0

Modern usage

Depending on the context, there may be different words used for the number zero, or the concept of zero. For the simple notion of lacking, the words "nothing" (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 as telephone numbers, street addresses, credit card numbers, military time, or years. For example, the area 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 as Canadian postal codes) may exclude the use of the letter O.^{[citation needed]}

Slang words for zero include "zip", "zilch", "nada", and "scratch".^[7] In the context of sports, "nil" is sometimes used, especially in British English. Several sports have specific words for a score of zero, such as "love" in tennis – from French l'œuf, "the egg" – and "duck" in cricket, a shortening of "duck's egg". "Goose egg" is another general slang term used for zero.^[7]

Ancient Near East

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Ancient Egyptian numerals were of base 10.^[8] They used hieroglyphs for the digits and were not positional. In one papyrus written around 1770 BC, a scribe recorded daily incomes and expenditures for the pharaoh's court, using the nfr hieroglyph to indicate cases where the amount of a foodstuff received was exactly equal to the amount disbursed. Egyptologist Alan Gardiner suggested that the nfr 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.^[9]

By the middle of the 2nd millennium BC, Babylonian mathematics had a sophisticated base 60 positional numeral system. The lack of a positional value (or zero) was indicated by a space between sexagesimal numerals. In a tablet unearthed at Kish (dating to as early as 700 BC), the scribe Bêl-bân-aplu used three hooks as a placeholder in the same Babylonian system.^[10] By 300 BC, a punctuation symbol (two slanted wedges) was repurposed as a placeholder.^[11][12]

The Babylonian positional numeral system differed from the later Hindu–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 a floating-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.^[13]

Pre-Columbian Americas

The Mesoamerican Long Count calendar developed in south-central Mexico and Central America required the use of zero as a placeholder within its vigesimal (base-20) positional numeral system. Many different glyphs, including the partial quatrefoil 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][14]

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,^[17] several centuries before the earliest known Long Count dates.^[18]

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 not to have influenced Old World numeral systems.^{[citation needed]}

Quipu, a knotted cord device, used in the Inca Empire and its predecessor societies in the Andean region to record accounting and other digital data, is encoded in a base ten positional system. Zero is represented by the absence of a knot in the appropriate position.^[19]

Classical antiquity

The ancient Greeks had no symbol for zero (μηδέν, pronounced 'midén'), and did not use a digit placeholder for it.^[20] According to mathematician Charles Seife, the ancient Greeks did begin to adopt the Babylonian placeholder zero for their work in astronomy after 500 BC, representing it with the lowercase Greek letter ό (όμικρον: omicron). However, after using the Babylonian placeholder zero for astronomical calculations they would typically convert the numbers back into Greek numerals. Greeks seemed to have a philosophical opposition to using zero as a number.^[21] 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 the conquests of Alexander.^[22][23]

Greeks seemed unsure about the status of zero as a number. Some of them asked themselves, "How can not being be?", leading to philosophical and, by the medieval period, religious arguments about the nature and existence of zero and the vacuum. The paradoxes of Zeno of Elea depend in large part on the uncertain interpretation of zero.^[24]

By AD 150, Ptolemy, influenced by Hipparchus and the Babylonians, was using a symbol for zero (—°)^[25][26] in his work on mathematical astronomy called the Syntaxis Mathematica, also known as the Almagest.^[27] This Hellenistic zero was perhaps the earliest documented use of a numeral representing zero in the Old World.^[28] Ptolemy used it many times in his Almagest (VI.8) for the magnitude of solar and lunar eclipses. It represented the value of both digits and minutes of immersion at first and last contact. Digits varied continuously from 0 to 12 to 0 as the Moon passed over the Sun (a triangular pulse), where twelve digits was the angular diameter of the Sun. Minutes of immersion was tabulated from 0′0″ to 31′20″ to 0′0″, where 0′0″ used the symbol as a placeholder in two positions of his sexagesimal positional numeral system,^[b] while the combination meant a zero angle. Minutes of immersion was also a continuous function ⁠1/12⁠ 31′20″ √d(24−d) (a triangular pulse with convex sides), where d was the digit function and 31′20″ was the sum of the radii of the Sun's and Moon's discs.^[29] 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 the overline, sometimes depicted as a large elongated 0-like omicron "Ο" or as omicron with overline "ō" instead of a dot with overline.^[30]

The earliest use of zero in the calculation of the Julian Easter occurred before AD 311, at the first entry in a table of epacts as preserved in an Ethiopic document for the years 311 to 369, using a Ge'ez word for "none" (English translation is "0" elsewhere) alongside Ge'ez numerals (based on Greek numerals), which was translated from an equivalent table published by the Church of Alexandria in Medieval Greek.^[31] This use was repeated in 525 in an equivalent table, that was translated via the Latin nulla ("none") by Dionysius Exiguus, alongside Roman numerals.^[32] When division produced zero as a remainder, nihil, meaning "nothing", was used. These medieval zeros were used by all future medieval calculators of Easter. The initial "N" was used as a zero symbol in a table of Roman numerals by Bede—or his colleagues—around AD 725.^[33]

In most cultures, 0 was identified before the idea of negative things (i.e., quantities less than zero) was accepted.^{[citation needed]}

China

The Sūnzĭ Suànjīng, of unknown date but estimated to be dated from the 1st to 5th centuries AD, describe how the 4th century BC Chinese counting rods system enabled one to perform decimal calculations. As noted in the Xiahou 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.^[35] The rods gave the decimal representation of a number, with an empty space denoting zero.^[34][36] The counting rod system is a positional notation system.^[37][38]

Zero was not treated as a number at that time, but as a "vacant position".^[39] Qín Jiǔsháo's 1247 Mathematical Treatise in Nine Sections is the oldest surviving Chinese mathematical text using a round symbol ‘〇’ for zero.^[40] The origin of this symbol is unknown; it may have been produced by modifying a square symbol.^[41] Chinese authors had been familiar with the idea of negative numbers by the Han dynasty (2nd century AD), as seen in The Nine Chapters on the Mathematical Art.^[42]

India

Pingala (c. 3rd or 2nd century BC),^[43] a Sanskrit prosody scholar,^[44] used binary sequences, in the form of short and long syllables (the latter equal in length to two short syllables), to identify the possible valid Sanskrit meters, a notation similar to Morse code.^[45] Pingala used the Sanskrit word śūnya explicitly to refer to zero.^[43]

The concept of zero as a written digit in the decimal place value notation was developed in India.^[47] 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.^[48] In 2017, researchers at the Bodleian Library reported radiocarbon dating results for three samples from the manuscript, indicating that they came from three different centuries: from AD 224–383, AD 680–779, and AD 885–993. It is not known how the birch bark fragments from different centuries forming the manuscript came to be packaged together. If the writing on the oldest birch bark fragments is as old as those fragments, it represents South Asia's oldest recorded use of a zero symbol. However, it is possible that the writing dates instead to the time period of the youngest fragments, AD 885–993. The latter dating has been argued to be more consistent with the sophisticated use of zero within the document, as portions of it appear to show zero being employed as a number in its own right, rather than only as a positional placeholder.^[46][49][50]

The Lokavibhāga, a Jain text on cosmology surviving in a medieval Sanskrit translation of the Prakrit original, which is internally dated to AD 458 (Saka era 380), uses a decimal place-value system, including a zero. In this text, śūnya ("void, empty") is also used to refer to zero.^[51]

The Aryabhatiya (c. 499), states sthānāt sthānaṁ daśaguṇaṁ syāt "from place to place each is ten times the preceding".^[52][53][54]

Rules governing the use of zero appeared in Brahmagupta's Brahmasputha Siddhanta (7th century), which states the sum of zero with itself as zero, and incorrectly describes division by zero in the following way:^[55][56]

A positive or negative number when divided by zero is a fraction with the zero as denominator. Zero divided by a negative or positive number is either zero or is expressed as a fraction with zero as numerator and the finite quantity as denominator. Zero divided by zero is zero.

Epigraphy

Sambor Inscription

The oldest, firmly dated use of zero as a decimal figure, found on the Sambor Inscription. The number "605" is written in Khmer numerals (top), referring to the year it was made: 605 Saka era (683 CE). The fragment, inscribed in Old Khmer, was once part of a temple doorway, and was found in Kratié province, Cambodia.

A black dot is used as a decimal placeholder in the Bakhshali manuscript, portions of which date from AD 224–993.^[46]

There are numerous copper plate inscriptions, with the same small O in them, some of them possibly dated to the 6th century, but their date or authenticity may be open to doubt.^[10]

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.^[57]

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 Chaturbhuj Temple, Gwalior, in India, dated AD 876.^[58][59]

Middle Ages

As a digit

In positional number systems (such as the usual decimal 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 in decimal fractions and in the decimal 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).^[69]

Elementary algebra

The number 0 is the smallest nonnegative integer, and the largest nonpositive integer. The natural number following 0 is 1 and no natural number precedes 0. The number 0 may or may not be considered a natural number,^[70][71] but it is an integer, and hence a rational number and a real number.^[72] All rational numbers are algebraic numbers, including 0. When the real numbers are extended to form the complex numbers, 0 becomes the origin of the complex plane.

The number 0 can be regarded as neither positive nor negative^[73] or, alternatively, both positive and negative^[74] and is usually displayed as the central number in a number line. Zero is even^[75] (that is, a multiple of 2), and is also an integer multiple of any other integer, rational, or real number. It is neither a prime number nor a composite 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.^[76] (However, the singleton set {0} is a prime ideal in the ring of the integers.)

The following are some basic 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 (or neutral element) with respect to 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 (no real number multiplied by 0 produces 1), a consequence of the previous rule.^[77]
• Exponentiation: x⁰ = ⁠x/x⁠ = 1, except that the case x = 0 is considered undefined in some contexts. For all positive real x, 0^x = 0.

The expression ⁠0/0⁠, which may be obtained in an attempt to determine the limit of an expression of the form ⁠f(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 mean that the limit sought is necessarily undefined; rather, it means that the limit of ⁠f(x)/g(x)⁠, if it exists, must be found by another method, such as l'Hôpital's rule.^[78]

The sum of 0 numbers (the empty sum) is 0, and the product of 0 numbers (the empty product) is 1. The factorial 0! evaluates to 1, as a special case of the empty product.^[79]

Other uses in mathematics

The role of 0 as the smallest counting number can be generalized or extended in various ways. In set theory, 0 is the cardinality of the empty set (notated as "{ }", "${\displaystyle \emptyset }$", 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 is defined to be the empty set.^[80] When this is done, the empty set is the von Neumann cardinal assignment for a set with no elements, which is the empty set. The cardinality function, applied to the empty set, returns the empty set as a value, thereby assigning it 0 elements.

Also in set theory, 0 is the lowest ordinal number, corresponding to the empty set viewed as a well-ordered set. In order theory (and especially its subfield lattice theory), 0 may denote the least element of a lattice or other partially ordered set.

The role of 0 as additive identity generalizes beyond elementary algebra. In abstract algebra, 0 is commonly used to denote a zero element, which is the identity element for addition (if defined on the structure under consideration) and an absorbing element for multiplication (if defined). (Such elements may also be called zero elements.) Examples include identity elements of additive groups and vector spaces. Another example is the zero function (or zero map) on a domain D. This is the constant function with 0 as its only possible output value, that is, it is the function f defined by f(x) = 0 for all x in D. As a function from the real numbers to the real numbers, the zero function is the only function that is both even and odd.

The number 0 is also used in several other ways within various branches of mathematics:

• A zero of a function f is a point x in the domain of the function such that f(x) = 0.
• In propositional logic, 0 may be used to denote the truth value false.
• In probability theory, 0 is the smallest allowed value for the probability of any event.^[81]
• Category theory introduces the idea of a zero object, often denoted 0, and the related concept of zero morphisms, which generalize the zero function.^[82]

Biology

In comparative zoology and cognitive science, recognition that some animals display awareness of the concept of zero leads to the conclusion that the capability for numerical abstraction arose early in the evolution of species.^[99]

Dating systems

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 forth.^[100]