0 (number)
Encyclopedia

0 (zero; icon ) is both a number
Number
A number is a mathematical object used to count and measure. In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers....

and the numerical digit
Numerical digit
A digit is a symbol used in combinations to represent numbers in positional numeral systems. The name "digit" comes from the fact that the 10 digits of the hands correspond to the 10 symbols of the common base 10 number system, i.e...

used to represent that number in numerals
Numeral system
A numeral system is a writing system for expressing numbers, that is a mathematical notation for representing numbers of a given set, using graphemes or symbols in a consistent manner....

.
It fulfills a central role in mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

In mathematics the additive identity of a set which is equipped with the operation of addition is an element which, when added to any element x in the set, yields x...

of the integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

s, real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

s, and many other algebra
Algebra
Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...

ic structures. As a digit, 0 is used as a placeholder in place value systems
Positional notation
Positional notation or place-value notation is a method of representing or encoding numbers. Positional notation is distinguished from other notations for its use of the same symbol for the different orders of magnitude...

. In the English language
English language
English is a West Germanic language that arose in the Anglo-Saxon kingdoms of England and spread into what was to become south-east Scotland under the influence of the Anglian medieval kingdom of Northumbria...

, 0 may be called zero, nought or (US) naught(icon), nil, or "o". Informal or slang terms for zero include zilch and zip. Ought or aught (icon) have also been used historically.

## Etymology

The word "zero" came via French zéro from Venetian
Venetian language
Venetian or Venetan is a Romance language spoken as a native language by over two million people, mostly in the Veneto region of Italy, where of five million inhabitants almost all can understand it. It is sometimes spoken and often well understood outside Veneto, in Trentino, Friuli, Venezia...

zero, which (together with cipher) came via Italian zefiro from Arabic صفر, ṣafira = "it was empty", ṣifr = "zero", "nothing
Nothing
Nothing is no thing, denoting the absence of something. Nothing is a pronoun associated with nothingness, is also an adjective, and an object as a concept in the Frege-Church ontology....

".

### Early history

By the middle of the 2nd millennium BC, the Babylonian mathematics
Babylonian mathematics
Babylonian mathematics refers to any mathematics of the people of Mesopotamia, from the days of the early Sumerians to the fall of Babylon in 539 BC. Babylonian mathematical texts are plentiful and well edited...

had a sophisticated sexagesimal positional numeral system. The lack of a positional value (or zero) was indicated by a space between sexagesimal numerals. By 300 BC, a punctuation symbol (two slanted wedges) was co-opted as a placeholder
Free variables and bound variables
In mathematics, and in other disciplines involving formal languages, including mathematical logic and computer science, a free variable is a notation that specifies places in an expression where substitution may take place...

in the same Babylonian system
Babylonian numerals
Babylonian numerals were written in cuneiform, using a wedge-tipped reed stylus to make a mark on a soft clay tablet which would be exposed in the sun to harden to create a permanent record....

. In a tablet unearthed at Kish
Kish (Sumer)
Kish is modern Tell al-Uhaymir , and was an ancient city of Sumer. Kish is located some 12 km east of Babylon, and 80 km south of Baghdad ....

(dating from about 700 BC), the scribe Bêl-bân-aplu wrote his zeros with three hooks, rather than two slanted wedges.

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.

Records show that the ancient Greeks
Ancient Greece
Ancient Greece is a civilization belonging to a period of Greek history that lasted from the Archaic period of the 8th to 6th centuries BC to the end of antiquity. Immediately following this period was the beginning of the Early Middle Ages and the Byzantine era. Included in Ancient Greece is the...

seemed unsure about the status of zero as a number. They asked themselves, "How can nothing be something?", leading to philosophical
Philosophy
Philosophy is the study of general and fundamental problems, such as those connected with existence, knowledge, values, reason, mind, and language. Philosophy is distinguished from other ways of addressing such problems by its critical, generally systematic approach and its reliance on rational...

and, by the Medieval period, religious arguments about the nature and existence of zero and the vacuum
Vacuum
In everyday usage, vacuum is a volume of space that is essentially empty of matter, such that its gaseous pressure is much less than atmospheric pressure. The word comes from the Latin term for "empty". A perfect vacuum would be one with no particles in it at all, which is impossible to achieve in...

Zeno's paradoxes are a set of problems generally thought to have been devised by Greek philosopher Zeno of Elea to support Parmenides's doctrine that "all is one" and that, contrary to the evidence of our senses, the belief in plurality and change is mistaken, and in particular that motion is...

of Zeno of Elea
Zeno of Elea
Zeno of Elea was a pre-Socratic Greek philosopher of southern Italy and a member of the Eleatic School founded by Parmenides. Aristotle called him the inventor of the dialectic. He is best known for his paradoxes, which Bertrand Russell has described as "immeasurably subtle and profound".- Life...

depend in large part on the uncertain interpretation of zero.

The concept of zero as a number and not merely a symbol for separation is attributed to India
where by the 9th century AD practical calculations were carried out using zero, which was treated like any other number, even in case of division. The Indian scholar Pingala
Pingala
Pingala is the traditional name of the author of the ' , the earliest known Sanskrit treatise on prosody.Nothing is known about Piṅgala himself...

(circa 5th-2nd century BC) used binary numbers
Binary numeral system
The binary numeral system, or base-2 number system, represents numeric values using two symbols, 0 and 1. More specifically, the usual base-2 system is a positional notation with a radix of 2...

in the form of short and long syllables (the latter equal in length to two short syllables), making it similar to Morse code
Morse code
Morse code is a method of transmitting textual information as a series of on-off tones, lights, or clicks that can be directly understood by a skilled listener or observer without special equipment...

. He and his contemporary Indian scholars used the Sanskrit
Sanskrit
Sanskrit , is a historical Indo-Aryan language and the primary liturgical language of Hinduism, Jainism and Buddhism.Buddhism: besides Pali, see Buddhist Hybrid Sanskrit Today, it is listed as one of the 22 scheduled languages of India and is an official language of the state of Uttarakhand...

word śūnya to refer to zero or void.

### History of zero

The Mesoamerican Long Count calendar
Mesoamerican Long Count calendar
The Mesoamerican Long Count calendar is a non-repeating, vigesimal and base-18 calendar used by several Pre-Columbian Mesoamerican cultures, most notably the Maya. For this reason, it is sometimes known as the Maya Long Count calendar...

developed in south-central Mexico and Central America required the use of zero as a place-holder within its vigesimal
Vigesimal
The vigesimal or base 20 numeral system is based on twenty .- Places :...

(base-20) positional numeral system. Many different glyphs, including this partial quatrefoil
Quatrefoil
The word quatrefoil etymologically means "four leaves", and applies to general four-lobed shapes in various contexts.-In heraldry:In heraldic terminology, a quatrefoil is a representation of a flower with four petals, or a leaf with four leaflets . It is sometimes shown "slipped", i.e. with an...

—were used as a zero symbol for these Long Count dates, the earliest of which (on Stela 2 at Chiapa de Corzo, Chiapas
Chiapas
Chiapas officially Estado Libre y Soberano de Chiapas is one of the 31 states that, with the Federal District, comprise the 32 Federal Entities of Mexico. It is divided in 118 municipalities and its capital city is Tuxtla Gutierrez. Other important cites in Chiapas include San Cristóbal de las...

) has a date of 36 BC. Since the eight earliest Long Count dates appear outside the Maya homeland, it is assumed that the use of zero in the Americas predated the Maya and was possibly the invention of the Olmec
Olmec
The Olmec were the first major Pre-Columbian civilization in Mexico. They lived in the tropical lowlands of south-central Mexico, in the modern-day states of Veracruz and Tabasco....

s. Many of the earliest Long Count dates were found within the Olmec heartland, although the Olmec civilization ended by the 4th century BC, several centuries before the earliest known Long Count dates.

Although zero became an integral part of Maya numerals
Maya numerals
Maya Numerals were a vigesimal numeral system used by the Pre-Columbian Maya civilization.The numerals are made up of three symbols; zero , one and five...

, it did not influence Old World
Old World
The Old World consists of those parts of the world known to classical antiquity and the European Middle Ages. It is used in the context of, and contrast with, the "New World" ....

numeral systems.

Quipu
Quipu
Quipus or khipus were recording devices used in the Inca Empire and its predecessor societies in the Andean region. A quipu usually consisted of colored, spun, and plied thread or strings from llama or alpaca hair. It could also be made of cotton cords...

, a knotted cord device, used in the Inca Empire
Inca Empire
The Inca Empire, or Inka Empire , was the largest empire in pre-Columbian America. The administrative, political and military center of the empire was located in Cusco in modern-day Peru. The Inca civilization arose from the highlands of Peru sometime in the early 13th century...

and its predecessor societies in the Andean
Andes
The Andes is the world's longest continental mountain range. It is a continual range of highlands along the western coast of South America. This range is about long, about to wide , and of an average height of about .Along its length, the Andes is split into several ranges, which are separated...

region to record accounting and other digital data, is encoded in a base ten
Decimal
The decimal numeral system has ten as its base. It is the numerical base most widely used by modern civilizations....

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

The use of a blank on a counting board to represent 0 dated back in India to 4th century BC.

In China, counting rods
Counting rods
Counting rods are small bars, typically 3–14 cm long, used by mathematicians for calculation in China, Japan, Korea, and Vietnam. They are placed either horizontally or vertically to represent any number and any fraction....

were used for decimal calculation since the 4th century BC including the use of blank spaces. Chinese mathematicians understood negative numbers and zero, some mathematicians used 無入, 空, 口 for the latter, until Gautama Siddha
Gautama Siddha
Gautama Siddha, astronomer, astrologer and compiler of Indian descent, known for leading the compilation of the Treatise on Astrology of the Kaiyuan Era during the Tang Dynasty. He was born in Chang'an, and his family was originally from India, according to a tomb stele uncovered in 1977 in Xi'an...

introduced the symbol 0. The Nine Chapters on the Mathematical Art
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 1st century CE...

, which was mainly composed in the 1st century AD, stated "[when subtracting] subtract same signed numbers, add differently signed numbers, subtract a positive number from zero to make a negative number, and subtract a negative number from zero to make a positive number."

Ptolemy
Claudius Ptolemy , was a Roman citizen of Egypt who wrote in Greek. He was a mathematician, astronomer, geographer, astrologer, and poet of a single epigram in the Greek Anthology. He lived in Egypt under Roman rule, and is believed to have been born in the town of Ptolemais Hermiou in the...

, influenced by Hipparchus
Hipparchus
Hipparchus, the common Latinization of the Greek Hipparkhos, can mean:* Hipparchus, the ancient Greek astronomer** Hipparchic cycle, an astronomical cycle he created** Hipparchus , a lunar crater named in his honour...

and the Babylonians, was using a symbol for zero (a small circle with a long overbar) within a sexagesimal numeral system otherwise using alphabetic Greek numerals
Greek numerals
Greek numerals are a system of representing numbers using letters of the Greek alphabet. They are also known by the names Ionian numerals, Milesian numerals , Alexandrian numerals, or alphabetic numerals...

. Because it was used alone, not just as a placeholder, this Hellenistic zero was perhaps the first documented use of a number zero in the Old World. However, the positions were usually limited to the fractional part of a number (called minutes, seconds, thirds, fourths, etc.)—they were not used for the integral part of a number. In later Byzantine
Byzantine Empire
The Byzantine Empire was the Eastern Roman Empire during the periods of Late Antiquity and the Middle Ages, centred on the capital of Constantinople. Known simply as the Roman Empire or Romania to its inhabitants and neighbours, the Empire was the direct continuation of the Ancient Roman State...

manuscripts of Ptolemy's Syntaxis Mathematica (also known as the Almagest), the Hellenistic zero had morphed into the Greek letter omicron
Omicron
Omicron is the 15th letter of the Greek alphabet. In the system of Greek numerals it has a value of 70. It is rarely used in mathematics because it is indistinguishable from the Latin letter O and easily confused with the digit 0...

(otherwise meaning 70).

Another zero was used in tables alongside Roman numerals by 525 (first known use by Dionysius Exiguus
Dionysius Exiguus
Dionysius Exiguus was a 6th-century monk born in Scythia Minor, modern Dobruja shared by Romania and Bulgaria. He was a member of the Scythian monks community concentrated in Tomis, the major city of Scythia Minor...

), but as a word, nulla meaning "nothing", not as a symbol. When division produced zero as a remainder, nihil, also meaning "nothing", was used. These medieval zeros were used by all future medieval computists
Computus
Computus is the calculation of the date of Easter in the Christian calendar. The name has been used for this procedure since the early Middle Ages, as it was one of the most important computations of the age....

(calculators of Easter
Easter
Easter is the central feast in the Christian liturgical year. According to the Canonical gospels, Jesus rose from the dead on the third day after his crucifixion. His resurrection is celebrated on Easter Day or Easter Sunday...

). The initial "N" was used as a zero symbol in a table of Roman numerals by Bede
Bede
Bede , also referred to as Saint Bede or the Venerable Bede , was a monk at the Northumbrian monastery of Saint Peter at Monkwearmouth, today part of Sunderland, England, and of its companion monastery, Saint Paul's, in modern Jarrow , both in the Kingdom of Northumbria...

or his colleague around 725.

In 498 AD, Indian mathematician and astronomer Aryabhata
Aryabhata
Aryabhata was the first in the line of great mathematician-astronomers from the classical age of Indian mathematics and Indian astronomy...

stated that "Sthanam sthanam dasa gunam" or place to place in ten times in value, which is the origin of the modern decimal-based place value notation.

The oldest known text to use a decimal place-value system
Positional notation
Positional notation or place-value notation is a method of representing or encoding numbers. Positional notation is distinguished from other notations for its use of the same symbol for the different orders of magnitude...

, including a zero, is the Jain text from India entitled the Lokavibhâga
Lokavibhaga
The Lokavibhaga is a Jain cosmological text originally composed in Prakrit by a Digambara monk, Sarvanandi, surviving in a Sanskrit version compiled by one Simhasuri...

, dated 458 AD. This text uses Sanskrit numeral words for the digits, with words for zero such as the Sanskrit word for "void" or "empty", shunya. The first known use of special glyph
Glyph
A glyph is an element of writing: an individual mark on a written medium that contributes to the meaning of what is written. A glyph is made up of one or more graphemes....

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 Chaturbhuja Temple
Chaturbhuja Temple
Chaturbhuj Temple, dedicated to Lord Vishnu, is situated at Orchha in Madhya Pradesh, India.Built in the year 875, during the reign of imperial Gurjara Pratihara dynasty, it is constructed within a later colonnade....

at Gwalior in India, dated 876 AD. There are many documents on copper plates, with the same small o in them, dated back as far as the sixth century AD, but their authenticity may be doubted.

The Hindu-Arabic numerals
Arabic numerals
Arabic numerals or Hindu numerals or Hindu-Arabic numerals or Indo-Arabic numerals are the ten digits . They are descended from the Hindu-Arabic numeral system developed by Indian mathematicians, in which a sequence of digits such as "975" is read as a numeral...

and the positional number system were introduced around 500 AD, and in 825 AD, it was introduced by a Persian
Persian people
The Persian people are part of the Iranian peoples who speak the modern Persian language and closely akin Iranian dialects and languages. The origin of the ethnic Iranian/Persian peoples are traced to the Ancient Iranian peoples, who were part of the ancient Indo-Iranians and themselves part of...

scientist, al-Khwārizmī, in his book on arithmetic. This book synthesized Greek and Hindu knowledge and also contained his own fundamental contribution to mathematics and science including an explanation of the use of zero.

It was only centuries later, in the 12th century, that the Arabic numeral system was introduced to the Western world
Western world
The Western world, also known as the West and the Occident , is a term referring to the countries of Western Europe , the countries of the Americas, as well all countries of Northern and Central Europe, Australia and New Zealand...

through Latin
Latin
Latin is an Italic language originally spoken in Latium and Ancient Rome. It, along with most European languages, is a descendant of the ancient Proto-Indo-European language. Although it is considered a dead language, a number of scholars and members of the Christian clergy speak it fluently, and...

translations of his Arithmetic.

## As a number

0 is the integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

immediately preceding 1. In most cultures
History of mathematics
The area of study known as the history of mathematics is primarily an investigation into the origin of discoveries in mathematics and, to a lesser extent, an investigation into the mathematical methods and notation of the past....

, 0 was identified before the idea of negative things (quantities) that go lower than zero was accepted. Zero is an even number, because it is divisible by 2. 0 is neither positive nor negative. By most definitions 0 is a natural number
Natural number
In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...

, and then the only natural number not to be positive.
Zero is a number which quantifies a count or an amount of null
Empty set
In mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality is zero. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced...

size.

The value, or number, zero is not the same as the digit zero, used in numeral system
Numeral system
A numeral system is a writing system for expressing numbers, that is a mathematical notation for representing numbers of a given set, using graphemes or symbols in a consistent manner....

s using positional notation
Positional notation
Positional notation or place-value notation is a method of representing or encoding numbers. Positional notation is distinguished from other notations for its use of the same symbol for the different orders of magnitude...

. Successive positions of digits have higher weights, so inside a numeral the digit zero is used 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, for example, in the number 02. In some instances, a leading zero
A leading zero is any 0 digits, that lead a number string in a positional notation. For example, James Bond's famous identifier, 007, has two leading zeros. Leading zeros occupy most significant digits, which could be left blank or omitted for the same numeric value...

may be used to distinguish a number.

## As a year label

In the BC calendar era
Calendar era
A calendar era is the year numbering system used by a calendar. For example, the Gregorian calendar numbers its years in the Western Christian era . The instant, date, or year from which time is marked is called the epoch of the era...

, the year 1 BC
1 BC
Year 1 BC was a common year starting on Friday or Saturday of the Julian calendar and a leap year starting on Thursday of the Proleptic Julian calendar...

is the first year before AD 1
1
Year 1 was a common year starting on Saturday or Sunday of the Julian calendar and a common year starting on Saturday of the Proleptic Julian calendar...

; no room is reserved for a year zero
Year zero
"Year zero" does not exist in the widely used Gregorian calendar or in its predecessor, the Julian calendar. Under those systems, the year 1 BC is followed by AD 1...

. By contrast, in astronomical year numbering
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 with positive numbers...

, the year 1 BC is numbered 0, the year 2 BC is numbered −1, and so on.

## Names and symbols

Persian people
The Persian people are part of the Iranian peoples who speak the modern Persian language and closely akin Iranian dialects and languages. The origin of the ethnic Iranian/Persian peoples are traced to the Ancient Iranian peoples, who were part of the ancient Indo-Iranians and themselves part of...

Abū ʿAbdallāh Muḥammad ibn Aḥmad ibn Yūsuf al-Kātib al-Khwārizmī, also referred to as al-Balkhī, was a tenth century Persian encyclopedist and the author of the early encyclopedia Mafātīḥ al-ʿulūm in the Arabic language....

, in his "Keys of the Sciences", remarked that if, in a calculation, no number appears in the place of tens, then a little circle should be used "to keep the rows". This circle the Arabs called صفر ṣifr, "empty". That was the earliest mention of the name ṣifr that eventually became zero.

Italian zefiro already meant "west wind" from Latin and Greek zephyrus
Anemoi
In Greek mythology, the Anemoi were Greek wind gods who were each ascribed a cardinal direction from which their respective winds came , and were each associated with various seasons and weather conditions...

; this may have influenced the spelling when transcribing Arabic ṣifr. The Italian mathematician Fibonacci
Fibonacci
Leonardo Pisano Bigollo also known as Leonardo of Pisa, Leonardo Pisano, Leonardo Bonacci, Leonardo Fibonacci, or, most commonly, simply Fibonacci, was an Italian mathematician, considered by some "the most talented western mathematician of the Middle Ages."Fibonacci is best known to the modern...

(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 zefiro in Italian, which was contracted to zero in Venetian.

As the decimal zero and its new mathematics spread from the Arab world to Europe in the Middle Ages
Middle Ages
The Middle Ages is a periodization of European history from the 5th century to the 15th century. The Middle Ages follows the fall of the Western Roman Empire in 476 and precedes the Early Modern Era. It is the middle period of a three-period division of Western history: Classic, Medieval and Modern...

, words derived from ṣifr and zephyrus came to refer to calculation, as well as to privileged knowledge and secret codes. According to Ifrah, "in thirteenth-century Paris, a 'worthless fellow' was called a '... cifre en algorisme', i.e., an 'arithmetical nothing'." From ṣifr also came French chiffre = "digit", "figure", "number", chiffrer = "to calculate or compute", chiffré = "encrypted". Today, the word in Arabic is still ṣifr, and cognates of ṣifr are common in the languages of Europe and southwest Asia.

The modern numerical digit 0 is usually written as a circle or ellipse. Traditionally, many print typefaces made the capital letter O
O
O is the fifteenth letter and a vowel in the basic modern Latin alphabet.The letter was derived from the Semitic `Ayin , which represented a consonant, probably , the sound represented by the Arabic letter ع called `Ayn. This Semitic letter in its original form seems to have been inspired by a...

more rounded than the narrower, elliptical digit 0. Typewriter
Typewriter
A typewriter is a mechanical or electromechanical device with keys that, when pressed, cause characters to be printed on a medium, usually paper. Typically one character is printed per keypress, and the machine prints the characters by making ink impressions of type elements similar to the pieces...

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 slashed zero
Slashed zero
The slashed zero is a representation of the number '0' , with a slash through it. In character encoding terms, it is an alternate glyph for the self-same zero character...

can be used to distinguish the number from the letter. The digit 0 with a dot in the center seems to have originated as an option on IBM 3270
IBM 3270
The IBM 3270 is a class of block oriented terminals made by IBM since 1972 normally used to communicate with IBM mainframes. As such, it was the successor to the IBM 2260 display terminal. Due to the text colour on the original models, these terminals are informally known as green screen terminals...

displays and has continued with the some modern computer typefaces such as Andalé Mono
Andale Mono
Andalé Mono is a monospaced sans-serif typeface designed by Steve Matteson for terminal emulation and software development environments, originally for the Taligent project by Apple Inc. and IBM...

. 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
FE-Schrift
' or ' has been the only typeface used on new vehicle registration plates of Germany since November 2000, except for plates issued to military-registered vehicles, which still use the former DIN 1451 typeface...

as used on 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.

### Rules of Brahmagupta

The rules governing the use of zero appeared for the first time in Brahmagupta
Brahmagupta
Brahmagupta was an Indian mathematician and astronomer who wrote many important works on mathematics and astronomy. His best known work is the Brāhmasphuṭasiddhānta , written in 628 in Bhinmal...

's book Brahmasputha Siddhanta
Brahmasphutasiddhanta
The main work of Brahmagupta, Brāhmasphuṭasiddhānta , written c.628, contains ideas including a good understanding of the mathematical role of zero, rules for manipulating both negative and positive numbers, a method for computing square roots, methods of solving linear and some quadratic...

(The Opening of the Universe)
, written in 628 AD. Here Brahmagupta considers not only zero, but negative numbers, and the algebraic rules for the elementary operations of arithmetic with such numbers. In some instances, his rules differ from the modern standard. Here are the rules of Brahmagupta:
• The sum of zero and a negative number is negative.
• The sum of zero and a positive number is positive.
• The sum of zero and zero is zero.
• The sum of a positive and a negative is their difference; or, if their absolute values are equal, zero.
• A positive or negative number when divided by zero
Division by zero
In mathematics, division by zero is division where the divisor is zero. Such a division can be formally expressed as a / 0 where a is the dividend . Whether this expression can be assigned a well-defined value depends upon the mathematical setting...

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.

In saying zero divided by zero is zero, Brahmagupta differs from the modern position. Mathematicians normally do not assign a value to this, whereas computers and calculators sometimes assign NaN
NaN
In computing, NaN is a value of the numeric data type representing an undefined or unrepresentable value, especially in floating-point calculations...

, which means "not a number." Moreover, non-zero positive or negative numbers when divided by zero are either assigned no value, or a value of unsigned infinity, positive infinity, or negative infinity. Once again, these assignments are not numbers, and are associated more with computer science than pure mathematics, where in most contexts no assignment is done.

### Zero as a decimal digit

Positional notation without the use of zero (using an empty space in tabular arrangements, or the word kha "emptiness") is known to have been in use in India from the 6th century. The earliest certain use of zero as a decimal positional digit dates to the 5th century mention in the text Lokavibhaga
Lokavibhaga
The Lokavibhaga is a Jain cosmological text originally composed in Prakrit by a Digambara monk, Sarvanandi, surviving in a Sanskrit version compiled by one Simhasuri...

. The glyph for the zero digit was written in the shape of a dot, and consequently called bindu
Bindu
Bindu is a Sanskrit term meaning "point" or "dot". The feminine case ending is bindi which denotes a small ornamental, devotional and/or mystical dot that is cosmetically applied or affixed to the forehead in Hinduism....

("dot"). The dot had been used in Greece during earlier ciphered numeral periods.

The Hindu-Arabic numeral system
Hindu-Arabic numeral system
The Hindu–Arabic numeral system or Hindu numeral system is a positional decimal numeral system developed between the 1st and 5th centuries by Indian mathematicians, adopted by Persian and Arab mathematicians , and spread to the western world...

(base 10) reached Europe in the 11th century, via the Iberian Peninsula
Iberian Peninsula
The Iberian Peninsula , sometimes called Iberia, is located in the extreme southwest of Europe and includes the modern-day sovereign states of Spain, Portugal and Andorra, as well as the British Overseas Territory of Gibraltar...

through Spanish Muslim
Muslim
A Muslim, also spelled Moslem, is an adherent of Islam, a monotheistic, Abrahamic religion based on the Quran, which Muslims consider the verbatim word of God as revealed to prophet Muhammad. "Muslim" is the Arabic term for "submitter" .Muslims believe that God is one and incomparable...

s, the Moors
Moors
The description Moors has referred to several historic and modern populations of the Maghreb region who are predominately of Berber and Arab descent. They came to conquer and rule the Iberian Peninsula for nearly 800 years. At that time they were Muslim, although earlier the people had followed...

, together with knowledge of astronomy
Astronomy
Astronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...

and instruments like the astrolabe
Astrolabe
An astrolabe is an elaborate inclinometer, historically used by astronomers, navigators, and astrologers. Its many uses include locating and predicting the positions of the Sun, Moon, planets, and stars, determining local time given local latitude and longitude, surveying, triangulation, and to...

, first imported by Gerbert of Aurillac. For this reason, the numerals came to be known in Europe as "Arabic numerals
Arabic numerals
Arabic numerals or Hindu numerals or Hindu-Arabic numerals or Indo-Arabic numerals are the ten digits . They are descended from the Hindu-Arabic numeral system developed by Indian mathematicians, in which a sequence of digits such as "975" is read as a numeral...

". The Italian mathematician Fibonacci
Fibonacci
Leonardo Pisano Bigollo also known as Leonardo of Pisa, Leonardo Pisano, Leonardo Bonacci, Leonardo Fibonacci, or, most commonly, simply Fibonacci, was an Italian mathematician, considered by some "the most talented western mathematician of the Middle Ages."Fibonacci is best known to the modern...

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 Hindus
Hinduism
Hinduism is the predominant and indigenous religious tradition of the Indian Subcontinent. Hinduism is known to its followers as , amongst many other expressions...

(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 algorism
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...

us
after the Persian mathematician al-Khwārizmī. The most popular was written by Johannes de Sacrobosco
Johannes de Sacrobosco
Johannes de Sacrobosco or Sacro Bosco was a scholar, monk, and astronomer who taught at the University of Paris and wrote the authoritative mediaeval astronomy text Tractatus de Sphaera.-Origins:Although described as English, his birthplace is unknown because Sacrobosco is...

, 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
The numeral system of ancient Rome, or Roman numerals, uses combinations of letters from the Latin alphabet to signify values. The numbers 1 to 10 can be expressed in Roman numerals as:...

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

### Elementary algebra

The number 0 is the smallest non-negative integer. The natural number
Natural number
In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...

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 a whole number and hence a rational number
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...

and a real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

(as well as an algebraic number
Algebraic number
In mathematics, an algebraic number is a number that is a root of a non-zero polynomial in one variable with rational coefficients. Numbers such as π that are not algebraic are said to be transcendental; almost all real numbers are transcendental...

and a complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

).

The number 0 is neither positive nor negative and appears in the middle of a number line
Number line
In basic mathematics, a number line is a picture of a straight line on which every point is assumed to correspond to a real number and every real number to a point. Often the integers are shown as specially-marked points evenly spaced on the line...

. It is neither a prime number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...

nor a composite number
Composite number
A composite number is a positive integer which has a positive divisor other than one or itself. In other words a composite number is any positive integer greater than one that is not a prime number....

. It cannot be prime because it has an infinite
Infinity
Infinity is a concept in many fields, most predominantly mathematics and physics, that refers to a quantity without bound or end. People have developed various ideas throughout history about the nature of infinity...

number of factors
Divisor
In mathematics, a divisor of an integer n, also called a factor of n, is an integer which divides n without leaving a remainder.-Explanation:...

and cannot be composite because it cannot be expressed by multiplying prime numbers (0 must always be one of the factors). Zero is, however, even (see parity of zero).

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
Identity element
In mathematics, an identity element is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them...

(or neutral element) with respect to addition
Addition is a mathematical operation that represents combining collections of objects together into a larger collection. It is signified by the plus sign . For example, in the picture on the right, there are 3 + 2 apples—meaning three apples and two other apples—which is the same as five apples....

.
• Subtraction: x − 0 = x and 0 − x = −x.
• Multiplication: x · 0 = 0 · x = 0.
• Division: = 0, for nonzero x. But is undefined, because 0 has no multiplicative inverse
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/b is b/a. For the multiplicative inverse of a real number, divide 1 by the...

(no real number multiplied by 0 produces 1), a consequence of the previous rule; see division by zero
Division by zero
In mathematics, division by zero is division where the divisor is zero. Such a division can be formally expressed as a / 0 where a is the dividend . Whether this expression can be assigned a well-defined value depends upon the mathematical setting...

.
• Exponentiation: x0 = x/x = 1, except that the case x = 0 may be left undefined in some contexts; see Zero to the zero power. For all positive real x, 0x = 0.

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
Limit of a function
In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input....

operator independently to both operands of the fraction, is a so-called "indeterminate form
Indeterminate form
In calculus and other branches of mathematical analysis, an indeterminate form is an algebraic expression obtained in the context of limits. Limits involving algebraic operations are often performed by replacing subexpressions by their limits; if the expression obtained after this substitution...

". 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
L'Hôpital's rule
In calculus, l'Hôpital's rule uses derivatives to help evaluate limits involving indeterminate forms. Application of the rule often converts an indeterminate form to a determinate form, allowing easy evaluation of the limit...

.

The sum of 0 numbers
Empty sum
In mathematics, an empty sum, or nullary sum, is a summation involving no terms at all. The value of any empty sum of numbers is conventionally taken to be zero...

is 0, and the product of 0 numbers
Empty product
In mathematics, an empty product, or nullary product, is the result of multiplying no factors. It is equal to the multiplicative identity 1, given that it exists for the multiplication operation in question, just as the empty sum—the result of adding no numbers—is zero, or the additive...

is 1. The factorial
Factorial
In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n...

0! evaluates to 1.

### Other branches of mathematics

• In set theory
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...

, 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 defined
Definition
A definition is a passage that explains the meaning of a term , or a type of thing. The term to be defined is the definiendum. A term may have many different senses or meanings...

to be the empty set. When this is done, the empty set is the Von Neumann cardinal assignment
Von Neumann cardinal assignment
The von Neumann cardinal assignment is a cardinal assignment which uses ordinal numbers. For a well-ordered set U, we define its cardinal number to be the smallest ordinal number equinumerous to U. More precisely:...

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
Ordinal number
In set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. They are usually identified with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals...

, corresponding to the empty set viewed as a well-ordered set
Well-order
In mathematics, a well-order relation on a set S is a strict total order on S with the property that every non-empty subset of S has a least element in this ordering. Equivalently, a well-ordering is a well-founded strict total order...

.
• In propositional logic
Propositional calculus
In mathematical logic, a propositional calculus or logic is a formal system in which formulas of a formal language may be interpreted as representing propositions. A system of inference rules and axioms allows certain formulas to be derived, called theorems; which may be interpreted as true...

, 0 may be used to denote the truth value false.
• In abstract algebra
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...

, 0 is commonly used to denote a zero element
Zero element
In mathematics, a zero element is one of several generalizations of the number zero to other algebraic structures. These alternate meanings may or may not reduce to the same thing, depending on the context.-Additive identities:...

, which is a neutral element
Identity element
In mathematics, an identity element is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them...

for addition (if defined on the structure under consideration) and an absorbing element
Absorbing element
In mathematics, an absorbing element is a special type of element of a set with respect to a binary operation on that set. The result of combining an absorbing element with any element of the set is the absorbing element itself. In semigroup theory, the absorbing element is called a zero element...

for multiplication (if defined).
• In lattice theory
Lattice (order)
In mathematics, a lattice is a partially ordered set in which any two elements have a unique supremum and an infimum . Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities...

, 0 may denote the bottom element
Greatest element
In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set is an element of S which is greater than or equal to any other element of S. The term least element is defined dually...

of a bounded lattice
Lattice (order)
In mathematics, a lattice is a partially ordered set in which any two elements have a unique supremum and an infimum . Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities...

.
• In category theory
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...

, 0 is sometimes used to denote an initial object of a category
Category (mathematics)
In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...

.
• In recursion theory
Recursion theory
Computability theory, also called recursion theory, is a branch of mathematical logic that originated in the 1930s with the study of computable functions and Turing degrees. The field has grown to include the study of generalized computability and definability...

, 0 can be used to denote the Turing degree
Turing degree
In computer science and mathematical logic the Turing degree or degree of unsolvability of a set of natural numbers measures the level of algorithmic unsolvability of the set...

of the partial computable functions
Computable function
Computable functions are the basic objects of study in computability theory. Computable functions are the formalized analogue of the intuitive notion of algorithm. They are used to discuss computability without referring to any concrete model of computation such as Turing machines or register...

.

### Related mathematical terms

• A zero of a function f is a point x in the domain of the function such that f(x) = 0. When there are finitely many zeros these are called the roots of the function. See also zero (complex analysis)
Zero (complex analysis)
In complex analysis, a zero of a holomorphic function f is a complex number a such that f = 0.-Multiplicity of a zero:A complex number a is a simple zero of f, or a zero of multiplicity 1 of f, if f can be written asf=g\,where g is a holomorphic function g such that g is not zero.Generally, the...

for zeros of a holomorphic function
Holomorphic function
In mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain...

.
• The zero function (or zero map) on a domain D is the constant function
Constant function
In mathematics, a constant function is a function whose values do not vary and thus are constant. For example the function f = 4 is constant since f maps any value to 4...

with 0 as its only possible output value, i.e., the function f defined by f(x) = 0 for all x in D. A particular zero function is a zero morphism
Zero morphism
In category theory, a zero morphism is a special kind of morphism exhibiting properties like those to and from a zero object.Suppose C is a category, and f : X → Y is a morphism in C...

in category theory; e.g., a zero map is the identity in the additive group of functions. The determinant
Determinant
In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...

on non-invertible square matrices
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...

is a zero map.
• Several branches of mathematics have zero element
Zero element
In mathematics, a zero element is one of several generalizations of the number zero to other algebraic structures. These alternate meanings may or may not reduce to the same thing, depending on the context.-Additive identities:...

s, which generalise either the property 0 + x = x, or the property 0 × x = 0, 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, on the Kelvin
Kelvin
The kelvin is a unit of measurement for temperature. It is one of the seven base units in the International System of Units and is assigned the unit symbol K. The Kelvin scale is an absolute, thermodynamic temperature scale using as its null point absolute zero, the temperature at which all...

temperature scale, zero is the coldest possible temperature (negative temperature
Negative temperature
In physics, certain systems can achieve negative temperatures; that is, their thermodynamic temperature can be a negative quantity. Negative temperatures can be expressed as negative numbers on the kelvin scale....

s exist but are not actually colder), whereas on the Celsius
Celsius
Celsius is a scale and unit of measurement for temperature. It is named after the Swedish astronomer Anders Celsius , who developed a similar temperature scale two years before his death...

scale, zero is arbitrarily defined to be at the freezing point
Melting point
The melting point of a solid 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 on pressure and is usually specified at standard atmospheric pressure...

of water. Measuring sound intensity in decibel
Decibel
The decibel is a logarithmic unit that indicates the ratio of a physical quantity relative to a specified or implied reference level. A ratio in decibels is ten times the logarithm to base 10 of the ratio of two power quantities...

s or phon
Phon
The phon was proposed in DIN 45631 and ISO 532 B as a unit of perceived loudness level LN for pure tones by S. S. Stevens.-Definition:The purpose of the phon scale is to compensate for the effect of frequency on the perceived loudness of tones...

s, the zero level is arbitrarily set at a reference value—for example, at a value for the threshold of hearing. In physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

, the zero-point energy
Zero-point energy
Zero-point energy is the lowest possible energy that a quantum mechanical physical system may have; it is the energy of its ground state. All quantum mechanical systems undergo fluctuations even in their ground state and have an associated zero-point energy, a consequence of their wave-like nature...

is the lowest possible energy
Energy
In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...

that a quantum mechanical
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

physical system
Physical system
In physics, the word system has a technical meaning, namely, it is the portion of the physical universe chosen for analysis. Everything outside the system is known as the environment, which in analysis is ignored except for its effects on the system. The cut between system and the world is a free...

may possess and is the energy of the ground state
Stationary state
In quantum mechanics, a stationary state is an eigenvector of the Hamiltonian, implying the probability density associated with the wavefunction is independent of time . This corresponds to a quantum state with a single definite energy...

of the system.

### Chemistry

Zero has been proposed as the atomic number
Atomic number
In chemistry and physics, the atomic number is the number of protons found in the nucleus of an atom and therefore identical to the charge number of the nucleus. It is conventionally represented by the symbol Z. The atomic number uniquely identifies a chemical element...

of the theoretical element tetraneutron
Tetraneutron
A tetraneutron is a hypothesised stable cluster of four neutrons. This cluster of particles is not supported by current models of nuclear forces...

. It has been shown that a cluster of four neutron
Neutron
The neutron is a subatomic hadron particle which has the symbol or , no net electric charge and a mass slightly larger than that of a proton. With the exception of hydrogen, nuclei of atoms consist of protons and neutrons, which are therefore collectively referred to as nucleons. The number of...

s may be stable enough to be considered an atom
Atom
The atom is a basic unit of matter that consists of a dense central nucleus surrounded by a cloud of negatively charged electrons. The atomic nucleus contains a mix of positively charged protons and electrically neutral neutrons...

in its own right. This would create an element
Chemical element
A chemical element is a pure chemical substance consisting of one type of atom distinguished by its atomic number, which is the number of protons in its nucleus. Familiar examples of elements include carbon, oxygen, aluminum, iron, copper, gold, mercury, and lead.As of November 2011, 118 elements...

with no proton
Proton
The proton is a subatomic particle with the symbol or and a positive electric charge of 1 elementary charge. One or more protons are present in the nucleus of each atom, along with neutrons. The number of protons in each atom is its atomic number....

s and no charge on its nucleus
Atomic nucleus
The nucleus is the very dense region consisting of protons and neutrons at the center of an atom. It was discovered in 1911, as a result of Ernest Rutherford's interpretation of the famous 1909 Rutherford experiment performed by Hans Geiger and Ernest Marsden, under the direction of Rutherford. The...

.

As early as 1926, Professor Andreas von Antropoff coined the term neutronium
Neutronium
Neutronium is a proposed name for a substance composed purely of neutrons. The word was coined by scientist Andreas von Antropoff in 1926 for the conjectured "element of atomic number zero" that he placed at the head of the periodic table...

for a conjectured form of matter
Matter
Matter is a general term for the substance of which all physical objects consist. Typically, matter includes atoms and other particles which have mass. A common way of defining matter is as anything that has mass and occupies volume...

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
Periodic table
The periodic table of the chemical elements is a tabular display of the 118 known chemical elements organized by selected properties of their atomic structures. Elements are presented by increasing atomic number, the number of protons in an atom's atomic nucleus...

. It was subsequently placed as a noble gas in the middle of several spiral representations of the periodic system for classifying the chemical elements.

## In 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 science
Computer science
Computer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...

programming languages such as Fortran
Fortran
Fortran is a general-purpose, procedural, imperative programming language that is especially suited to numeric computation and scientific computing...

and COBOL
COBOL
COBOL is one of the oldest programming languages. Its name is an acronym for COmmon Business-Oriented Language, defining its primary domain in business, finance, and administrative systems for companies and governments....

. However, in the late 1950s LISP
Lisp
A lisp is a speech impediment, historically also known as sigmatism. Stereotypically, people with a lisp are unable to pronounce sibilants , and replace them with interdentals , though there are actually several kinds of lisp...

introduced zero-based numbering for arrays while Algol 58
ALGOL 58
ALGOL 58, originally known as IAL, is one of the family of ALGOL computer programming languages. It was an early compromise design soon superseded by ALGOL 60...

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
Array data type
In computer science, an array type is a data type that is meant to describe a collection of elements , each selected by one or more indices that can be computed at run time by the program. Such a collection is usually called an array variable, array value, or simply array...

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 indexes parameters from 1 although Java
Java
Java is an island of Indonesia. With a population of 135 million , it is the world's most populous island, and one of the most densely populated regions in the world. It is home to 60% of Indonesia's population. The Indonesian capital city, Jakarta, is in west 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 value
Null (SQL)
Null is a special marker used in Structured Query Language to indicate that a data value does not exist in the database. Introduced by the creator of the relational database model, E. F. Codd, SQL Null serves to fulfill the requirement that all true relational database management systems support...

. 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
Ternary logic
In logic, a three-valued logic is any of several many-valued logic systems in which there are three truth values indicating true, false and some indeterminate third value...

. No longer is a condition either true or false, but it can be undetermined. Any computation including a null value delivers a null result. Asking for all records with value 0 or value not equal 0 will not yield all records, since the records with value null are excluded.

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
Compile time
In computer science, compile time refers to either the operations performed by a compiler , programming language requirements that must be met by source code for it to be successfully compiled , or properties of the program that can be reasoned about at compile time.The operations performed at...

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 "negative zero" distinct from zero. In some signed number representations
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 − sign. However, in computer hardware, numbers are represented in binary only without extra...

(but not the two's complement
Two's complement
The two's complement of a binary number is defined as the value obtained by subtracting the number from a large power of two...

representation used to represent integers in most computers today) and most floating point
Floating point
In computing, floating point describes a method of representing real numbers in a way that can support a wide range of values. Numbers are, in general, represented approximately to a fixed number of significant digits and scaled using an exponent. The base for the scaling is normally 2, 10 or 16...

number representations, zero has two distinct representations, one grouping it with the positive numbers and one with the negatives; this latter representation is known as negative zero.

## In other fields

• In some countries and some company phone networks, dialing 0 on a telephone places a call for operator assistance
Operator assistance
An operator-assisted call is one in which the calling party places a telephone call which requires an operator to provide some form of assistance in completing the call...

.
• DVDs that can be played in any region are sometimes referred to as being "region 0"
• Roulette
Roulette
Roulette is a casino game named after a French diminutive for little wheel. In the game, players may choose to place bets on either a single number or a range of numbers, the colors red or black, or whether the number is odd or even....

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
Formula One, also known as Formula 1 or F1 and referred to officially as the FIA Formula One World Championship, is the highest class of single seater auto racing sanctioned by the Fédération Internationale de l'Automobile . The "formula" designation in the name refers to a set of rules with which...

, 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 Hill
Damon Graham Devereux Hill OBE is a retired British racing driver. In 1996 Hill won the Formula One World Championship. As the son of the late Graham Hill, he is the only son of a world champion to win the title...

driving car 0, due to the reigning World Champion (Nigel Mansell
Nigel Mansell
Nigel Ernest James Mansell OBE is a British racing driver who won both the Formula One World Championship and the CART Indy Car World Series...

and Alain Prost
Alain Prost
Alain Marie Pascal Prost, OBE, Chevalier de la Légion d'honneur is a French racing driver. A four-time Formula One Drivers' Champion, Prost has won more titles than any driver except for Juan Manuel Fangio , and Michael Schumacher . From 1987 until 2001 Prost held the record for most Grand Prix...

respectively) not competing in the championship.

• Grammatical number
Grammatical number
In linguistics, grammatical number is a grammatical category of nouns, pronouns, and adjective and verb agreement that expresses count distinctions ....

• Number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...

• Peano axioms
Peano axioms
In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are a set of axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano...

• Zeroth
Zeroth
Zero-based numbering is numbering in which the initial element of a sequence is assigned the index 0, rather than the index 1 as is typical in everyday circumstances. Under zero-based numbering, the initial element is sometimes termed the zeroth element, rather than the first element; zeroth is a...

(Zero as an ordinal number
Ordinal number
In set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. They are usually identified with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals...

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• A History of Zero
• Zero Saga
• The History of Algebra
• Edsger W. Dijkstra: Why numbering should start at zero, EWD831 (PDF
Portable Document Format
Portable Document Format is an open standard for document exchange. This file format, created by Adobe Systems in 1993, is used for representing documents in a manner independent of application software, hardware, and operating systems....

of a handwritten manuscript)
• "My Hero Zero" Educational children's song in Schoolhouse Rock!
Schoolhouse Rock!
Schoolhouse Rock! is an American interstitial programming series of animated musical educational short films that aired during the Saturday morning children's programming on the U.S. television network ABC. The topics covered included grammar, science, economics, history, mathematics, and civics...

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