Cardinal0 zero o/oh nought nothing nilOrdinal0th zerothFactorization0displaystyle 0
DivisorsN/ARoman numeralN/ABinary0Octal0Duodecimal0Hexadecimal0

0 (zero) is both a number and also a numerical digit provided to stand for that number in numerals. Together a number, zero method nothing—an lack of various other values. The plays a central role in mathematics as the identity facet of the integers, genuine numbers, and also many other algebraic structures. As a digit, zero is used as a placeholder in place value systems. Historically, it was the critical digit come come right into use. In the English language, zero may likewise be dubbed nil as soon as a number, o/oh when a numeral, and also nought/naught in one of two people context.

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4 History5 In mathematics6 In science7 In computer science

0 as a number

0 is the integer that comes before the positive 1, and follows −1. In most (if not all) number systems, 0 to be identified prior to the idea of "negative integers" to be accepted.

Zero is one integer i beg your pardon quantifies a count or an amount of null size; that is, if the variety of your brother is zero, that method the very same thing as having actually no brothers, and also if something has actually a load of zero, it has no weight. If the difference in between the variety of pieces in two piles is zero, it method the 2 piles have actually an equal number of pieces. Prior to counting starts, the result can it is in assumed to it is in zero; that is the number of items counted before you count the first item and counting the first item bring the an outcome to one. And also if there space no item to it is in counted, zero remains the final result.

While mathematicians all expropriate zero as a number, some non-mathematicians would certainly say the zero is no a number, arguing one cannot have actually zero of something. Others host that if you have actually a financial institution balance that zero, you have a certain quantity that money in your account, specific none. It is that last view i m sorry is embraced by mathematicians and most others.

Almost all historians omit the year zero indigenous the proleptic Gregorian and Julian calendars, however astronomers encompass it in these very same calendars. However, the phrase Year Zero might be provided to describe any event considered so significant that it practically starts a brand-new time reckoning.

0 together a numeral


The modern-day numeral 0 is normally written as a circle or (rounded) rectangle. In old-style fonts with text figures, 0 is normally the same height as a small letter x.



On the seven-segment screens of calculators, watches, etc., 0 is generally written with 6 line segments, though on some historical calculator models that was created with four line segments. This different glyph has not recorded on.

It is essential to distinguish the number zero (as in the "zero brothers" instance above) from the numeral or digit zero, offered in numeral systems making use of positional notation. Succeeding positions of digits have greater values, for this reason the number zero is offered to skip a position and give proper value to the preceding and following digits. A zero number is not always necessary in a positional number system: bijective numeration provides a feasible counterexample.


The word zero comes v the Arabic literal meaning translation of the Sanskrit śūnya ( शून्य ), meaning void or empty, right into ṣifr (صفر) meaning empty or vacant. With transliteration this became zephyr or zephyrus in Latin. Words zephyrus currently meant "west wind" in Latin; the ideal noun Zephyrus was the roman god of the west wind (after the Greek god Zephyros). Through its brand-new use because that the concept of zero, zephyr came to mean a light breeze—"an nearly nothing."<1> This came to be zefiro in Italian, which was contracted come zero in Venetian, offering the modern-day English word.

As the Hindu decimal zero and also its brand-new mathematics spread out from the Arab people to Europe in the center Ages, words acquired from sifr and also zephyrus came to refer to calculation, and also to privileged understanding and mystery codes. Follow to Ifrah, "in thirteenth-century Paris, a "worthless fellow" was dubbed a… cifre en algorisme, i.e., one "arithmetical nothing.""<1> The Arabic root provided rise to the contemporary French chiffre, which way digit, figure, or number; chiffrer, to calculation or compute; and chiffré, encrypted; as well as to the English indigenous cipher. Listed below are some more examples:

Arabic: SifrCzech/Slovak: cifra, digit; šifra, cypherDanish: ciffer, digitDutch: cijfer, digitFrench: zéro, zeroGerman: Ziffer, digit, figure, numeral, cypherHindi: shunyaHungarian: nullaItalian: cifra, digit, numeral, cypher; zero, zeroKannada: sonneNorwegian: siffer, digit, numeral, cypher; null, zeroPersian: SefrPolish: cyfra, digit; szyfrować, come encrypt; zero, zeroPortuguese: cifra, figure, numeral, cypher, code; zero, zeroRussian: цифра (tsifra), digit, numeral; шифр (shifr) cypher, codeSlovenian: cifra, digitSpanish: cifra, figure, numeral, cypher, code; cero, zeroSwedish: siffra, numeral, sum, digit; chiffer, cypherSerbian: цифра (tsifra), digit, numeral; шифра (shifra) cypher, code; нула (nula), zeroTurkish: SıfırUrdu: Sifer, Anda, Zero

Note that zero in Greek is interpreted as Μηδέν (Mèden).


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Early background of zero

By the mid-second millennium B.C.E., the Babylonians had a innovative sexagesimal (base-60) positional character system. The absence of a positional value (or zero) was shown by a an are between sexagesimal numerals. By 300 B.C.E. A punctuation prize (two slanted wedges) was co-opted as a placeholder in the exact same Babylonian system. In a tablet unearthed in ~ Kish (dating from maybe as far back as 700 B.C.E.), the scribe Bêl-bân-aplu wrote his zeroes with three hooks, fairly than two slanted wedges.<2>

The Babylonian placeholder was no a true zero due to the fact that it was not provided alone. Nor to be it offered at the end of a number. For this reason numbers favor 2 and 120 (2×60), 3 and also 180 (3×60), 4 and 240 (4×60), etc. Looked the same since the larger numbers short a last sexagesimal placeholder. Just context can differentiate them.

Records present that the ancient Greeks seemed unsure about the status of zero as a number: they asked us "How can nothing be something?," top to exciting philosophical and, through the middle ages period, religious arguments around the nature and existence that zero and also the vacuum. The paradoxes that Zeno the Elea count in large part ~ above the uncertain interpretation of zero. (The ancient Greeks also questioned that 1 was a number.)

Early use of something choose zero through the Indian scholar Pingala (circa 5th-2nd century B.C.E.), implied at very first glance through his use of binary numbers, is only the modern-day binary depiction using 0 and also 1 applied to Pingala"s binary system, which offered short and also long rate (the latter equal in length to two short syllables), make it comparable to Morse code.<3><4> Nevertheless, he and other Indian scholar at the time supplied the Sanskrit word śūnya (the origin of words zero after a series of transliterations and a literal meaning translation) to refer to zero or void.<5>


History the zero

The long Count calendar developed in south-central Mexico forced the use of zero as a place-holder within its vigesimal (base-20) positional numeral system. A covering glyph—

—was provided as a zero symbol for these lengthy Count dates, the more quickly of which (on Stela 2 in ~ Chiapa de Corzo, Chiapas) has actually a day of 36 B.C.E. Because the eight earliest long Count dates show up outside the Maya homeland,<6> that is assumed the the usage of zero in the ameri predated the Maya and was perhaps the invention of the Olmecs. Indeed, many of the earliest long Count days were found within the Olmec heartland, return the fact that the Olmec civilization had come to an finish by the fourth century B.C.E., several centuries before the earliest known Long counting dates, argues against the zero being an Olmec invention.

Although zero became an integral part of Maya numerals, the of course did not affect Old civilization numeral systems.

By 130 C.E., Ptolemy, affected by Hipparchus and also the Babylonians, was using a symbol because that zero (a tiny circle through a lengthy overbar) within a sexagesimal numeral device otherwise using alphabetic Greek numerals. Due to the fact that it was provided alone, not just as a placeholder, this Hellenistic zero was perhaps the very first documented use of a number zero in the Old World. However, the positions to be usually minimal to the fractional part of a number (called minutes, seconds, thirds, fourths, etc.)—they were not provided for the integral component of a number.

Another zero was offered in tables alongside roman numerals through 525 (first known use by Dionysius Exiguus), but as a word, nulla, meaning nothing, no as a symbol. When division produced zero as a remainder, nihil, also an interpretation nothing, to be used. These middle ages zeros were supplied by every future middle ages computists (calculators of Easter). One isolated usage of their initial, N, was supplied in a table of roman numerals by Bede or a colleague around 725, as a zero symbol.

The oldest recognized text to use zero is the Jain text from India licensed has been granted the Lokavibhaaga, date 458 C.E.<1>

The very first indubitable figure of a symbol for zero appears in 876 in India top top a rock tablet in Gwalior. Files on copper plates, with the same small o in them, dated back as much as the sixth century C.E. Abound.<2>

Rules of Brahmagupta

The rule governing the usage of zero appeared for the an initial time in the publication Brahmasputha Siddhanta composed in 628 by Brahmagupta (598-670). Here, Brahmagupta considers not only zero yet also an adverse numbers, and the algebraic rules for the elementary work of arithmetic v such numbers. In part instances, his rules differ indigenous the contemporary standard. Brahamagupta"s rule are given below:<7>

The amount of two confident quantities is positiveThe sum of two an unfavorable quantities is negativeThe amount of zero and a an unfavorable number is negativeThe amount of a optimistic number and zero is positiveThe sum of zero and zero is zeroThe amount of a positive and a an adverse is your difference; or, if they room equal, zeroIn subtraction, the much less is to it is in taken native the greater, hopeful from positiveIn subtraction, the less is to be taken native the greater, an unfavorable from negativeWhen the better however, is subtracted native the less, the difference is reversedWhen hopeful is to be subtracted from negative, and an unfavorable from positive, they should be added togetherThe product of a an unfavorable quantity and also a confident quantity is negativeThe product of a negative quantity and also a an adverse quantity is positiveThe product of two positive, is positivePositive separated by hopeful or an unfavorable by an unfavorable is positivePositive divided by an adverse is negative. An adverse divided by positive is negativeA confident or negative number when divided by zero is a fraction with the zero as denominatorZero divided by a negative or positive number is either zero or is expressed as a fraction with zero together numerator and also the finite quantity as denominatorZero separated by zero is zero

In saying "zero split by zero is zero," Brahmagupta different from the modern position. Mathematicians usually do not assign a value, conversely, computers and calculators will periodically assign NaN, which method "not a number." Moreover, non-zero positive or an unfavorable numbers when separated by zero room either assigned no value, or a value of unsigned infinity, hopeful infinity, or an adverse infinity. Once again, this assignments space not numbers, and are associated more with computer system science 보다 pure mathematics, where in most contexts no assignment is made. (See division by zero)

Zero together a decimal digit

Positional notation without the usage of zero (using an empty room in tabular arrangements, or words kha "emptiness") is well-known to have remained in use in India indigenous the sixth century. The earliest particular use the zero as a decimal positional digit dates to the nine century. The glyph because that the zero number was written in the shape of a dot, and consequently referred to as bindu "dot."

The Hindu-Arabic numeral system reached Europe in the eleventh century, via the Iberian Peninsula through Spanish Muslims the Moors, together with knowledge that astronomy and also instruments favor the astrolabe, first imported through Gerbert the Aurillac (c. 940-1003). They happened known together "Arabic numerals." The Italian mathematician Leonardo of Pisa (c. 1170-1250), also called Fibonacci,* was important in happen the system right into European mathematics in 1202. Here, Leonardo states:

There, following my introduction, together a repercussion of marvelous accuse in the art, come the ripe digits the the Hindus, the knowledge of the art an extremely much appealed to me prior to all others, and also for it ns realized the all its elements were learned in Egypt, Syria, Greece, Sicily, and Provence, through their differing methods… however all this even, and also the algorism, and the arts of Pythagoras, I taken into consideration as almost a wrong in respect come the technique of the Hindus. (Modus Indorum)… The ripe 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.<8>

Here Leonardo the Pisa provides the word authorize "0," indicating it is choose a sign to carry out operations like addition or multiplication, yet he did not acknowledge zero as a number in its very own right.

In mathematics

Elementary algebra

Zero (0) is the lowest non-negative integer. The herbal number complying with zero is one and no herbal number comes before zero. Zero may or may not be counted together a organic number, depending upon the an interpretation of natural numbers.

In collection theory, the number zero is the cardinality that the empty set: if one does no have any kind of apples, climate one has zero apples. Therefore, in some cases, zero is characterized to be the empty set.

Zero is neither optimistic nor negative, neither a prime number nor a composite number, nor is that a unit.

The adhering to are some basic rules for taking care of the number zero. These rules apply for any facility number x, uneven otherwise stated.

Addition: x + 0 = 0 + x = x. (That is, 0 is an identity element with respect come addition.)Subtraction: x − 0 = x and 0 − x = − x.Multiplication: x · 0 = 0 · x = 0.Division: 0 / x = 0, for nonzero x. But x / 0 is undefined, because 0 has no multiplicative inverse, a an effect of the ahead rule. For confident x, together y in x / y approaches zero from confident values, its quotient boosts toward optimistic infinity, but as y philosophies zero from negative values, the quotient rises toward an unfavorable infinity. The various quotients confirms that department by zero is undefined.Exponentiation: x0 = 1, other than that the instance x = 0 may be left undefined in some contexts. For all hopeful real x, 0x = 0.The sum of 0 numbers is 0, and the product of 0 numbers is 1.

The expression "0/0" is one "indeterminate form." the does not simply average that that is undefined; rather, it way that if f(x) and also g(x) both technique 0 together x philosophies some number, climate f(x)/g(x) could approach any type of finite number or ∞ or −∞; it counts on which features f and g are. Watch L"Hopital"s rule.

Extended use of zero in mathematics

Zero is the identity facet in an additive team or the additive identification of a ring.A zero of a role is a allude in the domain the the duty whose picture under the duty is zero. When there space finitely numerous zeros this are dubbed the root of the function. View zero (complex analysis).The ide of "almost" difficult in probability. Much more generally, the ide of nearly nowhere in measure up theory. Because that instance: if one chooses a point on a unit heat interval <0,1) in ~ random, that is not impossible to pick 0.5 exactly, but the probability the you will certainly is zero.A zero duty (or zero map) is a continuous function v 0 as its only feasible output value; i.e., f(x) = 0 for every x defined. A particular zero role is a zero morphism in classification theory; e.g., a zero map is the identification in the additive group of functions. The determinant top top non-invertible square matrices is a zero map.Zero is one of three feasible return worths of the Möbius function. Passed an essence of the type x2 or x2y (for x > 1), the Möbius duty returns zero.Zero is the an initial Perrin number.

In science


The worth zero dram a special function for a huge number of physical quantities. For some quantities, the zero level is naturally identified from all other levels, whereas because that others that is an ext or much less arbitrarily chosen. Because that example, on the Kelvin temperature scale, zero is the coldest feasible temperature (negative temperature exist however are not actually colder), vice versa, on the Celsius scale, zero is arbitrarily identified to be at the freezing point of water. Measure sound strongness in decibels or phons, the zero level is arbitrarily set at a referral value—for example, in ~ a value for the threshold of hearing.


Zero has been proposed together the atomic variety of the theoretical aspect tetraneutronium. It has actually been shown that a cluster of 4 neutrons might be stable enough to be considered an atom in their own right. This would produce an element with no protons and no fee on the nucleus.

As at an early stage as 1926 Professor Andreas von Antropoff coined the hatchet neutronium because that a conjectured type of matter consisted of of neutrons through no protons, which he inserted as the chemical aspect of atom number zero at the head the his new version the the routine table. It was subsequently inserted as a noble gas in the center of numerous spiral depictions of the periodic system for classifying the chemistry elements. The is in ~ the facility of the chemical Galaxy (2005).

In computer system science

Numbering indigenous 1 or 0?

The most typical practice throughout human history has to be to start counting in ~ one. Nevertheless, in computer system science zero has come to be the standard beginning point. For example, in nearly all old programming languages, an array starts indigenous 1 by default. Together programming languages have developed, it has actually become more common that an array starts native zero by default, the "first" article in the selection being items 0. In particular, the popularity of the programming language "C" in the 1980s has made this approach common.

One factor for this convention is that modular arithmetic normally explains a set of N numbers as containing 0,1,2,… N-1 in order to contain the additive identity. Due to the fact that of this, many arithmetic principles (such together hash tables) are less elegant to express in password unless the selection starts in ~ zero.

In details cases, counting from zero improves the performance of various algorithms, such as in browsing or sorting arrays. Enhanced efficiency means that the algorithm bring away either less time, less resources, or both, to finish a provided task.

This situation can lead to some man in terminology. In a zero-based indexing scheme, the very first element is "element number zero"; likewise, the twelfth facet is "element number eleven." Therefore, an analogy native the ordinal numbers to the amount of objects numbered appears; the highest index that n objects will certainly be (n-1) and referred come the n:th element. For this reason, the very first element is often referred to as the zeroth element to get rid of any possible doubt.

Null value

In databases a ar can have a null value. This is tantamount to the field not having actually a value. Because that numeric fields it is no the worth zero. Because that text areas this is not empty nor the empty string. The visibility of null values leads come three-valued logic. No longer is a condition either true or false, however it deserve to be undetermined. Any computation consisting of a null value delivers a null result. Questioning for all records with worth 0 or worth not same 0 will certainly not yield every records, because the records with worth null room excluded.

Null pointer

A null pointer is a guideline in a computer program that does not point to any object or function, which method that once it appears in a regimen or code, it speak the computer to take no activity on the associated part of the code.

Negative zero

In some signed number depictions (but not the two"s complement representation predominant today) and most floating point number representations, zero has actually two unique representations, one group it through the confident numbers and one through the negatives; this last representation is known as an adverse zero. Depictions with an unfavorable zero can be troublesome, since the two zeros will certainly compare equal yet may it is in treated in different ways by part operations.

Distinguishing zero indigenous O


The oval-shaped zero and also circular letter O with each other came into use on modern character displays. The zero with a period in the facility seems to have originated together an option on IBM 3270 controllers (this has actually the problem that it looks choose the Greek letter Theta). The slashed zero, looking similar to the letter O other than the slash, is used in old-style ASCII graphic sets descended from the default typewheel top top the venerable ASR-33 Teletype. This format reasons problems due to the fact that of its similarity come the prize ∅, representing the north set, and also for certain Scandinavian language which use Ø together a letter.

The convention which has the letter O v a slash and the zero without was offered at IBM and a few other early on mainframe makers; this is even an ext problematic because that Scandinavians since it method two the their letter collide. Some Burroughs/Unisys equipment display screens a zero v a reversed slash. And also yet an additional convention usual on beforehand line printers left zero unornamented but added a tail or hook to the letter-O so the it resembled an inverted Q or cursive resources letter-O.

The typeface supplied on part European number plates because that cars distinguish the two symbols by make the zero quite egg-shaped and also the O an ext circular, yet most of every by slitting open up the zero top top the upper appropriate side, so the one is not closed any an ext (as in German plates). The typeface chosen is referred to as fälschungserschwerende Schrift (abbr.: FE Schrift), definition "unfalsifiable script." note that those offered in the unified Kingdom do not differentiate in between the 2 as there deserve to never be any ambiguity if the style is properly spaced.

In record writing one might not differentiate the 0 and O at all, or may add a slash across it in bespeak to display the difference, back this sometimes reasons ambiguity in regard come the symbol because that the null set.

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The prominence of the creation of the zero mark have the right to never be exaggerated. This offering to airy nothing, not just a neighborhood habitation and a name, a picture, a symbol, but valuable power, is the characteristics of the Hindu race from whence it sprang. The is favor coining the Nirvana right into dynamos. No single mathematical production has been much more potent because that the basic on-go the intelligence and also power. G. B. Halsted

…a profound and important idea which appears so simple to us now that we ignore its true merit. Yet its really simplicity and the good ease which it lent to every computations placed our arithmetic in the an initial rank of valuable inventions. Pierre-Simon Laplace

The point about zero is that we do not must use that in the operations of everyday life. Nobody goes out to to buy zero fish. The is in a method the many civilized of every the cardinals, and its use is only forced on us by the demands of cultivated modes the thought.Alfred north Whitehead

…a fine and wonderful refuge of the magnificent spirit—almost one amphibian between being and non-being. Gottfried Leibniz

In other fields


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