted and became abstract numerals. Between 2700 and 2000 BC, in Sumer, the round stylus was gradually replaced by a reed stylus that was used to press wedge-shaped cuneiform signs in clay. These cuneiform number signs resembled the round number signs they replaced and retained the additive
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  ...
of the round number signs. These systems gradually converged on a common
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� ...
number system; this was a place-value system consisting of only two impressed marks, the vertical wedge and the chevron, which could also represent fractions. This sexagesimal number system was fully developed at the beginning of the Old Babylonia period (about 1950 BC) and became standard in Babylonia.
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� ...
numerals were a
mixed radix Mixed radix numeral systems are non-standard positional numeral systems in which the numerical base varies from position to position. Such numerical representation applies when a quantity is expressed using a sequence of units that are each a mu ...
system that retained the alternating base 10 and base 6 in a sequence of cuneiform vertical wedges and chevrons. By 1950 BC, this was a
positional 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 decimal system). More generally, a positional system is a numeral system in which the ...
system. Sexagesimal numerals came to be widely used in commerce, but were also used in astronomical and other calculations. This system was exported from Babylonia and used throughout Mesopotamia, and by every Mediterranean nation that used standard Babylonian units of measure and counting, including the Greeks, Romans and Egyptians. Babylonian-style sexagesimal numeration is still used in modern societies to measure
time Time is the continued sequence of existence and events that occurs in an apparently irreversible succession from the past, through the present, into the future. It is a component quantity of various measurements used to sequence events, t ...
(minutes per hour) and
angle In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles a ...
s (degrees).


History of modern numbers

In China, armies and provisions were counted using modular tallies of
prime number 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 ...
s. Unique numbers of troops and measures of rice appear as unique combinations of these tallies. A great convenience of
modular arithmetic In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his bo ...
is that it is easy to multiply. This makes use of modular arithmetic for provisions especially attractive. Conventional tallies are quite difficult to multiply and divide. In modern times modular arithmetic is sometimes used in
digital signal processing Digital signal processing (DSP) is the use of digital processing, such as by computers or more specialized digital signal processors, to perform a wide variety of signal processing operations. The digital signals processed in this manner are a ...
. The oldest Greek system was that of the
Attic numerals The Attic numerals are a symbolic number notation used by the ancient Greeks. They were also known as Herodianic numerals because they were first described in a 2nd-century manuscript by Herodian; or as acrophonic numerals (from acrophony) ...
, but in the 4th century BC they began to use a quasidecimal alphabetic system (see
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 those ...
). Jews began using a similar system (
Hebrew numerals The system of Hebrew numerals is a quasi-decimal alphabetic numeral system using the letters of the Hebrew alphabet. The system was adapted from that of the Greek numerals in the late 2nd century BCE. The current numeral system is also known as t ...
), with the oldest examples known being coins from around 100 BC. The Roman empire used tallies written on wax, papyrus and stone, and roughly followed the Greek custom of assigning letters to various numbers. The Roman numerals system remained in common use in Europe until
positional 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 decimal system). More generally, a positional system is a numeral system in which the ...
came into common use in the 16th century. The
Maya Maya may refer to: Civilizations * Maya peoples, of southern Mexico and northern Central America ** Maya civilization, the historical civilization of the Maya peoples ** Maya language, the languages of the Maya peoples * Maya (Ethiopia), a populat ...
of Central America used a mixed base 18 and base 20 system, possibly inherited from the
Olmec The Olmecs () were the earliest known major Mesoamerican civilization. Following a progressive development in Soconusco, they occupied the tropical lowlands of the modern-day Mexican states of Veracruz and Tabasco. It has been speculated that ...
, including advanced features such as positional notation and a
zero 0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usuall ...
. They used this system to make advanced astronomical calculations, including highly accurate calculations of the length of the solar year and the orbit of
Venus Venus is the second planet from the Sun. It is sometimes called Earth's "sister" or "twin" planet as it is almost as large and has a similar composition. As an interior planet to Earth, Venus (like Mercury) appears in Earth's sky never fa ...
. The Incan Empire ran a large command economy using
quipu ''Quipu'' (also spelled ''khipu'') are recording devices fashioned from strings historically used by a number of cultures in the region of Andean South America. A ''quipu'' usually consisted of cotton or camelid fiber strings. The Inca people ...
, tallies made by knotting colored fibers. Knowledge of the encodings of the knots and colors was suppressed by the
Spanish Spanish might refer to: * Items from or related to Spain: **Spaniards are a nation and ethnic group indigenous to Spain **Spanish language, spoken in Spain and many Latin American countries **Spanish cuisine Other places * Spanish, Ontario, Can ...
conquistador Conquistadors (, ) or conquistadores (, ; meaning 'conquerors') were the explorer-soldiers of the Spanish and Portuguese Empires of the 15th and 16th centuries. During the Age of Discovery, conquistadors sailed beyond Europe to the Americas, ...
s in the 16th century, and has not survived although simple quipu-like recording devices are still used in the
Andean The Andes, Andes Mountains or Andean Mountains (; ) are the longest continental mountain range in the world, forming a continuous highland along the western edge of South America. The range is long, wide (widest between 18°S – 20°S ...
region. Some authorities believe that positional arithmetic began with the wide use of counting rods in China. The earliest written positional records seem to be
rod calculus Rod calculus or rod calculation was the mechanical method of algorithmic computation with counting rods in China from the Warring States to Ming dynasty before the counting rods were increasingly replaced by the more convenient and faster abacus. ...
results in China around 400. Zero was first used in India in the 7th century CE by
Brahmagupta Brahmagupta ( – ) was an Indian mathematician and astronomer. He is the author of two early works on mathematics and astronomy: the ''Brāhmasphuṭasiddhānta'' (BSS, "correctly established doctrine of Brahma", dated 628), a theoretical tr ...
. The modern positional Arabic numeral system was developed by mathematicians in India, and passed on to Muslim mathematicians, along with astronomical tables brought to
Baghdad Baghdad (; ar, بَغْدَاد , ) is the capital of Iraq and the second-largest city in the Arab world after Cairo. It is located on the Tigris near the ruins of the ancient city of Babylon and the Sassanid Persian capital of Ctesiphon ...
by an Indian ambassador around 773. From
India India, officially the Republic of India (Hindi: ), is a country in South Asia. It is the seventh-largest country by area, the second-most populous country, and the most populous democracy in the world. Bounded by the Indian Ocean on the so ...
, the thriving trade between Islamic sultans and Africa carried the concept to
Cairo Cairo ( ; ar, القاهرة, al-Qāhirah, ) is the capital of Egypt and its largest city, home to 10 million people. It is also part of the largest urban agglomeration in Africa, the Arab world and the Middle East: The Greater Cairo metro ...
. Arabic mathematicians extended the system to include
decimal fractions 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 ...
, and
Muḥammad ibn Mūsā al-Ḵwārizmī 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, astronom ...
wrote an important work about it in the 9th  century. The modern
Arabic numerals Arabic numerals are the ten numerical digits: , , , , , , , , and . They are the most commonly used symbols to write decimal numbers. They are also used for writing numbers in other systems such as octal, and for writing identifiers such as ...
were introduced to Europe with the translation of this work in the 12th century in Spain and
Leonardo of Pisa Fibonacci (; also , ; – ), also known as Leonardo Bonacci, Leonardo of Pisa, or Leonardo Bigollo Pisano ('Leonardo the Traveller from Pisa'), was an Italian mathematician from the Republic of Pisa, considered to be "the most talented Western ...
's ''Liber Abaci'' of 1201. In Europe, the complete Indian system with the zero was derived from the Arabs in the 12th century. The
binary system A binary system is a system of two astronomical bodies which are close enough that their gravitational attraction causes them to orbit each other around a barycenter ''(also see animated examples)''. More restrictive definitions require that thi ...
(base 2), was propagated in the 17th century by
Gottfried Leibniz Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mat ...
. Leibniz had developed the concept early in his career, and had revisited it when he reviewed a copy of the I Ching from China. Binary numbers came into common use in the 20th century because of computer applications.


Numerals in most popular systems


Additional numerals


See also

*
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 ...
*
Binary digit 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 ...
(
bit The bit is the most basic unit of information in computing and digital communications. The name is a portmanteau of binary digit. The bit represents a logical state with one of two possible values. These values are most commonly represented ...
), Quantum binary digit (
qubit In quantum computing, a qubit () or quantum bit is a basic unit of quantum information—the quantum version of the classic binary bit physically realized with a two-state device. A qubit is a two-state (or two-level) quantum-mechanical system, ...
) * Ternary digit ( trit), Quantum ternary digit (
qutrit A qutrit (or quantum trit) is a unit of quantum information that is realized by a 3-level quantum system, that may be in a superposition of three mutually orthogonal quantum states. The qutrit is analogous to the classical radix-3 trit, just as ...
) *
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 ...
(
dit DIT or dit may refer to: People * Dit name, an alternative family name, e.g., in French Canadian historical traditions * Dit Clapper (1907–1978), Canadian ice hockey player Information technology * Directory information tree * dit (unit), ...
) *
Hexadecimal digit 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 ...
(
Hexit Lindane, also known as ''gamma''-hexachlorocyclohexane (γ-HCH), gammaxene, Gammallin and benzene hexachloride (BHC), is an organochlorine chemical and an isomer of hexachlorocyclohexane that has been used both as an agricultural insecticide and ...
) * Natural digit ( nat, nit) * Naperian digit ( nepit) *
Significant digit 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 expres ...
*
Large numbers Large numbers are numbers significantly larger than those typically used in everyday life (for instance in simple counting or in monetary transactions), appearing frequently in fields such as mathematics, cosmology, cryptography, and statistical ...
*
Text figures Text figures (also known as non-lining, lowercase, old style, ranging, hanging, medieval, billing, or antique figures or numerals) are numerals designed with varying heights in a fashion that resembles a typical line of running text, hence 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 H ...
* History of large numbers * List of numeral system topics


Numeral notation in various scripts

*
Arabic numerals Arabic numerals are the ten numerical digits: , , , , , , , , and . They are the most commonly used symbols to write decimal numbers. They are also used for writing numbers in other systems such as octal, and for writing identifiers such as ...
*
Armenian numerals The system of Armenian numerals is a historic numeral system created using the majuscules (uppercase letters) of the Armenian alphabet. There was no notation for zero in the old system, and the numeric values for individual letters were added to ...
*
Babylonian numerals Assyro-Chaldean Babylonian cuneiform numerals were written in cuneiform, using a wedge-tipped 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. The Babylonians, who were fam ...
*
Balinese numerals The Balinese language has an elaborate decimal numeral system. Basic numerals The numerals 1–10 have basic, combining, and independent forms, many of which are formed through reduplication. The combining forms are used to form higher numbers. ...
*
Bengali numerals Bengali numerals ( bn, সংখ্যা ''sôṅkhya'', as, সংখ্যা ''xoiŋkha''), are the units of the numeral system, originating from the Indian subcontinent, used in Bengali, Sylheti, Chittagonian, Assamese, Bishnupriya Manipu ...
*
Burmese numerals Burmese numerals ( my, မြန်မာ ကိန်းဂဏန်းများ, ) are a set of numerals traditionally used in the , although Arabic numerals are also used. Burmese numerals follow the Hindu–Arabic numeral system commonl ...
*
Chinese numerals Chinese numerals are words and characters used to denote numbers in Chinese. Today, speakers of Chinese use three written numeral systems: the system of Arabic numerals used worldwide, and two indigenous systems. The more familiar indigenous sy ...
* Cistercian numerals * Dzongkha numerals *
Eastern Arabic numerals The Eastern Arabic numerals, also called Arabic-Hindu numerals or Indo–Arabic numerals, are the symbols used to represent numerical digits in conjunction with the Arabic alphabet in the countries of the Mashriq (the east of the Arab world), ...
*
Georgian numerals The Georgian numerals are the system of number names used in Georgian, a language spoken in the country of Georgia. The Georgian numerals from 30 to 99 are constructed using a base-20 system, similar to the scheme used in Basque, French for n ...
*
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 those ...
*
Gurmukhi numerals Gurmukhī ( pa, ਗੁਰਮੁਖੀ, , Shahmukhi: ) is an abugida developed from the Laṇḍā scripts, standardized and used by the second Sikh guru, Guru Angad (1504–1552). It is used by Punjabi Sikhs to write the language, commonly re ...
*
Hebrew numerals The system of Hebrew numerals is a quasi-decimal alphabetic numeral system using the letters of the Hebrew alphabet. The system was adapted from that of the Greek numerals in the late 2nd century BCE. The current numeral system is also known as t ...
* Hokkien numerals *
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 ...
*
Japanese numerals The Japanese numerals are the number names used in Japanese. In writing, they are the same as the Chinese numerals, and large numbers follow the Chinese style of grouping by 10,000. Two pronunciations are used: the Sino-Japanese (on'yomi) reading ...
*
Javanese numerals The Javanese language has a decimal numeral system with distinct words for the 'tweens' from 21 to 29, called ''likuran''. The basic numerals 1–10 have independent and combining forms, the latter derived via a suffix ''-ng''. The combining for ...
*
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. Numeral ...
*
Korean numerals The Korean language has two regularly used sets of numerals: a native Korean system and Sino-Korean system. The native Korean number system is used for general counting, like counting up to 99. It is also used to count people, hours, objects ...
*
Lao numerals Lao script or Akson Lao ( lo, ອັກສອນລາວ, links=no ) is the primary script used to write the Lao language and other minority languages in Laos. Its earlier form, the Tai Noi script, was also used to write the Isan language, b ...
* Mayan numerals * Mongolian numerals *
Quipu ''Quipu'' (also spelled ''khipu'') are recording devices fashioned from strings historically used by a number of cultures in the region of Andean South America. A ''quipu'' usually consisted of cotton or camelid fiber strings. The Inca people ...
*
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 ...
*
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 ...
*
Sinhala numerals Sinhala numerals, are the units of the numeral system, originating from the Indian subcontinent, used in Sinhala language in modern-day Sri Lanka. Numerals or numerations around Kandyan Kingdom It had been found that five different types of nu ...
*
Suzhou numerals The Suzhou numerals, also known as ' (), is a numeral system used in China before the introduction of Arabic numerals. The Suzhou numerals are also known as ' (), ' (), ' (), ' () and ' (). History The Suzhou numeral system is the only survivin ...
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Tamil numerals This article is about the number words of the Tamil language, as well as the dedicated symbols for them used in the Tamil script. Basic numbering Zero Old Tamil possesses a special numerical character for zero ''(see Old Tamil numerals ...
*
Thai numerals Thai numerals ( th, เลขไทย, , ) are a set of numerals traditionally used in Thailand, although the Arabic numerals are more common due to extensive westernization of Thailand in the modern Rattanakosin period. Thai numerals follow the ...
*
Vietnamese numerals Historically Vietnamese has two sets of numbers: one is etymologically native Vietnamese; the other uses Sino-Vietnamese vocabulary. In the modern language the native Vietnamese vocabulary is used for both everyday counting and mathematical pur ...


References

{{DEFAULTSORT:Numerical Digit Numeral systems