0(zero) is both anumber^[2]and the numerical digit used to represent that number in numerals. The number 0 fulfills a central role inmathematicsas the additive identity of theintegers,real numbers, and many other algebraic structures. As a digit, 0 is used as a placeholder in place value systems. Names for the number 0 in English includezero,nought(UK),naught(US) (/nɔːt/),nil, or—in contexts where at least one adjacent digit distinguishes it from the letter “O”—ohoro(/oʊ/). Informal or slang terms for zero includezilchandzip.^[3]Oughtandaught(/ɔːt/),^[4]as well ascipher,^[5]have also been used historically.^[6]

The wordzerocame into the English language via FrenchzérofromItalianzero, Italian contraction of Venetianzeveroform of Italianzefiroviaṣafiraorṣifr.^[7]In pre-Islamic time the wordṣifr(Arabic صفر) had the meaning “empty”.^[8]Sifrevolved to mean zero when it was used to translateśūnya(Sanskrit: शून्य) from India.^[8]The first known English use ofzerowas in 1598.^[9]

The Italian mathematician Fibonacci (c. 1170–1250), who grew up in North Africa and is credited with introducing the decimal system to Europe, used the termzephyrum. This becamezefiroin Italian, and was then contracted tozeroin Venetian. The Italian wordzefirowas already in existence (meaning “west wind” from Latin and Greekzephyrus) and may have influenced the spelling when transcribing Arabicṣifr.^[10]

There are different words used for the number or concept of zero depending on the context. For the simple notion of lacking, the wordsnothingandnoneare often used. Sometimes the wordsnought,naughtandaught^[11]are used. Several sports have specific words for zero, such asnilinassociation football(soccer),lovein tennis, andduckin cricket. It is often calledohin the context of telephone numbers. Slang words for zero includezip,zilch,nada, andscratch.Duck eggandgoose eggare also slang for zero.^[12]

Ancient Egyptian numerals werebase 10. They used hieroglyphs 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.^[13]

By the middle of the 2nd millennium BC, the Babylonian mathematics had a sophisticated sexagesimal positional numeral system. The lack of a positional value (or zero) was indicated by aspacebetween sexagesimal numerals. By 300 BC, a punctuation symbol (two slanted wedges) was co-opted as a placeholder in the same Babylonian system. In a tablet unearthed at Kish (dating from about 700 BC), the scribe Bêl-bân-aplu wrote his zeros with three hooks, rather than two slanted wedges.^[14]

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.

The Mesoamerican Long Count calendar developed in south-central Mexico and Central America required the use of zero as a place-holder within its vigesimal (base-20) positional numeral system. Many different glyphs, including this partial quatrefoil—[[INLINE_IMAGE|//upload.wikimedia.org/wikipedia/commons/thumb/1/14/MAYA-g-num-0-inc-v1.svg/30px-MAYA-g-num-0-inc-v1.svg.png|//upload.wikimedia.org/wikipedia/commons/thumb/1/14/MAYA-g-num-0-inc-v1.svg/45px-MAYA-g-num-0-inc-v1.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/14/MAYA-g-num-0-inc-v1.svg/60px-MAYA-g-num-0-inc-v1.svg.png 2x|small illustration of a partial quatrefoil in right half, whitespace in left half|h25|w30]]—were used as a zero symbol for these Long Count dates, the earliest of which (on Stela 2 at Chiapa de Corzo, Chiapas) has a date of 36 BC.^[1]

Since the eight earliest Long Count dates appear outside the Maya homeland,^[15]it is generally believed that the use of zero in the Americas predated the Maya and was possibly the invention of the Olmecs.^[16]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, with a different, empty tortoise-like “shell shape” used for many depictions of the “zero” numeral, it is assumed to have not influenced Old World numeral systems.

Quipu, a knotted cord device, used in theInca Empireand its predecessor societies in the Andean region to record accounting and other digital data, is encoded in abase tenpositional system. Zero is represented by the absence of a knot in the appropriate position.

Theancient Greekshad no symbol for zero (μηδέν), and did not use a digit placeholder for it.^[17]They seemed unsure about the status of zero as a number. They asked themselves, “How can nothingbesomething?”, leading to philosophical and, by the medieval period, religious arguments about the nature and existence of zero and thevacuum. The paradoxes of Zeno of Elea depend in large part on the uncertain interpretation of zero.

By 130 AD,Ptolemy, influenced by Hipparchus and the Babylonians, was using a symbol for zero (a small circle with a long overbar) in his work on mathematical astronomy called theSyntaxis Mathematica, also known as theAlmagest. The way in which it is used can be seen in his table of chords in that book. Ptolemy’s zero was used within a sexagesimal numeral system otherwise using alphabetic Greek numerals. Because it was used alone, not just as a placeholder, this Hellenistic zero was perhaps the earliest documented use of a numeral representing zero in the Old World.^[18]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, indicating a concept perhaps better expressed as “none”, rather than “zero” in the modern sense. In laterByzantinemanuscripts of Ptolemy’sAlmagest, the Hellenistic zero had morphed into the Greek letter omicron (otherwise meaning 70).

Another zero was used in tables alongside Roman numerals by 525 (first known use by Dionysius Exiguus), but as a word,nullameaning “nothing”, not as a symbol.^[19]When division produced zero as a remainder,nihil, also 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 byBedeor his colleagues around 725.

TheSūnzĭ Suànjīng, of unknown date but estimated to be dated from the 1st to 5th centuries AD, and Japanese records dated from the 18th century, describe how the c. 4th century BC Chinese counting rods system enables one to perform decimal calculations. According toA History of Mathematics, the rods “gave the decimal representation of a number, with an empty space denoting zero.”^[20]The counting rod system is considered a positional notation system.^[21]

In AD 690, Empress Wǔ promulgated Zetian characters, one of which was “〇”. The symbol 0 for denoting zero is a variation of this character.

Zero was not treated as a number at that time, but as a “vacant position”.^[22]Qín Jiǔsháo’s 1247Mathematical Treatise in Nine Sectionsis the oldest surviving Chinese mathematical text using a round symbol for zero.^[23]Chinese authors had been familiar with the idea of negative numbers by the Han Dynasty (2nd century AD), as seen inThe Nine Chapters on the Mathematical Art.^[24]

Pingala (c. 3rd/2nd century BC^[25]), a Sanskrit prosody scholar,^[26]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.^[27]Pingala used theSanskritwordśūnyaexplicitly to refer to zero.^[28]

It was considered that the earliest text to use a decimal place-value system, including a zero, is theLokavibhāga, a Jain text oncosmologysurviving in a medieval Sanskrit translation of the Prakrit original, which is internally dated to AD 458 (Saka era 380). In this text,śūnya(“void, empty”) is also used to refer to zero.^[29]

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.^[30]In 2017 three samples from the manuscript were shown byradiocarbon datingto come from three different centuries: from 224-383 AD, 680-779 AD, and 885-993 AD, making it the world’s oldest recorded use of the zero symbol. It is not known how the birch bark fragments from different centuries that form the manuscript came to be packaged together.^[31][32][33]

The origin of the modern decimal-based place value notation can be traced to theAryabhatiya(c. 500), which statessthānāt sthānaṁ daśaguṇaṁ syāt“from place to place each is ten times the preceding.”^{[34][34][35][36]}The concept of zero as a digit in the decimal place value notation was developed inIndia, presumably as early as during the Gupta period (c. 5th century), with the oldest unambiguous evidence dating to the 7th century.^[37]

The rules governing the use of zero appeared for the first time in Brahmagupta’sBrahmasputha Siddhanta(7th century). This work considers not only zero, but also 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, specifically the definition of the value of zero divided by zero as zero.^[38]

There are numerous copper plate inscriptions, with the same smalloin them, some of them possibly dated to the 6th century, but their date or authenticity may be open to doubt.^[14]

A stone tablet found in the ruins of a temple near Sambor on the Mekong, Kratié Province, Cambodia, includes the inscription of “605” in Khmer numerals (a set of numeral glyphs for the Hindu–Arabic numeral system). The number is the year of the inscription in the Saka era, corresponding to a date of AD 683.^[39]

The first known use of special glyphs for the decimal digits that includes the indubitable appearance of a symbol for the digit zero, a small circle, appears on a stone inscription found at the Chaturbhuja Temple atGwaliorin India, dated 876.^[40][41]Zero is also used as a placeholder in the Bakhshali manuscript, portions of which date from AD 224–383.^[42]

TheArabic-language inheritance of science was largely Greek,^[43]followed by Hindu influences.^[44]In 773, at Al-Mansur’s behest, translations were made of many ancient treatises including Greek, Roman, Indian, and others.

In AD 813, astronomical tables were prepared by aPersianmathematician, Muḥammad ibn Mūsā al-Khwārizmī, using Hindu numerals;^[44]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.^[45]This book was later translated intoLatinin the 12th century under the titleAlgoritmi 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 meaning any arithmetic based on decimals.^[44]

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 calledṣifr.^[46]

The Hindu–Arabic numeral system (base 10) reached Europe in the 11th century, viaAl-Andalusthrough Spanish Muslims, theMoors, together with knowledge ofastronomyand instruments like the astrolabe, first imported by Gerbert of Aurillac. For this reason, the numerals came to be known in Europe as “Arabic numerals”. The Italian mathematician Fibonacci or Leonardo of Pisa was instrumental in bringing the system into European mathematics in 1202, stating:

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 calledalgorismusafter 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 beprintedin 1488. Until the late 15th century, Hindu–Arabic numerals seem to have predominated among mathematicians, while merchants preferred to use theRoman numerals. In the 16th century, they became commonly used in Europe.

0is theintegerimmediately preceding 1. Zero is an even number^[49]because it is divisible by 2 with no remainder. 0 is neither positive nor negative.^[50]Many definitions^[51]include 0 as anatural number, and then the only natural number not to be positive. Zero is a number which quantifies a count or an amount of null size. In most cultures, 0 was identified before the idea of negative things, or quantities less than zero, was accepted.

The value, ornumber, zero is not the same as thedigitzero, used in numeral systems using positional notation. 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 may be used to distinguish a number.

The number 0 is the smallest non-negative integer. Thenatural numberfollowing 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 arational numberand areal number(as well as an algebraic number and acomplex number).

The number 0 is neither positive nor negative and is usually displayed as the central number in a number line. It is neither a prime number nor a composite number. It cannot be prime because it has aninfinitenumber of factors, and cannot be composite because it cannot be expressed as a product of prime numbers (0 must always be one of the factors).^[52]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 numberx, unless otherwise stated.

The expression 0/0, which may be obtained in an attempt to determine the limit of an expression of the formf(x)/g(x) as a result of applying the lim operator independently to both operands of the fraction, is a so-called “indeterminate form”. That does not simply mean that the limit sought is necessarily undefined; rather, it means that the limit off(x)/g(x), if it exists, must be found by another method, such as l’Hôpital’s rule.

The sum of 0 numbers (theempty sum) is 0, and the product of 0 numbers (theempty product) is 1. The factorial 0! evaluates to 1, as a special case of the empty product.

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

Zero has been proposed as the atomic number of the theoretical element tetraneutron. It has been shown that a cluster of fourneutronsmay be stable enough to be considered an atom in its own right. This would create an element with no protons and no charge on itsnucleus.

As early as 1926, Andreas von Antropoff coined the term neutronium for a conjectured form ofmattermade up of neutrons with no protons, which he placed as the chemical element of atomic number zero at the head of his new version of the periodic table. It was subsequently placed as a noble gas in the middle of several spiral representations of the periodic system for classifying the chemical elements.

The most common practice throughout human history has been to start counting at one, and this is the practice in early classiccomputer programminglanguages such asFortranand COBOL. However, in the late 1950s LISP introduced zero-based numbering for arrays while Algol 58 introduced completely flexible basing for array subscripts (allowing any positive, negative, or zero integer as base for array subscripts), and most subsequent programming languages adopted one or other of these positions. For example, the elements of an array are numbered starting from 0 in C, so that for an array ofnitems the sequence of array indices runs from 0 ton−1. 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 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.^[53]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. No longer is a condition eithertrueorfalse, but it can beundetermined. Any computation including a null value delivers a null result.

A null pointer is a pointer in a computer program that does not point to any object or function. In C, the integer constant 0 is converted into the null pointer at compile time when it appears in a pointer context, and so 0 is a standard way to refer to the null pointer in code. However, the internal representation of the null pointer may be any bit pattern (possibly different values for different data types).

In mathematics −0 = +0 = 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, zero has two distinct representations, a positive one grouped with the positive numbers and a negative one grouped with the negatives; this kind of dual representation is known as signed zero, with the latter form sometimes called negative zero. These representations include the signed magnitude and one’s complement binary integer representations (but not the two’s complement binary form used in most modern computers), and most floating point number representations (such as IEEE 754 and IBM S/390 floating point formats).

In binary, 0 represents the value for “off”, which means no electricity flow.^[54]

Zero is the value of false in many programming languages.

The Unix epoch (the date and time associated with a zero timestamp) begins the midnight before the first of January 1970.^[55][56][57]

The MacOS epoch and Palm OS epoch (the date and time associated with a zero timestamp) begins the midnight before the first of January 1904.^[58]

ManyAPIsandoperating systemsthat 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 or warning conditions.

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.^[59]Typewriters originally made no distinction in shape between O and 0; some models did not even have a separate key for the digit 0. The distinction came into prominence on modern character displays.^[59]

A slashed zero 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 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. Sometimes the digit 0 is used either exclusively, or not at all, to avoid confusion altogether.

In the BC calendar era, the year 1 BC is the first year before AD 1; there is not a year zero. By contrast, in astronomical year numbering, the year 1 BC is numbered 0, the year 2 BC is numbered −1, and so on.^[60]