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# Zero

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### Zero

1. 1. INTRODUTIONZero, written 0, is both a number and the numericaldigit used to represent that number in numerals. Itplays a central role in mathematics as the additiveidentity of the integers, real numbers, and many otheralgebraic structures. As a digit, 0 is used as aplaceholder in place value systems. In the Englishlanguage, 0 may be called zero, nought or (US) naught(both pronounced /ˈn(ˈt)/, nil, or "o". Informal orslang terms for zero include zilch and zip. Ought oraught (both pronounced /ˈ (ˈt)/, have also been used.
2. 2. AS A NUMBER0 is the integer immediately preceding 1. In most cultures, 0 wasidentified before the idea of negative things that go lower thanzero was accepted. Zero is an even number, because it is divisibleby 2. 0 is neither positive nor negative. By some definitions 0 isalso a natural number, and then the only natural number not to bepositive.[citation needed] Zero is a number which quantifies acount or an amount of null size.The value, or number, zero is not the same as the digit zero, usedin numeral systems using positional notation. Successive positionsof digits have higher weights, so inside a numeral the digit zero isused to skip a position and give appropriate weights to thepreceding and following digits. A zero digit is not always necessaryin a positional number system, for example, in the number 02. Insome instances, a leading zero may be used to distinguish a
3. 3. Early historyBy the middle of the 2nd millennium BC, the Babylonianmathematics had a sophisticated sexagesimal positional numeralsystem. The lack of a positional value (or zero) was indicated by aspace between sexagesimal numerals. By 300 BC, a punctuationsymbol (two slanted wedges) was co-opted as a placeholder in thesame Babylonian system. In a tablet unearthed at Kish (dating fromabout 700 BC), the scribe Bêl-bân-aplu wrote his zeros with threehooks, rather than two slanted wedges. The Babylonian placeholderwas not a true zero because it was not used alone. Nor was it used atthe end of a number. Thus numbers like 2 and 120 (2×60), 3 and 180(3×60), 4 and 240 (4×60), looked the same because the largernumbers lacked a final sexagesimal placeholder. Only context coulddifferentiate them. Records show that the ancient Greeks seemedunsure about the status of zero as a number. They asked themselves,"How can nothing be something?",
4. 4. Early historyleading to philosophical and, by the Medieval period, religiousarguments about the nature and existence of zero and the vacuum.The paradoxes of Zeno of Elea depend in large part on the uncertaininterpretation of zero. The concept of zero as a number and notmerely a symbol for separation is attributed to India where by the9th century AD practical calculations were carried out usingzero, which was treated like any other number, even in case ofdivision. The Indian scholar Pingala (circa 5th-2nd century BC) usedbinary numbers in the form of short and long syllables (the latterequal in length to two short syllables), making it similar to Morsecode. He and his contemporary Indian scholars used the Sanskritword śūnya to refer to zero or void.
5. 5. History of zeroThe back of Olmec Stela C from Tres Zapotes, the second oldest LongCount date yet discovered. The numerals 7.16.6.16.18 translate toSeptember, 32 BC (Julian). The glyphs surrounding the date arethought to be one of the few surviving examples of Epi-Olmec script.The Mesoamerican Long Count calendar developed in south-centralMexico and Central America required the use of zero as a place-holder within its vigesimal (base-20) positional numeral system.Many different glyphs, including this partial quatrefoil——were usedas a zero symbol for these Long Count dates, the earliest of which (onStela 2 at Chiapa de Corzo, Chiapas) has a date of 36 BC. Since theeight earliest Long Count dates appear outside the Maya homeland,it is assumed that the use of zero in the Americas predated the Mayaand was possibly the invention of the Olmecs. Many of the earliestLong Count dates were found within the Olmec heartland, althoughthe Olmec civilization ended by the 4th century BC, several centuriesbefore the earliest known Long Count dates.Although zero became an integral part of Maya numerals, it did notinfluence Old World numeral systems.
6. 6. History of zeroQuipu, a knotted cord device, used in the Inca Empire and itspredecessor societies in the Andean region to record accounting andother digital data, is encoded in a base ten positional system. Zero isrepresented by the absence of a knot in the appropriate position. Theuse of a blank on a counting board to represent 0 dated back in Indiato 4th century BC.In China, counting rods were used for decimal calculation since the4th century BC including the use of blank spaces. Chinesemathematicians understood negative numbers and zero, somemathematicians used 無入, 空, 口 for the latter, until GautamaSiddha introduced the symbol 0. The Nine Chapters on theMathematical Art, which was mainly composed in the 1st century AD,stated "[when subtracting] subtract same signed numbers, adddifferently signed numbers, subtract a positive number from zero tomake a negative number, and subtract a negative number from zeroto make a positive number."
7. 7. History of zeroBy 130 AD, Ptolemy, influenced by Hipparchus and theBabylonians, was using a symbol for zero (a small circle with a longoverbar) within a sexagesimal numeral system otherwise usingalphabetic Greek numerals. Because it was used alone, not just as aplaceholder, this Hellenistic zero was perhaps the first documenteduse of a number zero in the Old World. However, the positions wereusually limited to the fractional part of a number (calledminutes, seconds, thirds, fourths, etc.)—they were not used for theintegral part of a number.Another zero was used in tables alongside Roman numerals by 525(first known use by Dionysius Exiguous), but as a word, nulla meaning"nothing", not as a symbol. When division produced zero as aremainder, nihil, also meaning "nothing", was used. These medievalzeros were used by all future medieval computists (calculators of
8. 8. History of zeroThe initial "N" was used as a zero symbol in a table of Romannumerals by Bede or his colleague around 725. In 498 AD, Indianmathematician and astronomer Aryabhata stated that "Sthanamsthanam dasa gunam" or place to place in ten times in value, which isthe origin of the modern decimal-based place value notation.The oldest known text to use a decimal place-value system, includinga zero, is the Jain text from India entitled the Lokavibhâga, dated 458AD. This text uses Sanskrit numeral words for the digits, with wordssuch as the Sanskrit word for void for zero.The first known use of special glyphs for the decimal digits thatincludes the indubitable appearance of a symbol for the digit zero, asmall circle, appears on a stone inscription found at the ChaturbhujaTemple at Gwalior in India, dated 876 AD.[22][23] There are manydocuments on copper plates, with the same small o in them, datedback as far as the sixth century AD, but their authenticity may bedoubted.
9. 9. Zero as a decimalPositional notation without the use of zero (using an empty space intabular arrangements, or the word kha "emptiness") is known tohave been in use in India from the 6th century. The earliest certainuse of zero as a decimal positional digit dates to the 5th centurymention in the text Lokavibhaga. The glyph for the zero digit waswritten in the shape of a dot, and consequently called bindu ("dot").The dot had been used in Greece during earlier ciphered numeralperiods .The Hindu-Arabic numeral system (base 10) reached Europe inthe 11th century, via the Iberian Peninsula through SpanishMuslims, the Moors, together with knowledge of astronomy andinstruments like the astrolabe, first imported by Gerbert of Aurillac. Forthis reason, the numerals came to be known in Europe as "Arabicnumerals". The Italian mathematician Fibonacci or Leonardo of Pisa wasinstrumental in bringing the system into European mathematics in 1202.
10. 10. Zero as a decimalAfter my fathers appointment by his homeland as state official in thecustoms house of Bugia for the Pisan merchants who thronged to it,he took charge; and in view of its future usefulness and convenience,had me in my boyhood come to him and there wanted me to devotemyself to and be instructed in the study of calculation for some days. There, following my introduction, as a consequence of marvelousinstruction in the art, to the nine digits of the Hindus, the knowledgeof the art very much appealed to me before all others, and for it Irealized that all its aspects were studied in Egypt, Syria, Greece, Sicily,and Provence, with their varying methods; and at these placesthereafter, while on business. I pursued my study in depth andlearned the give-and-take of disputation
11. 11. Zero as a decimalBut all this even, and the algorism, as well as the art of Pythagoras, Iconsidered as almost a mistake in respect to the method of theHindus (Modus Indorum). Therefore, embracing more stringentlythat method of the Hindus, and taking stricter pains in its study, whileadding certain things from my own understanding and inserting alsocertain things from the niceties of Euclids geometric art.I have striven to compose this book in its entirety as understandablyas I could, dividing it into fifteen chapters. Almost everything which Ihave introduced I have displayed with exact proof, in order that thosefurther seeking this knowledge, with its pre-eminent method, mightbe instructed, and further, in order that the Latin people might notbe discovered to be without it, as they have been up to now.
12. 12. Zero as a decimalIf I have perchance omitted anything more or less proper ornecessary, I beg indulgence, since there is no one who is blamelessand utterly provident in all things. The nine Indian figures are: 9 8 7 65 4 3 2 1. With these nine figures, and with the sign 0 ... any numbermay be written.Here Leonardo of Pisa uses the phrase "sign0", indicating it is like a sign to do operations like addition ormultiplication. From the 13th century, manuals on calculation(adding, multiplying, extracting roots, etc.) became common inEurope where they were called algorismus after the Persianmathematician al-Khwārizmī. The most popular was written byJohannes de Sacrobosco, about 1235 and was one of the earliestscientific books to be printed in 1488. Until the late 15thcentury, Hindu-Arabic numerals seem to have predominated amongmathematicians, while merchants preferred to use the Romannumerals. In the 16th century, they became commonly used inEurope.