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0 ('''zero''') is both a Number and a Numeral . Zero is the last numeral to be created in most numerical systems, and has sometimes been represented only by a gap. In the English language, zero may also be called '''nil''' when a number, '''o'''/'''oh''' when a numeral, and '''nought'''/'''naught''' in either context. 0 as a number 0 is the Integer that precedes the positive 1 , and follows −1 . Zero first appeared as a number in Brahmagupta 's work dated to 628 . Prior to that Babylonians used a space marker that played one of the functions of zero. Babylonians did not have a special symbol for zero. In most (if not all) numerical systems, 0 was identified before the idea of 'negative integers' was accepted. Zero is an integer which quantifies a count or an amount of Null size; that is, if the number of your brothers is zero, that means the same thing as having no brothers, and if something has a weight of zero, it has no weight. If the difference between the number of pieces in two piles is zero, it means the two piles have an equal number of pieces. Before counting starts, the result can be assumed to be zero; that is the number of items counted before you count the first item and counting the first item brings the result to one. And if there are no items to be counted, zero is remains the final result. While mathematicians all accept zero as a number, some others would say that zero is not a number, arguing one cannot have zero of something. Others hold that if you have zero dollars in your bank account, you have a specific quantity of money in your account, namely none. It is that latter view which is accepted by mathematicians and most others. Almost all Historian s omit the Year Zero from the Proleptic Gregorian and Julian Calendar s, but Astronomer s include it in these same calendars. However, the phrase Year Zero may be used to describe any event considered so significant that it virtually starts a new time reckoning. 0 as a numeral The modern numeral 0 is normally written as a circle or (rounded) rectangle. On the Seven-segment Display s of calculators, watches, etc., 0 is usually written with six line segments, though on some historical calculator models it was written with four line segments. This variant glyph has not caught on. Early Europeans hesitated to consider zero as a numeral or number. Leonardo of Pisa or Fibonacci says the following in 1202 AD when the Indian number system arrived in Europe.
Here Leonardo of Pisa uses the word sign "0", indicating it is like a sign to do operations like addition or multiplication. He did not recognize zero as a number on its own right. It is important to distinguish the ''number'' zero (as in the "zero brothers" example above) from the ''numeral'' or ''digit'' zero, used in Numeral System s using Positional Notation . Successive positions of digits have higher values, so the digit zero is used to skip a position and give appropriate value to the preceding and following digits. The Babylonian Numeral System used two narrow slanting wedges, similar to //, for the equivalent of a positional zero numeral starting in about 400BC . A zero digit is not always necessary in a positional number system: Bijective Numeration provides a possible counterexample. In old-style fonts with Text Figures , 0 is usually the same height as a lowercase x. Etymology The word ''zero'' comes ultimately from the Arabic ' (صفر''') meaning ''empty'' or ''vacant'', a literal translation of the India n Sanskrit '''' meaning ''void'' or ''empty''. Through transliteration this became ''zephyr'' or ''zephyrus'' in Latin . The word ''zephyrus'' already meant "west wind" in Latin; the proper noun Zephyrus was the Roman god of the west wind (after the Greek god Zephyros). With its new use for the concept of zero, zephyr came to mean a light breeze—"an ''almost'' nothing" (Ifrah 2000; see References). The word ''zephyr'' survives with this meaning in English today. The Italian mathematician Fibonacci (c.1170-1250), who grew up in Arab North Africa and is credited with introducing the Hindu decimal system to Europe, used the term ''zephyrum''. This became ''zefiro'' in Italian, which was contracted to ''zero'' in the Venetian dialect, giving the modern English word. As the decimal zero and its new mathematics spread through a Europe that was still in the Middle Ages , words derived from ''sifr'' and ''zephyrus'' came to refer to calculation, as well as to privileged knowledge and secret codes. According to Ifrah (2000), "in thirteenth-century Paris, a 'worthless fellow' was called a... ''cifre en algorisme'', i.e., an 'arithmetical nothing.' " ('' Algorithm '' is also a borrowing from the Arabic, in this case from the name of the 9th Century mathematician Muḥammad Ibn Mūsā Al-Ḵwārizmī .) The Arabic root gave rise to the modern French ''chiffre'', which means digit, figure, or number; ''chiffrer'', to calculate or compute; and ''chiffré'', encrypted; as well as to the English word '' Cipher ''. Today, the word in Arabic is still ''sifr'', and cognates of ''sifr'' are common throughout the languages of Europe. A few additional examples follow.
Note that zero in Greek is translated as ''Μηδέν'' (Meiden). History Prehistory of zero By the mid ). However, the Babylonian placeholder is not the same as a true number zero, considered as a quantity, and the Babylonian system of digits is not quite the same as a true base 60 system using a zero digit, since the Babylonians did not have a symbol for zero. The so-called Babylonian zero is a separation mark that came between two place value numbers. Babylonians did have a 60 based place value notation, but they were not able to differentiate between numbers as 120 and 2, 3 and 180, 4 and 240, etc. They simply could not differentiate between numbers that required a zero at the end. They simply did not have a zero. All they had was a separation mark for numbers that separated different place value numbers from each other. Records show that the 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. (The ancient Greeks even questioned that 1 was a number.) In . In Pingala's system, four short syllables meant one, not zero. Nevertheless, he and other Indian scholars at the time used the Sanskrit word ''shunya'' (the origin of the word ''zero'' after a series of transliterations and a literal translation) to refer to zero or ''void''. {Link without Title} . History of zero In the Bakhshali Manuscript , whose date is uncertain but which is claimed by some to be quite early, zero is symbolized and used as a number; if the early dating is accepted, it would predate Brahmagupta. In 498 AD, Hindu astronomer and mathematician Aryabhata stated that "Stanam stanam dasa gunam" or place to place in ten times in value, which may be the origin of the modern decimal based place value notation; his positional number system included a zero in his letter code for numerals (which allowed him to express numbers as words) in his mathematical astronomy text ''Aryabhatiya''. The first unambiguous appearance of the mathematical zero is in Brahmagupta 's Brahmasphuta Siddhanta , along with consideration of negative numbers and the algebraic rules discussed below. The late Olmec people of south-central Mexico began to use a zero digit (a shell glyph) in the New World possibly by the 4th Century BC but certainly by 40 BC , which became an integral part of Maya Numerals , but did not influence Old World numeral systems. By 130 , Ptolemy , influenced by Hipparchus and the Babylonians, was using a symbol for zero (a small circle with a long overbar) within a sexagesimal numeral system otherwise using alphabetic Greek Numerals . Because it was used alone, not as just a placeholder, this Hellenistic Zero was one of the first ''documented'' uses of a digit zero in the Old World . In later Byzantine manuscripts of his ''Syntaxis Mathematica'' (''Almagest''), the Hellenistic zero had morphed into the Greek Letter Omicron (otherwise meaning 70). Another zero was used in tables alongside Roman Numerals by 525 (first known use by Dionysius Exiguus ), but as a word, ''nulla'' meaning ''nothing'', not as a symbol; this usage, more or less contemporary with Aryabhata, might represent a concept of true, mathematical zero, though not so clearly as in the case of Brahmagupta. When division produced zero as a remainder, ''nihil'', also meaning ''nothing'', was used. These medieval zeros were used by all future medieval Computists (calculators of Easter ). An isolated use of their initial, N, was used in a table of Roman numerals by Bede or a colleague about 725 , a zero symbol. By the 7th century, when Brahmagupta lived, some concept of zero had clearly reached Cambodia , and documentation shows the idea later spreading to China and the Islam ic world. The Rules of Brahmagupta Zero as a number appeared for the first time in in 1817 .)
In saying zero divided by zero is zero, Brahmagupta differs from the modern position. Mathematicians normally do not assign a value, whereas computers and calculators will sometimes assign NaN , which means "not a number." Moreover, non-zero positive or negative numbers when divided by zero are either assigned no value, or a value of unsigned infinity, positive infinity, or negative infinity. Once again, these assignments are not numbers, and are associated more with computer science than pure mathematics, where in most contexts no assignment is made. (See Division By Zero ) Zero as a decimal digit See also: History Of The Hindu-Arabic Numeral System . Positional notation without the use of zero (using an empty space in tabular arrangements, or the word ''kha'' "emptiness") is known to have been in use in India upto the 6th Century . The earliest certain use of zero as a positional digit dates to the 7th Century . The glyph for the zero digit was written in the shape of a dot, and consequently called '''' "dot". The Hindu-Arabic Numeral System reached Europe in the 11th Century , via Andalusia , together with knowledge of Astronomy and instruments like the Astrolabe , first imported by Gerbert Of Aurillac . They came to be known as " Arabic Numerals ". The Italian mathematician Fibonacci was instrumental in bringing the system into European mathematics around 1200. From the 13th century, manuals on calculation (adding, multiplying, extracting roots etc.) became common in Europe where they were called '' Algorimus '' in deference to the Persian mathematician Muḥammad Ibn Mūsā Al-Ḵwārizmī . The most popular was written by John of Sacrobosco and was one of the earliest scientific books to be printed in 1488. Hindu-Arabic numerals until the late 15th century seem to have predominated among mathematicians, while merchants preferred to use the abacus instead, and it was only from the 16th century that they became common knowledge in Europe. In mathematics Elementary algebra Zero (0) is the lowest Non-negative Integer . The Natural Number following zero is One and no natural number precedes zero. Zero may or may not be counted as a natural number, depending on the definition of natural numbers. Mathematical operations involving zero were first described by Brahmasphutasiddhanta in the 7th century. In from Set Theory , zero 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. The Cardinality Function , applied to the empty set, returns the empty set as a value, thereby assigning it zero elements. Zero is neither positive nor negative, neither a Prime Number nor a Composite Number , nor is it a Unit . If zero is excluded from the Rational Number s, the Real Number s or the Complex Number s, the remaining numbers form an Abelian Group . The following are some basic rules for dealing with the number zero. These rules apply for any Complex Number ''x'', unless otherwise stated.
The expression "0/0" is an " Indeterminate Form ". That does not simply mean that it is undefined; rather, it means that if ''f''(''x'') and ''g''(''x'') both approach 0 as ''x'' approaches some number, then ''f''(''x'')/''g''(''x'') could approach any finite number or ∞ or −∞; it depends on which functions ''f'' and ''g'' are. See L'Hopital's Rule . The sum of 0 numbers is 0, and The Product Of 0 Numbers Is 1 . Extended use of ''zero'' in mathematics
In physics The value zero plays a special role for a large number of physical quantities. For some quantities, the zero level is naturally distinguished from all other levels, where as it for others is more or less arbitrarily chosen. For example, on the Kelvin temperature scale, zero is the coldest possible temperature ( Negative Temperature s exist but are not actually colder), where as on the Celsius scale, zero is arbitrarily defined to be at the Freezing Point of water. Measuring sound intensity in Decibel s or Phon s, the zero level is arbitrarily set at a reference value, e.g. at a value for the threshold of hearing. In computer science Numbering from 1 or 0? People usually number things starting from one, not zero. Yet in Computer Science zero has become the popular indication for a starting point. For example, in almost all old Programming Language s, an Array starts from 1 By Default . As programming languages have developed, it has become more common that an array starts from zero by default. And the first item in the array is item 0. (Note: the word "first" is unrelated to the number 1.) In particular, the popularity of the programming language "C" in the 80s has made this approach common. One reason for this convention is that Modular Arithmetic normally describes a set of N numbers as containing 0,1,2,...N-1 in order to contain the additive identity. Because of this, many arithmetic concepts (such as hash tables) are less elegant to express in code unless the array starts at zero. Another reason to use zero-based array indices is that it can improve efficiency under certain circumstances. To illustrate, suppose ''a'' is the Memory Address of the first element of an array, and ''i'' is the index of the desired element. In this fairly typical scenario, it is quite common to want the address of the desired element. If the index numbers count from 1, the desired address is computed by this expression: : where ''s'' is the size of each element. In contrast, if the index numbers count from 0, the expression becomes this: : This simpler expression can be more efficient to compute in certain situations. Note, however, that a language wishing to index arrays from 1 could simply adopt the convention that every "array address" is represented by ; that is, rather than using the address of the first array element, such a language would use the address of an imaginary element located immediately before the first actual element. The indexing expression for a 1-based index would be the following: : Hence, the efficiency benefit of zero-based indexing is not inherent, but is an artifact of the decision to represent an array by the address of its first element. This situation can lead to some confusion in terminology. In a zero-based indexing scheme, the First element is "element number zero"; likewise, the twelfth element is "element number eleven". For this reason, the first element is often referred to as the '' Zeroth '' element to eliminate any possible doubt (though, strictly speaking, this is unnecessary and arguably incorrect, since the meanings of the Ordinal Number s are not ambiguous). Null value In databases a field can have a null value. This is equivalent to the field not having a value. 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 to three-valued logic. No longer is a condition either true or false, but it can be undetermined. Any computation including a null value delivers a null result. Asking for all records with value 0 or value not equal 0 will not yield all records, since the records with value null are excluded. Null pointer A '' Null Pointer '' is a pointer in a computer program that does not point to any object or function. In C , the integer constant 0 is converted into the null pointer at Compile Time when it appears in a pointer context, and so 0 is a standard way to refer to the null pointer in code. However, the internal representation of the null pointer may be any bit pattern (possibly different values for different data types), and has no particular association with zero. Negative zero See Also: -0 In some Signed Number Representations (but not the Two's Complement representation predominant today) and most Floating Point Number representations, zero has two distinct representations, one grouping it with the positive numbers and one with the negatives; this latter representation is known as Negative Zero . Representations with negative zero can be troublesome, because the two zeroes will compare equal but may be treated differently by some operations. Distinguishing zero from O The oval-shaped zero and circular letter O together came into use on modern character displays. The zero with a dot in the centre seems to have originated as an option on IBM 3270 controllers (this has the problem that it looks like the Greek Letter Theta ). The Slashed Zero , looking identical to the letter O other than the slash, is used in old-style ASCII graphic sets descended from the default Typewheel on the venerable ASR-33 Teletype . This format causes problems because of its similarity to the symbol ∅, representing the Empty Set , aswell as for certain Scandinavian Languages which use Ø as a letter. The convention which has the letter O with a slash and the zero without was used at IBM and a few other early mainframe makers; this is even more problematic for Scandinavia ns because it means two of their letters collide. Some Burroughs / Unisys equipment displays a zero with a ''reversed'' slash. And yet another convention common on early line Printers left zero unornamented but added a tail or hook to the letter-O so that it resembled an inverted Q or cursive capital letter-O. The typeface used on some European number plates for do not differentiate between the two as there can never be any ambiguity if the design is correctly spaced. In paper writing one may not distinguish the 0 and O at all, or may add a slash across it in order to show the difference, although this sometimes causes ambiguity in regard to the symbol for the Null Set . Quotes : "The importance of the creation of the zero mark can never be exaggerated. This giving to airy nothing, not merely a local habitation and a name, a picture, a symbol, but helpful power, is the characteristic of the Hindu race from whence it sprang. It is like coining the Nirvana into dynamos. No single mathematical creation has been more potent for the general on-go of intelligence and power." '' G.B. Halsted '' In other fields for 0]]
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