0 (number)

2008/9 Schools Wikipedia Selection. Related subjects: Mathematics

0
0 1 2 3 4 5 6 7 8 9 → List of numbers — Integers 0 10 20 30 40 50 60 70 80 90 →
Cardinal	0, zero, "oh" (IPA:  [oʊ]), nought, naught, nil, null
Ordinal	0th, zeroth
Factorization	0
Divisors	all numbers
Roman numeral	N/A
Arabic	٠ and 0
Bengali	০
Devanāgarī	०
Chinese	零
Japanese numeral	〇
Khmer	០
Thai	๐
Binary	0
Octal	0
Duodecimal	0
Hexadecimal	0

0 (zero) is both a number and the 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. In the English language, zero may also be called null or nil when a number, "oh" (IPA:  [oʊ]) or cipher (archaic) when a numeral, and nought or naught in either context.

0 as a number

0 is the integer between 1 and −1. In most systems, 0 was identified before the idea of 'negative integers' was accepted. Zero is an even number. 0 is neither positive nor negative.

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 accept zero as a number, some non-mathematicians would say that zero is not a number, arguing that one cannot have zero of something (for example, 'zero oranges'). Others hold that if one has a bank balance of zero, one has a specific quantity of money in that account, namely none.

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 serves as a new base point in time.

0 as a digit

The modern numerical digit 0 is usually written as a circle, an ellipse, or a rounded rectangle. While the height of the 0 character is the same as the other digits in most modern typefaces, in typefaces with text figures the character is often less tall ( x-height).

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. The latter is less common than the former.

The value, or number, zero (as in the "zero brothers" example above) is not the same as the digit zero, used in numeral systems using positional notation. Successive positions of digits have higher weights, so inside a numeral the digit zero is used to skip a position and give appropriate weights to the preceding and following digits. A zero digit is not always necessary in a positional number system: numeration provides a possible counterexample.

Distinguishing zero from O

The oval (i. e. narrower) zero and more nearly circular letter O together came into prominence on modern character displays, though the distinction had already been present in some print typefaces. The zero with a dot in the centre seems to have originated as an option on IBM 3270 displays - a variation used a short vertical bar instead of the dot; in theory this could be confused with the Greek letter Theta on a badly focused display, but in practice there was no confusion because theta was not (then) a displayable character. An alternative, the slashed zero (looking similar to the letter O other than the slash), was primarily used in hand-written coding sheets before transcription to punched cards or tape, and is also used in old-style ASCII graphic sets descended from the default typewheel on the ASR-33 Teletype. This form is similar to the symbol , representing the empty set, as well as to the letter Ø used in several Scandinavian languages.

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. Another convention used on some 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 ().

Some fonts, especially on screens, made the zero more angular (i. e. more closely resembling a rectangle), and the letter O more rounded. The typeface used on some European number plates for cars distinguishes the two symbols partially in this manner, but mostly by slitting open the zero on the upper right side, so that 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 the typefaces used on United Kingdom numberplates do not differentiate between the two as there can never be any ambiguity if the design is correctly spaced (the vehicle "numbers" are allocated in a manner that avoids any such confusion); the same applies to UK postcodes.

Sometimes the number zero is used either exclusively, or not at all, to avoid confusion altogether. For example, confirmation numbers used by Southwest Airlines use only the letters O and I instead of 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. 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"." (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.

History

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. In a tablet unearthed at Kish (dating from perhaps as far back as 700 BC), the scribe Bêl-bân-aplu wrote his zeroes with three hooks, rather than two slanted wedges.

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), et al., 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 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.

Early use of something like zero by the Indian scholar Pingala (circa 5th- 2nd 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), making it similar to Morse code. Nevertheless, he and other Indian scholars at the time used the Sanskrit word śūnya (the origin of the word zero after a series of transliterations and a literal translation) to refer to zero or void.

History of zero

The use of a blank on a counting board to represent 0 dated back in India to 4th century BC. 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. Many different glyphs, including this partial quatrefoil——were 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. Since the eight earliest Long Count dates appear outside the Maya homeland, 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 BC, 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.

In China, counting rods were used for calculation since the 4th century BCE and Chinese mathematicians understood negative numbers and zero, though they had no symbol for the latter. The Nine Chapters on the Mathematical Art, which was mainly composed in the 1st century CE, stated "[when subtracting] subtract same signed numbers, add differently signed numbers, subtract a positive number from zero to make a negative number, and subtract a negative number from zero to make a positive number."

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.

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, 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.

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:

• 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 done

Zero as a decimal digit

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 bindu ("dot").

The Indian numeral system (base 10) reached Europe in the 11th century, via the Iberian Peninsula through Spanish Muslims the Moors, together with knowledge of astronomy and instruments like the astrolabe, first imported by Gerbert of Aurillac. So in Europe 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.

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 in 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 least non-negative integer. The natural number following zero is one and no natural number precedes zero. Zero may or may not be considered a natural number, but it is a whole number and hence a rational number and a real number (as well as an algebraic number and a complex number).

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. It is, however, even (see evenness of zero). 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 (elementary) rules for dealing with the number zero. These rules apply for any real or complex number x, unless otherwise stated.

• Addition: x + 0 = 0 + x = x. That is, 0 is an identity element (or neutral 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. It is also said that ^x/₀ equals unsigned infinity, see division by zero.
• Exponentiation: x⁰ = 1, except that the case x = 0 may be left undefined in some contexts; see Zero to the zero power. 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 the limit of ^f(x)/_g(x) is determined by the particular functions f and g as they both approach 0. As x approaches some number, the limit may approach any finite number, 0, ∞, or −∞, depending on the specific behaviour of the functions. See l'Hôpital'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.
• 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 the probability that you will get is zero.
• 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 of the form x² or x²y (for x > 1, x and y are both integers), the Möbius function returns zero.
• Zero is the first Perrin number.

In science

Physics

The value zero plays a special role for many physical quantities. For some quantities, the zero level is naturally distinguished from all other levels, whereas for others it 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), whereas 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—for example, at a value for the threshold of hearing. See also Zero-point energy.

Chemistry

Zero has been proposed as the atomic number of the theoretical element tetraneutron. 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?...

The most common practice throughout human history has been to start counting at one. Nevertheless, in computer science zero is often used as the 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, the "first" index in the array being 0. In particular, the popularity of the C programming language in the 1980s 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 indexes is that this 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 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:

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, [0,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 [1,0]). 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". Therefore, an analogy from the ordinal numbers to the quantity of objects numbered appears; the highest index of n objects will be (n-1) and referred to the n:th element. 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

Because of − 0 = 0 = + 0, both -0 and +0 represent the exact same number in mathematics and there is no “negative zero” distinct from zero. But 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. However, they are necessary to achieve numerical accuracy in many critical problems, as illustrated in Prof. W. Kahan's article "Branch Cuts for Complex Elementary Functions, or Much Ado About Nothing's Sign Bit" in The State of the Art in Numerical Analysis, (eds. Iserles and Powell), Clarendon Press, Oxford, 1987. In Mathematical Analysis the difference between -0 and +0 is greater: if you divide a positive number to a very small positive number (such as +0.00000001) you will get a really big number as a result. The smaller the dividend is, the bigger the result. But if you divide the same number to a very small negative number (such as -0.00000001) you will get a really big negative number. If these two very small positive, respectively negative numbers are further pushed towards zero, but not actually reaching it (+0 and -0), the result will be +infinite respectively -infinite which is quite a big difference.

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: Anton Bruckner wrote a Symphony No. 0 in D minor and a Symphony No. 00; Alfred Schnittke also wrote a Symphony No. 0.
• In tarot, card No. 0 is the Fool
• Roulette wheels usually feature a "0" space (and sometimes also a "00" space), whose presence is ignored when calculating payoffs (thereby allowing the house to win in the long run).
• A chronological prequel of a series may be numbered as 0.

Quotations

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

Dividing by zero...allows you to prove, mathematically, anything in the universe. You can prove that 1+1=42, and from there you can prove that J. Edgar Hoover is a space alien, that William Shakespeare came from Uzbekistan, or even that the sky is polka-dotted. (See appendix A for a proof that Winston Churchill was a carrot.) Charles Seife, from: Zero: The Biography of a Dangerous Idea

...a profound and important idea which appears so simple to us now that we ignore its true merit. But its very simplicity and the great ease which it lent to all computations put our arithmetic in the first rank of useful inventions. Pierre-Simon Laplace

The point about zero is that we do not need to use it in the operations of daily life. No one goes out to buy zero fish. It is in a way the most civilized of all the cardinals, and its use is only forced on us by the needs of cultivated modes of thought. Alfred North Whitehead

...a fine and wonderful refuge of the divine spirit--almost an amphibian between being and non-being. Gottfried Leibniz