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Number 0

0 (zero) is both a number and a numerical digit used to represent that number in numerals. It plays a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures. As a digit, zero is used as a placeholder in place value systems. Historically, it was the last digit to come into use. In the English language, zero may also be called null or nil when a number,

Contents

  • 1 0 as a number
  • 2 0 as a numeral

     

    • 2.1 Distinguishing zero from O
  • 3 Etymology
  • 4 History
  • 4.1 History of zero
  • 4.2 Rules of Brahmagupta
  • 4.3 Zero as a decimal digit
  • 5.1 Elementary algebra
  • 5.2 Extended use of zero in mathematics
  • 6.1 Physics
  • 6.2 Chemistry
  • 7.1 Numbering from 1 or 0?
  • 7.2 Null value
  • 7.3 Null pointer
  • 7.4 Negative zero
  • 5 In mathematics
  • 6 In science
  • 7 In computer science
  • 8 In other fields
  • 9 Quotations
  • 10 Notes

0 as a number

0 is the integer that precedes the positive 1, and follows −1. In most (if not all) numerical systems, 0 was identified before the idea of 'negative integers' was accepted. It means "courageous one" in hieroglyphics.

Zero is a number 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 remains the final result.

While mathematicians all accept zero as a number, some non-mathematicians would say that zero is not a number, arguing that one cannot have zero of something. Others hold that if one has a bank balance of zero, one has a specific quantity of money in that account, namely none. It is that latter view which is accepted by mathematicians and most others.

Almost all historians omit the year zero from the proleptic Gregorian and Julian calendars, but astronomers 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, ellipse or (rounded) rectangle. While the shape of the 0 character has an ascender in most modern typefaces, in typefaces with text figures the character usually is of x-height, as shown in the image to the right.

 

 

On the seven-segment displays 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.

It is important to distinguish the number zero (as in the "zero brothers" example above) from the numeral or digit zero, used in numeral systems 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. A zero digit is not always necessary in a positional number system: bijective numeration provides a possible counterexample.

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 \emptyset, representing the empty set, as well 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 Scandinavians 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.

An official German license plate showing zeros

The typeface used on some European number plates for cars distinguish the two symbols by making the zero rather egg-shaped and the O more circular, but most of all by slitting open the zero on the upper right side, so the circle is not closed any more (as in German plates). The typeface chosen is called fälschungserschwerende Schrift (abbr.: FE Schrift), meaning "script which is harder to falsify". Note that those used in the United Kingdom 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 .

Sometimes the number zero used exclusively or not used at all to avoid confusion altogether. For example, confirmation numbers used by Southwest Airlines uses only the letters O and I instead the numbers 0 and 1.

Etymology

The word "zero" came via French zéro from Venetian language zero, which (together with "cipher") came via Italian zefiro from Arabic صفر, şafira = "it was empty", şifr = "zero", "nothing", which was used to translate Sanskrit śūnya ( शून्य ), meaning void or empty.

Italian zefiro already meant "west wind" from Latin and Greek zephyrus; this may have influenced the spelling when transcribing Arabic şifr.[1] 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 Venetian, giving the modern English word.

As the Hindu decimal zero and its new mathematics spread from the Arab world to Europe 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, "in thirteenth-century Paris, a 'worthless fellow' was called a "… cifre en algorisme", i.e., an "arithmetical nothing"."[1] (Algorithm is also a borrowing from the Arabic, in this case from the name of the 9th century mathematician al-Khwarizmi.) From şifr also came French chiffre = "digit", "figure", "number", chiffrer = "to calculate or compute", chiffré= "encrypted". Today, the word in Arabic is still sifr, and cognates of sifr are common throughout the languages of Europe and southwest Asia.

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History of zero

The Mesoamerican Long Count calendar developed in south-central Mexico required the use of zero as a place-holder within its vigesimal (base-20) positional numeral system. A shell glyph was 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.[6] Since the eight earliest Long Count dates appear outside the Maya homeland,[7] it is assumed that the use of zero in the Americas predated the Maya and was possibly the invention of the Olmecs. Indeed, many of the earliest Long Count dates were found within the Olmec heartland, although the fact that the Olmec civilization had come to an end by the 4th century BCE, several centuries before the earliest known Long Count dates, argues against the zero being an Olmec discovery.

Although zero became an integral part of Maya numerals, it of course 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 just as a placeholder, this Hellenistic zero was perhaps the first documented use of a number zero in the Old World. However, the positions were usually limited to the fractional part of a number (called minutes, seconds, thirds, fourths, etc.)—they were not used for the integral part of a number. 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. 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.

In 498 AD, Indian mathematician and astronomer Aryabhata stated that "Sthanam sthanam dasa gunam" or place to place in ten times in value, which may be the origin of the modern decimal based place value notation.[8]

The oldest known text to use zero is the Jain text from India entitled the Lokavibhaaga, dated 458 AD. it was first introduced to the world centuries later by Al-Khwarizmi, a Persian mathematician, astronomer and geographer . He was the founder of several branches and basic concepts of mathematics. In the words of Philip Hitti, Al Khawarizmi's contribution to mathematics influenced mathematical thought to a greater extent. His work on algebra initiated the subject in a systematic form and also developed it to the extent of giving analytical solutions of linear and quadratic equations, which established him as the founder of Algebra. The very name Algebra has been derived from his famous book Al-Jabr wa-al-Muqabilah.

His arithmetic synthesized Greek and Hindu knowledge and also contained his own contribution of fundamental importance to mathematics and science. Thus, he explained the use of zero, a numeral of fundamental importance developed by the Indians. And 'algorithm' or 'algorizm' is named after him.

The first apparent appearance of a symbol for zero appears in 876 in India on a stone tablet in Gwalior. Documents on copper plates, with the same small o in them, dated back as far as the sixth century AD, abound.[10]

Rules of Brahmagupta

The rules governing the use of zero appeared for the first time in Brahmagupta's book Brahmasputha Siddhanta, written in 628. Here Brahmagupta considers not only zero, but negative numbers, and the algebraic rules for the elementary operations of arithmetic with such numbers. In some instances, his rules differ from the modern standard. Here are the rules of Brahamagupta:[11]

  • The sum of zero and a negative number is negative
  • The sum of zero and a positive number is positive
  • The sum of zero and zero is zero.
  • The sum of a positive and a negative is their difference; or, if they are equal, zero
  • 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.

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.

 

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Further Reading:
http://en.wikipedia.org/wiki/Zero
http://en.wikipedia.org/wiki/Number

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