# Zero: Wikis

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# Encyclopedia

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0 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, 0 is used as a placeholder in place value systems. In the English language, 0 may be called zero, oh, null, nil, "o" or nought, dependent on dialect and context.[1]

## As number

0 is the integer preceding 1. In most systems, 0 was identified before the idea of negative things that go lower than zero was accepted. Zero is an even number.[2] 0 is neither positive nor negative. By some definitions 0 is also a natural number, and then the only natural number not to be positive.

Zero is a number which quantifies a count or an amount of null size. 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.

## As digit

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

On the seven-segment displays of calculators, watches, and household appliances, 0 is usually written with six line segments, though on some historical calculator models it was written with four line segments.

The value, or number, zero 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, for example, in the number 02.

In rare instances, a leading 0 may distinguish a number. This appears in roulette in the United States, where '00' is distinct from '0' (a wager on '0' will not win if the ball lands in '00', and vice versa). Sports where competitors are numbered follow this as well; a stock car numbered '07' would be considered distinct from one numbered '7'. This is most common with single-digit numbers.

### Distinguishing the digit 0 from the letter O

Traditionally, many print typefaces made the capital letter O more rounded than narrower, elliptical digit 0.[3] Typewriters originally made no distinction in shape between O and 0; some models did not even have a separate key for the digit 0. The distinction came into prominence on modern character displays.[3]

The digit 0 with a dot in the centre seems to have originated as an option on IBM 3270 displays. Its appearance has continued with the Microsoft Windows typeface Andalé Mono. One variation used a short vertical bar instead of the dot. 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 and very little used anyway.

An alternative, the slashed zero (looking similar to the letter O except for 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 $\emptyset$, or "∅" (Unicode character U+2205), representing the empty set, as well as to the letter Ø used in several Scandinavian languages. Some Burroughs/Unisys equipment displays a digit 0 with a reversed slash.

The opposing convention that has the letter O with a slash and the digit 0 without was advocated by SHARE, a prominent IBM user group,[3] and recommended by IBM for writing FORTRAN programs,[4] and by a few other early mainframe makers; this is even more problematic for Scandinavians because it means two of their letters collide. Others advocated the opposite convention,[3] including IBM for writing Algol programs.[4] Another convention used on some early line printers left digit 0 unornamented but added a tail or hook to the capital O so that it resembled an inverted Q or cursive capital letter-O ($\mathcal O$).[3]

Some fonts designed for use with computers made one of the capital-O–digit-0 pair more rounded and the other more angular (closer to a rectangle). The Texas Instruments TI-99/4A computer featured a more angular capital O and a more rounded digit 0, whereas others made the choice the other way around.

German license plate with slit zeros

The typeface used on most European vehicle registration plates distinguishes the two symbols partially in this manner (having a more rectangular or wider shape for the capital O than the digit 0), but in several countries a further distinction is made by slitting open the digit 0 on the upper right side (as in German plates using the fälschungserschwerende Schrift, "harder-to-falsify script").

Sometimes the digit 0 is used either exclusively, or not at all, to avoid confusion altogether. For example, confirmation numbers used by Southwest Airlines use only the capital letters O and I instead of the digits 0 and 1, while Canadian postal codes use only the digits 1 and 0 and never the capital letters O and I, although letters and numbers always alternate.

## Names

In 976 Muhammad ibn Musa al-Khwarizmi, in his Keys of the Sciences, remarked that if, in a calculation, no number appears in the place of tens, a little circle should be used "to keep the rows." This circle the Arabs called sifr.[5]

The word "zero" came via French zéro from Venetian zero, which (together with cipher) came via Italian zefiro from Arabic صفر, ṣafira = "it was empty", ṣifr = "zero", "nothing".[6]

Italian zefiro already meant "west wind" from Latin and Greek zephyrus; this may have influenced the spelling when transcribing Arabic ṣifr.[7] The Italian mathematician Fibonacci (c.1170-1250), who grew up in Arab North Africa and is credited with introducing the decimal system to Europe, used the term zephyrum. This became zefiro in Italian, which was contracted to zero in Venetian.

As the decimal zero and its new mathematics spread from the Arab world to Europe in the Middle Ages, words derived from ṣifr 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"."[7] From ṣifr also came French chiffre = "digit", "figure", "number", chiffrer = "to calculate or compute", chiffré = "encrypted". Today, the word in Arabic is still ṣifr, and cognates of ṣifr are common in the languages of Europe and southwest Asia.

## History

### Early history

By the middle of the 2nd millennium BC, the Babylonian mathematics 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 about 700 BC), the scribe Bêl-bân-aplu wrote his zeros with three hooks, rather than two slanted wedges.[8]

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), 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.

The concept of zero as a number and not merely a symbol for separation is attributed to India where by the 9th century AD practical calculations were carried out using zero, which was treated like any other number, even in case of division.[9][10] The Indian scholar Pingala (circa 5th-2nd century BC) used binary numbers in the form of short and long syllables (the latter equal in length to two short syllables), making it similar to Morse code.[11][12] He and his contemporary Indian scholars used the Sanskrit word śūnya to refer to zero or void.

### History of zero

The back of Olmec Stela C from Tres Zapotes, the second oldest Long Count date yet discovered. The numerals 7.16.6.16.18 translate to September, 32 BC (Julian). The glyphs surrounding the date are thought to be one of the few surviving examples of Epi-Olmec script.

The Mesoamerican Long Count calendar developed in south-central Mexico and Central America required the use of zero as a place-holder within its vigesimal (base-20) positional numeral system. Many different glyphs, including this partial quatrefoil—were 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.[13] Since the eight earliest Long Count dates appear outside the Maya homeland,[14] it is assumed that the use of zero in the Americas predated the Maya and was possibly the invention of the Olmecs. Many of the earliest Long Count dates were found within the Olmec heartland, although the Olmec civilization ended by the 4th century BC, several centuries before the earliest known Long Count dates.

Although zero became an integral part of Maya numerals, it did not influence Old World numeral systems.

Quipu, a knotted cord device, used in the Inca Empire and its predecessor societies in the Andean region to record accounting and other digital data, is encoded in a base ten positional system. Zero is represented by the absence of a knot in the appropriate position.

The use of a blank on a counting board to represent 0 dated back in India to 4th century BC.[15]

In China, counting rods were used for calculation since the 4th century BC. Chinese mathematicians understood negative numbers and zero, though they had no symbol for the latter,[16] until the work of Song Dynasty mathematician Qin Jiushao in 1247 established a symbol for zero in China.[17] The Nine Chapters on the Mathematical Art, which was mainly composed in the 1st century AD, 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."[18]

By 130 AD, 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 Ptolemy's Syntaxis Mathematica (also known as the 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 the 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.[19]

The oldest known text to use a decimal place-value system, including a zero, is the Jain text from India entitled the Lokavibhâga, dated 458 AD. This text uses Sanskrit numeral words for the digits, with words such as the Sanskrit word for void for zero.[20] The first known use of special glyphs for the decimal digits that includes the indubitable appearance of a symbol for the digit zero, a small circle, appears on a stone inscription found at the Chaturbhuja Temple at Gwalior in India, dated 876 AD.[21][22] There are many documents on copper plates, with the same small o in them, dated back as far as the sixth century AD, but their authenticity may be doubted.[8]

The Hindu-Arabic numerals and the positional number system were introduced around 500 AD, and it was introduced by a Persian scientist, Al-Khwarizmi.[5] Al-Khwarizmi's book on arithmetic. This book synthesized Greek and Hindu knowledge and also contained his own fundamental contribution to mathematics and science including an explanation of the use of zero.

It was only centuries later, in the 12th century, that the Arabic numeral system was introduced to the Western world through Latin translations of his Arithmetic.

### Rules of Brahmagupta

The rules governing the use of zero appeared for the first time in Brahmagupta's book Brahmasputha Siddhanta (The Opening of the Universe),[23] 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 Brahmagupta:[23]

• 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 their absolute values 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 to this, whereas computers and calculators 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 5th century mention in the text Lokavibhaga. The glyph for the zero digit was written in the shape of a dot, and consequently called bindu ("dot"). The dot had been used in Greece during earlier ciphered numeral periods.

The Hindu-Arabic 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. For this reason, the numerals came to be known in Europe 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.[24][25]

Here Leonardo of Pisa uses the phrase "sign 0," indicating it is like a sign to do operations like addition or multiplication. 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 Johannes de Sacrobosco, about 1235 and was one of the earliest scientific books to be printed in 1488. Until the late 15th century, Hindu-Arabic numerals seem to have predominated among mathematicians, while merchants preferred to use the Roman numerals. In the 16th century, they became commonly used in Europe.

## In mathematics

### Elementary algebra

The number 0 is the least non-negative integer. The natural number following 0 is 1 and no natural number precedes 0. The number 0 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). Zero is, while a purely real number, also a pure imaginary number because it lies on both the real and imaginary axes on the complex plane.[1]

The number 0 is neither positive nor negative, neither a prime number nor a composite number, nor is it a unit. It is, however, even (see parity of zero).

The following are some basic (elementary) rules for dealing with the number 0. 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: 0x = 0, for nonzero x. But x0 is undefined, because 0 has no multiplicative inverse, a consequence of the previous rule; see division by zero.
• Exponentiation: x0 = x/x = 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, 0x = 0.

The expression 00, which may be obtained in an attempt to determine the limit of an expression of the form f(x)g(x) as a result of applying the lim operator independently to both operands of the fraction, is a so-called "indeterminate form". That does not simply mean that the limit sought is necessarily undefined; rather, it means that the limit of f(x)g(x), if it exists, must be found by another method, such as l'Hôpital's rule.

The sum of 0 numbers is 0, and the product of 0 numbers is 1. The factorial 0! evaluates to 1.

## 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. In physics, the zero-point energy is the lowest possible energy that a quantum mechanical physical system may possess and is the energy of the ground state of the system.

### 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 its 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 the vast majority of programming languages, the elements of an array are numbered starting from 0 by default. The popularity of the C programming language in the 1980s has made this approach common.

One advantage of this convention is in the use of modular arithmetic. Every integer is congruent modulo N to one of the numbers 0, 1, 2, ..., N − 1, where N ≥ 1. Because of this, many arithmetic concepts (such as hash tables) are more elegantly expressed in code when the array starts at zero.

A second advantage of 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:

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

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′ = as; 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 advantage 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 is 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 nth element. For this reason, the first element is often referred to as the zeroth element to avoid confusion.

### 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).

(Note that on most common architectures, the null pointer is represented internally the same way an integer of the same byte width having a value of zero is represented, so C compilers on such systems perform no actual conversion.)

### Negative zero

In mathematics − 0 = 0 = + 0, both −0 and +0 represent the exact same number, i.e., there is no “negative zero” distinct from zero. In some signed number representations (but not the two's complement representation used to represent integers in most computers 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.

## In other fields

• In some countries and some company phone networks, 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.
• 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.
• In Formula One, if the reigning World Champion no longer competes in Formula One in the year following their victory in the title race, 0 is given to one of the drivers of the team that the reigning champion won the title with. This happened in 1993 and 1994, with Damon Hill driving car 0, due to the reigning World Champion (Nigel Mansell and Alain Prost respectively) not competing in the championship.
• In the educational series Schoolhouse Rock!, the song My Hero, Zero is about the use of zero as a placeholder. The song explains that by appending zeros to a number, it is multiplied by 10 for each one added. This enables mathematicians to create numbers as large as needed.

## 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, 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

## Notes

1. ^ Catherine Soanes, ed (2001) (Hardback). The Oxford Dictionary, Thesaurus and Wordpower Guide. Maurice Waite, Sara Hawker (2nd ed.). New York, United States: Oxford University Press. ISBN 978-0-19-860373-3.
2. ^ Lemma B.2.2, The integer 0 is even and is not odd, in Penner, Robert C. (1999). Discrete Mathematics: Proof Techniques and Mathematical Structures. World Scientific. pp. 34. ISBN 9810240880.
3. ^ a b c d e R. W. Bemer. "Towards standards for handwritten zero and oh: much ado about nothing (and a letter), or a partial dossier on distinguishing between handwritten zero and oh". Communications of the ACM, Volume 10, Issue 8 (August 1967), pp. 513–518.
4. ^ a b Bo Einarsson and Yurij Shokin. Fortran 90 for the Fortran 77 Programmer. Appendix 7: "The historical development of Fortran"
5. ^ a b Will Durant, 'The Story of Civilization', Volume 4, The Age of Faith, pp. 241.
6. ^ Merriam Webster online Dictionary
7. ^ a b Georges Ifrah. The Universal History of Numbers: From Prehistory to the Invention of the Computer. Wiley (2000). ISBN 0-471-39340-1.
8. ^ a b Kaplan, Robert. (2000). The Nothing That Is: A Natural History of Zero. Oxford: Oxford University Press.
9. ^ Bourbaki, Nicolas (1998). Elements of the History of Mathematics. Berlin, Heidelberg, and New York: Springer-Verlag. 46. ISBN 3540647678.
10. ^ Britannica Concise Encyclopedia (2007), entry algebra
11. ^ Binary Numbers in Ancient India
12. ^ Math for Poets and Drummers (pdf, 145KB)
13. ^ No long count date actually using the number 0 has been found before the 3rd century AD, but since the long count system would make no sense without some placeholder, and since Mesoamerican glyphs do not typically leave empty spaces, these earlier dates are taken as indirect evidence that the concept of 0 already existed at the time.
14. ^ Diehl, p. 186
15. ^ Robert Temple, The Genius of China, A place for zero; ISBN 1-85375-292-4
16. ^ Wáng, Qīngxiáng (1999), Sangi o koeta otoko (The man who exceeded counting rods), Tokyo: Tōyō Shoten, ISBN 4-88595-226-3
17. ^ Needham, Joseph (1986). Science and Civilization in China: Volume 3, Mathematics and the Sciences of the Heavens and the Earth. Taipei: Caves Books, Ltd. Page 43.
18. ^ The statement in Chinese, found in Chapter 8 of The Nine Chapters on the Mathematical Art is 正負術曰: 同名相除，異名相益，正無入負之，負無入正之。其異名相除，同名相益，正無入正之，負無入負之。The word 無入 used here, for which zero is the standard translation by mathematical historians, literally means: no entry. The full Chinese text can be found at wikisource:zh:九章算術.
19. ^ Aryabhatiya of Aryabhata, translated by Walter Eugene Clark.
20. ^ Ifrah, Georges (2000), p. 416.
21. ^ Feature Column from the AMS
22. ^ Ifrah, Georges (2000), p. 400.
23. ^ a b Algebra with Arithmetic of Brahmagupta and Bhaskara, translated to English by Henry Thomas Colebrooke, London1817
24. ^ Sigler, L., Fibonacci’s Liber Abaci. English translation, Springer, 2003.
25. ^ Grimm, R.E., "The Autobiography of Leonardo Pisano", Fibonacci Quarterly 11/1 (February 1973), pp. 99-104.

## References

• Barrow, John D. (2001) The Book of Nothing, Vintage. ISBN 0-09-928845-1.
• Diehl, Richard A. (2004) The Olmecs: America's First Civilization, Thames & Hudson, London.
• Ifrah, Georges (2000) The Universal History of Numbers: From Prehistory to the Invention of the Computer, Wiley. ISBN 0-471-39340-1.
• Kaplan, Robert (2000) The Nothing That Is: A Natural History of Zero, Oxford: Oxford University Press.
• Seife, Charles (2000) Zero: The Biography of a Dangerous Idea, Penguin USA (Paper). ISBN 0-14-029647-6.
• Bourbaki, Nicolas (1998). Elements of the History of Mathematics. Berlin, Heidelberg, and New York: Springer-Verlag. ISBN 3540647678.
• Isaac Asimov article "nothing counts" in "Asimov on Numbers" Pocket Books, 1978

# Travel guide

Up to date as of January 14, 2010

### From Wikitravel

There is more than one place called Zero:

### United States

• Zero (Iowa) - A town in the state of Iowa.
• Zero (Mississippi) - A town in the state of Mississippi.
• Zero (Montana) - A town in the state of Montana.

# 1911 encyclopedia

Up to date as of January 14, 2010

### From LoveToKnow 1911

ZERO, the figure 0 in the Arabic notation for numbers, nought, cipher. The Arabic name for the figure was sifr, which meant literally an empty thing. The old Latin writers on arithmetic translated or transliterated the Arabic word as zephyrum; this in Ital. became zefiro, contracted to zero, borrowed by F. zero, whence it came late into English. The Spanish form cifra, more closely resembling the original Arabic, gave O. Fr. cifre, mod. chiffre, also used in the sense of monogram, and English "cipher" which is thus a doublet. In physics, the term is applied to a point with which phenomena are quantitatively compared, especially to a point of a graduated instrument between a positive and negative or ascending and descending scale, as in the scales of temperature.

 << Zermatt

# Strategy wiki

Up to date as of January 23, 2010
(Redirected to Defender article)

### From StrategyWiki, the free strategy guide and walkthrough wiki

Defender
Developer(s) Williams
Publisher(s) Williams
Designer(s) Eugene Jarvis
Release date(s)
Genre(s) Shooter
System(s) Arcade, Apple II, Atari 8-bit, Atari 2600, Atari 5200, Intellivision, ColecoVision, Commodore 64/128, Commodore VIC-20, MS-DOS, TI-99/4A, Xbox Live Arcade, Entex Adventure Vision, GameTap
Players 1-2
"Zero" redirects here. For other uses, see Zero (disambiguation).

Defender is considered one of the most challenging arcade games ever made. One look at the controls was enough to send many players in search of a simpler game. But those who were brave enough to attempt it found a compelling and addictive game, with an incredibly rich level of strategy. Defender was the brain child of early (and continuing) game developer Eugene Jarvis. Eugene wished to make a video game that immersed the player more than the myriad of Space Invader clones that flooded the markets, and by many experts' accounts, he succeeded.

Defender featured a beautifully horizontal scrolling world, something that was a rarity in its day, and made Defender unique among space shooters. Because the "world" was larger than the screen, Defender also featured a scanner at the top of the screen to give players indications of what was taking place off screen. In addition to the space craft's primary laser weapon, it had a limited supply of smart bombs that could wipe out everything on the screen, and a hyperspace ability. The control scheme for the game matched the complexity of the abilities available to the ship.

The Defender's job is to protect the remnants of humanity from a marauding band of alien invaders who wish to abduct the human race. Landers swoop down to the planet terrain and tractor beam a human to them, and then rise in to the sky with it. A cry for help alerts the player to fly to its rescue as quickly as possible. Either the player shoots the Lander and rescues the human, or the Lander merges with the human at the top of the screen and becomes a deadly mutant. A grim fate awaits the player who fails to keep humanity safe.

The break-neck pace of the action made the game an easy choice for Atari to buy the home licensing rights. They brought a conversion to the 2600 that was generally well received, despite it's short comings. But the version created for the 8-bit family of Atari computers was lauded as one of the best. Defender was also one of four games produced for the Entex Adventure Vision, a stand alone system that accepted cartridges. In fact, it was the system's pack-in game. Atarisoft handled the conversion of the game to several other home console and computer platforms. A few years later, the sequel Stargate would be released, but due to licensing issues, it was commonly known as Defender II. Defender was also released for download on the Xbox Live Arcade in 2006.

## Story

The story is simple: You are the last line of defense against an invading legion of aliens who are determined to abduct all of the remaining humans to create an unstoppable army of hyrbid mutants. You pilot the only space craft capable of undoing their plans and protecting the fate of human life.

## Gameplay summary

• You pilot the spaceship Defender with the controls. The joystick controls your height. Use Thrust and Reverse to control your position over the world.
• You must destroy every enemy to advance to the next stage.
• You must defend the Humanoid population from being abducted by Landers.
• Your ship can fire a laser, launch a smart bomb that eliminates every visible enemy, or hyperspace.
• If a Lander abducts a Humanoid, fly over as quickly as possible, destroy the Lander, capture the Humanoid before it hits the ground, and safely return it.
• If a Lander makes it to the top of the screen with a Humanoid, it becomes a Mutant.
• If all of the Humanoids are capture, the world explodes and every stage is a Mutant stage until after the next fifth round.
• All Humanoids are restored after every fifth round.

# Gaming

Up to date as of February 01, 2010

### From Wikia Gaming, your source for walkthroughs, games, guides, and more!

Zero is the name of a character from the popular Mega Man X series. Zero is a blonde-haired, laser sabre-wielding, robot. He debuted in Mega Man X as the slick, somewhat stand-offish ally to the protagonist, Mega Man X. He has been dubbed as the Knuckles to X's Sonic. Zero had not been invented in the classic Mega Man series and is therefore not included in any of them.

Zero was later given his own series of games in the Game Boy Advance. These games, the Mega Man Zero series, are currently hovering around Mega Man Zero 4.

In the SNK fighting game SvC Chaos: SNK vs. Capcom, Zero appears as an unlockable character instead of Mega Man.

## Appearances

 This Mega Man-related article is a stub. You can help by adding to it. Stubs are articles that writers have begun work on, but are not yet complete enough to be considered finished articles.

# Simple English

Zero is a special number. If there are zero things, there are not any things. There are none. For example, if John has zero hats, that means he does not have a hat at all. The term "zeroes" also designates the values of functions at zero.

Order zeroth

## Symbol

The symbol for the number zero is "0".

## Arithmetic with Zero

Zero is a special number. That is because:

• Adding a number to zero results in that number. For example, adding zero to three gives three. In symbols:
3 + 0 = 3

• Subtracting zero from a number always gives that number. For example, subtracting zero from three gives three. In symbols:
3 − 0 = 3

• Subtracting a positive number from zero always makes that number negative. In symbols:
0 − 3 = −3

• Multiplying a number by zero always gives zero. For example, multiplying forty-three by zero gives zero. In symbols:
43 × 0 = 0

• Dividing zero by a number always gives zero. For example, dividing zero by forty three gives zero. In symbols:
0 ÷ 43 = 0

43 ÷ 0 has no answer.

• Zero Divided by zero has no answer. In symbols:
0 ÷ 0 has no answer.


## History of zero

The idea of zero was first thought about in Babylon, India and in Central America at different times. Some places and countries did not know about a zero, which may have made it harder for those people to do mathematics.

Over hundreds of years the idea of zero was passed from country to country. From India and Babylon to other places, like Greece, Persia and the Arab parts of the world. The Europeans learned about zero from the Arabs.

## The place of zero as a number

Zero is almost never used as a place number (ordinal number). This means that it is not used like 1, 2, or 3 to indicate the order, or place, of something, like 1st, 2nd, or 3rd.

Any number divided by itself equals one, except if that number is zero. In symbols:

0 ÷ 0 = "not a number."

In time, zero means "now". For example, a person is counting down the time to the start of something, such as a foot race or when a rocket takes off. The count is: "three, two, one, zero (or go)". Zero is the exact time of the start of the race or when the rocket takes off into the sky.

## 0 as a number

 Cardinal 0 zero o/oh nought naught nil null Ordinal 0th zeroth Factorization $0$ Divisors N/A Roman numeral N/A Japanese numeral 〇 Binary 0 Octal 0 Duodecimal 0 Hexadecimal 0

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 means 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 say 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 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. The four-segment 0 is not common.

It is important to know that the number zero (as in the "zero brothers" example above) is not the same as 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 numerical digit zero

0 (zero) is also used as a numerical digit used to represent that number in numerals. It is used to hold the place of that digit, because correct placing of digits affects a numeral's value.

Examples:

• In the numeral 10, which stands for one times ten and zero units (or ones).
• In the numeral 100, which stands for one times a hundred plus zero tens plus zero units.

## Telling Zero and the Letter O Apart

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 begun as a choice on IBM 3270 controllers (this has the problem that it looks like the Greek letter Theta). The slashed zero, looking like the letter O with a diagonal line drawn inside it, is used in old-style ASCII graphic sets that came from the default typewheel on the well known ASR-33 Teletype. This format causes problems because of it looks like the symbol $\emptyset$, representing the empty set, as well as for certain Scandinavian languages which use Ø as a letter.

The rule 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 of a problem for Scandinavians because it looks like two of their letters at the same time. Some Burroughs/Unisys computers display a zero with a backwards slash. And yet another convention common on early line printers left zero without any extra dots or slashes but added a tail or hook to the letter-O so that it resembled an inverted Q or cursive capital letter-O.

The letters 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 cutting open the zero on the upper right side, so the circle is not closed any more (as in German plates). The style of letters chosen is called [[:Image:|fälschungserschwerende Schrift]] (abbr.: FE Schrift), meaning "script which is harder to falsify". But those used in the United Kingdom do not make the letter o and the number 0 look different from each other because there can never be any mistake if the letters are correctly spaced.

In paper writing you do not have to make the 0 and O look different at all, or you may add a slash across the zero in order to show the difference, although this sometimes causes mistakes in the number 0.

## References

• Barrow, John D. (2001) The Book of Nothing, Vintage. ISBN 0-09-928845-1.
• Diehl, Richard A. (2004) The Olmecs: America's First Civilization, Thames & Hudson, London.
• Ifrah, Georges (2000) The Universal History of Numbers: From Prehistory to the Invention of the Computer, Wiley. ISBN 0-471-39340-1.
• Kaplan, Robert (2000) The Nothing That Is: A Natural History of Zero, Oxford: Oxford University Press.
• Seife, Charles (2000) Zero: The Biography of a Dangerous Idea, Penguin USA (Paper). ISBN 0-14-029647-6.