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0 (zero) is 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 can ...
, and the
numerical digit A numerical digit (often shortened to just digit) is a single symbol used alone (such as "2") or in combinations (such as "25"), to represent number A number is a mathematical object used to count, measure, and label. The original example ...
used to represent that number in numerals. It fulfills a central role in
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
as the
additive identity In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mode ...
of the
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 of ...
s,
real number In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
s, and many other
algebraic structure In mathematics, an algebraic structure consists of a nonempty Set (mathematics), set ''A'' (called the underlying set, carrier set or domain), a collection of operation (mathematics), operations on ''A'' (typically binary operations such as addit ...
s. As a digit, 0 is used as a placeholder in place value systems.
Names for the number 0 in English "Zero" is the usual name for the number 0 in English. In British English "nought" is also used. In American English "naught" is used occasionally for zero, but (as with British English) "naught" is more often used as an archaic word for nothing. "N ...
include zero, nought (UK), naught (US; ), nil, or—in contexts where at least one adjacent digit distinguishes it from the letter "O"—oh or o (). Informal or slang terms for zero include zilch and zip. ''Ought'' and ''aught'' (), as well as ''cipher'', have also been used historically.


Etymology

The word ''zero'' came into the English language via French from the Italian , a contraction of the Venetian form of Italian via ''ṣafira'' or ''ṣifr''. In pre-Islamic time the word (Arabic ) had the meaning "empty". evolved to mean zero when it was used to translate ( sa, शून्य) from
India India, officially the Republic of India (Hindi: ), is a country in South Asia. It is the List of countries and dependencies by area, seventh-largest country by area, the List of countries and dependencies by population, second-most populous ...
.See: * Smithsonian Institution, , Annual Report of the Board of Regents of the Smithsonian Institution; Harvard University Archives, Quote="Sifr occurs in the meaning of "empty" even in the pre-Islamic time. ... Arabic sifr in the meaning of zero is a translation of the corresponding India sunya."; * Jan Gullberg (1997), Mathematics: From the Birth of Numbers, W.W. Norton & Co., , p. 26, Quote = ''Zero derives from Hindu sunya – meaning void, emptiness – via Arabic sifr, Latin cephirum, Italian zevero.''; * Robert Logan (2010), The Poetry of Physics and the Physics of Poetry, World Scientific, , p. 38, Quote = "The idea of sunya and place numbers was transmitted to the Arabs who translated sunya or "leave a space" into their language as sifr." The first known English use of ''zero'' was in 1598. The Italian mathematician
Fibonacci Fibonacci (; also , ; – ), also known as Leonardo Bonacci, Leonardo of Pisa, or Leonardo Bigollo Pisano ('Leonardo the Traveller from Pisa'), was an Italians, Italian mathematician from the Republic of Pisa, considered to be "the most talente ...
(c. 1170–1250), who grew up in North Africa and is credited with introducing the decimal system to Europe, used the term ''zephyrum''. This became in Italian, and was then contracted to in Venetian. The Italian word was already in existence (meaning "west wind" from Latin and Greek ) and may have influenced the spelling when transcribing Arabic .


Modern usage

Depending on the context, there may be different words used for the number zero, or the concept of zero. For the simple notion of lacking, the words "nothing" and "none" are often used. Sometimes, the word "nought" or "naught" is used. It is often called "oh" in the context of telephone numbers. Slang words for zero include "zip", "zilch", "nada", and "scratch."'Aught' synonyms
Thesaurus.com – Retrieved April 2013.
"Nil" is used for many sports in
British English British English (BrE, en-GB, or BE) is, according to Oxford Dictionaries, " English as used in Great Britain, as distinct from that used elsewhere". More narrowly, it can refer specifically to the English language in England, or, more broa ...
. Several sports have specific words for a score of zero, such as "
love Love encompasses a range of strong and positive emotional and mental states, from the most sublime virtue or good habit, the deepest Interpersonal relationship, interpersonal affection, to the simplest pleasure. An example of this range of ...
" in
tennis Tennis is a List of racket sports, racket sport that is played either individually against a single opponent (singles (tennis), singles) or between two teams of two players each (doubles (tennis), doubles). Each player uses a tennis racket th ...
– from French ''l'oeuf'', "the egg" – and "
duck Duck is the common name for numerous species of waterfowl in the family (biology), family Anatidae. Ducks are generally smaller and shorter-necked than swans and goose, geese, which are members of the same family. Divided among several subfam ...
" in
cricket Cricket is a Bat-and-ball games, bat-and-ball game played between two teams of eleven players on a cricket field, field at the centre of which is a cricket pitch, pitch with a wicket at each end, each comprising two Bail (cricket), bails b ...
, a shortening of "duck's egg"; "goose egg" is another general slang term used for zero.


History


Ancient Near East

Ancient Egyptian numerals were of
base 10 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 ...
. They used
hieroglyphs A hieroglyph (Ancient Greek, Greek for "sacred carvings") was a Character (symbol), character of the Egyptian hieroglyphs, ancient Egyptian writing system. logogram, Logographic scripts that are pictographic in form in a way reminiscent of ancien ...
for the digits and were not positional. By 1770 BC, the Egyptians had a symbol for zero in accounting texts. The symbol nfr, meaning beautiful, was also used to indicate the base level in drawings of tombs and pyramids, and distances were measured relative to the base line as being above or below this line. By the middle of the
2nd millennium BC The 2nd millennium BC spanned the years 2000 BC to 1001 BC. In the Ancient Near East, it marks the transition from the Middle to the Late Bronze Age. The Ancient Near Eastern cultures are well within the historical era: The first half of the m ...
, the
Babylonian mathematics Babylonian mathematics (also known as ''Assyro-Babylonian mathematics'') are the mathematics developed or practiced by the people of Mesopotamia, from the days of the early Sumerians to the centuries following the fall of Babylon in 539 BC. Babyl ...
had a sophisticated
sexagesimal Sexagesimal, also known as base 60 or sexagenary, is a numeral system with 60 (number), sixty as its radix, base. It originated with the ancient Sumerians in the 3rd millennium BC, was passed down to the ancient Babylonians, and is still used—in ...
positional numeral system. The lack of a positional value (or zero) was indicated by a ''space'' between sexagesimal numerals. In a tablet unearthed at Kish (dating to as early as 700 BC), the scribe Bêl-bân-aplu used three hooks as a placeholder in the same Babylonian system.Kaplan, Robert. (2000). ''The Nothing That Is: A Natural History of Zero''. Oxford: Oxford University Press. By 300 BC, a punctuation symbol (two slanted wedges) was co-opted to serve as this placeholder. The Babylonian placeholder was not a true zero because it was not used alone, nor was it used at the end of a number. Thus numbers like 2 and 120 (2×60), 3 and 180 (3×60), 4 and 240 (4×60) looked the same, because the larger numbers lacked a final sexagesimal placeholder. Only context could differentiate them.


Pre-Columbian Americas

The Mesoamerican Long Count calendar developed in south-central Mexico and Central America required the use of zero as a placeholder within its
vigesimal A vigesimal () or base-20 (base-score) numeral system is based on 20 (number), twenty (in the same way in which the decimal, decimal numeral system is based on 10 (number), ten). ''wikt:vigesimal#English, Vigesimal'' is derived from the Latin ad ...
(base-20) positional numeral system. Many different glyphs, including this partial
quatrefoil A quatrefoil (anciently caterfoil) is a decorative element consisting of a symmetrical shape which forms the overall outline of four partially overlapping circles of the same diameter. It is found in art, architecture, heraldry and traditional ...
——were used as a zero symbol for these Long Count dates, the earliest of which (on Stela 2 at Chiapa de Corzo,
Chiapas Chiapas (; Tzotzil language, Tzotzil and Tzeltal language, Tzeltal: ''Chyapas'' ), officially the Free and Sovereign State of Chiapas ( es, Estado Libre y Soberano de Chiapas), is one of the states that make up the Political divisions of Mexico, ...
) has a date of 36 BC. Since the eight earliest Long Count dates appear outside the Maya homeland, it is generally believed that the use of zero in the Americas predated the Maya and was possibly the invention of the
Olmec The Olmecs () were the earliest known major Mesoamerican civilization. Following a progressive development in Soconusco, Veracruz, Soconusco, they occupied the tropical lowlands of the modern-day Mexican states of Veracruz and Tabasco. It has b ...
s. Many of the earliest Long Count dates were found within the Olmec heartland, although the Olmec civilization ended by the , several centuries before the earliest known Long Count dates. Although zero became an integral part of
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 notation, positional numeral system. The numerals are made up of three symbols; Zero number#The ...
, with a different, empty
tortoise Tortoises () are reptiles of the family Testudinidae of the order Testudines (Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally a dialect ...
-like " shell shape" used for many depictions of the "zero" numeral, it is assumed not to have influenced
Old World The "Old World" is a term for Afro-Eurasia that originated in Europe , after Europeans became aware of the existence of the Americas. It is used to contrast the continents of Africa, Europe, and Asia, which were previously thought of by thei ...
numeral systems.
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 South America is a continent entirely in the Western Hemisphere and mostl ...
, a knotted cord device, used in the
Inca Empire The Inca Empire (also Quechuan and Aymaran spelling shift, known as the Incan Empire and the Inka Empire), called ''Tawantinsuyu'' by its subjects, (Quechuan languages, Quechua for the "Realm of the Four Parts",  "four parts together" ) wa ...
and its predecessor societies 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 South America is a continent entirely in the Western ...
region to record accounting and other digital data, is encoded in a
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 An integer is the number zero (), a positive natural number (, , , etc.) or a negative in ...
positional system. Zero is represented by the absence of a knot in the appropriate position.


Classical antiquity

The
ancient Greeks Ancient Greece ( el, Ἑλλάς, Hellás) was a northeastern Mediterranean Sea, Mediterranean civilization, existing from the Greek Dark Ages of the 12th–9th centuries BC to the end of Classical Antiquity, classical antiquity ( AD 600), th ...
had no symbol for zero (μηδέν), and did not use a digit placeholder for it. They seemed unsure about the status of zero as a number. They asked themselves, "How can nothing ''be'' something?", leading to philosophical and, by the
medieval In the history of Europe, the Middle Ages or medieval period lasted approximately from the late 5th to the late 15th centuries, similar to the post-classical period of global history. It began with the fall of the Western Roman Empire ...
period, religious arguments about the nature and existence of zero and the
vacuum A vacuum is a space devoid of matter. The word is derived from the Latin adjective ''vacuus'' for "vacant" or "Void (astronomy), void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Ph ...
. The
paradoxes A paradox is a logically self-contradictory statement or a statement that runs contrary to one's expectation. It is a statement that, despite apparently valid reasoning from true premises, leads to a seemingly self-contradictory or a logically u ...
of
Zeno of Elea Zeno of Elea (; grc, wikt:Ζήνων, Ζήνων ὁ Ἐλεᾱ́της; ) was a pre-Socratic Greek philosopher of Magna Graecia and a member of the Eleatic School founded by Parmenides. Aristotle called him the inventor of the dialectic. He i ...
depend in large part on the uncertain interpretation of zero. By AD 150,
Ptolemy Claudius Ptolemy (; grc-gre, wikt:Πτολεμαῖος, Πτολεμαῖος, ; la, Claudius Ptolemaeus; AD) was a mathematician, astronomer, astrologer, geographer, and music theorist, who wrote about a dozen scientific Treatise, treatis ...
, influenced by
Hipparchus Hipparchus (; el, wikt:Ἵππαρχος, Ἵππαρχος, ''Hipparkhos'';  BC) was a Ancient astronomy, Greek astronomer, geographer, and mathematician. He is considered the founder of trigonometry, but is most famous for his incidenta ...
and the
Babylonia Babylonia (; Akkadian: , ''māt Akkadī'') was an ancient Akkadian-speaking state and cultural area based in the city of Babylon in central-southern Mesopotamia (present-day Iraq and parts of Syria). It emerged as an Amorites, Amorite-ruled ...
ns, was using a symbol for zero () in his work on mathematical astronomy called the ''Syntaxis Mathematica'', also known as the ''
Almagest The ''Almagest'' is a 2nd-century Greek-language mathematical and astronomical treatise on the apparent motions of the stars and planetary paths, written by Claudius Ptolemy ( ). One of the most influential scientific texts in history, it ca ...
''. This Hellenistic zero was perhaps the earliest documented use of a numeral representing zero in the Old World. Ptolemy used it many times in his ''Almagest'' (VI.8) for the magnitude of and
lunar eclipse A lunar eclipse occurs when the Moon moves into the Earth's shadow. Such alignment occurs during an eclipse season, approximately every six months, during the full moon phase, when the Moon's orbital plane is closest to Ecliptic, the plane of t ...

lunar eclipse
s. It represented the value of both digits and
minutes Minutes, also known as minutes of meeting (abbreviation MoM), protocols or, informally, notes, are the instant written record of a meeting or hearing (law), hearing. They typically describe the events of the meeting and may include a list of atte ...
of immersion at first and last contact. Digits varied continuously from as the Moon passed over the Sun (a triangular pulse), where twelve digits was the
angular diameter The angular diameter, angular size, apparent diameter, or apparent size is an angular distance describing how large a sphere or circle appears from a given point of view. In the vision sciences, it is called the visual angle, and in optics, it is ...

angular diameter
of the Sun. Minutes of immersion was tabulated from , where 0′0″ used the symbol as a placeholder in two positions of his
sexagesimal Sexagesimal, also known as base 60 or sexagenary, is a numeral system with 60 (number), sixty as its radix, base. It originated with the ancient Sumerians in the 3rd millennium BC, was passed down to the ancient Babylonians, and is still used—in ...
positional numeral system, while the combination meant a zero angle. Minutes of immersion was also a continuous function (a triangular pulse with
convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytope, ...

convex
sides), where d was the digit function and 31′20″ was the sum of the radii of the Sun's and Moon's discs. Ptolemy's symbol was a placeholder as well as a number used by two continuous mathematical functions, one within another, so it meant zero, not none. The earliest use of zero in the calculation of the Julian Easter occurred before AD311, at the first entry in a table of epacts as preserved in an document for the years AD311 to 369, using a word for "none" (English translation is "0" elsewhere) alongside Ge'ez numerals (based on Greek numerals), which was translated from an equivalent table published by the
Church of Alexandria The Church of Alexandria in Egypt is the Christian Church headed by the Patriarch of Alexandria. It is one of the Pentarchy, original Apostolic Sees of Christianity, alongside Rome, Antioch, Constantinople and Jerusalem. Tradition holds that the C ...
in
Medieval Greek Medieval Greek (also known as Middle Greek, Byzantine Greek, or Romaic) is the stage of the Greek language between the end of classical antiquity in the 5th–6th centuries and the end of the Middle Ages, conventionally dated to the Fall of Co ...
.. The pages in this edition have numbers six less than the same pages in the original edition. This use was repeated in AD525 in an equivalent table, that was translated via the Latin ''nulla'' or "none" by
Dionysius Exiguus Dionysius Exiguus (Latin language, Latin for "Dionysius the Humble", Ancient Greek, Greek: Διονύσιος; – ) was a 6th-century Eastern Roman monk born in Scythia Minor (Roman province), Scythia Minor. He was a member of a community of S ...

Dionysius Exiguus
, alongside
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 ...

Roman numerals
. When division produced zero as a remainder, ''nihil'', meaning "nothing", was used. These medieval zeros were used by all future medieval calculators of Easter. The initial "N" was used as a zero symbol in a table of Roman numerals by
Bede Bede ( ; ang, Bǣda , ; 672/326 May 735), also known as Saint Bede, The Venerable Bede, and Bede the Venerable ( la, Beda Venerabilis), was an English monk at the monastery of St Peter and its companion monastery of St Paul in the Kingdom o ...

Bede
—or his colleagues—around AD 725.C. W. Jones, ed., ''Opera Didascalica'', vol. 123C in ''Corpus Christianorum, Series Latina''.


China

The '' Sūnzĭ Suànjīng'', of unknown date but estimated to be dated from the 1st to , and Japanese records dated from the 18th century, describe how the Chinese
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 ...
system enabled one to perform decimal calculations. As noted in Xiahou Yang's Suanjing (425–468 AD) that states that to multiply or divide a number by 10, 100, 1000, or 10000, all one needs to do, with rods on the counting board, is to move them forwards, or back, by 1, 2, 3, or 4 places, According to ''A History of Mathematics'', the rods "gave the decimal representation of a number, with an empty space denoting zero." The counting rod system is considered a
positional notation Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any radix, base of the Hindu–Arabic numeral system (or decimal, decimal system). More generally, a positional system is a numeral syste ...
system. In AD 690, Empress Wǔ promulgated Zetian characters, one of which was "〇"; originally meaning 'star', it subsequently came to represent zero. Zero was not treated as a number at that time, but as a "vacant position". Qín Jiǔsháo's 1247 ''
Mathematical Treatise in Nine Sections The ''Mathematical Treatise in Nine Sections'' () is a mathematical text written by Chinese Southern Song dynasty mathematician Qin Jiushao in the year 1247. The mathematical text has a wide range of topics and is taken from all aspects of the ...
'' is the oldest surviving Chinese mathematical text using a round symbol for zero. Chinese authors had been familiar with the idea of negative numbers by the
Han Dynasty The Han dynasty (, ; ) was an imperial dynasty of China (202 BC – 9 AD, 25–220 AD), established by Emperor Gaozu of Han, Liu Bang (Emperor Gao) and ruled by the House of Liu. The dynasty was preceded by the short-lived Qin dynasty (22 ...

Han Dynasty
, as seen in ''
The Nine Chapters on the Mathematical Art ''The Nine Chapters on the Mathematical Art'' () is a Chinese mathematics book, composed by several generations of scholars from the 10th–2nd century BCE, its latest stage being from the 2nd century CE. This book is one of the earliest sur ...
''.Struik, Dirk J. (1987). ''A Concise History of Mathematics''. New York: Dover Publications. pp. 32–33. "''In these matrices we find negative numbers, which appear here for the first time in history.''"


India

Pingala Acharya Pingala ('; c. 3rd2nd century Common Era, BCE) was an ancient Indian poet and Indian mathematics, mathematician, and the author of the ' (also called the ''Pingala-sutras''), the earliest known treatise on Sanskrit prosody. The ' is a wo ...
(c. 3rd/2nd century BC), a
Sanskrit prosody Sanskrit prosody or Chandas refers to one of the six Vedangas, or limbs of Vedic studies.James Lochtefeld (2002), "Chandas" in The Illustrated Encyclopedia of Hinduism, Vol. 1: A-M, Rosen Publishing, , page 140 It is the study of Metre (poetr ...
scholar, used binary numbers in the form of short and long syllables (the latter equal in length to two short syllables), a notation similar to
Morse code Morse code is a method used in telecommunication to Character encoding, encode Written language, text characters as standardized sequences of two different signal durations, called ''dots'' and ''dashes'', or ''dits'' and ''dahs''. Morse cod ...

Morse code
. Pingala used the
Sanskrit Sanskrit (; attributively , ; nominalization, nominally , , ) is a classical language belonging to the Indo-Aryan languages, Indo-Aryan branch of the Indo-European languages. It arose in South Asia after its predecessor languages had Trans-cul ...

Sanskrit
word '' śūnya'' explicitly to refer to zero. The concept of zero as a written digit in the decimal place value notation was developed in
India India, officially the Republic of India (Hindi: ), is a country in South Asia. It is the List of countries and dependencies by area, seventh-largest country by area, the List of countries and dependencies by population, second-most populous ...

India
, presumably as early as during the
Gupta period The Gupta Empire was an Outline of ancient India, ancient Indian empire which existed from the early 4th century CE to late 6th century CE. At its zenith, from approximately 319 to 467 CE, it covered much of the Indian subcontinent. This period ...
, with the oldest unambiguous evidence dating to the 7th century.Bourbaki, Nicolas ''Elements of the History of Mathematics'' (1998), p. 46. ''Britannica Concise Encyclopedia'' (2007), entry "Algebra" A symbol for zero, a large dot likely to be the precursor of the still-current hollow symbol, is used throughout the , a practical manual on arithmetic for merchants. In 2017, three samples from the manuscript were shown by
radiocarbon dating Radiocarbon dating (also referred to as carbon dating or carbon-14 dating) is a method for determining the age of an object containing organic material by using the properties of radiocarbon, a radioactive isotope of carbon. The method was ...
to come from three different centuries: from AD 224–383, AD 680–779, and AD 885–993, making it South Asia's oldest recorded use of the zero symbol. It is not known how the
birch A birch is a thin-leaved deciduous hardwood tree of the genus ''Betula'' (), in the family Betulaceae, which also includes alders, hazels, and hornbeams. It is closely related to the beech-oak family Fagaceae. The genus ''Betula'' contains 30 to ...

birch
bark fragments from different centuries forming the manuscript came to be packaged together. The '' Lokavibhāga'', a
Jain Jainism ( ), also known as Jain Dharma, is an Indian religions, Indian religion. Jainism traces its spiritual ideas and history through the succession of twenty-four tirthankaras (supreme preachers of ''Dharma''), with the first in the current ...

Jain
text on
cosmology Cosmology () is a branch of physics and metaphysics dealing with the nature of the universe. The term ''cosmology'' was first used in English in 1656 in Thomas Blount (lexicographer), Thomas Blount's ''Glossographia'', and in 1731 taken up in ...
surviving in a medieval Sanskrit translation of the
Prakrit The Prakrits (; sa, prākṛta; psu, 𑀧𑀸𑀉𑀤, ; pka, ) are a group of vernacular Middle Indo-Aryan languages that were used in the Indian subcontinent The Indian subcontinent is a list of the physiographic regions of the w ...
original, which is internally dated to AD 458 (
Saka era The Shaka era (IAST The International Alphabet of Sanskrit Transliteration (IAST) is a transliteration scheme that allows the lossless romanisation of Indic scripts as employed by Sanskrit Sanskrit (; attributively , ; nominalizati ...
380), uses a decimal
place-value system Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any radix, base of the Hindu–Arabic numeral system (or decimal, decimal system). More generally, a positional system is a numeral syste ...
, including a zero. In this text, '' śūnya'' ("void, empty") is also used to refer to zero. The ''
Aryabhatiya ''Aryabhatiya'' ( IAST: ') or ''Aryabhatiyam'' ('), a Sanskrit astronomical treatise, is the '' magnum opus'' and only known surviving work of the 5th century Indian mathematician Aryabhata Aryabhata (ISO 15919, ISO: ) or Aryabhata I (47 ...
'' (c. 500), states ''sthānāt sthānaṁ daśaguṇaṁ syāt'' "from place to place each is ten times the preceding."''Aryabhatiya of Aryabhata'', translated by Walter Eugene Clark. Rules governing the use of zero appeared in
Brahmagupta Brahmagupta ( – ) was an Indian Indian mathematics, mathematician and Indian astronomy, astronomer. He is the author of two early works on mathematics and astronomy: the ''Brāhmasphuṭasiddhānta'' (BSS, "correctly established Siddhanta, do ...

Brahmagupta
's '' Brahmasputha Siddhanta'' (7th century), which states the sum of zero with itself as zero, and incorrectly
division by zero In mathematics, division by zero is division (mathematics), division where the divisor (denominator) is 0, zero. Such a division can be formally expression (mathematics), expressed as \tfrac, where is the dividend (numerator). In ordinary ari ...
as:''Algebra with Arithmetic of Brahmagupta and Bhaskara''
translated to English by Henry Thomas Colebrooke (1817) London
A positive or negative number when divided by zero is a fraction with the zero as denominator. Zero divided by a negative or positive number is either zero or is expressed as a fraction with zero as numerator and the finite quantity as denominator. Zero divided by zero is zero.


Epigraphy

There are numerous copper plate inscriptions, with the same small ''o'' in them, some of them possibly dated to the 6th century, but their date or authenticity may be open to doubt. A stone tablet found in the ruins of a temple near Sambor on the
Mekong The Mekong or Mekong River is a trans-boundary river in East Asia and Southeast Asia. It is the world's List of rivers by length, twelfth longest river and List of longest rivers of Asia, the third longest in Asia. Its estimated length is , ...

Mekong
,
Kratié Province Kratié (also transliterated Kracheh) may refer to: *Kratié (town), a town in Kratié Commune, Cambodia *Kratié Commune, a commune in Kratié District, Cambodia *Kratié District, a district in Kratié Province, Cambodia *Kratié Province, a provi ...
, Cambodia, includes the inscription of "605" in Khmer numerals (a set of numeral glyphs for the Hindu–Arabic numeral system). The number is the year of the inscription in the
Saka era The Shaka era (IAST The International Alphabet of Sanskrit Transliteration (IAST) is a transliteration scheme that allows the lossless romanisation of Indic scripts as employed by Sanskrit Sanskrit (; attributively , ; nominalizati ...
, corresponding to a date of AD 683.George Cœdès, Cœdès, George, "A propos de l'origine des chiffres arabes," Bulletin of the School of Oriental Studies, University of London, Vol. 6, No. 2, 1931, pp. 323–328. Diller, Anthony, "New Zeros and Old Khmer," The Mon-Khmer Studies Journal, Vol. 25, 1996, pp. 125–132. The first known use of special glyphs for the decimal digits that includes the indubitable appearance of a symbol for the digit zero, a small circle, appears on a stone inscription found at the Chaturbhuj Temple, Gwalior, in India, dated 876. Zero is also used as a placeholder in the , portions of which date from AD 224–383.


Middle Ages


Transmission to Islamic culture

The Arabic-language inheritance of science was largely Greece, Greek, followed by Hindu influences.Will Durant (1950), ''The Story of Civilization'', Volume 4, The Age of Faith: Constantine to Dante – A.D. 325–1300, Simon & Schuster, , p. 241, Quote = "The Arabic inheritance of science was overwhelmingly Greek, but Hindu influences ranked next. In 773, at Mansur's behest, translations were made of the ''Siddhantas'' – Indian astronomical treatises dating as far back as 425 BC; these versions may have the vehicle through which the "Arabic" numerals and the zero were brought from India into Islam. In 813, al-Khwarizmi used the Hindu numerals in his astronomical tables." In 773, at Al-Mansur's behest, translations were made of many ancient treatises including Greek, Roman, Indian, and others. In AD 813, astronomical tables were prepared by a Persian people, Persian mathematician, Muḥammad ibn Mūsā al-Khwārizmī, using Hindu numerals; and about 825, he published a book synthesizing Greek and Hindu knowledge and also contained his own contribution to mathematics including an explanation of the use of zero. This book was later translated into Latin in the 12th century under the title ''Algoritmi de numero Indorum''. This title means "al-Khwarizmi on the Numerals of the Indians". The word "Algoritmi" was the translator's Latinization of Al-Khwarizmi's name, and the word "Algorithm" or "Algorism" started to acquire a meaning of any arithmetic based on decimals. Muhammad ibn Ahmad al-Khwarizmi, in 976, stated that if no number appears in the place of tens in a calculation, a little circle should be used "to keep the rows". This circle was called ''ṣifr''.


Transmission to Europe

The Hindu–Arabic numeral system (base 10) reached Western Europe in the 11th century, via Al-Andalus, through Spanish Muslims, the Moors, together with knowledge of classical astronomy and instruments like the astrolabe; Pope Sylvester II, Gerbert of Aurillac is credited with reintroducing the lost teachings into Catholic Europe. For this reason, the numerals came to be known in Europe as "Arabic numerals". The Italian mathematician
Fibonacci Fibonacci (; also , ; – ), also known as Leonardo Bonacci, Leonardo of Pisa, or Leonardo Bigollo Pisano ('Leonardo the Traveller from Pisa'), was an Italians, Italian mathematician from the Republic of Pisa, considered to be "the most talente ...
or Leonardo of Pisa was instrumental in bringing the system into European mathematics in 1202, stating:
After my father's appointment by his homeland as state official in the customs house of Bugia for the Pisan merchants who thronged to it, he took charge; and in view of its future usefulness and convenience, had me in my boyhood come to him and there wanted me to devote myself to and be instructed in the study of calculation for some days. There, following my introduction, as a consequence of marvelous instruction in the art, to the nine digits of the Hindus, the knowledge of the art very much appealed to me before all others, and for it I realized that all its aspects were studied in Egypt, Syria, Greece, Sicily, and Provence, with their varying methods; and at these places thereafter, while on business. I pursued my study in depth and learned the give-and-take of disputation. But all this even, and the algorism, as well as the art of Pythagoras, I considered as almost a mistake in respect to the method of the Hinduism, Hindus (Modus Indorum). Therefore, embracing more stringently that method of the Hindus, and taking stricter pains in its study, while adding certain things from my own understanding and inserting also certain things from the niceties of Euclid's geometric art. I have striven to compose this book in its entirety as understandably as I could, dividing it into fifteen chapters. Almost everything which I have introduced I have displayed with exact proof, in order that those further seeking this knowledge, with its pre-eminent method, might be instructed, and further, in order that the Latin people might not be discovered to be without it, as they have been up to now. If I have perchance omitted anything more or less proper or necessary, I beg indulgence, since there is no one who is blameless and utterly provident in all things. The nine Indian figures are: 9 8 7 6 5 4 3 2 1. With these nine figures, and with the sign 0  ... any number may be written.
Here Leonardo of Pisa uses the phrase "sign 0", indicating it is like a sign to do operations like addition or multiplication. From the 13th century, manuals on calculation (adding, multiplying, extracting roots, etc.) became common in Europe where they were called ''algorismus'' after the Persian mathematician al-Khwārizmī. The most popular was written by Johannes de Sacrobosco, about 1235 and was one of the earliest scientific books to be ''printed'' in 1488. Until the late 15th century, Hindu–Arabic numerals seem to have predominated among mathematicians, while merchants preferred to use the Roman numerals. In the 16th century, they became commonly used in Europe.


Mathematics

0 is the
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 of ...
immediately preceding 1 (number), 1. Parity of zero, Zero is an even number because it is divisible by 2 (number), 2 with no remainder. 0 is neither positive nor negative, or both positive and negative. Many definitions include 0 as a natural number, in which case it is the only natural number that is not positive. Zero is a number which quantifies a count or an amount of Empty set, null size. In most History of mathematics, cultures, 0 was identified before the idea of negative things (i.e., quantities less than zero) was accepted. As a value or a ''number'', zero is not the same as the ''digit'' zero, used in numeral systems with
positional notation Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any radix, base of the Hindu–Arabic numeral system (or decimal, decimal system). More generally, a positional system is a numeral syste ...
. Successive positions of digits have higher weights, so the digit zero is used inside a numeral to skip a position and give appropriate weights to the preceding and following digits. A zero digit is not always necessary in a positional number system (e.g., the number 02). In some instances, a leading zero may be used to distinguish a number.


Elementary algebra

The number 0 is the smallest non-negative integer. The natural number following 0 is 1 and no natural number precedes 0. The number 0 Natural number#History of natural numbers and the status of zero, may or may not be considered a natural number, but it is an integer, and hence a rational number and a
real number In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
(as well as an algebraic number and a complex number). The number 0 is neither positive nor negative, and is usually displayed as the central number in a number line. It is neither a prime number nor a composite number. It cannot be prime because it has an infinity, infinite number of divisor, factors, and cannot be composite because it cannot be expressed as a product of prime numbers (as 0 must always be one of the factors). Zero is, however, Parity (mathematics), even (i.e. a multiple of 2, as well as being a multiple of any other integer, rational, or real number). The following are some basic (elementary) rules for dealing with the number 0. These rules apply for any real or complex number ''x'', unless otherwise stated. * Addition: ''x'' + 0 = 0 + ''x'' = ''x''. That is, 0 is an identity element (or neutral element) with respect to addition. * Subtraction: ''x'' − 0 = ''x'' and 0 − ''x'' = −''x''. * Multiplication: ''x'' · 0 = 0 · ''x'' = 0. * Division: = 0, for nonzero ''x''. But is Defined and undefined, undefined, because 0 has no multiplicative inverse (no real number multiplied by 0 produces 1), a consequence of the previous rule. * Exponentiation: ''x''0 = = 1, except that the case ''x'' = 0 may be left undefined in some 0 to the power of 0, contexts. For all positive real ''x'', . The expression , which may be obtained in an attempt to determine the limit of an expression of the form as a result of applying the limit of a function, lim operator independently to both operands of the fraction, is a so-called "indeterminate form". That does not simply mean that the limit sought is necessarily undefined; rather, it means that the limit of , if it exists, must be found by another method, such as l'Hôpital's rule. The sum of 0 numbers (the ''empty sum'') is 0, and the product of 0 numbers (the ''empty product'') is 1. The factorial 0! evaluates to 1, as a special case of the empty product.


Other branches of mathematics

* In set theory, 0 is the cardinality of the empty set: if one does not have any apples, then one has 0 apples. In fact, in certain axiomatic developments of mathematics from set theory, 0 is ''definition, defined'' to be the empty set. When this is done, the empty set is the von Neumann cardinal assignment for a set with no elements, which is the empty set. The cardinality function, applied to the empty set, returns the empty set as a value, thereby assigning it 0 elements. * Also in set theory, 0 is the lowest ordinal number, corresponding to the empty set viewed as a well-order, well-ordered set. * In propositional calculus, propositional logic, 0 may be used to denote the truth value false. * In abstract algebra, 0 is commonly used to denote a zero element, which is a Identity element, neutral element for addition (if defined on the structure under consideration) and an absorbing element for multiplication (if defined). * In lattice (order), lattice theory, 0 may denote the Greatest element, bottom element of a Lattice (order), bounded lattice. * In category theory, 0 is sometimes used to denote an initial and terminal objects, initial object of a category (mathematics), category. * In recursion theory, 0 can be used to denote the Turing degree of the computable function, partial computable functions.


Related mathematical terms

* A root of a function, zero of a function ''f'' is a point ''x'' in the domain of the function such that . When there are finitely many zeros these are called the roots of the function. This is related to zero (complex analysis), zeros of a holomorphic function. * The zero function (or zero map) on a domain ''D'' is the constant function with 0 as its only possible output value, i.e., the function ''f'' defined by for all ''x'' in ''D''. The zero function is the only function that is both Even function, even and Odd function, odd. A particular zero function is a zero morphism in category theory; e.g., a zero map is the identity in the additive group of functions. The determinant on non-invertible Matrix (mathematics), square matrices is a zero map. * Several branches of mathematics have zero elements, which generalize either the property , or the property or both.


Physics

The value zero plays a special role for many physical quantities. For some quantities, the zero level is naturally distinguished from all other levels, whereas for others it is more or less arbitrarily chosen. For example, for an Thermodynamic temperature, absolute temperature (as measured in kelvins), absolute zero, zero is the lowest possible value (negative temperatures are defined, but negative-temperature systems are not actually colder). This is in contrast to for example temperatures on the Celsius scale, where zero is arbitrarily defined to be at the Melting point, freezing point of water. Measuring sound intensity in decibels or phons, the zero level is arbitrarily set at a reference value—for example, at a value for the threshold of hearing. In physics, the zero-point energy is the lowest possible energy that a quantum mechanics, quantum mechanical physical system may possess and is the energy of the Stationary state, ground state of the system.


Chemistry

Zero has been proposed as the atomic number of the theoretical element tetraneutron. It has been shown that a cluster of four neutrons may be stable enough to be considered an atom in its own right. This would create an chemical element, element with no protons and no charge on its atomic nucleus, nucleus. As early as 1926, Andreas von Antropoff coined the term neutronium for a conjectured form of matter made up of neutrons with no protons, which he placed as the chemical element of atomic number zero at the head of his new version of the periodic table. It was subsequently placed as a noble gas in the middle of several spiral representations of the periodic system for classifying the chemical elements.


Computer science

The most common practice throughout human history has been to start counting at one, and this is the practice in early classic computer programming languages such as Fortran and COBOL. However, in the late 1950s LISP introduced zero-based numbering for arrays while Algol 58 introduced completely flexible basing for array subscripts (allowing any positive, negative, or zero integer as base for array subscripts), and most subsequent programming languages adopted one or other of these positions. For example, the elements of an Array data type, array are numbered starting from 0 in C (computer language), C, so that for an array of ''n'' items the sequence of array indices runs from 0 to . This permits an array element's location to be calculated by adding the index directly to address of the array, whereas 1-based languages precalculate the array's base address to be the position one element before the first. There can be confusion between 0- and 1-based indexing, for example Java's JDBC indexes parameters from 1 although Java (programming language), Java itself uses 0-based indexing. In databases, it is possible for a field not to have a value. It is then said to have a null (SQL), null value. For numeric fields it is not the value zero. For text fields this is not blank nor the empty string. The presence of null values leads to Ternary logic, three-valued logic. No longer is a condition either ''true'' or ''false'', but it can be ''undetermined''. Any computation including a null value delivers a null result. A null pointer is a pointer in a computer program that does not point to any object or function. In C, the integer constant 0 is converted into the null pointer at compile time when it appears in a pointer context, and so 0 is a standard way to refer to the null pointer in code. However, the internal representation of the null pointer may be any bit pattern (possibly different values for different data types). In mathematics both −0 and +0 represent exactly the same number, i.e., there is no "positive zero" or "negative zero" distinct from zero. However, in some computer hardware signed number representations, zero has two distinct representations, a positive one grouped with the positive numbers and a negative one grouped with the negatives; this kind of dual representation is known as signed zero, with the latter form sometimes called negative zero. These representations include the signed magnitude and one's complement binary integer representations (but not the two's complement binary form used in most modern computers), and most floating point number representations (such as IEEE floating point, IEEE 754 and IBM hexadecimal floating-point, IBM S/390 floating point formats). In binary, 0 represents the value for "off", which means no electricity flow. Zero is the value of false in many programming languages. The Unix epoch (the date and time associated with a zero timestamp) begins the midnight before the first of January 1970. The Classic Mac OS epoch (computing), epoch and Palm OS epoch (the date and time associated with a zero timestamp) begins the midnight before the first of January 1904. Many Application programming interface, APIs and operating systems that require applications to return an integer value as an exit status typically use zero to indicate success and non-zero values to indicate specific error code, error or warning conditions. Programmers often use a Slashed zero#Usage, slashed zero to avoid confusion with the letter "O".


Other fields

* In zoology, comparative zoology and cognitive science, recognition that some animals display awareness of the concept of zero leads to the conclusion that the capability for numerical abstraction arose early in the evolution of species. * In telephony, pressing 0 is often used for dialling out of a Business telephone system, company network or to a different Trunk prefix, city or region, and 00 is used for dialling International call prefix, abroad. In some countries, dialling 0 places a call for operator assistance. * DVDs that can be played in any region are sometimes referred to as being "region 0" * Roulette wheels usually feature a "0" space (and sometimes also a "00" space), whose presence is ignored when calculating payoffs (thereby allowing the house to win in the long run). * In Formula One, if the reigning List of Formula One World Drivers' Champions, World Champion no longer competes in Formula One in the year following their victory in the title race, 0 is given to one of the drivers of the team that the reigning champion won the title with. This happened in 1993 and 1994, with Damon Hill driving car 0, due to the reigning World Champion (Nigel Mansell and Alain Prost respectively) not competing in the championship. * On the U.S. Interstate Highway System, in most states exits are numbered based on the nearest milepost from the highway's western or southern terminus within that state. Several that are less than half a mile (800 m) from state boundaries in that direction are numbered as Exit 0.


Symbols and representations

The modern numerical digit 0 is usually written as a circle or ellipse. Traditionally, many print typefaces made the capital letter O more rounded than the narrower, elliptical digit 0. Typewriters originally made no distinction in shape between O and 0; some models did not even have a separate key for the digit 0. The distinction came into prominence on modern character Visual display unit, displays. A slashed zero (0\!\!\!) can be used to distinguish the number from the letter (mostly used in computing, navigation and in the military). The digit 0 with a dot in the center seems to have originated as an option on IBM 3270 displays and has continued with some modern computer typefaces such as Andalé Mono, and in some airline reservation systems. One variation uses a short vertical bar instead of the dot. Some fonts designed for use with computers made one of the capital-O–digit-0 pair more rounded and the other more angular (closer to a rectangle). A further distinction is made in FE-Schrift, falsification-hindering typeface as used on Vehicle registration plates of Germany, German car number plates by slitting open the digit 0 on the upper right side. Sometimes the digit 0 is used either exclusively, or not at all, to avoid confusion altogether.


Year label

In the Before Christ, BC calendar era, the year 1 BC is the first year before AD 1; there is not a year zero. By contrast, in astronomical year numbering, the year 1 BC is numbered 0, the year 2 BC is numbered −1, and so forth.


See also

*
Brahmagupta Brahmagupta ( – ) was an Indian Indian mathematics, mathematician and Indian astronomy, astronomer. He is the author of two early works on mathematics and astronomy: the ''Brāhmasphuṭasiddhānta'' (BSS, "correctly established Siddhanta, do ...

Brahmagupta
* Division by zero * Grammatical number * Gwalior Fort * Mathematical constant * Number theory * Peano axioms * Signed zero * 0th (disambiguation), Zeroth (zero as an ordinal number)


Notes


References


Bibliography

* * * *


Historical studies

* * * * *


External links


Searching for the World's First Zero



Zero Saga


* Edsger W. Dijkstra
Why numbering should start at zero
EWD831 (Portable Document Format, PDF of a handwritten manuscript) * * * {{DEFAULTSORT:0 (Number) 0 (number), Elementary arithmetic Integers, 00 Indian inventions