0 (zero), alternatively called naught, nil, nada, ought, zilch, zip, nothing or nought, is both a number and a numeral. It was the last numeral to be created in most numerical systems, as it is not a counting number (which is to say, one begins counting at the number 1) and was in many eras and places represented only by a gap or mark very different from the other numerals. In linguistics, cardinal numbers is the name given to number words that are used for quantity (one, two, three), as opposed to ordinal numbers, words that are used for order (first, second, third). ... Commonly, ordinal numbers, or ordinals for short, are numbers used to denote the position in an ordered sequence: first, second, third, fourth, etc. ... The zeroth item is the initial item of a sequence, if that sequence is numbered beginning from zero rather than one. ... In mathematics, factorization or factoring is the decomposition of an object (for example, a number, a polynomial, or a matrix) into a product of other objects, or factors, which when multiplied together give the original. ... In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. ... The system of Roman numerals is a numeral system originating in ancient Rome, and was adapted from Etruscan numerals. ... The binary numeral system (base 2 numerals) represents numeric values using two symbols, typically 0 and 1. ... The octal numeral system is the base-8 number system, and uses the digits 0 to 7. ... A duodecimal multiplication table The duodecimal (also known as base-twelve or dozenal) system is a numeral system using twelve as its base. ... In mathematics and computer science, hexadecimal, or simply hex, is a numeral system with a radix or base of 16 usually written using the symbols 0–9 and A–F or a–f. ... Number is the current mathematics collaboration of the week! Please help improve it to featured article standard. ... A numeral is a symbol or group of symbols that represents a number. ...

0 as a number

0 is the integer that precedes the positive 1, and follows -1. Zero first appeared as a number in Brahmagupta's work dated to 628. Prior to that Babylonians used a space marker that played one of the functions of zero. Babylonians did not have a special symbol for zero. In most (if not all) numerical systems, 0 was identified before the idea of 'negative integers' was accepted. The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ... Look up one in Wiktionary, the free dictionary. ... In mathematics, −1 is the integer greater than negative two (−2) and less than 0. ... Brahmagupta (ब्रह्मगुप्त) (598_668) was an Indian mathematician and astronomer. ... Events Khusro II of Persia overthrown Pippin of Landen becomes Mayor of the Palace Brahmagupta writes the Brahmasphutasiddhanta Births Deaths Empress Suiko of Japan Theodelinda, queen of the Lombards Categories: 628 ...

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. KK Null, a Japanese musician Null, a special value in computer programming. ...

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

Almost all historians omit the year zero from the proleptic Gregorian and Julian calendars, but astronomers include it in these same calendars. However, the phrase Year Zero may be used to describe any event considered so significant that it virtually starts a new time reckoning. A historian is a person who studies history. ... A year zero does not exist in the Christian Era and thus also does not currently exist in the calculation of times in most cultures. ... The proleptic Gregorian calendar is produced by extending the Gregorian Calendar to dates preceding its official introduction in 1582. ... The proleptic Julian calendar is produced by extending the Julian calendar to dates preceding its official introduction in 45 BC. Historians since Bede have traditionally represented the years preceding AD 1 as 1 BC, 2 BC, etc. ... An astronomer or astrophysicist is a scientist whose area of research is astronomy or astrophysics. ... The term Year Zero, applied to the takeover of Cambodia in 1975 by the Khmer Rouge, is an analogy to the Year One of the French Revolutionary Calendar. ...

0 as a numeral

The modern numeral 0 is normally written as a circle or (rounded) rectangle. On the seven-segment displays of calculators, watches, etc., 0 is usually written with six line segments (at right), though on some historical calculator models it was written with four line segments. This variant glyph has not caught on. Early Europeans hesitated to consider zero as a numeral. Leonardo of Pisa or Fibonacci says the following in 1,202 AD when the Indian number system arrived Europe. Download high resolution version (832x1564, 18 KB) I, the creator of this image, hereby release it into the public domain. ... A seven-segment display (sometimes written as 7-segment display) is a form of display that predates the now ubiquitous dot-matrix displays. ...

“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, 2003 and Grimm 1973 see refernces).

Here Leonardo of Pisa uses the word sign "0", indicating it is like a sign to do operations like addition or multiplication. He did not recognize zero as a number on its own right.

It is important to distinguish the number zero (as in the "zero brothers" example above) from the numeral or digit zero, used in numeral systems using positional notation. Successive positions of digits have higher values, so the digit zero is used to skip a position and give appropriate value to the preceding and following digits. The Babylonian numeral system used two narrow slanting wedges, similar to //, for the equivalent of a positional zero numeral starting in about 400BC. A numeral is a symbol or group of symbols that represents a number. ... Positional notation is a system in which each position has a value represented by a unique symbol or character. ... Babylonian numerals were written in cuneiform, using a wedge-tipped reed stylus to make a mark on a soft clay tablet which would be exposed in the sun to harden to create a permanent record. ... (5th century BC - 4th century BC - 3rd century BC - other centuries) (2nd millennium BC - 1st millennium BC - 1st millennium AD) // Events Invasion of the Celts into Ireland Battle of the Allia and subsequent Gaulish sack of Rome 383 BCE Second Buddhist Councel at Vesali. ...

A zero digit is not always necessary in a positional number system: decimal without a zero provides a possible counterexample. Decimal without a zero is a base ten positional numeral system which does not use a digit to represent zero; it instead has a digit to represent ten, such as X. As with conventional decimal, each digit position represents a power of ten, so for example 123 is one hundred...

In fonts with text figures, 0 is usually the same height as an uppercase X, for example, . Hoefler Text, a contemporary font, uses text figures. ... Image File history File links The numerals 0, 3 and 6 written in text figures. ...

Etymology

The word zero comes ultimately from the Arabic ṣifr (صفر) meaning empty or vacant, a literal translation of the Indian Sanskrit śūnya meaning void or empty. Through transliteration this became zephyr or zephyrus in Latin. The word zephyrus already meant "west wind" in Latin; the proper noun Zephyrus was the Roman god of the west wind (after the Greek god Zephyros). With its new use for the concept of zero, zephyr came to mean a light breeze—"an almost nothing" (Ifrah 2000; see References). The word zephyr survives with this meaning in English today. The Italian mathematician Fibonacci (c.1170-1250), who grew up in Arab North Africa and is credited with introducing the Hindu decimal system to Europe, used the term zephyrum. This became zefiro in Italian, which was contracted to zero in the Venetian dialect, giving the modern English word. Arabic can mean: From or related to Arabia From or related to the Arabs The Arabic language; see also Arabic grammar The Arabic alphabet, used for expressing the languages of Arabic, Persian, Malay ( Jawi), Kurdish, Panjabi, Pashto, Sindhi and Urdu, among others. ... Sanskrit ( संस्कृतम्) is an Indo-European Classical language of India and a liturgical language of Hinduism, Buddhism, and Jainism. ... Latin is an ancient Indo-European language originally spoken in the region around Rome called Latium. ... Zephyr and Hyakinth; Attic red figure cup from Tarquinia, circa 480 BCE. Boston Museum of Fine Arts. ... The English language is a West Germanic language that originates in England. ... Portrait of Fibonacci, probably not authentic Leonardo of Pisa or Leonardo Pisano (Pisa, c. ...

As the decimal zero and its new mathematics spread through a Europe that was still in the Middle Ages, words derived from sifr and zephyrus came to refer to calculation, as well as to privileged knowledge and secret codes. According to Ifrah (2000), "in thirteenth-century Paris, a 'worthless fellow' was called a... cifre en algorisme, i.e., an 'arithmetical nothing.' " (Algorithm is also a borrowing from the Arabic, in this case from the name of the 9th century mathematician 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. The Middle Ages formed the middle period in a traditional schematic division of European history into three ages: the classical civilization of Antiquity, the Middle Ages, and modern times, beginning with the Renaissance. ... Flowcharts are often used to represent algorithms. ... As a means of recording the passage of time the 9th century was that century that lasted from 801 to 900. ... Soviet postage stamp commemorating the 1200th anniversary of Muhammad al‑Khwarizmi in 1983. ... This article is about algorithms for encryption and decryption. ...

• French: zéro, zero
• German: Ziffer, digit, figure, numeral, cypher
• Hindi: Shunya, numeral, sum, digit
• Italian: cifra, digit, numeral, cypher; zero, zero
• Polish: cyfra, digit; szyfrować, to encrypt
• Portuguese: zero
• Spanish: cifra, figure, numeral, cypher, code; cero, zero
• Swedish: siffra, numeral, sum, digit

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

History

Prehistory of zero

By the mid 2nd millennium BC, the Babylonians had a sophisticated sexagesimal positional numeral system. The lack of a positional value (or zero) was indicated by a space between sexagesimal numerals. By 300 BC a punctuation symbol (two slanted wedges) was co-opted as a placeholder in the same Babylonian system. However, "... a tablet found at Kish ... thought to date from around 700 BC, uses three hooks to denote an empty place in the positional notation. Other tablets dated from around the same time use a single hook for an empty place" ([1] and natural number). (3rd millennium BC – 2nd millennium BC – 1st millennium BC – other millennia) // Events To grasp the spirit of the 2nd millennium BC, we must divide it in two parts, for there is a period of change around its middle so important that it creates two separate sub-millennia. First half (2000... Babylonian mathematics refers to any mathematics of the peoples of Mesopotamia, from the days of the early Sumerians to the fall of Babylon in 539 BC. In contrast to the sparcity of sources in Egyptian mathematics, our knowledge of Babylonian mathematics is derived from some 400 clay tablets unearthed since... The sexagesimal (base-sixty) is a numeral system with sixty as the base. ... Centuries: 4th century BC - 3rd century BC - 2nd century BC Decades: 350s BC 340s BC 330s BC 320s BC 310s BC - 300s BC - 290s BC 280s BC 270s BC 260s BC 250s BC Years: 305 BC 304 BC 303 BC 302 BC 301 BC - 300 BC - 299 BC 298 BC... For the World of Warcraft ex-NPC, see Captain Placeholder. ... Babylonian numerals were written in cuneiform, using a wedge-tipped reed stylus to make a mark on a soft clay tablet which would be exposed in the sun to harden to create a permanent record. ... Kish [kish] (Tall al-Uhaymir) was an ancient city of Sumer, now in central Iraq. ... Centuries: 9th century BC - 8th century BC - 7th century BC Decades: 750s BC 740s BC 730s BC 720s BC 710s BC - 700s BC - 690s BC 680s BC 670s BC 660s BC 650s BC Events and Trends 708 BC - Spartan immigrants found Taras (Tarentum, the modern Taranto) colony in southern Italy. ... In mathematics, a natural number is either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). The former definition is generally used in number theory, while the latter is preferred in set theory. ...

However, the Babylonian placeholder is not the same as a true number zero, considered as a quantity, and the Babylonian system of digits is not quite the same as a true base 60 system using a zero digit, since the Babylonians did not have a symbol for zero. The so-called Babylonian zero is a separation mark that came between two place value numbers. Babylonians did have a 60 based place value notation. Babyonians were not able to differentiate bewteen 120 and 2 or 3 and 180 or 4 and 240 etc. They simply could not differentiate between numbers that required a zero at the end. They simply did not have a zero. All what they had was a separation mark for numbers that separated different place value numbers from each other.

Records show that the ancient Greeks seemed unsure about the status of zero as a number: they asked themselves "how can 'nothing' be something?", leading to interesting philosophical and, by the Medieval period, religious arguments about the nature and existence of zero and the vacuum. The paradoxes of Zeno of Elea depend in large part on the uncertain interpretation of zero. (The ancient Greeks even questioned that 1 was a number.) Ancient Greece is the term used to describe the Greek-speaking world in ancient times. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... For other uses, see vacuum cleaner and Vacuum (musical group). ... Zenos paradoxes are a set of paradoxes devised by Zeno of Elea to support Parmenides doctrine that all is one and that contrary to the evidence of our senses, the belief in plurality and change is mistaken, and in particular that motion is nothing but an illusion. ... Zeno of Elea should not be confused with Zeno of Citium. ... Look up one in Wiktionary, the free dictionary. ...

In ancient India, the linguist Panini (5th century BC) used the null (zero or shunya) operator in the Ashtadhyayi, his algebraic grammar of the Sanskrit language. Another early use of something like zero by the Indian scholar Pingala (circa 5th-3rd century BC), implied at first glance by his use of binary numbers, is only the modern binary representation using 0 and 1 of Pingala's binary system, which used short and long syllables (the latter equal in length to two short syllables) as described in Math for Poets and Drummers (pdf), making it similar to Morse code. In Pingala's system, four short syllables meant one, not zero. Nevertheless, he and other Indian scholars at the time used the Sanskrit word shunya (the origin of the word zero after a series of transliterations) to refer to zero or void. [2]. Prehistory The prehistory of India goes back to the old Stone age (Palaeolithic). ... The following is a list of linguists, those who study linguistics. ... Panini can refer to: Pāṇini, the 5th century BC Sanskrit grammarian Panini (sandwich), a type of Italian sandwich Panini (stickers), a brand of collectible stickers Giovanni Paolo Panini, an Italian artist This is a disambiguation page — a navigational aid which lists pages that might otherwise share the same title. ... (6th century BC - 5th century BC - 4th century BC - other centuries) (2nd millennium BC - 1st millennium BC - 1st millennium AD) The 5th and 6th centuries BC are a period of philosophical brilliance among advanced civilizations. ... KK Null, a Japanese musician Null, a special value in computer programming. ... The Ashtadhyayi (Ạṣtādhyāyī, meaning eight chapters) is the earliest known grammar of Sanskrit, and one of the first works on descriptive linguistics, generative linguistics, or linguistics altogether. ... Algebra is the current mathematics collaboration of the week! Please help improve it to featured article standard. ... Sanskrit ( संस्कृतम्) is an Indo-European Classical language of India and a liturgical language of Hinduism, Buddhism, and Jainism. ... Pingala (पिङ्गल) is the author of the Chhandah-shastra, the Sanskrit book on meters, or long syllables. ... (6th century BC - 5th century BC - 4th century BC - other centuries) (2nd millennium BC - 1st millennium BC - 1st millennium AD) The 5th and 6th centuries BC are a period of philosophical brilliance among advanced civilizations. ... // Events The first two Punic Wars between Carthage and Rome over dominance in western Mediterranean Rome conquers Spain Gaulish migration to Macedonia, Thrace and Galatia 281 BCE Antiochus I Soter, on the assassination of his father Seleucus becomes emperor of the Seleucid empire. ... 1922 Chart of the Morse Code Letters and Numerals Morse code is a method for transmitting information, using standardized sequences of short and long marks or pulses — commonly known as dots and dashes — for the letters, numerals and special characters of a message. ... Sanskrit ( संस्कृतम्) is an Indo-European Classical language of India and a liturgical language of Hinduism, Buddhism, and Jainism. ...

History of zero

In the Bakhshali Manuscript, whose date is uncertain but which is claimed by some to be quite early, zero is symbolized and used as a number; if the early dating is accepted, it would predate Brahmagupta. In 498 AD, Hindu astronomer and mathematician Aryabhata stated that "Stanam stanam dasa gunam" or place to place in ten times in value, which may be the origin of the modern decimal based place value notation; his positional number system included a zero in his letter code for numerals (which allowed him to express numbers as words) in his mathematical astronomy text Aryabhatiya. The first unambiguous appearance of the mathematical zero is in Brahmagupta's Brahmasphuta Siddhanta, along with consideration of negative numbers and the algebraic rules discussed below. The Bakhshali Manuscript is a mathematical manuscript written on birch bark which was found near the village of Bakhshali in what is now Pakistan in 1881. ... Aryabhata (आर्यभट) Ä€ryabhaá¹­a) (476 - 550) is the first of the great astronomers of the classical age of India. ... Brahmagupta (ब्रह्मगुप्त) (598_668) was an Indian mathematician and astronomer. ... The main work of Brahmagupta, Brahmasphutasiddhanta (The Opening of the Universe), written in 628, contains some remarkably advanced ideas, including a good understanding of the mathematical role of zero, rules for manipulating both positive and negative numbers, a method for computing square roots, methods of solving linear and some quadratic...

The late Olmec people of south-central Mexico began to use a zero digit (a shell glyph) in the New World possibly by the 4th century BC but certainly by 40 BC, which became an integral part of Maya numerals, but did not influence Old World numeral systems. Monument 1, an Olmec colossal head at La Venta The Olmec were an ancient people living in the tropical lowlands of south-central Mexico, roughly in what are the modern-day states of Veracruz and Tabasco on the Isthmus of Tehuantepec. ... (5th century BC - 4th century BC - 3rd century BC - other centuries) (2nd millennium BC - 1st millennium BC - 1st millennium AD) // Events Invasion of the Celts into Ireland Battle of the Allia and subsequent Gaulish sack of Rome 383 BCE Second Buddhist Councel at Vesali. ... Centuries: 2nd century BC - 1st century BC - 1st century Decades: 90s BC 80s BC 70s BC 60s BC 50s BC - 40s BC - 30s BC 20s BC 10s BC 0s BC 10s BC Years: 45 BC 44 BC 43 BC 42 BC 41 BC 40 BC 39 BC 38 BC 37... Mayan numerals. ...

By 130, Ptolemy, influenced by Hipparchus and the Babylonians, was using a symbol for zero (a small circle with a long overbar) within a sexagesimal numeral system otherwise using alphabetic Greek numerals. Because it was used alone, not as just a placeholder, this Hellenistic zero was one of the first documented uses of a digit zero in the Old World. In later Byzantine manuscripts of his Syntaxis Mathematica (Almagest), the Hellenistic zero had morphed into the Greek letter omicron (otherwise meaning 70). For other uses, see number 130. ... Claudius Ptolemaeus (Greek: ; ca. ... Hipparchus (Greek Ἳππαρχος) (ca. ... Greek numerals are a system of representing numbers using letters of the Greek alphabet. ... Greek numerals are a system of representing numbers using letters of the Greek alphabet. ... The Old World consists of those parts of Earth known to Europeans before the voyages of Christopher Columbus; it includes Europe, Asia, and Africa (collectively known as Africa-Eurasia), plus surrounding islands. ... Byzantine Empire (Greek: ) is the term conventionally used since the 19th century to describe the Greek-speaking Roman Empire during the Middle Ages, centered at its capital in Constantinople. ... Note: This article contains special characters. ... Omicron (upper case Ο, lower case ο, literally small o) is the 15th letter of the Greek alphabet. ...

Another zero was used in tables alongside Roman numerals by 525 (first known use by Dionysius Exiguus), but as a word, nulla meaning nothing, not as a symbol; this usage, more or less contemporary with Aryabhata, might represent a concept of true, mathematical zero, though not so clearly as in the case of Brahmagupta. When division produced zero as a remainder, nihil, also meaning nothing, was used. These medieval zeros were used by all future medieval computists (calculators of Easter). An isolated use of their initial, N, was used in a table of Roman numerals by Bede or a colleague about 725, a zero symbol. The system of Roman numerals is a numeral system originating in ancient Rome, and was adapted from Etruscan numerals. ... Events Bernicia settled by the Angles Ethiopia conquers Yemen The Daisan river, a tributary of the Euphrates, floods Edessa and within a couple of hours fills the entire city except for the highest parts. ... Dionysius Exiguus (Dennis the Little, meaning humble) (c. ... Computus (Latin for computation) is the calculation of the date of Easter in the Christian calendar. ... Easter is the most important religious holiday of the Christian liturgical year, observed in March, April, or May to celebrate the resurrection of Jesus, which Christians believe occurred after his death by crucifixion in AD 27-33 (see Good Friday). ... Bede depicted in an early medieval manuscript Depiction of Bede from the Nuremberg Chronicle, 1493. ... Events Births Deaths Wihtred, king of Kent Categories: 725 ...

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. Islam (Arabic: ; ( (help· info)), submission (to the will of God)) is a monotheistic faith, considered one of the Abrahamic religions, and the worlds second-largest religion. ...

The Rules of Brahmagupta

Zero as a number appeared for the first time in Brahmagupta's book Brahmasputha Siddhanta. Here Brahmagupta considers not only zero, but negative numbers, and the algebraic rules for the elementary operations of arithmetic with such numbers. In some instances, his rules differ from the modern standard. Here are the rules of Brahamagupta: (From the English translation by Henry Thomas Colebrooke in 1817.) Brahmagupta (ब्रह्मगुप्त) (598_668) was an Indian mathematician and astronomer. ... The main work of Brahmagupta, Brahmasphutasiddhanta (The Opening of the Universe), written in 628, contains some remarkably advanced ideas, including a good understanding of the mathematical role of zero, rules for manipulating both positive and negative numbers, a method for computing square roots, methods of solving linear and some quadratic... Henry Thomas Colebrooke (June 15, 1765 - March 18, 1837) was an English orientalist. ... 1817 was a common year starting on Wednesday (see link for calendar). ...

• 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. In computing, NaN (Not a Number) is a value or symbol that is usually produced as the result of an operation on invalid input operands, especially in floating-point calculations. ...

Zero as a decimal digit

See also: History of the Hindu-Arabic numeral system.

Positional notation without the use of zero (using an empty space in tabular arrangements, or the word kha "emptiness") is known to have been in use in India upto the 6th century. The earliest certain use of zero as a positional digit dates to the 7th century. The glyph for the zero digit was written in the shape of a dot, and consequently called bindu "dot". // Origins The Hindu-Arabic numeral system originated from the Hindu numeral system, which is a pure place value system, that requires a zero. ... This Buddhist stela from China, Northern Wei period, was built in the early 6th century. ... // Overview Events The Roman-Persian Wars end. ...

The Hindu-Arabic numeral system reached Europe in the 11th century, via Andalusia, together with knowledge of astronomy and instruments like the astrolabe, first imported by Gerbert of Aurillac. They came to be known as "Arabic numerals". The Italian mathematician Fibonacci was instrumental in bringing the system into European mathematics around 1200. From the 13th century, manuals on calculation (adding, multiplying, extracting roots etc.) became common in Europe where they were called Algorimus in deference to the Persian mathematician Al-Khwarizmi. The most popular was written by John of Sacrobosco and was one of the earliest scientific books to be printed in 1488. Hindu-Arabic numerals until the late 15th century seem to have predominated among mathematicians, while merchants preferred to use the abacus instead, and it was only from the 16th century that they became common knowledge in Europe. The Hindu-Arabic numeral system (also called Algorism) is a positional decimal numeral system documented from the 9th century. ... World map showing Europe Europe is conventionally considered one of the seven continents of Earth which, in this case, is more a cultural and political distinction than a physiogeographic one. ... As a means of recording the passage of time, the 11th century was that century which lasted from 1001 to 1100. ... Motto: Dominator Hercules Fundator Andalucía por sí, para España y la humanidad (Andalusia for herself, for Spain, and for humanity) Capital Seville Area  â€“ Total  â€“ % of Spain Ranked 2nd  87 268 km²  17,2% Population  â€“ Total (2003)  â€“ % of Spain  â€“ Density Ranked 1st  7 478 432  17,9%  85,70... Lunar astronomy: the large crater is Daedalus, photographed by the crew of Apollo 11 as they circled the Moon in 1969. ... A 16th century astrolabe. ... Gerbert of Aurillac, later known as pope Silvester II, (or Sylvester II), (ca. ... Numerals sans serif Arabic numerals, also known as Hindu-Arabic numerals, Indian numerals, Hindu numerals, European numerals, and Western numerals, are the most common symbolic representation of numbers around the world. ... Portrait of Fibonacci, probably not authentic Leonardo of Pisa or Leonardo Pisano (Pisa, c. ... Abu Abdullah Muhammad bin Musa al-Khwarizmi Algorism is the name for the Indo-Arabic decimal system of writing and working with numbers, in which symbols (the ten digits 0 through 9) are used to describe values using a place-value system (positional notation), where each symbol has ten times... Soviet postage stamp commemorating the 1200th anniversary of Muhammad al‑Khwarizmi in 1983. ... Johannes de Sacrobosco or Sacro Bosco (John of Holywood, c. ...

In mathematics

Elementary algebra

Zero (0) is the lowest non-negative integer. The natural number following zero is one and no natural number precedes zero. Zero may or may not be counted as a natural number, depending on the definition of natural numbers. Mathematical operations involving zero were first described by Brahmasphutasiddhanta in the 7th century. A negative number is a number that is less than zero, such as −3. ... The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ... In mathematics, a natural number is either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). The former definition is generally used in number theory, while the latter is preferred in set theory. ... Look up one in Wiktionary, the free dictionary. ... The main work of Brahmagupta, Brahmasphutasiddhanta (The Opening of the Universe), written in 628, contains some remarkably advanced ideas, including a good understanding of the mathematical role of zero, rules for manipulating both positive and negative numbers, a method for computing square roots, methods of solving linear and some quadratic...

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. Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... In mathematics, the cardinality of a set is a measure of the number of elements of the set. There are two approaches to cardinality – one which compares sets directly using bijections, injections, and surjections, and another which uses cardinal numbers. ... In mathematics and more specifically set theory, the empty set is the unique set which contains no elements. ... Euclid, detail from The School of Athens by Raphael. ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... A definition delimits or describes the meaning of a concept or term by stating the essential properties of the entities or objects denoted by that concept or term. ... The von Neumann cardinal assignment is a cardinal assignment which uses ordinal numbers. ... In mathematics, the cardinality of a set is a measure of the number of elements of the set. There are two approaches to cardinality – one which compares sets directly using bijections, injections, and surjections, and another which uses cardinal numbers. ...

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. In mathematics, a prime number (or a prime) is a natural number that has exactly two distinct positive divisors, one and itself. ... A composite number is a positive integer which has a positive divisor other than one or itself. ... In mathematics, a unit in a ring R is an element u such that there is v in R with uv = vu = 1R. That is, u is an invertible element of the multiplicative monoid of R. The units of R form a group U(R) under multiplication, the group of... In mathematics, a rational number (or informally fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... Wikibooks Algebra has more about this subject: Complex numbers In mathematics, a complex number is an expression of the form where a and b are real numbers, and i represents the imaginary unit, i2 = −1. ... In mathematics, an abelian group, also called a commutative group, is a group (G, *) such that a * b = b * a for all a and b in G. Abelian groups are named after Niels Henrik Abel. ...

The following are some basic rules for dealing with the number zero. These rules apply for any complex number x, unless otherwise stated. Wikibooks Algebra has more about this subject: Complex numbers In mathematics, a complex number is an expression of the form where a and b are real numbers, and i represents the imaginary unit, i2 = −1. ...

• 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. In mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a binary operation on that set. ... 3 + 2 with apples, a popular choice in textbooks Addition is the basic operation of arithmetic. ... In mathematics, defined and undefined are used to explain whether expressions have meaningful, sensible output. ... In mathematics, defined and undefined are used to explain whether expressions have meaningful, sensible output. ... In mathematical analysis, and in particular in elementary calculus, certain expressions are indeterminate forms and must be treated as symbolic only, until more careful discussion has taken place. ... In calculus, lHôpitals rule uses derivatives to help compute limits with indeterminate forms. ...

The sum of 0 numbers is 0, and the product of 0 numbers is 1. In mathematics, an empty product, or nullary product, is the result of multiplying no numbers. ...

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.

An additive group is a group, and any group can be written as an additive group, so the adjective additive does not describe a class of groups, but rather the notation used to write the group operation. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have similar (but not identical) properties to those familiar from the integers. ... In mathematics, a root (or a zero) of a function f is an element x in the domain of f such that f(x) = 0. ... In complex analysis, a zero of a holomorphic function f is a complex number a such that f(a) = 0. ... Table of Geometry, from the 1728 Cyclopaedia. ... 2-dimensional renderings (ie. ... A spatial point is an entity with a location in space but no extent (volume, area or length). ... Wikibooks has more about this subject: Trigonometry Trigonometry (from the Greek trigonon = three angles and metro = measure) is a branch of mathematics dealing with angles, triangles and trigonometric functions such as sine, cosine and tangent. ... The word probability derives from the Latin probare (to prove, or to test). ... In measure theory (a branch of mathematical analysis), one says that a property holds almost everywhere if the set of elements for which the property does not hold is a null set, i. ... In mathematics, a measure is a function that assigns a number, e. ... In mathematics a constant function is a function whose values do not vary and thus are constant. ... In category theory, a zero morphism is a special kind of trivial morphism. ... Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ... The classical Möbius function is an important multiplicative function in number theory and combinatorics. ...

In physics

The value zero plays a special role for a large number of physical quantities. For some quantities, the zero level is naturally distinguished from all other levels, where as it for others is more or less arbitrarily chosen. For example, on the kelvin temperature scale, zero is the coldest possible temperature (negative temperatures exist but are not actually colder), where as on the celsius scale, zero is arbitrarily defined to be at the freezing point of water. Measuring sound intensity in decibels or phons, the zero level is arbitrarily set at a reference value, e.g. at a value for the threshold of hearing. The kelvin (symbol: K) is the SI unit of temperature, and is one of the seven SI base units. ... Temperature is the physical property of a system which underlies the common notions of hot and cold; the material with the higher temperature is said to be hotter. ... A degree Celsius (°C) is a unit of temperature named after the Swedish astronomer Anders Celsius (1701-1744), who first proposed a similar system in 1742. ... The melting point of a solid is the temperature at which it changes state from solid to liquid. ... The decibel (dB) is a measure of the ratio between two quantities, and is used in a wide variety of measurements in acoustics, physics and electronics. ... Fig. ...

In computer science

Numbering from 1 or 0?

Human beings usually number things starting from one, not zero. Yet in computer science zero has become the popular indication for a starting point. For example, in almost all old programming languages, an array starts from 1 by default, which is natural for humans. As programming languages have developed, it has become more common that an array starts from zero by default (zeroth, or zero-based). Computer science is the study of the theoretical foundations of information and computation and their implementation and application in computer systems. ... Computer code (HTML with JavaScript) in a tool that uses syntax highlighting (colors) to help the developer see the purpose of each piece of code. ... In computer programming, an array, also known as a vector or list (for one-dimensional arrays) or a matrix (for two-dimensional arrays), is one of the simplest data structures. ... Look up one in Wiktionary, the free dictionary. ... Default is the name of a number of quite different concepts. ... Binomial name Homo sapiens Linnaeus, 1758 Subspecies Homo sapiens idaltu† Homo sapiens sapiens Homo (genus). ... The zeroth item is the initial item of a sequence, if that sequence is numbered beginning from zero rather than one. ...

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. Modular arithmetic is a system of arithmetic for integers, where numbers wrap around after they reach a certain value — the modulus. ...

Another reason to use zero-based array indices is that it can improve efficiency under certain circumstances. To illustrate, suppose a is the memory address of the first element of an array, and i is the index of the desired element. In this fairly typical scenario, it is quite common to want the address of the desired element. If the index numbers count from 1, the desired address is computed by this expression: In computer science, a memory address is a unique identifier for a memory location at which a CPU or other device can store a piece of data for later retrieval. ...

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.

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). FIRST Logo FIRST, or For Inspiration and Recognition of Science and Technology, is an organization founded by inventor Dean Kamen in 1989 in order to develop ways to excite students about engineering and technology. ... The zeroth item is the initial item of a sequence, if that sequence is numbered beginning from zero rather than one. ... Commonly, ordinal numbers, or ordinals for short, are numbers used to denote the position in an ordered sequence: first, second, third, fourth, etc. ...

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. It has been suggested that Software pointer be merged into this article or section. ... The C Programming Language, Brian Kernighan and Dennis Ritchie, the original edition that served for many years as an informal specification of the language The C programming language is a standardized imperative computer programming language developed in the early 1970s by Dennis Ritchie for use on the Unix operating system. ... In computer science, compile time, as opposed to runtime, is the time when a compiler compiles code written in a programming language into an executable form. ...

Negative zero

Main article: -0

In some signed number representations (but not the two's complement representation predominant today) and most floating point number representations, zero has two distinct representations, one grouping it with the positive numbers and one with the negatives; this latter representation is known as negative zero. Representations with negative zero can be troublesome, because the two zeroes will compare equal but may be treated differently by some operations. −0 is the representation of negative zero, a number that exists in computing in some signed number representations for integers and in most floating point number representations. ... Comparison of Integer Representations in 4-bit In mathematics, negative numbers in any base are represented in the usual way, by prefixing them with a − sign. ... Twos complement is the most popular method of signifying negative integers in computer science. ... A floating-point number is a digital representation for a number in a certain subset of the rational numbers, and is often used to approximate an arbitrary real number on a computer. ... This article is being considered for deletion in accordance with Wikipedias deletion policy. ...

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 for certain Scandinavian languages which use Ø as a letter. Ive created this image myself. ... O is the fifteenth letter of the Latin alphabet. ... Clemson Universitys library catalog as displayed in a 3270 emulation program The IBM 3270 is a class of terminals made by IBM (known as Display Devices) normally used to communicate with IBM mainframes. ... Note: This article contains special characters. ... Theta (upper case Θ, lower case θ or ) is the eighth letter of the Greek alphabet. ... The slashed zero looks just like a regular letter O or number 0 (zero), but it has a slash through it. ... Teletype machines in World War II A teleprinter (teletypewriter, teletype or TTY) is a now largely obsolete electro-mechanical typewriter which can be used to communicate typed messages from point to point through a simple electrical communications channel, often just a pair of wires. ... The North Germanic languages (also Scandinavian languages or Nordic languages) is a branch of the Germanic languages spoken in Scandinavia, parts of Finland and on the Faroe Islands and Iceland. ... Ø, ø is a vowel and a letter used in the Danish, Faroese and Norwegian alphabets. ...

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. See also the Nordic countries. ... William Seward Burroughs (1857-1898), US inventor William S. Burroughs (1914-1997), author and grandson of William Seward Burroughs Edgar Rice Burroughs (1875-1950), American author of Tarzan fame The Burroughs Corporation began in 1886 as the American Arithmometer Company in St. ... Unisys Corporation NYSE: UIS is a provider of information technology services and solutions with operations across the world. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... Q is the seventeenth letter of the Latin alphabet. ...

The typeface used on some European number plates for cars distinguish the two symbols by making the O rather egg-shaped and the zero more circular, but most of all by slitting open the zero on the upper right side, so the circle is not closed any more (as in German plates). The typeface chosen is called fälschungserschwerende Schrift (abbr.: FE Schrift), meaning "unfalsifiable script". Note that those used in the United Kingdom do not differentiate between the two as there can never be any ambiguity if the design is correctly spaced. A small variety of cars, the most popular kind of automobile. ... ImageMetadata File history File links Download high resolution version (1627x356, 67 KB) Beschreibung Deutsches Kfz-Kennzeichen für Behördenfahrzeuge, Nummernbereich 3xx, Zulassungsbezirk Erlangen Kennzeichen selbst fotografiert am 2005-09-25 German number plate for official cars, Code 3xx, registration office Erlangen Picture taken by myself on 2005-09-25...

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. In measure theory, a null set is a set that is negligible for the purposes of the measure in question. ...

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

George Bruce Halsted (November 25, 1853-March 16, 1922) was a mathematician who explored foundations of geometry and introduced Non-Euclidean geometry into the United States through his own work and his many important translations. ...

In other fields

International maritime signal flag for 0

• 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".

Image File history File links ICS_Zero. ... Image File history File links ICS_Zero. ... The system of international maritime signal flags is a way of representing individual letters of the alphabet on ships or in nautical situations. ... PREMIER - first The braille system, named after Louis Braille, is a method that is widely used by blind people to read and write. ... The letter J is the tenth of the Latin alphabet; it was the last to be added to that alphabet. ... It has been suggested that Dual layer recording be merged into this article or section. ...

See also

• Negative and non-negative numbers
• Nothing
• Null
• Number theory
• Slashed zero
• Nullar
• Division by zero

A negative number is a number that is less than zero, such as −3. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... KK Null, a Japanese musician Null, a special value in computer programming. ... Number theory is the formal study of numbers. ... The slashed zero looks just like a regular letter O or number 0 (zero), but it has a slash through it. ... In grammar, nullar number refers to where nouns take a special form when referring to zero objects. ... In mathematics, a division is called a division by zero if the divisor is zero. ...

References

• The Universal History of Numbers: From Prehistory to the Invention of the Computer. Georges Ifrah. Wiley (2000)
• A Brief History of Zero - Kristen McQuillin, July 1997 (revised January 2004)
• A history of Zero
• Zero Saga
• The Discovery of the Zero
• Charles Seife (2000). "Zero: The Biography of a Dangerous Idea". Publisher: Penguin USA (Paper). ISBN 0140296476

Algebra with Arithmetic of Brahmagupta and Bhaskara by Henry Thomas Colebrooke, London 1817 The Free On-line Dictionary of Computing (FOLDOC) is an on-line, searchable encyclopedic dictionary of computing subjects. ... GNU logo The GNU Free Documentation License (GNU FDL or simply GFDL) is a copyleft license for free content, designed by the Free Software Foundation (FSF) for the GNU project. ...

Grimm, R.E., "The Autobiography of Leonardo Pisano", Fibonacci Quarterly, Vol. 11, No. 1, February 1973, pp. 99-104.

Sigler, L., “Fibonacci’s Liber Abaci”, English translation, Springer, 2003.

Further reading

• The Book of Nothing, John D. Barrow, Vintage (July, 2001), ISBN 0-09-928845-1

Aryabhatiya of Aryabhata, translated by Walter Eugene Clark