0
0 is a number representing an empty quantity. Adding 0 to any number leaves that number unchanged; in mathematical terminology, 0 is the additive identity of the integers, rational numbers, real numbers, and complex numbers, as well as other algebraic structures. Multiplying any number by 0 results in 0, and consequently dividing by 0 is generally considered to be undefined in arithmetic.
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, 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. Informal or slang terms for 0 include zilch and zip. Historically, ought, aught, 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 via ṣafira or ṣifr. In pre-Islamic time the word had the meaning "empty". evolved to mean zero when it was used to translate from India. The earliest known use of zero as a Loanword in English literature was 1598.The Italian mathematician Fibonacci, 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 in Venetian. The Italian word Wikt:zefiro was already in existence and may have influenced the spelling when transcribing Arabic.
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" and "none" are often used. The British English words "nought" or "naught", and "nil" 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 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.
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, 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. The lack of a positional value was indicated by a space between sexagesimal numerals. In a tablet unearthed at Kish, the scribe Bêl-bân-aplu used three hooks as a placeholder in the same Babylonian system. By, a punctuation symbol was repurposed as a placeholder. Significantly, however, these placeholder signs for zero invented by the Babylonians possessed no numerical or conceptual value of their own—they appear as mathematical syntax, not as number. “Therefore, they cannot be interpreted as representations of the concept or the number zero.”
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.
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 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 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, 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, 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 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 AD150, 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 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, where twelve digits was the angular diameter of the Sun. Minutes of immersion was tabulated from 00 to 3120 to 00, where 00 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 , where d was the digit function and 3120 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 AD311, 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" alongside Geʽez 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 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 AD725.
China
The Sūnzĭ Suànjīng, of unknown date but estimated to be dated from the 1st to, describe how the Chinese counting rods system enabled one to perform positional decimal calculations. As noted in the Xiahou Yang Suanjing, 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, 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".