0 (number)

This page is about the number and digit 0. For other uses of 0 or "zero", see 0 (disambiguation)

{| width="250" border="1" style="float: right; background: #feffff; border-collapse: collapse; margin: 0.5em" Cardinal 0
zero
o/oh
nought
naught
nil Ordinal 0th
zeroth Factorization

0

Divisors N/A Roman numeral N/A Binary 0 Octal 0 Duodecimal 0 Hexadecimal 0

0 (zero) is both a number — or, more precisely, a numeral representing a number — and a numerical digit. Zero is the last digit to be incorporated in most numeral systems. 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. 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 remains the final result.

While mathematicians all accept zero as a number, some non-mathematicians would say that zero is not a number, arguing one cannot have zero of something. Others hold that if you have a bank balance of zero, 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. In old-style fonts with text figures, 0 is usually the same height as a lowercase x.

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.

Etymology

The word zero comes ultimately from the Arabic ' (صفر) meaning empty or vacant, a literal translation of the 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."Georges Ifrah. The Universal History of Numbers: From Prehistory to the Invention of the Computer. Wiley (2000). ISBN 0471393401. 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, "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 al-Khwarizmi.) 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
Catalan; xifra, digit, figure, cypher; zero, zero; xifrar, to encode, to number
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
Dutch: cijfer, digit

Note that zero in Greek is translated as Μηδέν (Mèden).

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".A History of Zero

The Babylonian placeholder was not a true zero because it was not used alone. Nor was it used at the end of a number. Thus numbers like 2 and 120 (2×60), 3 and 180 (3×60), 4 and 240 (4×60), etc. looked the same because the larger numbers lacked a final sexagesimal placeholder. Only context could differentiate them.

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 applied to Pingala's binary system, which used short and long syllables (the latter equal in length to two short syllables),Math for Poets and Drummers (pdf) making it similar to Morse code. 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. Zero story 1 Zero story 2

History of zero

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, within a vigesimal (base-20) positional numeral system. It 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 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, Indian mathematician and astronomer 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.Aryabhatiya of Aryabhata, translated by Walter Eugene Clark. In the Bakhshali Manuscript, whose date is uncertain but which is believed by some scholars to pre-date Aryabhata, zero is symbolized and used as a number; if the early dating is accepted, it would also predate Brahmagupta.

The first unambiguous use of a decimal zero and the rules governing its use appear in Brahmagupta's Brahmasphuta Siddhanta, along with consideration of negative numbers and the algebraic rules discussed below. 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.

Rules of Brahmagupta

The rules governing the use of zero 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:Henry Thomas Colebrooke. Algebra with Arithmetic of Brahmagupta and Bhaskara. London 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 positive, 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. (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 from the 6th century. The earliest certain use of zero as a decimal positional digit dates to the 9th 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 or Leonardo of Pisa was instrumental in bringing the system into European mathematics in 1202, stating:

After my father's appointment by his homeland as state official in the customs 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 devote myself to and be instructed in the study of calculation for some days. There, following my introduction, as a consequence of marvelous instruction in the art, to the nine digits of the Hindus, the knowledge of the art very much appealed to me before all others, and for it I realized that all its aspects were studied in Egypt, Syria, Greece, Sicily, and Provence, with their varying methods; and at these places thereafter, while on business. I pursued my study in depth and learned the give-and-take of disputation. But all this even, and the algorism, as well as the art of Pythagoras, I considered as almost a mistake in respect to the method of the Hindus. (Modus Indorum). Therefore, embracing more stringently that method of the Hindus, and taking stricter pains in its study, while adding certain things from my own understanding and inserting also certain things from the niceties of Euclid's geometric art. I have striven to compose this book in its entirety as understandably as I could, dividing it into fifteen chapters. Almost everything which I have introduced I have displayed with exact proof, in order that those further seeking this knowledge, with its pre-eminent method, might be instructed, and further, in order that the 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.Sigler, L., Fibonacci’s Liber Abaci. English translation, Springer, 2003.Grimm, R.E., "The Autobiography of Leonardo Pisano", Fibonacci Quarterly 11/1 (February 1973), pp. 99-104.

Here Leonardo of Pisa uses the word sign "0", indicating it is like a sign to do operations like addition or multiplication, but he did not recognize zero as a number on its own right. From the 13th century, manuals on calculation (adding, multiplying, extracting roots etc.) became common in Europe where they were called algorimus after the Persian mathematician al-Khwarizmi. The most popular was written by John of Sacrobosco about 1235 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. 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.

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 under multiplication.

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, 0^x = 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., $f(x) = 0$ 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 science

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.

Chemistry

Zero has been proposed as the atomic number of the theoretical element tetraneutronium. It has been shown that a cluster of four neutrons may be stable enough to be considered an atom in their own right. This would create an element with no protons and no charge on its nucleus.

As early as 1926 Professor Andreas von Antropoff coined the term neutronium for a conjectured form of matter made up of neutrons with no protons, which he placed as the chemical element of atomic number zero at the head of his new version of the periodic table. It was subsequently placed as a noble gas in the middle of several spiral representations of the periodic system for classifying the chemical elements. It is at the centre of the Chemical Galaxy (2005).

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 languages, 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.

A second 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:

a + s \times (i-1) \,\!

where s is the size of each element. In contrast, if the index numbers count from 0, the expression becomes this:

a + s \times i \,\!

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 $a'=a-s$ ; 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:

a' + s \times i \,\!

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.

A third reason is that ranges are more elegantly expressed as the half-open interval, $n)$ , as opposed to the closed interval, $[1, n$ , because empty ranges often occur as input to algorithms (which would be tricky to express with the closed interval without resorting to obtuse conventions like $0$ ). On the other hand, closed intervals occur in mathematics because it's often necessary to calculate the terminating condition (which would be impossible in some cases because the half-open interval isn't always a closed set) which would have a subtraction by 1 everywhere.

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.

(Note that on most common architectures, the null pointer is represented internally by the integer 0, so C compilers on such systems perform no actual conversion).

Negative zero

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 zeros 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, 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.

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 Deutsches Kfz-Kennzeichen für Behördenfahrzeuge (Nummernbereich 3).jpg). The typeface chosen is called Bild:FE-Buchstaben.png (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"
In classical music, 0 is very rarely used as a number for a composition, the only two examples on the fringes of the standard repertoire probably being Anton Bruckner's Symphony No. 0 in D minor and Alfred Schnittke's Symphony No. 0
In tarot, card No. 0 is "the Fool"

Historical years

B.C. 1000, 2000, etc.; A.D. 1000, 2000, etc.

References

Charles Seife. Zero: The Biography of a Dangerous Idea. Publisher: Penguin USA (Paper), 2000. ISBN 0140296476.
John D. Barrow. The Book of Nothing. Vintage (July, 2001). ISBN 0-09-928845-1.

External links

Elementary arithmetic | Famous numbers | Integers

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