0 (zero) is a

, Thesaurus.com – Retrieved April 2013. "Nil" is used for many sports in

translated to English by Henry Thomas Colebrooke (1817) London

^{0} = = 1, except that the case ''x'' = 0 may be left undefined in some

Searching for the World's First Zero

Zero Saga

* Edsger W. Dijkstra

Why numbering should start at zero

EWD831 (

number
A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers c ...

representing an empty quantity
Quantity or amount is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value multiple of a uni ...

. In place-value notation such as the Hindu–Arabic numeral system
The Hindu–Arabic numeral system or Indo-Arabic numeral system Audun HolmeGeometry: Our Cultural Heritage 2000 (also called the Hindu numeral system or Arabic numeral system) is a positional decimal numeral system, and is the most common syste ...

, 0 also serves as a placeholder numerical digit
A numerical digit (often shortened to just digit) is a single symbol used alone (such as "2") or in combinations (such as "25"), to represent numbers in a positional numeral system. The name "digit" comes from the fact that the ten digits (Latin ...

, which works by multiplying digits to the left of 0 by 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 ...

, usually by 10. As a number, 0 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, the additive identity of a set that is equipped with the operation of addition is an element which, when added to any element ''x'' in the set, yields ''x''. One of the most familiar additive identities is the number 0 from eleme ...

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 ...

s, real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one- dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...

s, and other algebraic structure
In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set o ...

s.
Common 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. " ...

are ''zero'', ''nought'', ''naught'' (), ''nil''. In contexts where at least one adjacent digit distinguishes it from the letter O, the number is sometimes pronounced as ''oh'' or ''o'' (). Informal or slang
Slang is vocabulary (words, phrases, and linguistic usages) of an informal register, common in spoken conversation but avoided in formal writing. It also sometimes refers to the language generally exclusive to the members of particular in-gr ...

terms for 0 include ''zilch'' and ''zip''. Historically, ''ought'', ''aught'' (), and ''cipher'', have also been used.
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.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 mathematicianFibonacci
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 ...

(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 reading out a string of digits, such astelephone number
A telephone number is a sequence of digits assigned to a landline telephone subscriber station connected to a telephone line or to a wireless electronic telephony device, such as a radio telephone or a mobile telephone, or to other devices ...

s, street addresses, credit card numbers, military time, or years (e.g. the area code
A telephone numbering plan is a type of numbering scheme used in telecommunication to assign telephone numbers to subscriber telephones or other telephony endpoints. Telephone numbers are the addresses of participants in a telephone network, reac ...

201 would be pronounced "two oh one"; a year such as 1907 is often pronounced "nineteen oh seven"). The presence of other digits, indicating that the string contains only numbers, avoids confusion with the letter O. For this reason, systems that include strings with both letters and numbers (e.g. Canadian postal code
A Canadian postal code (french: code postal) is a six-character string that forms part of a postal address in Canada. Like British, Irish and Dutch postcodes, Canada's postal codes are alphanumeric. They are in the format ''A1A 1A1'', where ...

s) may exclude the use of the letter O.
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 broadly, ...

. 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 affection, to the simplest pleasure. An example of this range of meanings is that the love ...

" in tennis
Tennis is a racket sport that is played either individually against a single opponent ( singles) or between two teams of two players each ( doubles). Each player uses a tennis racket that is strung with cord to strike a hollow rubber ball co ...

– from French ''l'oeuf'', "the egg" – and "duck
Duck is the common name for numerous species of waterfowl in the family Anatidae. Ducks are generally smaller and shorter-necked than swans and geese, which are members of the same family. Divided among several subfamilies, they are a form ...

" 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 ofbase 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 numera ...

. They used hieroglyphs
A hieroglyph (Greek for "sacred carvings") was a character of the ancient Egyptian writing system. Logographic scripts that are pictographic in form in a way reminiscent of ancient Egyptian are also sometimes called "hieroglyphs". In Neoplatoni ...

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 mil ...

, 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. Baby ...

had a sophisticated base 60
Sexagesimal, also known as base 60 or sexagenary, is a numeral system with sixty as its base. It originated with the ancient Sumerians in the 3rd millennium BC, was passed down to the ancient Babylonians, and is still used—in a modified form ...

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 itsvigesimal
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) positional numeral system. Many different glyphs, including the 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 and 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 32 federal entities of Mexico. It comprises 124 municipalities ...

) 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, they occupied the tropical lowlands of the modern-day Mexican states of Veracruz and Tabasco. It has been speculated that ...

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 numeral system. The numerals are made up of three symbols; zero (a shell), one (a dot) and ...

, with a different, empty tortoise
Tortoises () are reptiles of the family Testudinidae of the order Testudines (Latin: ''tortoise''). Like other turtles, tortoises have a shell to protect from predation and other threats. The shell in tortoises is generally hard, and like oth ...

-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, a knotted cord device, used in the Inca Empire
The Inca Empire (also known as the Incan Empire and the Inka Empire), called ''Tawantinsuyu'' by its subjects, ( Quechua for the "Realm of the Four Parts", "four parts together" ) was the largest empire in pre-Columbian America. The admi ...

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. The range is long, wide (widest between 18°S – 20°S l ...

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 and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral ...

positional system. Zero is represented by the absence of a knot in the appropriate position.
Classical antiquity

Theancient Greeks
Ancient Greece ( el, Ἑλλάς, Hellás) was a northeastern Mediterranean civilization, existing from the Greek Dark Ages of the 12th–9th centuries BC to the end of classical antiquity ( AD 600), that comprised a loose collection of cult ...

had no symbol for zero (μηδέν), and did not use a digit placeholder for it. According to mathematician Charles Seife, the ancient Greeks did begin to adopt the Babylonian placeholder zero for their work in astronomy after 500 BC, representing it with the lowercase Greek letter ''ό'' (''όμικρον'') or omicron. However, after using the Babylonian placeholder zero for astronomical calculations they would typically convert the numbers back into 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 ...

. Greeks seemed to have a philosophical opposition to using zero as a number. Other scholars give the Greek partial adoption of the Babylonian zero a later date, with the scientist Andreas Nieder giving a date of after 400 BC and the mathematician Robert Kaplan dating it after the conquests of Alexander.
Greeks seemed unsure about the status of zero as a number. Some of them asked themselves, "How can not being be?", 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". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Physicists often di ...

. The paradoxes of Zeno of Elea
Zeno of Elea (; grc, 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 is best known fo ...

depend in large part on the uncertain interpretation of zero.
By AD 150, Ptolemy
Claudius Ptolemy (; grc-gre, Πτολεμαῖος, ; la, Claudius Ptolemaeus; AD) was a mathematician, astronomer, astrologer, geographer, and music theorist, who wrote about a dozen scientific treatises, three of which were of import ...

, influenced by Hipparchus
Hipparchus (; el, Ἵππαρχος, ''Hipparkhos''; BC) was a Greek astronomer, geographer, and mathematician. He is considered the founder of trigonometry, but is most famous for his incidental discovery of the precession of the equi ...

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 Amorite-ruled state ...

ns, was using a symbol for zero () in his work on mathematical astronomy
Theoretical astronomy is the use of analytical and computational models based on principles from physics and chemistry to describe and explain astronomical objects and astronomical phenomena. Theorists in astronomy endeavor to create theoretical ...

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 canon ...

''. 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 solar 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 the plane of the Earth' ...

s. It represented the value of both digit
Digit may refer to:
Mathematics and science
* Numerical digit, as used in mathematics or computer science
** Hindu-Arabic numerals, the most common modern representation of numerical digits
* Digit (anatomy), the most distal part of a limb, such ...

s 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. They typically describe the events of the meeting and may include a list of attendees, a state ...

of immersion at first and last contact. Digits varied continuously from 0 to 12 to 0 as the Moon passed over the Sun (a triangular pulse), where twelve digits was the angular diameter of the Sun. Minutes of immersion was tabulated from 00 to 3120 to 00, where 00 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 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 ...

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 polytop ...

sides), where d was the digit function and 3120 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 epact
The epact ( la, epactae, from grc, ἐπακται ἡμεραι () = added days), used to be described by medieval computists as the age of a phase of the Moon in days on 22 March; in the newer Gregorian calendar, however, the epact is reckone ...

s as preserved in an Ethiopic document for the years AD311 to 369, using a Ge'ez 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 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 Ottoman c ...

.. 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 for "Dionysius the Humble", Greek: Διονύσιος; – ) was a 6th-century Eastern Roman monk born in Scythia Minor. He was a member of a community of Scythian monks concentrated in Tomis (present day Constanța ...

, 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, ...

. 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 ...

—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 Chinesecounting 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 ...

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 base of the Hindu–Arabic numeral system (or decimal system). More generally, a positional system is a numeral system in which the ...

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 t ...

'' 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 Liu Bang (Emperor Gao) and ruled by the House of Liu. The dynasty was preceded by the short-lived Qin dynasty (221–207 BC) and a w ...

, 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 BCE) was an ancient Indian poet and mathematician, and the author of the ' (also called the ''Pingala-sutras''), the earliest known treatise on Sanskrit prosody.
The ' is a work of eight chapters in the lat ...

(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 poetic metr ...

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 encode text characters as standardized sequences of two different signal durations, called ''dots'' and ''dashes'', or ''dits'' and ''dahs''. Morse code is named after Samuel Morse, one of ...

. Pingala used the Sanskrit
Sanskrit (; attributively , ; nominally , , ) is a classical language belonging to the Indo-Aryan branch of the Indo-European languages. It arose in South Asia after its predecessor languages had diffused there from the northwest in the l ...

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.Bourbaki, Nicolas ''Elements of the History of Mathematics'' (1998), p. 46 A symbol for zero, a large dot likely to be the precursor of the still-current hollow symbol, is used throughout the Bakhshali manuscript, 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 de ...

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 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 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 time cycle being ...

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's ''Glossographia'', and in 1731 taken up in Latin by German philosophe ...

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 from around the 3rd century BCE to the 8th century CE. The term Prakrit is usu ...

original, which is internally dated to AD 458 (Saka era
The Shaka era (IAST: Śaka, Śāka) is a historical Hindu calendar era (year numbering), the epoch (its year zero) of which corresponds to Julian year 78.
The era has been widely used in different regions of India as well as in SE Asia.
Hist ...

380), uses a decimal place-value 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 ...

, 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. Philosopher of astronomy Roger Billard estimates that ...

'' (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 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 trea ...

'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 where the divisor (denominator) is zero. Such a division can be formally expressed as \tfrac, where is the dividend (numerator). In ordinary arithmetic, the expression has no meaning, as there is ...

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

A black dot is used as a decimal placeholder in the Bakhshali manuscript, portions of which date from AD 224–993. 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 theMekong
The Mekong or Mekong River is a trans-boundary river in East Asia and Southeast Asia. It is the world's twelfth longest river and the third longest in Asia. Its estimated length is , and it drains an area of , discharging of water annual ...

, Kratié Province, Cambodia
Cambodia (; also Kampuchea ; km, កម្ពុជា, UNGEGN: ), officially the Kingdom of Cambodia, is a country located in the southern portion of the Indochinese Peninsula in Southeast Asia, spanning an area of , bordered by Thailan ...

, includes the inscription of "605" in 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.
Nume ...

(a set of numeral glyphs for the Hindu–Arabic numeral system
The Hindu–Arabic numeral system or Indo-Arabic numeral system Audun HolmeGeometry: Our Cultural Heritage 2000 (also called the Hindu numeral system or Arabic numeral system) is a positional decimal numeral system, and is the most common syste ...

). The number is the year of the inscription in the Saka era
The Shaka era (IAST: Śaka, Śāka) is a historical Hindu calendar era (year numbering), the epoch (its year zero) of which corresponds to Julian year 78.
The era has been widely used in different regions of India as well as in SE Asia.
Hist ...

, corresponding to a date of AD 683. 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 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 ...

s 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.
Middle Ages

Transmission to Islamic culture

TheArabic
Arabic (, ' ; , ' or ) is a Semitic language spoken primarily across the Arab world.Semitic languages: an international handbook / edited by Stefan Weninger; in collaboration with Geoffrey Khan, Michael P. Streck, Janet C. E.Watson; Walte ...

-language inheritance of science was largely Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece, a country in Southern Europe:
*Greeks, an ethnic group.
*Greek language, a branch of the Indo-European language family.
**Proto-Greek language, the assumed last common ancestor ...

, 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, "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
Abū Jaʿfar ʿAbd Allāh ibn Muḥammad al-Manṣūr (; ar, أبو جعفر عبد الله بن محمد المنصور; 95 AH – 158 AH/714 CE – 6 October 775 CE) usually known simply as by his laqab Al-Manṣūr (المنصور) w ...

'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 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
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 ...

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

TheHindu–Arabic numeral system
The Hindu–Arabic numeral system or Indo-Arabic numeral system Audun HolmeGeometry: Our Cultural Heritage 2000 (also called the Hindu numeral system or Arabic numeral system) is a positional decimal numeral system, and is the most common syste ...

(base 10) reached Western Europe in the 11th century, via Al-Andalus
Al-Andalus translit. ; an, al-Andalus; ast, al-Ándalus; eu, al-Andalus; ber, ⴰⵏⴷⴰⵍⵓⵙ, label=Berber, translit=Andalus; ca, al-Àndalus; gl, al-Andalus; oc, Al Andalús; pt, al-Ândalus; es, al-Ándalus () was the Mu ...

, through Spanish Muslim
Muslims ( ar, المسلمون, , ) are people who adhere to Islam, a monotheistic religion belonging to the Abrahamic tradition. They consider the Quran, the foundational religious text of Islam, to be the verbatim word of the God of Abrah ...

s, the Moors
The term Moor, derived from the ancient Mauri, is an exonym first used by Christian Europeans to designate the Muslim inhabitants of the Maghreb, the Iberian Peninsula, Sicily and Malta during the Middle Ages.
Moors are not a distinct or ...

, together with knowledge of classical astronomy
Astronomy () is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution. Objects of interest include planets, moons, stars, nebulae, galaxi ...

and instruments like the astrolabe
An astrolabe ( grc, ἀστρολάβος ; ar, ٱلأَسْطُرلاب ; persian, ستارهیاب ) is an ancient astronomical instrument that was a handheld model of the universe. Its various functions also make it an elaborate incli ...

; Gerbert of Aurillac
Pope Sylvester II ( – 12 May 1003), originally known as Gerbert of Aurillac, was a French-born scholar and teacher who served as the bishop of Rome and ruled the Papal States from 999 to his death. He endorsed and promoted study of Arab and Gr ...

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 Italian mathematician from the Republic of Pisa, considered to be "the most talented Western ...

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 theHere 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 ''Hindus Hindus (; ) are people who religiously adhere to Hinduism. Jeffery D. Long (2007), A Vision for Hinduism, IB Tauris, , pages 35–37 Historically, the term has also been used as a geographical, cultural, and later religious identifier for ...(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.

algorism
Algorism is the technique of performing basic arithmetic by writing numbers in place value form and applying a set of memorized rules and facts to the digits. One who practices algorism is known as an algorist. This positional notation system h ...

us'' 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
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, ...

. In the 16th century, they became commonly used in Europe.
Mathematics

0 is theinteger
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 ...

immediately preceding 1. Zero is an even number because it is divisible by 2 with no remainder. 0 is neither positive nor negative, or both positive and negative. Many definitions include 0 as a natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...

, 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 null size. In most cultures
Culture () is an umbrella term which encompasses the social behavior, institutions, and norms found in human societies, as well as the knowledge, beliefs, arts, laws, customs, capabilities, and habits of the individuals in these groups.Ty ...

, 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 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 symb ...

s with 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 ...

. 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
A leading zero is any 0 digit that comes before the first nonzero digit in a number string in positional notation.. For example, James Bond's famous identifier, 007, has two leading zeros. Any zeroes appearing to the left of the first non-zero di ...

may be used to distinguish a number.
Elementary algebra

The number 0 is the smallestnon-negative
In mathematics, the sign of a real number is its property of being either positive, negative, or zero. Depending on local conventions, zero may be considered as being neither positive nor negative (having no sign or a unique third sign), or it ...

integer. The natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...

following 0 is 1 and no natural number precedes 0. The number 0 may or may not be considered a natural number, but it is an integer, and hence a rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ratio ...

and a real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one- dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...

(as well as an algebraic number
An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, (1 + \sqrt)/2, is an algebraic number, because it is a root of the p ...

and a complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...

).
The number 0 is neither positive nor negative, and is usually displayed as the central number in a number line
In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a poin ...

. It is neither a 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 ...

nor a composite number
A composite number is a positive integer that can be formed by multiplying two smaller positive integers. Equivalently, it is a positive integer that has at least one divisor other than 1 and itself. Every positive integer is composite, prime, o ...

. It cannot be prime because it has an infinite number of 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, 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
In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...

(or neutral element) with respect to addition.
* Subtraction
Subtraction is an arithmetic operation that represents the operation of removing objects from a collection. Subtraction is signified by the minus sign, . For example, in the adjacent picture, there are peaches—meaning 5 peaches with 2 taken ...

: ''x'' − 0 = ''x'' and 0 − ''x'' = −''x''.
* Multiplication
Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being addi ...

: ''x'' · 0 = 0 · ''x'' = 0.
* Division
Division or divider may refer to:
Mathematics
*Division (mathematics), the inverse of multiplication
* Division algorithm, a method for computing the result of mathematical division
Military
*Division (military), a formation typically consisting ...

: = 0, for nonzero ''x''. But is undefined, because 0 has no multiplicative inverse
In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a fraction ''a''/'' ...

(no real number multiplied by 0 produces 1), a consequence of the previous rule.
* Exponentiation
Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to r ...

: ''x''contexts
''Contexts'': ''Understanding People in their Social Worlds'' is a quarterly peer-reviewed academic journal and an official publication of the American Sociological Association. It is designed to be a more accessible source of sociological ideas ...

. 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 lim operator independently to both operands of the fraction, is a so-called "indeterminate form
In calculus and other branches of mathematical analysis, limits involving an algebraic combination of functions in an independent variable may often be evaluated by replacing these functions by their limits; if the expression obtained after this s ...

". That does not 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
In mathematics, an empty product, or nullary product or vacuous product, is the result of multiplying no factors. It is by convention equal to the multiplicative identity (assuming there is an identity for the multiplication operation in question ...

'') is 1. The factorial
In mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial:
\begin
n! &= n \times (n-1) \times (n-2) \ ...

0! evaluates to 1, as a special case of the empty product.
Other branches of mathematics

* Inset theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concer ...

, 0 is the cardinality
In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...

of the empty set
In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in othe ...

: 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 '' 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
In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets.
A finite set can be enumerated by successively labeling each element with the least n ...

, corresponding to the empty set viewed as a well-ordered set.
* In propositional logic
Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations ...

, 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 neutral element for addition (if defined on the structure under consideration) and an absorbing element for multiplication (if defined).
* In lattice theory
A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra. It consists of a partially ordered set in which every pair of elements has a unique supremum (also called a least upper boun ...

, 0 may denote the bottom element of a bounded lattice
A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra. It consists of a partially ordered set in which every pair of elements has a unique supremum (also called a least upper bo ...

.
* In category theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cat ...

, 0 is sometimes used to denote an initial object of a category
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
*Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce)
...

.
* In recursion theory, 0 can be used to denote the Turing degree of the partial computable functions.
Related mathematical terms

* A ''zero of a function
In mathematics, a zero (also sometimes called a root) of a real-, complex-, or generally vector-valued function f, is a member x of the domain of f such that f(x) ''vanishes'' at x; that is, the function f attains the value of 0 at x, or e ...

'' ''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 zeros of a holomorphic function
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex deri ...

.
* The zero function (or zero map) on a domain ''D'' is the constant function
In mathematics, a constant function is a function whose (output) value is the same for every input value. For example, the function is a constant function because the value of is 4 regardless of the input value (see image).
Basic propertie ...

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 and 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 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 anabsolute temperature
Thermodynamic temperature is a quantity defined in thermodynamics as distinct from kinetic theory or statistical mechanics.
Historically, thermodynamic temperature was defined by Kelvin in terms of a macroscopic relation between thermodynamic wo ...

(as measured in kelvin
The kelvin, symbol K, is the primary unit of temperature in the International System of Units (SI), used alongside its prefixed forms and the degree Celsius. It is named after the Belfast-born and University of Glasgow-based engineer and phys ...

s), 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, usual ...

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 freezing point
The melting point (or, rarely, liquefaction point) of a substance is the temperature at which it changes state from solid to liquid. At the melting point the solid and liquid phase exist in equilibrium. The melting point of a substance depends ...

of water. Measuring sound intensity in decibel
The decibel (symbol: dB) is a relative unit of measurement equal to one tenth of a bel (B). It expresses the ratio of two values of a power or root-power quantity on a logarithmic scale. Two signals whose levels differ by one decibel have a p ...

s or phon
The phon is a logarithmic unit of loudness level for tones and complex sounds. Loudness is measured in sone which is a linear unit. Human sensitivity to sound is variable across different frequencies; therefore, although two different tones ma ...

s, the zero level is arbitrarily set at a reference value—for example, at a value for the threshold of hearing. In physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which rel ...

, the zero-point energy is the lowest possible energy that a quantum mechanical
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, qu ...

physical system
A physical system is a collection of physical objects.
In physics, it is a portion of the physical universe chosen for analysis. Everything outside the system is known as the environment. The environment is ignored except for its effects on the ...

may possess and is the energy of the ground state
The ground state of a quantum-mechanical system is its stationary state of lowest energy; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state. ...

of the system.
Chemistry

Zero has been proposed as theatomic number
The atomic number or nuclear charge number (symbol ''Z'') of a chemical element is the charge number of an atomic nucleus. For ordinary nuclei, this is equal to the proton number (''n''p) or the number of protons found in the nucleus of every ...

of the theoretical element tetraneutron. It has been shown that a cluster of four neutron
The neutron is a subatomic particle, symbol or , which has a neutral (not positive or negative) charge, and a mass slightly greater than that of a proton. Protons and neutrons constitute the nuclei of atoms. Since protons and neutrons behave ...

s may be stable enough to be considered an atom
Every atom is composed of a nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons.
Every solid, liquid, ga ...

in its own right. This would create an element
Element or elements may refer to:
Science
* Chemical element, a pure substance of one type of atom
* Heating element, a device that generates heat by electrical resistance
* Orbital elements, parameters required to identify a specific orbit of ...

with no proton
A proton is a stable subatomic particle, symbol , H+, or 1H+ with a positive electric charge of +1 ''e'' elementary charge. Its mass is slightly less than that of a neutron and 1,836 times the mass of an electron (the proton–electron ma ...

s and no charge on its nucleus.
As early as 1926, Andreas von Antropoff coined the term neutronium for a conjectured form of matter
In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of atoms, which are made up of interacting subatomic parti ...

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
The periodic table, also known as the periodic table of the (chemical) elements, is a rows and columns arrangement of the chemical elements. It is widely used in chemistry, physics, and other sciences, and is generally seen as an icon of ch ...

. 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 classiccomputer programming
Computer programming is the process of performing a particular computation (or more generally, accomplishing a specific computing result), usually by designing and building an executable computer program. Programming involves tasks such as anal ...

languages such as Fortran and COBOL
COBOL (; an acronym for "common business-oriented language") is a compiled English-like computer programming language designed for business use. It is an imperative, procedural and, since 2002, object-oriented language. COBOL is primarily us ...

. However, in the late 1950s LISP
A lisp is a speech impairment in which a person misarticulates sibilants (, , , , , , , ). These misarticulations often result in unclear speech.
Types
* A frontal lisp occurs when the tongue is placed anterior to the target. Interdental lispin ...

introduced zero-based numbering
Zero-based numbering is a way of numbering in which the initial element of a sequence is assigned the index 0, rather than the index 1 as is typical in everyday ''non-mathematical'' or ''non-programming'' circumstances. Under zero-base ...

for arrays while Algol 58
ALGOL 58, originally named IAL, is one of the family of ALGOL computer programming languages. It was an early compromise design soon superseded by ALGOL 60. According to John Backus
The Zurich ACM-GAMM Conference had two principal motives in prop ...

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
An array is a systematic arrangement of similar objects, usually in rows and columns.
Things called an array include:
{{TOC right
Music
* In twelve-tone and serial composition, the presentation of simultaneous twelve-tone sets such that the ...

are numbered starting from 0 in 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
Java Database Connectivity (JDBC) is an application programming interface (API) for the programming language Java, which defines how a client may access a database. It is a Java-based data access technology used for Java database connectivity. I ...

indexes parameters from 1 although Java
Java (; id, Jawa, ; jv, ꦗꦮ; su, ) is one of the Greater Sunda Islands in Indonesia. It is bordered by the Indian Ocean to the south and the Java Sea to the north. With a population of 151.6 million people, Java is the world's most ...

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 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 three-valued logic
In logic, a three-valued logic (also trinary logic, trivalent, ternary, or trilean, sometimes abbreviated 3VL) is any of several many-valued logic systems in which there are three truth values indicating ''true'', ''false'' and some indeterminate ...

. 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
In computing, a null pointer or null reference is a value saved for indicating that the pointer or reference does not refer to a valid object. Programs routinely use null pointers to represent conditions such as the end of a list of unknown leng ...

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
In computer science, compile time (or compile-time) describes the time window during which a computer program is compiled.
The term is used as an adjective to describe concepts related to the context of program compilation, as opposed to concept ...

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, −0 and +0 is equivalent to 0; 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
In computing, signed number representations are required to encode negative numbers in binary number systems.
In mathematics, negative numbers in any base are represented by prefixing them with a minus sign ("−"). However, in RAM or CPU regist ...

, 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
In computing, signed number representations are required to encode negative numbers in binary number systems.
In mathematics, negative numbers in any base are represented by prefixing them with a minus sign ("−"). However, in RAM or CPU regist ...

and one's complement binary integer representations (but not the two's complement
Two's complement is a mathematical operation to reversibly convert a positive binary number into a negative binary number with equivalent (but negative) value, using the binary digit with the greatest place value (the leftmost bit in big- endia ...

binary form used in most modern computers), and most floating-point
In computing, floating-point arithmetic (FP) is arithmetic that represents real numbers approximately, using an integer with a fixed precision, called the significand, scaled by an integer exponent of a fixed base. For example, 12.345 can be r ...

number representations (such as IEEE 754
The IEEE Standard for Floating-Point Arithmetic (IEEE 754) is a technical standard for floating-point arithmetic established in 1985 by the Institute of Electrical and Electronics Engineers (IEEE). The standard addressed many problems found in ...

and 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
Current Unix time ()
Unix time is a date and time representation widely used in computing. It measures time by the number of seconds that have elapsed since 00:00:00 UTC on 1 January 1970, the beginning of the Unix epoch, less adjustments ...

(the date and time associated with a zero timestamp) begins the midnight before the first of January 1970.
The Classic Mac OS
Mac OS (originally System Software; retronym: Classic Mac OS) is the series of operating systems developed for the Macintosh family of personal computers by Apple Computer from 1984 to 2001, starting with System 1 and ending with Mac OS 9. The ...

epoch
In chronology and periodization, an epoch or reference epoch is an instant in time chosen as the origin of a particular calendar era. The "epoch" serves as a reference point from which time is measured.
The moment of epoch is usually decided by ...

and Palm OS
Palm OS (also known as Garnet OS) was a mobile operating system initially developed by Palm, Inc., for personal digital assistants (PDAs) in 1996. Palm OS was designed for ease of use with a touchscreen-based graphical user interface. It is prov ...

epoch (the date and time associated with a zero timestamp) begins the midnight before the first of January 1904.
Many APIs
Apis or APIS may refer to:
*Apis (deity), an ancient Egyptian god
* Apis (Greek mythology), several different figures in Greek mythology
* Apis (city), an ancient seaport town on the northern coast of Africa
** Kom el-Hisn, a different Egyptian ci ...

and operating system
An operating system (OS) is system software that manages computer hardware, software resources, and provides common services for computer programs.
Time-sharing operating systems schedule tasks for efficient use of the system and may also ...

s 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
An error (from the Latin ''error'', meaning "wandering") is an action which is inaccurate or incorrect. In some usages, an error is synonymous with a mistake. The etymology derives from the Latin term 'errare', meaning 'to stray'.
In statistics ...

or warning conditions.
Programmers often use a slashed zero to avoid confusion with the letter " O".
Other fields

* In 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 company network or to a different city or region, and 00 is used for dialling 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
Roulette is a casino game named after the French word meaning ''little wheel'' which was likely developed from the Italian game Biribi''.'' In the game, a player may choose to place a bet on a single number, various groupings of numbers, the ...

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
Formula One (also known as Formula 1 or F1) is the highest class of international racing for open-wheel single-seater formula racing cars sanctioned by the Fédération Internationale de l'Automobile (FIA). The World Drivers' Championship, ...

, if the reigning 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
Damon Graham Devereux Hill, (born 17 September 1960) is a British former professional racing driver from England and the 1996 Formula One World Champion. He is the son of Graham Hill, and, along with Nico Rosberg, one of two sons of a Formula ...

driving car 0, due to the reigning World Champion (Nigel Mansell
Nigel Ernest James Mansell, (; born 8 August 1953) is a British retired racing driver who won both the Formula One World Championship (1992) and the CART Indy Car World Series (1993). Mansell was the reigning F1 champion when he moved over ...

and Alain Prost
Alain Marie Pascal Prost (; born 24 February 1955) is a French retired racing driver and Formula One team owner. A four-time Formula One World Drivers' Champion, from 1987 until 2001 he held the record for most Grand Prix victories until Mich ...

respectively) not competing in the championship.
* On the U.S. Interstate Highway System
The Dwight D. Eisenhower National System of Interstate and Defense Highways, commonly known as the Interstate Highway System, is a network of controlled-access highways that forms part of the National Highway System in the United States. T ...

, 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.Typewriter
A typewriter is a mechanical or electromechanical machine for typing characters. Typically, a typewriter has an array of keys, and each one causes a different single character to be produced on paper by striking an inked ribbon selective ...

s 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 displays
A display device is an output device for presentation of information in visual or tactile form (the latter used for example in tactile electronic displays for blind people). When the input information that is supplied has an electrical signal ...

.
A slashed zero ($0\backslash !\backslash !\backslash !$) 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 falsification-hindering typeface as used on German car number plates by slitting open the digit 0 on the upper right side. In some systems either the letter O or the numeral 0, or both, are excluded from use, to avoid confusion.
Year label

In the BCcalendar era
A calendar era is the period of time elapsed since one ''epoch'' of a calendar and, if it exists, before the next one. For example, it is the year as per the Gregorian calendar, which numbers its years in the Western Christian era (the Coptic ...

, the year 1 BC is the first year before AD 1; there is not a year zero
A year zero does not exist in the Anno Domini (AD) calendar year system commonly used to number years in the Gregorian calendar (nor in its predecessor, the Julian calendar); in this system, the year is followed directly by year . However, the ...

. By contrast, in astronomical year numbering
Astronomical year numbering is based on AD/ CE year numbering, but follows normal decimal integer numbering more strictly. Thus, it has a year 0; the years before that are designated with negative numbers and the years after that are designated ...

, the year 1 BC is numbered 0, the year 2 BC is numbered −1, and so forth.
See also

*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 trea ...

* Aryabhata
Aryabhata (ISO: ) or Aryabhata I (476–550 CE) was an Indian mathematician and astronomer of the classical age of Indian mathematics and Indian astronomy. He flourished in the Gupta Era and produced works such as the ''Aryabhatiya'' (which ...

* Division by zero
In mathematics, division by zero is division where the divisor (denominator) is zero. Such a division can be formally expressed as \tfrac, where is the dividend (numerator). In ordinary arithmetic, the expression has no meaning, as there is ...

* Grammatical number
In linguistics, grammatical number is a grammatical category of nouns, pronouns, adjectives and verb agreement that expresses count distinctions (such as "one", "two" or "three or more"). English and other languages present number categories of ...

* Gwalior Fort
The Gwalior Fort commonly known as the ''Gwāliiyar Qila'', is a hill fort near Gwalior, Madhya Pradesh, India. The fort has existed at least since the 10th century, and the inscriptions and monuments found within what is now the fort campus in ...

* Mathematical constant
A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a symbol (e.g., an alphabet letter), or by mathematicians' names to facilitate using it across multiple mathematical problems. Const ...

* Number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Math ...

* Peano axioms
In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearly ...

* Signed zero
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 (

PDF
Portable Document Format (PDF), standardized as ISO 32000, is a file format developed by Adobe in 1992 to present documents, including text formatting and images, in a manner independent of application software, hardware, and operating systems. ...

of a handwritten manuscript)
*
*
*
{{DEFAULTSORT:0 (Number)
Elementary arithmetic
00
Indian inventions