As a numerical digit, 0 plays a crucial role in decimal notation: it indicates that the power 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 in place-value notations that uses a base other than ten, such as binary and hexadecimal.
The modern use of 0 in this manner derives from Indian mathematics that was transmitted to Europe via medieval Islamic mathematicians and popularized by Fibonacci. It was independently used by the Maya.
Common names for the number 0 in English include zero, nought, naught (/nɔːt/), and nil. In contexts where at least one adjacent digit distinguishes it from the letter O, the number is sometimes pronounced as oh or o (/oʊ/). Informal or slang terms for 0 include zilch and zip. Historically, ought, aught (/ɔːt/), and cipher have also been used.
Etymology
The word zero came into the English language via French zéro from the Italian zero, a contraction of the Venetian zevero form of Italian zefiro, itself borrowed from Arabic ṣafira or ṣifr.
In pre-Islamic time the word ṣifr (Arabic صفر) had the meaning "empty". Sifr evolved to mean zero when it was used to translate śūnya (Sanskrit: शून्य) from India. The earliest known use of zero as a loanword in English literature was 1598.
The Italian mathematician Fibonacci (c. 1170 – c. 1250), who grew up in North Africa and is credited with introducing the decimal system to Europe, used the term zephyrum.
This became zefiro in Italian, and was then contracted to zero via the Venetian form zevero. The Italian word zefiro was already in existence (meaning "west wind" from Latin and Greek Zephyrus) and may have influenced the spelling when transcribing Arabic ṣifr.
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 English words "nought" or "naught", "nil", and null are also synonymous.
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 postcodes in the UK) may exclude the use of the letter O.
Slang words for zero include "zip", "zilch", "nada", and "scratch". 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 – possibly 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.
Mathematics
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.
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).
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, but it is an integer, and hence a rational number and a real number. Zero is even (that is, a multiple of 2), and is also an integer multiple of any other integer. It is neither a prime number nor a composite number.
The number 0 is regarded as neither positive nor negative, and is usually displayed as the origin of a number line. When the real numbers are extended to form the complex numbers, 0 becomes the origin of the complex plane.
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. The converse also holds: If x · y = 0 then x=0 or y=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.
- Exponentiation: x0 = x/x = 1, except that the case x = 0 is considered undefined in some contexts. For all positive real x, 0x = 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.
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.
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 "{ }", "", 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. When this is done, the empty set is the von Neumann cardinal assignment for a set with no elements, which is the empty set.
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). 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.
- Category theory introduces the idea of a zero object, often denoted 0, and the related concept of zero morphisms, which generalize the zero function.
History
Ancient Near East
Ancient Egyptian numerals were of base 10. 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.
By the middle of the 2nd millennium BC, Babylonian mathematics had a sophisticated base 60 positional numeral system, but a positional value of zero was indicated by a space between 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.
By 300 BC, a punctuation symbol (two slanted wedges) was repurposed as a placeholder. Significantly, however, these placeholder signs were not considered a numerical value, they were never used alone “Therefore, they cannot be interpreted as representations of the concept or the number zero.” They were also not written at the end of a number (so 1 and 60 and 60×60 were all written as 
).
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.
Since the eight earliest Long Count dates appear outside the Maya homeland, it is generally believed that the use of zero in the Americas predated the Maya and was possibly the invention of the Olmecs. 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 not to have influenced Old World numeral systems.
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.
Classical antiquity
The earliest confidently cited exemplar of the Greek use of the Hellenistic zero appears in Hipparchus in 140 CE.
The archaic Greece had no symbol for zero (μηδέν, pronounced mēdén), and did not use a digit placeholder for it. According to mathematician Charles Seife, after the Babylonian placeholder zero shows up sometime shortly after 500 BC, Greek astronomers began to use the lowercase Greek letter ό (όμικρον: omicron) as a placeholder or representation of ground-level/null degree value.
However, after using the Babylonian placeholder zero for astronomical calculations they would typically convert the numbers back into Greek numerals. As with the rejection of infinitesimals by Pythagoras, Greeks appear to maintain to a philosophical opposition to using zero as a number. "The whole of the Greek universe rested on this pillar: There is no void." Nieder dates the appearance of zero in Greek astronomical texts after 400 BC and mathematician Robert Kaplan further specifies that it must have been after the conquests of Alexander.
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.
By AD 150, Ptolemy, influenced by Hipparchus and the Babylonians, was using a symbol for zero () in his work on mathematical astronomy called the Syntaxis Mathematica, also known as the Almagest.
This Hellenistic zero was perhaps the earliest documented use of a numeral representing zero in the Old World. 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, 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.
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.
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 Geʽez word for "none" (English translation is "0" elsewhere) alongside Geʽez numerals (based on Greek numerals), which was translated from an equivalent table published by the Church of Alexandria in Medieval Greek.
This use was repeated in 525 in an equivalent table, that was translated via the Latin nulla ("none") by Dionysius Exiguus, alongside Roman numerals. 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.
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 positional 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.
The rods gave the decimal representation of a number, with an empty space denoting zero. A circa 190 AD, manual, the "Supplementary Notes on the Art of Figures", by Xu Yue, also outlines the techniques to add, subtract, multiply, and divide numbers, containing zero values in a decimal power, on counting devices, that include counting rods, and abacus.
Chinese authors had been familiar with the idea of negative numbers, and decimal fractions, by the Han dynasty (2nd century AD), as seen in The Nine Chapters on the Mathematical Art. Qín Jiǔsháo's 1247 Mathematical Treatise in Nine Sections is the oldest surviving Chinese mathematical text using a round symbol '〇' for zero.
The origin of this symbol is unknown; it may have been produced by modifying a square symbol. Zero was not treated as a number at that time, but as a "vacant position".
Chinese Epigraphy
A variety of Chinese characters have been used, through history, to represent zero: 空, 零, 洞, 〇.