0
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Assamese, Bengali ০ | | |||
Maya numerals | 𝋠 | |||
Morse code | _ _ _ _ _ |
0 (zero) is a
As a
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
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 in Venetian. 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.[4]
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.[5][6]
It is often called "oh" in the context of reading out a string of digits, such as
Slang words for zero include "zip", "zilch", "nada", and "scratch".
History
Ancient Near East
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Ancient
By the middle of the 2nd millennium BC,
The Babylonian positional numeral system differed from the later
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,
Although zero became an integral part of
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.[citation needed]
Classical antiquity
The ancient Greeks had no symbol for zero (μηδέν, pronounced 'midén'), and did not use a digit placeholder for it.[17] 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.[18] 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.[19][20]
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
By AD 150,
The earliest use of zero in the calculation of the
In most cultures, 0 was identified before the idea of negative things (i.e., quantities less than zero) was accepted.
China
The Sūnzĭ Suànjīng, of unknown date but estimated to be dated from the 1st to 5th centuries AD, and Japanese records dated from the 18th century, 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.[31] According to A History of Mathematics, the rods "gave the decimal representation of a number, with an empty space denoting zero".[30] The counting rod system is considered a positional notation system.[32]
Zero was not treated as a number at that time, but as a "vacant position".[33] Qín Jiǔsháo's 1247 Mathematical Treatise in Nine Sections is the oldest surviving Chinese mathematical text using a round symbol 〇 for zero.[34] The origin of this symbol is unknown; it may have been borrowed from Indian sources or produced by modifying a square symbol.[35] 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.[36]
India
The concept of zero as a written digit in the decimal place value notation was developed in India.[40] 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.[41] In 2017, three samples from the manuscript were shown by radiocarbon dating to come from three different centuries: from AD 224–383, AD 680–779, and AD 885–993, making it South Asia's oldest recorded use of the zero symbol. It is not known how the birch bark fragments from different centuries forming the manuscript came to be packaged together.[42][43][44]
The
The Aryabhatiya (c. 500), states sthānāt sthānaṁ daśaguṇaṁ syāt "from place to place each is ten times the preceding".[46][47][48]
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:[49][50]
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
A black dot is used as a decimal placeholder in the Bakhshali manuscript, portions of which date from AD 224–993.[42]
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
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 876.[52][53]
Middle Ages
Transmission to Islamic culture
The Arabic-language inheritance of science was largely Greek,[54] followed by Hindu influences.[55] In 773, at Al-Mansur's behest, translations were made of many ancient treatises including Greek, Roman, Indian, and others.
In AD 813, astronomical tables were prepared by a
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 called ṣifr.[57]
Transmission to Europe
The
After my father's appointment by
Latin 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.[58]
From the 13th century, manuals on calculation (adding, multiplying, extracting roots, etc.) became common in Europe where they were called algorismus after the Persian mathematician
Symbols and representations
Today, the numerical digit 0 is usually written as a circle or ellipse. Traditionally, many print
A slashed 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 on IBM 3270 displays and has continued with some modern computer typefaces such as Andalé 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 one of the capital-O–digit-0 pair more rounded and the other more angular (closer to a rectangle). A further distinction is made in falsification-hindering typeface as used on German 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.
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
Elementary algebra
The number 0 is the smallest
The number 0 can be regarded as neither positive nor negative
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.
- undefined, because 0 has no multiplicative inverse (no real number multiplied by 0 produces 1), a consequence of the previous rule.[71]
- 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.[72]
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.[73]
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: 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.[74] 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
The role of 0 as additive identity generalizes beyond elementary algebra. In
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 valuefalse.
- In probability theory, 0 is the smallest allowed value for the probability of any event.[75]
- zero object, often denoted 0, and the related concept of zero morphisms, which generalize the zero function.[76]
Physics
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 an absolute temperature (typically measured in kelvins), 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 the freezing point of water.[77][78] Measuring sound intensity in decibels or phons, the zero level is arbitrarily set at a reference value—for example, at a value for the threshold of hearing. In physics, the zero-point energy is the lowest possible energy that a quantum mechanical physical system may possess and is the energy of the ground state of the system.
Computer science
Modern computers store information in binary, that is, using an "alphabet" that contains only two symbols, usually chosen to be "0" and "1". Binary coding is convenient for digital electronics, where "0" and "1" can stand for the absence or presence of electrical current in a wire.[79] Computer programmers typically use high-level programming languages that are more easily intelligible to humans than the binary instructions that are directly executed by the central processing unit. 0 plays various important roles in high-level languages. For example, a Boolean variable stores a value that is either true or false, and 0 is often the numerical representation of false.[80]
0 also plays a role in
There can be confusion between 0- and 1-based indexing; for example, Java's
In C, a byte containing the value 0 serves to indicate where a string of characters ends. Also, 0 is a standard way to refer to a null pointer in code.[82]
In databases, it is possible for a field not to have a value. It is then said to have a
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 hardware
An
Many
Programmers often use a slashed zero to avoid confusion with the letter "O".[89]
Other fields
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.[90]
Dating systems
In the
See also
Notes
- ^ 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 in Greek numerals from 0 to 59, where 31 was written λα meaning 30+1, and 20 was written κ meaning 20.
References
- ISBN 978-0-486-27096-8. Retrieved 5 January 2016.from the original on 7 March 2012. Retrieved 4 March 2012.
- "zero, n." OED Online. Oxford University Press. December 2011. Archived
French zéro (1515 in Hatzfeld & Darmesteter) or its source Italian zero, for *zefiro, < Arabic çifr
The idea of sunya and place numbers was transmitted to the Arabs who translated sunya or "leave a space" into their language as sifr.
zero was regarded as a number in India ... whereas the Chinese employed a vacant position
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
The Arabic inheritance of science was overwhelmingly Greek, but Hindu influences ranked next. In 773, at Mansur's behest, translations were made of the Siddhantas – 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.
zero neither prime nor composite
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.
In the B.C./A.D. scheme there is no year zero. After 31 December 1 BC came 1 January AD 1. ... If you object to that no-year-zero scheme, then don't use it: use the astronomer's counting scheme, with negative year numbers.