0 (number)
From Free net encyclopedia
- This page is about the number and numeral 0. For other uses of 0 or "zero", see 0 (disambiguation)
| Cardinal | 0 zero nought naught nil |
| Ordinal | 0th zeroth |
| Factorization | <math> 0 </math> |
| Divisors | N/A |
| Roman numeral | N/A |
| Binary | 0 |
| Octal | 0 |
| Duodecimal | 0 |
| Hexadecimal | 0 |
0 (zero) is both a number and a numeral. The last numeral to be created in most numerical systems, zero is not a counting number (counting begins with the number 1) and in many eras and places is represented only by a gap or mark very different from counting numbers. In the English language, zero may also be called nil when a number, o/oh when a numeral, and nought/naught in either context.
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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.
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 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 or (rounded) rectangle. Image:7-segment abcdef.svgImage:7-segment cdeg.svg 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. Early Europeans hesitated to consider zero as a numeral. Leonardo of Pisa or Fibonacci says the following in 1,202 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 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. 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. Image:TextFigs036.png
Etymology
The word zero comes ultimately from the Arabic ṣifr (صفر) meaning empty or vacant, a literal translation of the Indian Sanskrit Template:IAST 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.
- French: zéro, zero
- German: Ziffer, digit, figure, numeral, cypher
- Italian: cifra, digit, numeral, cypher; zero, zero
- Russian: tzifra, digit, numeral; shifr cypher, code
- Polish: cyfra, digit; szyfrować, to encrypt; zero, zero
- Portuguese: zero, zero; dígito, digit; número, number; algarismo, figure, numeral
- Spanish: cifra, figure, numeral, cypher, code; cero, zero
- Swedish: siffra, numeral, sum, digit; chiffer, cypher
Note that zero in Greek is translated as Μηδέν (Meiden).
History
Prehistory of zero
By the mid 2nd millennium BC, the Babylonians had a sophisticated sexagesimal positional numeral system. The lack of a positional value (or zero) was indicated by a space between sexagesimal numerals. By 300 BC a punctuation symbol (two slanted wedges) was co-opted as a placeholder in the same Babylonian system. However, "... a tablet found at Kish ... thought to date from around 700 BC, uses three hooks to denote an empty place in the positional notation. Other tablets dated from around the same time use a single hook for an empty place" ([1] and natural number).
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 bewteen 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 ancient Greeks seemed unsure about the status of zero as a number: they asked themselves "How can nothing be something?", leading to interesting 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. (The ancient Greeks even questioned that 1 was a number.)
In ancient India, the linguist Panini (5th century BC) used the null (zero or shoonya) operator in the Ashtadhyayi, his algebraic grammar of the Sanskrit language. Another early use of something like zero by the Indian scholar Pingala (circa 5th-3rd century BC), implied at first glance by his use of binary numbers, is only the modern binary representation using 0 and 1 of Pingala's binary system, which used short and long syllables (the latter equal in length to two short syllables) as described in Math for Poets and Drummers (pdf), making it similar to Morse code. 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. [2].
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 Islamic world.
The Rules of Brahmagupta
Zero as a number appeared for the first time in Brahmagupta's book Brahmasputha Siddhanta. 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: (From the English translation by Henry Thomas Colebrooke in 1817.)
- The sum of two positive quantities is positive
- The sum of two negative quantities is negative
- The sum of zero and a negative number is negative
- The sum of a positive number and zero 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
- In subtraction, the less is to be taken from the greater, positive from positive
- In subtraction, the less is to be taken from the greater, negative from negative
- When the greater however, is subtracted from the less, the difference is reversed
- When positive is to be subtracted from negative, and negative from positive, they must be added together
- The product of a negative quantity and a positive quantity is negative
- The product of a negative quantity and a negative quantity is positive
- The product of two positives is positive
- Positive divided by positive or negative by negative is positive
- Positive divided by negative is negative. Negative divided by positive is negative
- 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.
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 Template:IAST "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 set theory, the number zero is the cardinality of the empty set: if one does not have any apples, then one has zero apples. In fact, in certain axiomatic developments of mathematics 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 numbers, the real numbers or the complex numbers, 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.
- Addition: x + 0 = 0 + x = x. (That is, 0 is an identity element with respect to addition.)
- Subtraction: x − 0 = x and 0 − x = − x.
- Multiplication: x · 0 = 0 · x = 0.
- Division: 0 / x = 0, for nonzero x. But x / 0 is undefined, because 0 has no multiplicative inverse, a consequence of the previous rule. For positive x, as y in x / y approaches zero from positive values, its quotient increases toward positive infinity, but as y approaches zero from negative values, the quotient increases toward negative infinity. The different quotients confirms that division by zero is undefined.
- Exponentiation: x0 = 1, except that the case x = 0 may be left undefined in some contexts. For all positive real x, 0x = 0.
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
- Zero is the identity element in an additive group or the additive identity of a ring.
- A zero of a function is a point in the domain of the function whose image under the function is zero. When there are finitely many zeros these are called the roots of the function. See zero (complex analysis).
- In geometry, the dimension of a point is 0.
- In trigonometry, sin 0 = 0, tan 0 = 0, arcsin 0 = 0, and arctan 0 = 0.
- The concept of "almost" impossible in probability. More generally, the concept of almost nowhere in measure theory. For instance: if one chooses a point on a unit line interval [0,1) at random, it is not impossible to choose 0.5 exactly, but there is a probability of zero that you will.
- A zero function (or zero map) is a constant function with 0 as its only possible output value; i.e., <math>f(x) = 0</math> for all x defined. A particular zero function is a zero morphism in category theory; e.g., a zero map is the identity in the additive group of functions. The determinant on non-invertible square matrices is a zero map.
- Zero is one of three possible return values of the Möbius function. Passed an integer x2 or x2y, the Möbius function returns zero.
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 temperatures 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 decibels or phons, 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?
Human beings 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 languages, an array starts from 1 by default, which is natural for humans. As programming languages have developed, it has become more common that an array starts from zero by default (zeroth, or zero-based).
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:
- <math>a + s \times (i-1) \,\!</math>
where s is the size of each element. In contrast, if the index numbers count from 0, the expression becomes this:
- <math>a + s \times i \,\!</math>
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 <math>a'=a-s</math>; 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:
- <math>a' + s \times i \,\!</math>
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 numbers 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
Template:Main 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
Image:Zero o comparison.PNG 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 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.
The typeface used on some European number plates for cars distinguish the two symbols by making the O rather egg-shaped and the zero 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 "unfalsifiable script". 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.
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
- In some countries, dialing 0 on a telephone places a call for operator assistance.
- In Braille, the numeral 0 has the same dot configuration as the letter J.
- DVDs that can be played in any region are sometimes referred to as being "region 0".
See also
References
- The Universal History of Numbers: From Prehistory to the Invention of the Computer. Georges Ifrah. Wiley (2000). ISBN 0471393401.
- Charles Seife (2000). Zero: The Biography of a Dangerous Idea. Publisher: Penguin USA (Paper). ISBN 0140296476.
- Algebra with Arithmetic of Brahmagupta and Bhaskara by Henry Thomas Colebrooke, London 1817.
- Grimm, R.E., "The Autobiography of Leonardo Pisano", Fibonacci Quarterly, Vol. 11, No. 1, February 1973, pp. 99-104.
- Sigler, L., “Fibonacci’s Liber Abaci”, English translation, Springer, 2003.
- The Book of Nothing, John D. Barrow, Vintage (July, 2001), ISBN 0-09-928845-1.
- Aryabhatiya of Aryabhata, translated by Walter Eugene Clark.
External links
- A Brief History of Zero - Kristen McQuillin, July 1997 (revised January 2004)
- A history of Zero
- Zero Saga
- The Discovery of the Zero
- This article was originally based on material from the Free On-line Dictionary of Computing, which is licensed under the GFDL.
ca:Zero cs:Nula da:0 (tal) de:Null et:Null es:Cero eo:Nulo fr:0 (nombre) gl:Cero ko:0 ia:Zero it:Zero he:0 (מספר) la:0 lt:0 (skaičius) nl:0 (getal) ja:0 nn:0 pl:0 (liczba) pt:Zero ro:0 (cifră) ru:0 (число) simple:Zero sl:0 (število) fi:0 (luku) sv:0 (tal) tl:0 (bilang) th:0 yi:נול zh:0