As anumerical digit, 0 plays a crucial role indecimal notation: it indicates that thepower of ten corresponding to the place containing a 0 does not contribute to the total. For example, "205" in decimal means two hundreds, no tens, and five ones. The same principle applies inplace-value notations that uses a base other than ten, such asbinary andhexadecimal. The modern use of 0 in this manner derives fromIndian mathematics that was transmitted to Europe viamedieval Islamic mathematicians and popularized byFibonacci. It was independently used by theMaya.
Commonnames for the number 0 in English includezero,nought,naught (/nɔːt/), andnil. In contexts where at least one adjacent digit distinguishes it from theletter O, the number is sometimes pronounced asoh oro (/oʊ/). Informal orslang terms for 0 includezilch andzip. Historically,ought,aught (/ɔːt/), andcipher have also been used.
The wordzero came into the English language viaFrenchzéro from theItalianzero, a contraction of the Venetianzevero form of Italianzefiro viaṣafira orṣifr.[1] In pre-Islamic time the wordṣifr (Arabicصفر) had the meaning "empty".[2]Sifr evolved to mean zero when it was used to translateśūnya (Sanskrit:शून्य) from India.[2] The earliest known use ofzero as aLoanword inEnglish literature was 1598.[3]
Depending on the context, there may be different words used for the number zero, or the concept of zero. For the simple notion of lacking, the words "nothing" (although this is not accurate) and "none" are often used. The British English words"nought" or "naught", and "nil" are also synonymous.[5][6]
It is often called "oh" in the context of reading out a string of digits, such astelephone numbers,street addresses,credit card numbers,military time, or years. For example, thearea code 201 may be pronounced "two oh one", and the year 1907 is often pronounced "nineteen oh seven". The presence of other digits, indicating that the string contains only numbers, avoids confusion with the letter O. For this reason, systems that include strings with both letters and numbers (such aspostcodes in the UK) may exclude the use of the letter O.[7]
Slang words for zero include "zip", "zilch", "nada", and "scratch".[8] In the context of sports, "nil" is sometimes used, especially inBritish English. Several sports have specific words for a score of zero, such as "love" intennis – possibly from Frenchl'œuf, "the egg" – and "duck" incricket, a shortening of "duck's egg". "Goose egg" is another general slang term used for zero.[8]
AncientEgyptian numerals were ofbase 10.[9] They usedhieroglyphs for the digits and were notpositional. Inone papyrus written around1770 BC, a scribe recorded daily incomes and expenditures for thepharaoh's court, using thenfr hieroglyph to indicate cases where the amount of a foodstuff received was exactly equal to the amount disbursed. EgyptologistAlan Gardiner suggested that thenfr hieroglyph was being used as a symbol for zero. The same symbol 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.[10]
By the middle of the 2nd millennium BC,Babylonian mathematics had a sophisticatedbase 60 positional numeral system. The lack of a positional value (or zero) was indicated by aspace betweensexagesimal numerals. In a tablet unearthed atKish (dating to as early as700 BC), the scribe Bêl-bân-aplu used three hooks as aplaceholder in the sameBabylonian system.[11] By300 BC, a punctuation symbol (two slanted wedges) was repurposed as a placeholder.[12][13]
The Babylonian positional numeral system differed from the laterHindu–Arabic system in that it did not explicitly specify the magnitude of the leading sexagesimal digit, so that for example the lone digit 1 () might represent any of 1, 60, 3600 = 602, etc., similar to the significand of afloating-point number but without an explicit exponent, and so only distinguished implicitly from context. The zero-like placeholder mark was only ever used in between digits, but never alone or at the end of a number.[14]
Pre-Columbian Americas
Maya numeral zero
TheMesoamerican Long Count calendar developed in south-central Mexico and Central America required the use of zero as a placeholder within itsvigesimal (base-20) positional numeral system. Many different glyphs, including the partialquatrefoil 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.[a][15]
Since the eight earliest Long Count dates appear outside the Maya homeland,[16] it is generally believed that the use of zero in the Americas predated the Maya and was possibly the invention of theOlmecs.[17] Many of the earliest Long Count dates were found within the Olmec heartland, although the Olmec civilization ended by the4th century BC,[18] several centuries before the earliest known Long Count dates.[19]
Although zero became an integral part ofMaya numerals, with a different, emptytortoise-like "shell shape" used for many depictions of the "zero" numeral, it is assumed not to have influencedOld World numeral systems.[20]
Quipu, a knotted cord device, used in theInca Empire and its predecessor societies in theAndean region to record accounting and other digital data, is encoded in abase ten positional system. Zero is represented by the absence of a knot in the appropriate position.[21]
Classical antiquity
Theancient Greeks had no symbol for zero (μηδέν, pronouncedmēdén), and did not use a digit placeholder for it.[22] According to mathematicianCharles Seife, the ancient Greeks did begin to adopt the Babylonian placeholder zero for their work inastronomy after 500 BC, representing it with the lowercase Greek letterό (όμικρον:omicron). However, after using the Babylonian placeholder zero for astronomical calculations they would typically convert the numbers back intoGreek numerals. Greeks seemed to have a philosophical opposition to using zero as a number.[23] Other scholars give the Greek partial adoption of the Babylonian zero a later date, with neuroscientist Andreas Nieder giving a date of after 400 BC and mathematician Robert Kaplan dating it after theconquests of Alexander.[24][25]
Greeks seemed unsure about the status of zero as a number. Some of them asked themselves, "How can not being be?", leading to philosophical and, by themedieval period, religious arguments about the nature and existence of zero and thevacuum. Theparadoxes ofZeno of Elea depend in large part on the uncertain interpretation of zero.[26]
Example of the early Greek symbol for zero (lower right corner) from a 2nd-century papyrus
By AD150,Ptolemy, influenced byHipparchus and theBabylonians, was using a symbol for zero (—°)[27][28] in his work onmathematical astronomy called theSyntaxis Mathematica, also known as theAlmagest.[29] ThisHellenistic zero was perhaps the earliest documented use of a numeral representing zero in the Old World.[30] Ptolemy used it many times in hisAlmagest (VI.8) for the magnitude ofsolar andlunar eclipses. It represented the value of bothdigits andminutes of immersion at first and last contact. Digits varied continuously from 0 to 12 to 0 as the Moon passed over the Sun (a triangular pulse), where twelve digits was theangular diameter of the Sun. Minutes of immersion was tabulated from 0′0″ to 31′20″ to 0′0″, where 0′0″ used the symbol as a placeholder in two positions of hissexagesimal positional numeral system,[b] while the combination meant a zero angle. Minutes of immersion was also a continuous function1/12 31′20″√d(24−d) (a triangular pulse withconvex sides), where d was the digit function and 31′20″ was the sum of the radii of the Sun's and Moon's discs.[31] Ptolemy's symbol was a placeholder as well as a number used by two continuous mathematical functions, one within another, so it meant zero, not none. Over time, Ptolemy's zero tended to increase in size and lose theoverline, sometimes depicted as a large elongated 0-like omicron "Ο" or as omicron with overline "ō" instead of a dot with overline.[32]
The earliest use of zero in the calculation of theJulian Easter occurred before AD311, at the first entry in a table ofepacts as preserved in anEthiopic document for the years 311 to 369, using aGeʽez word for "none" (English translation is "0" elsewhere) alongside Geʽez numerals (based on Greek numerals), which was translated from an equivalent table published by theChurch of Alexandria inMedieval Greek.[33] This use was repeated in 525 in an equivalent table, that was translated via the Latinnulla ("none") byDionysius Exiguus, alongsideRoman numerals.[34] When division produced zero as a remainder,nihil, meaning "nothing", was used. These medieval zeros were used by all future medievalcalculators of Easter. The initial "N" was used as a zero symbol in a table of Roman numerals byBede—or his colleagues—around AD725.[35]
In mostcultures, 0 was identified before the idea of negative things (i.e., quantities less than zero) was accepted.[citation needed]
China
This is a depiction of zero expressed in Chinesecounting rods, based on the example provided byA History of Mathematics. An empty space is used to represent zero.[36]
TheSūnzĭ Suànjīng, of unknown date but estimated to be dated from the 1st to5th centuries AD, describe how the 4th century BC Chinesecounting rods system enabled one to performpositional decimal calculations.[37][38] As noted in theXiahou Yang Suanjing (425–468 AD), to multiply or divide a number by 10, 100, 1000, or 10000, all one needs to do, with rods on the counting board, is to move them forwards, or back, by 1, 2, 3, or 4 places.[39] The rods gave the decimal representation of a number, with an empty space denoting zero.[36][40] A circa 190 AD, manual, the "Supplementary Notes on the Art of Figures", byXu Yue, also outlines the techniques to add, subtract, multiply, and divide numbers, containing zero values in a decimal power, oncounting devices, that include counting rods, and abacus.[41][42] Chinese authors had been familiar with the idea of negative numbers, and decimal fractions, by theHan dynasty(2nd century AD), as seen inThe Nine Chapters on the Mathematical Art.[43]Qín Jiǔsháo's 1247Mathematical Treatise in Nine Sections is the oldest surviving Chinese mathematical text using a round symbol '〇' for zero.[44] The origin of this symbol is unknown; it may have been produced by modifying a square symbol.[45] Zero was not treated as a number at that time, but as a "vacant position".[46]
Chinese Epigraphy
A variety ofChinese characters have been used, through history, to represent zero: 空, 零, 洞, 〇.
TheLokavibhāga, aJain text oncosmology surviving in a medieval Sanskrit translation of thePrakrit original, which is internally dated to AD 458 (Saka era 380), uses a decimalplace-value system, including a zero. In this text,śūnya ("void, empty") is also used to refer to zero.[52]
A positive or negative number when divided by zero is a fraction with the zero as denominator. Zero divided by a negative or positive number is either zero or is expressed as a fraction with zero as numerator and the finite quantity as denominator. Zero divided by zero is zero.
Bhāskara II's, 12th century,Līlāvatī instead proposed that division by zero results in an infinite quantity,[58]
A quantity divided by zero becomes a fraction the denominator of which is zero. This fraction is termed an infinite quantity. In this quantity consisting of that which has zero for its divisor, there is no alteration, though many may be inserted or extracted; as no change takes place in the infinite and immutable God when worlds are created or destroyed, though numerous orders of beings are absorbed or put forth.
Early Asian Epigraphy
Sambor Inscription
The oldest, firmly dated use of zero as a decimal figure, found on the Sambor Inscription. The number "605" is written inKhmer numerals (top), referring to the year it was made:605 Saka era (683 CE). The fragment, inscribed inOld Khmer, was once part of a temple doorway, and was found inKratié province,Cambodia.
There are numerous copper plate inscriptions, with the same smallO in them, some of them possibly dated to the 6th century, but their date or authenticity may be open to doubt.[11]
The first known use of specialglyphs 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 theChaturbhuj Temple, Gwalior, in India, dated AD 876.[60][61]
A symbol for zero, a black dot, is used throughout theBakhshali manuscript, a practical manual on arithmetic for merchants. TheBodleian Library reportedradiocarbon dating results for six folio from the manuscript, indicating that they came from different centuries, but date the manuscript to AD 799 – 1102.[50]
TheArabic-language inheritance of science was largelyGreek,[62] followed by Hindu influences.[63] In 773, atAl-Mansur's behest, translations were made of many ancient treatises including Greek, Roman, Indian, and others.
In AD 813, astronomical tables were prepared by aPersian mathematician,Muḥammad ibn Mūsā al-Khwārizmī, using Hindu numerals;[63] 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.[64] This book was later translated intoLatin in 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 to acquire a meaning of any arithmetic based on decimals.[63]
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.[65]
Transmission to Europe
TheHindu–Arabic numeral system (base 10) reached Western Europe in the 11th century, viaAl-Andalus, through SpanishMuslims, theMoors, together with knowledge ofclassical astronomy and instruments like theastrolabe.Gerbert of Aurillac is credited with reintroducing the lost teachings into Catholic Europe. For this reason, the numerals came to be known in Europe as "Arabic numerals". The Italian mathematicianFibonacci or Leonardo of Pisa was instrumental in bringing the system into European mathematics in 1202, stating:
After my father's appointment byhis homeland as state official in the customs house ofBugia 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 thealgorism, as well as the art ofPythagoras, I considered as almost a mistake in respect to the method of theHindus [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 ofEuclid'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 theLatin 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.[66]
From the 13th century, manuals on calculation (adding, multiplying, extracting roots, etc.) became common in Europe where they were calledalgorismus after the Persian mathematicianal-Khwārizmī. One popular manual was written byJohannes de Sacrobosco in the early 1200s and was one of the earliest scientific books to beprinted, in 1488.[67][68] The practice of calculating on paper using Hindu–Arabic numerals only gradually displaced calculation by abacus and recording withRoman numerals.[69] In the 16th century, Hindu–Arabic numerals became the predominant numerals used in Europe.[67]
Today, the numerical digit 0 is usually written as a circle or ellipse. Traditionally, many printtypefaces made the capital letterO more rounded than the narrower, elliptical digit 0.[70]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 characterdisplays.[70]
Aslashed zero () is often used to distinguish the number from the letter (mostly in computing, navigation and in the military, for example). The digit 0 with a dot in the center seems to have originated as an option onIBM 3270 displays and has continued with some modern computer typefaces such asAndalé 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 the "0" character more squared at the edges, like a rectangle, and the "O" character more rounded. A further distinction is made infalsification-hindering typeface as used onGerman car number plates by slitting open the digit 0 on the upper right side. In some systems either the letter O or the numeral 0, or both, are excluded from use, to avoid confusion.
The concept of zero plays multiple roles in mathematics: as a digit, it is an important part of positional notation for representing numbers, while it also plays an important role as a number in its own right in many algebraic settings.
In positional number systems (such as the usualdecimal notation for representing numbers), the digit 0 plays the role of a placeholder, indicating that certain powers of the base do not contribute. For example, the decimal number 205 is the sum of two hundreds and five ones, with the 0 digit indicating that no tens are added. The digit plays the same role indecimal fractions and in thedecimal representation of other real numbers (indicating whether any tenths, hundredths, thousandths, etc., are present) and in bases other than 10 (for example, in binary, where it indicates which powers of 2 are omitted).[71]
The number 0 can be regarded as neither positive nor negative[75] or, alternatively, both positive and negative[76] and is usually displayed as the central number in anumber line. Zero iseven[77] (that is, a multiple of 2), and is also aninteger multiple of any other integer, rational, or real number. It is neither aprime number nor acomposite number: it is not prime because prime numbers are greater than 1 by definition, and it is not composite because it cannot be expressed as the product of two smaller natural numbers.[78] (However, thesingleton set {0} is aprime ideal in thering of the integers.)
5+0=5 illustrated with collections of dots.
The following are some basic rules for dealing with the number 0. These rules apply for any real or complex numberx, unless otherwise stated.
Addition:x + 0 = 0 +x =x. That is, 0 is anidentity element (or neutral element) with respect to addition.
Exponentiation:x0 =x/x = 1, except thatthe casex = 0 is considered undefined in some contexts. For all positive realx,0x = 0.
The expression0/0, 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 thelim operator independently to both operands of the fraction, is a so-called "indeterminate form". That does not 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 asl'Hôpital's rule.[80]
The sum of 0 numbers (theempty sum) is 0, and the product of 0 numbers (theempty product) is 1. Thefactorial 0! evaluates to 1, as a special case of the empty product.[81]
Other uses in mathematics
The empty set has zero elements
The role of 0 as the smallest counting number can be generalized or extended in various ways. Inset theory, 0 is thecardinality of theempty set (notated as "{ }", "", or "∅"): if one does not have any apples, then one has 0 apples. In fact, in certain axiomatic developments of mathematics from set theory, 0 isdefined to be the empty set.[82] When this is done, the empty set is thevon Neumann cardinal assignment for a set with no elements, which is the empty set. The cardinality function, applied to the empty set, returns the empty set as a value, thereby assigning it 0 elements.
The role of 0 as additive identity generalizes beyond elementary algebra. Inabstract algebra, 0 is commonly used to denote azero element, which is theidentity element for addition (if defined on the structure under consideration) and anabsorbing element for multiplication (if defined). (Such elements may also be calledzero elements.) Examples include identity elements ofadditive groups andvector spaces. Another example is thezero function (orzero map) on a domainD. This is theconstant function with 0 as its only possible output value, that is, it is the functionf defined byf(x) = 0 for allx inD. As a function from the real numbers to the real numbers, the zero function is the only function that is botheven andodd.
The number 0 is also used in several other ways within various branches of mathematics:
Azero of a functionf is a pointx in the domain of the function such thatf(x) = 0.
The value zero plays a special role for many physical quantities. For some quantities, the zero level is naturally distinguished from all other levels, whereas for others it is more or less arbitrarily chosen. For example, for anabsolute temperature (typically measured inkelvins),zero is the lowest possible value. (Negative temperatures can be defined for some physical systems, but negative-temperature systems are not actually colder.) This is in contrast to temperatures on the Celsius scale, for example, where zero is arbitrarily defined to be at thefreezing point of water.[85][86] Measuring sound intensity indecibels orphons, the zero level is arbitrarily set at a reference value—for example, at a value for the threshold of hearing. Inphysics, thezero-point energy is the lowest possible energy that aquantum mechanicalphysical system may possess and is the energy of theground state of the system.
Computer science
Modern computers store information inbinary, that is, using an "alphabet" that contains only two symbols, usually chosen to be "0" and "1". Binary coding is convenient fordigital electronics, where "0" and "1" can stand for the absence or presence of electrical current in a wire.[87]Computer programmers typically usehigh-level programming languages that are more intelligible to humans than thebinary instructions that are directly executed by thecentral processing unit. 0 plays various important roles in high-level languages. For example, aBoolean variable stores a value that is eithertrue orfalse, and 0 is often the numerical representation offalse.[88]
0 also plays a role inarray indexing. The most common practice throughout human history has been to start counting at one, and this is the practice in early classic programming languages such asFortran andCOBOL.[89] However, in the late 1950sLISP introducedzero-based numbering for arrays whileAlgol 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.[citation needed] For example, the elements of an array are numbered starting from 0 inC, so that for an array ofn items the sequence of array indices runs from 0 ton−1.[90]
There can be confusion between 0- and 1-based indexing; for example, Java'sJDBC indexes parameters from 1 althoughJava itself uses 0-based indexing.[91]
In C, abyte containing the value 0 serves to indicate where astring of characters ends. Also, 0 is a standard way to refer to anull pointer in code.[92]
In databases, it is possible for a field not to have a value. It is then said to have anull value.[93] 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 tothree-valued logic. No longer is a condition eithertrue orfalse, but it can beundetermined. Any computation including a null value delivers a null result.[94]
In mathematics, there is no "positive zero" or "negative zero" distinct from zero; both −0 and +0 represent exactly the same number. However, in some computer hardwaresigned 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 assigned zero, with the latter form sometimes called negative zero. These representations include thesigned magnitude andones' complement binary integer representations (but not thetwo's complement binary form used in most modern computers), and mostfloating-point number representations (such asIEEE 754 andIBM S/360 floating-point formats).
Anepoch, in computing terminology, is the date and time associated with a zero timestamp. TheUnix epoch begins the midnight before the first of January 1970.[95][96][97] TheClassic Mac OS epoch andPalm OS epoch begin the midnight before the first of January 1904.[98]
ManyAPIs andoperating systems that require applications to return an integer value as anexit status typically use zero to indicate success and non-zero values to indicate specificerror or warning conditions.[99]
Programmers often use aslashed zero to avoid confusion with the letter "O".[100]
Other fields
Biology
Incomparative zoology andcognitive science, recognition that some animals display awareness of the concept of zero leads to the conclusion that the capability for numerical abstraction arose early in theevolution of species.[101]
^No long count date actually using the number 0 has been found before the 3rd century AD, but since the long count system would make no sense without some placeholder, and since Mesoamerican glyphs do not typically leave empty spaces, these earlier dates are taken as indirect evidence that the concept of 0 already existed at the time.
^Each place in Ptolemy's sexagesimal system was written inGreek numerals from0 to 59, where 31 was written λα meaning 30+1, and 20 was written κ meaning 20.
Harper, Douglas (2011)."Zero".Etymonline. Archived fromthe original on 3 July 2017."figure which stands for naught in the Arabic notation," also "the absence of all quantity considered as quantity", c. 1600, from Frenchzéro or directly from Italianzero, from Medieval Latinzephirum, from Arabicsifr "cipher", translation of Sanskritsunya-m "empty place, desert, naught.
"zero, n."OED Online.Oxford University Press. December 2011.Archived from the original on 7 March 2012. Retrieved4 March 2012.French zéro (1515 in Hatzfeld & Darmesteter) or its source Italian zero, for *zefiro, < Arabic çifr.
Smithsonian Institution.Oriental Elements of Culture in the Occident, p. 518, atGoogle Books. Annual Report of the Board of Regents of the Smithsonian Institution; Harvard University Archives. "Sifr occurs in the meaning of "empty" even in the pre-Islamic time. ... Arabic sifr in the meaning of zero is a translation of the corresponding India sunya."
Gullberg, Jan (1997).Mathematics: From the Birth of Numbers.W.W. Norton & Co.ISBN978-0-393-04002-9. p. 26:Zero derives from Hindu sunya – meaning void, emptiness – via Arabic sifr, Latin cephirum, Italian zevero.
Logan, Robert (2010).The Poetry of Physics and the Physics of Poetry. World Scientific.ISBN978-981-4295-92-5. p. 38:The idea of sunya and place numbers was transmitted to the Arabs who translated sunya or "leave a space" into their language as sifr.
^Lumpkin, Beatrice (2002). "Mathematics Used in Egyptian Construction and Bookkeeping".The Mathematical Intelligencer.24 (2):20–25.doi:10.1007/BF03024613.S2CID120648746.
^O'Connor, J. J.; Robertson, E. F. (2000)."Zero".Maths History. University of St Andrews.Archived from the original on 21 September 2021. Retrieved7 September 2021.
^Cyphers, Ann (2014), Renfrew, Colin; Bahn, Paul (eds.),"The Olmec, 1800–400 BCE",The Cambridge World Prehistory, Cambridge: Cambridge University Press, pp. 1005–1025,ISBN978-0-521-11993-1, retrieved13 August 2024.
^Birchak, Gabrielle (3 June 2025)."Mayan Mathematics".Math! Science! History!™. Retrieved30 September 2025.Historian Georges Ifrah calls the Mayan use of zero "one of the most striking inventions ever to emerge in a mathematical culture isolated from the Old World."
^Wallin, Nils-Bertil (19 November 2002)."The History of Zero".YaleGlobal online. The Whitney and Betty Macmillan Center for International and Area Studies at Yale. Archived fromthe original on 25 August 2016. Retrieved1 September 2016.
^Huggett, Nick (2019)."Zeno's Paradoxes". In Zalta, Edward N. (ed.).The Stanford Encyclopedia of Philosophy (Winter 2019 ed.). Metaphysics Research Lab, Stanford University.Archived from the original on 10 January 2021. Retrieved9 August 2020.
^O'Connor, J. J.; Robertson, E. F."A history of Zero". MacTutor History of Mathematics.Archived from the original on 7 April 2020. Retrieved28 March 2020.
^Neugebauer, Otto (2016) [1979].Ethiopic Astronomy and Computus (Red Sea Press ed.). Red Sea Press. pp. 25, 53, 93, 183, Plate I.ISBN978-1-56902-440-9.. The pages in this edition have numbers six less than the same pages in the original edition.
^Eberhard-Bréard, Andrea (2008), "Mathematics in China", in Selin, Helaine (ed.),Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures, Dordrecht: Springer Netherlands, pp. 1371–1378,doi:10.1007/978-1-4020-4425-0_9453,ISBN978-1-4020-4425-0.
^Struik, Dirk J. (1987).A Concise History of Mathematics. New York: Dover Publications. pp. 32–33. "In these matrices we find negative numbers, which appear here for the first time in history."
^abPlofker, Kim (2009).Mathematics in India. Princeton University Press. pp. 54–56.ISBN978-0-691-12067-6.In the Chandah-sutra of Pingala, dating perhaps the third or second century BC, [ ...] Pingala's use of a zero symbol [śūnya] as a marker seems to be the first known explicit reference to zero. ... In the Chandah-sutra of Pingala, dating perhaps the third or second century BC, there are five questions concerning the possible meters for any value "n". [ ...] The answer is (2)7 = 128, as expected, but instead of seven doublings, the process (explained by the sutra) required only three doublings and two squarings – a handy time saver where "n" is large. Pingala's use of a zero symbol as a marker seems to be the first known explicit reference to zero.
^O'Connor, J. J.; Robertson, E. F. (2000)."Aryabhata the Elder".School of Mathematics and Statistics, University of St. Andrews. Scotland.Archived from the original on 11 July 2015. Retrieved26 May 2013.
^Casselman, Bill."All for Nought".ams.org. University of British Columbia), American Mathematical Society.Archived from the original on 6 December 2015. Retrieved20 December 2015.
^abcDurant, Will (1950).The Story of Civilization, Volume IV, The Age of Faith: Constantine to Dante – A.D. 325–1300. Simon & Schuster. p. 241:The Arabic inheritance of science was overwhelmingly Greek, but Hindu influences ranked next. In 773, at Mansur's behest, translations were made of theSiddhantas – Indian astronomical treatises dating as far back as 425 BC; these versions may have the vehicle through which the "Arabic" numerals and the zero were brought from India into Islam. In 813, al-Khwarizmi used the Hindu numerals in his astronomical tables.
^Durant 1950, p. 241: "In 976, Muhammad ibn Ahmad, in hisKeys of the Sciences, remarked that if, in a calculation, no number appears in the place of tens, a little circle should be used "to keep the rows". This circle the Mosloems calledṣifr, "empty" whence our cipher".
Sigler, Laurence (2003).Fibonacci's Liber Abaci: A Translation into Modern English of Leonardo Pisano's Book of Calculation. Sources and Studies in the History of Mathematics and Physical Sciences. Translated by Sigler, Laurence E. Springer.doi:10.1007/978-1-4613-0079-3.ISBN978-1-4613-0079-3.
^abBemer, R. W. (1967). "Towards standards for handwritten zero and oh: much ado about nothing (and a letter), or a partial dossier on distinguishing between handwritten zero and oh".Communications of the ACM.10 (8):513–518.doi:10.1145/363534.363563.S2CID294510.
^"Null values and the nullable type".IBM. 12 December 2018.Archived from the original on 23 November 2021. Retrieved23 November 2021.In regard to services, sending a null value as an argument in a remote service call means that no data is sent. Because the receiving parameter is nullable, the receiving function creates a new, uninitialized value for the missing data then passes it to the requested service function.
