0 (zero) is both a number and the numerical digit used to represent that number in numerals. It plays a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures. As a digit, zero is used as a placeholder in place value systems. In the English language, zero may also be called null or nil when a number, "oh" (IPA: [oʊ]), "goose egg", or cipher (archaic) when a numeral, and nought or naught^[1] in either context. Aleph-0, the smallest infinite cardinal In mathematics, cardinal numbers, or cardinals for short, are generalized numbers used to measure the cardinality (size) of sets. ... In linguistics, ordinal numbers are the words representing the rank of a number with respect to some order, in particular order or position (i. ... The zeroth item is the initial item of a sequence, if that sequence is numbered beginning from zero rather than one. ... ... 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. ... Arabic redirects here. ... Bangla redirects here. ... () is an abugida script used to write several Indo-Aryan languages, including Sanskrit, Hindi, Gujarati,Marathi, Sindhi, Bihari, Bhili, Marwari, Konkani, Bhojpuri, Pahari (Garhwali and Kumaoni), Santhali, Nepali, Newari, Tharu and sometimes Kashmiri and Romani. ... The system of Japanese numerals is the system of number names used in the Japanese language. ... Khmer numerals are the numerals used in the Khmer language of Cambodia. ... The binary numeral system, or base-2 number system, is a numeral system that represents numeric values using two symbols, usually 0 and 1. ... The octal numeral system, or oct for short, is the base-8 number system, and uses the digits 0 to 7. ... The duodecimal (also known as base-12 or dozenal) system is a numeral system using twelve as its base. ... In mathematics and computer science, hexadecimal, base-16, 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. ... For other uses, see Number (disambiguation). ... In mathematics and computer science, a numerical digit is a symbol, e. ... A numeral is a symbol or group of symbols that represents a number. ... For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... The additive identity of a number n is the number which, when added to n will yield n. ... Not to be confused with Natural number. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... This article is about the branch of mathematics. ... The place value system is a method of writing numbers with a base 10 numerical system. ... The English language is a West Germanic language that originates in England. ... In language, an archaism is the deliberate use of an older form that has fallen out of current use. ...

0 as a number

0 is the integer preceding 1. In most systems, 0 was identified before the idea of 'negative integers' was accepted. Zero is an even number.^[2] 0 is neither positive nor negative. Not to be confused with Natural number. ... This article is about the number one. ... Zero objects, divided into two equal groups. ...

Zero is a number which quantifies a count or an amount of null size; that is, if the number of your brothers is zero, that means the same thing as having no brothers, and if something has a weight of zero, it has no weight. If the difference between the number of pieces in two piles is zero, it means the two piles have an equal number of pieces. Before counting starts, the result can be assumed to be zero; that is the number of items counted before you count the first item and counting the first item brings the result to one. And if there are no items to be counted, zero remains the final result. Look up null in Wiktionary, the free dictionary. ...

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. For other uses, see Historian (disambiguation). ... For the political notion, see Year Zero (political notion). ... 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. ... Galileo is often referred to as the Father of Modern Astronomy. ... 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 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 less tall (x-height). Image File history File links The numerals 0, 3 and 6 written in text figures. ... In mathematics and computer science, a numerical digit is a symbol, e. ... “Font” redirects here. ... Hoefler Text, a contemporary font, uses hanging or old style text figures. ... In typography, the x-height or corpus size refers to the height of the lowercase letter x in any font, which is usually the same for a, c, e, m, n, o, r, s, u, v, w, and z. ...

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. Image File history File links 7-segment_cdeg. ... Image File history File links 7-segment_abcdef. ... A seven-segment display (abbreviation: 7-seg(ment) display), less commonly known as a seven-segment indicator, is a form of display device that is an alternative to the more complex dot-matrix displays. ...

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. This article is about different methods of expressing numbers with symbols. ... A positional notation or place-value notation system is a numeral system in which each position is related to the next by a constant multiplier, a common ratio, called the base or radix of that numeral system. ...

Distinguishing the digit 0 from the letter O

Traditionally, standard typewriters made no distinction in shape between the letter O and the digit 0; some models did not even have a separate key for the digit 0. The oval (i.e. narrower) digit 0 and more nearly circular letter O together came into prominence on modern character displays, though the distinction was already present in some print typefaces.^[3] Image File history File links No higher resolution available. ... Look up O, o in Wiktionary, the free dictionary. ...

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 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. Clemson Universitys library catalog displayed in a 3270 emulation program The IBM 3270 is a class of terminals made by IBM since 1972 (known as display devices) normally used to communicate with IBM mainframes. ... Windows redirects here. ... This page contains special characters. ... Look up Θ, θ in Wiktionary, the free dictionary. ...

An alternative, the slashed zero (looking similar to the letter O other than the slash), was primarily used in hand-written coding sheets before transcription to punched cards or tape, and is also used in old-style ASCII graphic sets descended from the default typewheel on the ASR-33 Teletype. This form is similar to the symbol , or "∅" (Unicode character U+2205), representing the empty set, as well as to the letter Ø used in several Scandinavian languages. 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 for TeleTYpe/TeleTYpewriter) 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 empty set is the set containing no elements. ... // For the similarly named Danish land, see Ø, Denmark. ... 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. ...

The 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] Some Burroughs/Unisys equipment displays a digit 0 with a reversed slash. 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 ().^[3] Fortran (previously FORTRAN[1]) is a general-purpose[2], procedural,[3] imperative programming language that is especially suited to numeric computation and scientific computing. ... For other uses, see Scandinavia (disambiguation). ... It has been suggested that ALGOL object code be merged into this article or section. ... 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), based in Blue Bell, Pennsylvania, United States, and incorporated in Delaware[3], is a global provider of information technology services and solutions. ... Fragment of lineprinter cylinder with the type of % The line printer is a form of high speed impact printer in which one line of type is printed at a time. ... This article is about the Latin alphabet letter. ...

An official German license plate showing zeros

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. 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... 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... 1979 TI-99/4 with RF modulator, optional Speech Synthesizer, keyboard overlays, and a cartridge. ...

The typeface used on most European number plates for cars 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). This typeface is called fälschungserschwerende Schrift (abbr.: FE Schrift) in Germany, meaning "script which is harder to falsify". Typefaces used on United Kingdom plates do not differentiate between the two as there can never be any ambiguity if the design is correctly spaced (the vehicle "numbers" are allocated in a manner that avoids any such confusion). The same applies to postal codes in the United Kingdom. Car redirects here. ... FE-Schrift or fälschungserschwerende Schrift (tamper-hindering font) is the typeface that has been used since November 2000 on vehicle registration plates in Germany. ...

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. This article is about the American airline. ... A Canadian postal code is a string of six characters that forms part of a postal address in Canada. ...

Etymology

The word "zero" came via French zéro from Venetian zero, which (together with "cipher") came via Italian zefiro from Arabic صفر, şafira = "it was empty", şifr = "zero", "nothing", which was used to translate Sanskrit śūnya ( शून्य ), meaning void or empty. A sign in Venetian reading Here we also speak Venetian Venetian or Venetan is a Romance language spoken by over five million people,[1] mostly in the Veneto region of Italy. ... Arabic redirects here. ... Sanskrit ( , for short ) is a classical language of India, a liturgical language of Hinduism, Buddhism, Sikhism, and Jainism, and one of the 23 official languages of India. ... Śūnyatā, शून्यता (Sanskrit), Suññatā (Pāli), stong pa nyid (Tibetan), Kuu, 空 (Japanese) qoÉ£usun (Mongolian), generally translated into English as Emptiness or Voidness, is a concept of central importance in the teaching of the Buddha, as a direct realization of Sunyata is required to achieve liberation from the cycle of...

Italian zefiro already meant "west wind" from Latin and Greek zephyrus; this may have influenced the spelling when transcribing Arabic şifr.^[5] The Italian mathematician Fibonacci (c.1170-1250), who grew up in Arab North Africa and is credited with introducing the Hindu decimal system to Europe, used the term zephyrum. This became zefiro in Italian, which was contracted to zero in Venetian, the modern English word. Zephyr and Hyakinth; Attic red figure cup from Tarquinia, circa 480 BCE. Boston Museum of Fine Arts. ... For the number sequence, see Fibonacci number. ... A sign in Venetian reading Here we also speak Venetian Venetian or Venetan is a Romance language spoken by over five million people,[1] mostly in the Veneto region of Italy. ...

As the Hindu decimal zero and its new mathematics spread from the Arab world to Europe in the Middle Ages, words derived from sifr and zephyrus came to refer to calculation, as well as to privileged knowledge and secret codes. According to Ifrah, "in thirteenth-century Paris, a 'worthless fellow' was called a "... cifre en algorisme", i.e., an "arithmetical nothing"."^[5] From şifr also came French chiffre = "digit", "figure", "number", chiffrer = "to calculate or compute", chiffré= "encrypted". Today, the word in Arabic is still sifr, and cognates of sifr are common in the languages of Europe and southwest Asia. This article discusses the adherents of Hinduism. ... 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. ...

History

By the middle of the 2nd millennium BC, the Babylonians had a sophisticated sexagesimal positional numeral system. The lack of a positional value (or zero) was indicated by a space between sexagesimal numerals. By 300 BC, a punctuation symbol (two slanted wedges) was co-opted as a placeholder in the same Babylonian system. In a tablet unearthed at Kish (dating from about 700 BC), the scribe Bêl-bân-aplu wrote his zeroes with three hooks, rather than two slanted wedges.^[6] The 2nd millennium BC marks the transition from the Middle to the Late Bronze Age. ... Babylonian clay tablet YBC 7289 with annotations. ... 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. ...

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 term ancient Greece refers to the periods of Greek history in Classical Antiquity, lasting ca. ... For other uses, see Philosophy (disambiguation). ... Look up Vacuum in Wiktionary, the free dictionary. ... “Arrow paradox” redirects here. ... Zeno of Elea (pronounced , Greek: Ζήνων ὁ Ἐλεάτης) (ca. ...

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.^{[7] [8]} He and his contemporary Indian scholars used the Sanskrit word śūnya to refer to zero or void.^[9] Pingala (पिङ्गल ) is the supposed author of the Chandas shastra (, also Chandas sutra ), a Sanskrit treatise on prosody considered one of the Vedanga. ... The 5th century BC started the first day of 500 BC and ended the last day of 401 BC. // The Parthenon of Athens seen from the hill of the Pnyx to the west. ... (2nd millennium BC - 1st millennium BC - 1st millennium) The 2nd century BC started on January 1, 200 BC and ended on December 31, 101 BC. // Coin of Antiochus IV. Reverse shows Apollo seated on an omphalos. ... The binary or base-two numeral system is a system for representing numbers in which a radix of two is used; that is, each digit in a binary numeral may have either of two different values. ... 1922 Chart of the Morse Code Letters and Numerals Morse code is a method for transmitting telegraphic information, using standardized sequences of short and long elements to represent the letters, numerals, punctuation and special characters of a message. ... Sanskrit ( , for short ) is a classical language of India, a liturgical language of Hinduism, Buddhism, Sikhism, and Jainism, and one of the 23 official languages of India. ...

The back of Stela C from Tres Zapotes, an Olmec archaeological site
This is 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 what is thought to be one of the few surviving examples of Epi-Olmec script.

Tres Zapotes is a Mesoamerican archaeological site located in the south-central Gulf Lowlands of Mexico in the Papaloapan river plain. ... Epi-Olmec (after Olmec) is a Mesoamerican writing system in use in the Isthmus of Tehuantepec from perhaps 500 BCE to 500 CE, although there is disagreement on these dates. ...

History of zero

The use of a blank on a counting board to represent 0 dated back in India to 4th century BC^[10]. The Mesoamerican Long Count calendar developed in south-central Mexico required the use of zero as a place-holder within its vigesimal (base-20) positional numeral system. Many different glyphs, including this partial quatrefoil— —were used as a zero symbol for these Long Count dates, the earliest of which (on Stela 2 at Chiapa de Corzo, Chiapas) has a date of 36 BC.^[11] Since the eight earliest Long Count dates appear outside the Maya homeland,^[12] 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. Long Count redirects here. ... The vigesimal or base-20 numeral system is based on twenty (in the same way in which the ordinary decimal numeral system is based on ten). ... Image File history File links MAYA-g-num-0-inc-v1. ... Location within Mexico Municipalities of Chiapas Country Mexico Capital Municipalities 118 Largest City Tuxtla Gutiérrez Government  - Governor Juan José Sabines Guerrero (PRD)  - Federal Deputies PRI: 7 PRD: 5  - Federal Senators PRI: 1 PRD: 1 PVEM: 1 Area Ranked 8th  - Total 74,211 km² (28,653 sq mi) Population (2005... Olmec stone head The Olmec were an ancient people living in the tropical lowlands of south-central Mexico, roughly what would now be the Veracruz and Tabasco regions of the Mexican isthmus. ...

Although zero became an integral part of Maya numerals, it did not influence Old World numeral systems. Mayan numerals. ... For other uses, see Old World (disambiguation). ...

In China, counting rods were used for calculation since the 4th century BCE. Chinese mathematicians understood negative numbers and zero, though they had no symbol for the latter.^[13] The Nine Chapters on the Mathematical Art, which was mainly composed in the 1st century CE, stated "[when subtracting] subtract same signed numbers, add differently signed numbers, subtract a positive number from zero to make a negative number, and subtract a negative number from zero to make a positive number."^[14] The counting rods (Traditional Chinese: , Simplified Chinese: , pinyin: chou2) were used by ancient Chinese before the invention of the abacus. ... (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 Kingdom of Macedon conquers Persian empire Romans build first aqueduct Chinese use bellows The Scythians are beginning to be absorbed into the Sarmatian... The Nine Chapters on the Mathematical Art (九章算術) is a Chinese mathematics book, probably composed in the 1st century AD, but perhaps as early as 200 BC. This book is the earliest surviving mathematical text from China that has come down to us by being copied by scribes and (centuries later... (Redirected from 1st century CE) (1st century BC - 1st century - 2nd century - other centuries) The 1st century was that century which lasted from 1 to 99. ...

By 130, Ptolemy, influenced by Hipparchus and the Babylonians, was using a symbol for zero (a small circle with a long overbar) within a sexagesimal numeral system otherwise using alphabetic Greek numerals. Because it was used alone, not just as a placeholder, this Hellenistic zero was perhaps the first documented use of a number zero in the Old World. However, the positions were usually limited to the fractional part of a number (called minutes, seconds, thirds, fourths, etc.)—they were not used for the integral part of a number. In later Byzantine manuscripts of Ptolemy's Syntaxis Mathematica (also known as the Almagest), the Hellenistic zero had morphed into the Greek letter omicron (otherwise meaning 70). For other uses, see number 130. ... This article is about the geographer, mathematician and astronomer Ptolemy. ... For the Athenian tyrant, see Hipparchus (son of Pisistratus). ... 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. ... Not to be confused with Natural number. ... Byzantine redirects here. ... This page contains special characters. ... Look up Ο, ο in Wiktionary, the free dictionary. ...

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. Roman numerals are a numeral system originating in ancient Rome, 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. ... This article is about the Christian festival. ... For other uses, see Bede (disambiguation). ... Events Births Deaths Wihtred, king of Kent Categories: 725 ...

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.^[15] For other uses, see Aryabhata (disambiguation). ...

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 (see also the section Etymology above).^[16] 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 CE.^[17][18] 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.^[19] Positional notation or place-value notation is a numeral system in which each position is related to the next by a constant multiplier called the base (or radix) of that numeral system. ... Sanskrit ( , for short ) is a classical language of India, a liturgical language of Hinduism, Buddhism, Sikhism, and Jainism, and one of the 23 official languages of India. ... variant glyphs representing the character a (allographs of a) in the Zapfino typeface. ... , Gwalior   is a city in Madhya Pradesh in India. ...

The Indian numerals and the positional number system were introduced to the Islamic civilization by Al-Khwarizmi, the founder of several branches and basic concepts of mathematics. In the words of Philip Hitti, Al-Khwarizmi's contribution to mathematics influenced mathematical thought to a greater extent. His work on algebra initiated the subject in a systematic form and also developed it to the extent of giving analytical solutions of linear and quadratic equations, which established him as the founder of Algebra. The word algebra is derived from the title of his famous book Al-Jabr wa-al-Muqabilah, and the word algorithm is derived from his name. ... Soviet postage stamp commemorating the 1200th anniversary of Muhammad al‑Khwarizmi in 1983. ... A page from the book (Arabic for The Compendious Book on Calculation by Completion and Balancing), also known under a shorter name spelled as Hisab al-jabr w’al-muqabala, Kitab al-Jabr wa-l-Muqabala and other transliterations) is a mathematical book written approximately 820 AD by the Persian...

Al-Khwarizmi's book on arithmetic synthesized Greek and Hindu knowledge and also contained his own fundamental contribution to mathematics and science. Thus, he explained the use of zero, a numeral of fundamental importance developed by the Indians.

It was only centuries later, in the 12th century, that the Indian numeral system was introduced to the Western world through Latin translations of his Arithmetic. Occident redirects here. ... For other uses, see Latins and Latin (disambiguation). ...

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), ^[20] 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:^[21] Brahmagupta (ब्रह्मगुप्त) ( ) (589–668) 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... 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 ...

• The sum of zero and a negative number is negative
• The sum of zero and a positive number is positive
• The sum of zero and zero is zero.
• The sum of a positive and a negative is their difference; or, if they are equal, zero.
• A positive or negative number when divided by zero is a fraction with the zero as denominator.
• Zero divided by a negative or positive number is either zero or is expressed as a fraction with zero as numerator and the finite quantity as denominator.
• Zero divided by zero is zero.

In saying zero divided by zero is zero, Brahmagupta differs from the modern position. Mathematicians normally do not assign a value, whereas computers and calculators 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. 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 from the 6th century. The earliest certain use of zero as a decimal positional digit dates to the 9th century. The glyph for the zero digit was written in the shape of a dot, and consequently called bindu ("dot"). The Hindu-Arabic numeral system originaju from the Hindu numeral system, which is a pure place value system, that requires a zero. ... The 6th century is the period from 501 - 600 in accordance with the Julian calendar in the Christian Era. ... As a means of recording the passage of time the 9th century was the century that lasted from 801 to 900. ... Bindu is an Indian concept that signifies action (as in worship or prayer) - a concept that can take the meaning of a single dot. ...

The Indian numeral system (base 10) reached Europe in the 11th century, via the Iberian Peninsula through Spanish Muslims the Moors, together with knowledge of astronomy and instruments like the astrolabe, first imported by Gerbert of Aurillac. 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: The Hindu-Arabic numeral system (also called Algorism) is a positional decimal numeral system documented from the 9th century. ... The Iberian Peninsula, or Iberia, is located in the extreme southwest of Europe, and includes modern day Spain, Portugal, Andorra and Gibraltar. ... A Muslim is a believer in or follower of Islam. ... For other uses, see moor. ... For other uses, see Astronomy (disambiguation). ... A 16th century astrolabe. ... Gerbert of Aurillac, later known as pope Silvester II, (or Sylvester II), (ca. ... For other uses, see Arabic numerals (disambiguation). ... For the number sequence, see Fibonacci number. ...

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.^[22][23] This article is about the Hindu religion; for other meanings of the word, see Hindu (disambiguation). ...

Here Leonardo of Pisa uses the word sign "0", indicating it is like a sign to do operations like addition or multiplication, but he did not recognize zero as a number in its own right. From the 13th century, manuals on calculation (adding, multiplying, extracting roots, etc.) became common in Europe where they were called algorimus after the Persian mathematician al-Khwarizmi. The most popular was written by John of Sacrobosco about 1235 and was one of the earliest scientific books to be printed in 1488. 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. Algorism comprises all of the rules of performing arithmetic computations using a decimal system for representing numbers in which numbers written using ten symbols having the values 0 through 9 are combined using a place-value system (positional notation), where each symbol has ten times the weight of the one... Johannes de Sacrobosco or Sacro Bosco (John of Holywood, c. ... Roman numerals are a numeral system originating in ancient Rome, adapted from Etruscan numerals. ...

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). A negative number is a number that is less than zero, such as −2. ... Not to be confused with Natural number. ... In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ... This article is about the number one. ... In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ...

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 evenness of zero). In mathematics, a prime number (or a prime) is a natural number greater than 1 which has exactly two distinct natural number divisors: 1 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, the parity of an object refers to whether it is even or odd. ... Zero objects, divided into two equal groups. ...

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. In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... A complex number can be visually represented as a pair of numbers forming a vector on a diagram called an Argand diagram In mathematics, the complex numbers are the extension of the real numbers obtained by adjoining an imaginary unit, denoted i, which satisfies:[1] Every complex number can be...

• Addition: x + 0 = 0 + x = x. That is, 0 is an identity element (or neutral element) with respect to addition.
• Subtraction: x − 0 = x and 0 − x = − x.
• Multiplication: x · 0 = 0 · x = 0.
• Division: ⁰/_x = 0, for nonzero x. But ^x/₀ is undefined, because 0 has no multiplicative inverse, a consequence of the previous rule; see division by zero. In the real numbers, for positive x, as y in ^x/_y approaches 0 from the positive side, the quotient increases indefinitely toward positive infinity, but as y approaches 0 from the negative side, the quotient tends toward negative infinity.
• Exponentiation: x⁰ = 1, except that the case x = 0 may be left undefined in some contexts; see Zero to the zero power. For all positive real x, 0^x = 0.

The expression ⁰/₀, 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. For other uses, see identity (disambiguation). ... 3 + 2 = 5 with apples, a popular choice in textbooks[1] This article is about addition in mathematics. ... In mathematics, defined and undefined are used to explain whether expressions have meaningful, sensible output. ... For the album by Hux Flux, see Division by Zero (album). ... In mathematics, defined and undefined are used to explain whether expressions have meaningful, sensible output. ... “Exponent” redirects here. ... Wikibooks Calculus has a page on the topic of Limits In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements as... In calculus and other branches of mathematical analysis, an indeterminate form is an algebraic expression obtained in the context of limits. ... In calculus, lHôpitals rule (also spelled lHospital) 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, the empty sum, or nullary sum, is the result of adding no numbers. ... In mathematics, an empty product, or nullary product, is the result of multiplying no numbers. ...

Other branches of mathematics

• In set theory, 0 is the cardinality of the empty set: if one does not have any apples, then one has 0 apples. In fact, in certain axiomatic developments of mathematics from set theory, 0 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 0 elements.
• Also in set theory, 0 is the least ordinal number, corresponding to the empty set viewed as a well-ordered set.
• In propositional logic, 0 may be used to denote the truth value false.
• In abstract algebra, 0 is commonly used to denote a zero element, which is a neutral element for addition (if defined on the structure under consideration) and an absorbing element for multiplication (if defined).
• In lattice theory, 0 may denote the bottom element of a bounded lattice.
• In category theory, 0 is sometimes used to denote an initial object of a category.

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 and injections, and another which uses cardinal numbers. ... The empty set is the set containing no elements. ... For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... For other uses, see Definition (disambiguation). ... 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 and injections, and another which uses cardinal numbers. ... In set theory, ordinal, ordinal number, and transfinite ordinal number refer to a type of number introduced by Georg Cantor in 1897, to accommodate infinite sequences and to classify sets with certain kinds of order structures on them. ... In mathematics, a well-order (or well-ordering) on a set S is a total order on S with the property that every non-empty subset of S has a least element in this ordering. ... Propositional logic or sentential logic is the logic of propositions, sentences, or clauses. ... In logic, a truth value, or truth-value, is a value indicating to what extent a statement is true. ... Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ... In mathematics, a zero element is the element of an additive group, ring, field, module, or monoid that is an additive identity element. ... 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. ... See lattice for other mathematical as well as non-mathematical meanings of the term. ... In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set is an element of S which is greater than or equal to any other element of S. The term least element is defined dually. ... See lattice for other mathematical as well as non-mathematical meanings of the term. ... In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ... In mathematics, an initial object of a category C is an object I in C such that to every object X in C, there exists precisely one morphism I → X. The dual notion is that of a terminal object: T is terminal, if to every object X in C there... In mathematics, categories allow one to formalize notions involving abstract structure and processes that preserve structure. ...

Other uses of zero in mathematics

• A zero of a function f is a point x in the domain of the function such that f(x) = 0. When there are finitely many zeros these are called the roots of the function. See also zero (complex analysis) for zeros of a holomorphic function.
• The zero function (or zero map) on a domain D is the constant function with 0 as its only possible output value, i.e., the function f defined by f(x) = 0 for all x in D. 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.

This article is about the zeroes of a function. ... 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. ... Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ... 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. ... In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ... In algebra, a determinant is a function depending on n that associates a scalar, det(A), to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ... In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, a table consisting of abstract quantities that can be added and multiplied. ...

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. For other uses, see Kelvin (disambiguation). ... In physics, certain systems can achieve negative temperatures; that is, their thermodynamic temperature can be a negative quantity. ... For other uses, see Celsius (disambiguation). ... Freezing point can refer to several things: For the chemistry term, see Melting point. ... For other uses, see Decibel (disambiguation). ... Fig. ... A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... 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. ... Fig. ... A physical system is a system that is comprised of matter and energy. ... In physics, the ground state of a quantum mechanical system is its lowest-energy state. ...

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. See also: List of elements by atomic number In chemistry and physics, the atomic number (also known as the proton number) is the number of protons found in the nucleus of an atom. ... A tetraneutron is a hypothesised stable cluster of four neutrons. ... Properties In physics, the neutron is a subatomic particle with no net electric charge and a mass of 940 MeV/c² (1. ... For other uses, see Atom (disambiguation). ... The periodic table of the chemical elements A chemical element, or element, is a type of atom that is distinguished by its atomic number; that is, by the number of protons in its nucleus. ... For alternative meanings see proton (disambiguation). ... The nucleus of an atom is the very small dense region, of positive charge, in its centre consisting of nucleons (protons and neutrons). ...

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). Neutronium is a term used in science fiction and popular literature to refer to an extremely dense phase of matter composed primarily of neutrons. ... This article is about matter in physics and chemistry. ... The Periodic Table redirects here. ... The Chemical Galaxy Longman Version The Chemical Galaxy is a new periodic table designed by Philip Stewart in November 2004 based on the cyclical nature of characteristics of the chemical elements (which depend principally on the valence electrons). ...

In computer science

Numbering from 1 or 0?...

The most common practice throughout human history has been to start counting at one. Nevertheless, in computer science zero is often used as the starting point. For example, in almost all old programming languages, an array starts from 1 by default. As programming languages have developed, it has become more common that an array starts from zero by default, the "first" index in the array being 0. In particular, the popularity of the C programming language in the 1980s has made this approach common. Computer science, or computing science, is the study of the theoretical foundations of information and computation and their implementation and application in computer systems. ... A programming language is an artificial language that can be used to control the behavior of a machine, particularly a computer. ... For the microarray in genetics, see SNP array. ... This article is about the number one. ... Default in computer science refers to a default setting or value automatically assigned to a programme or device. ... C is a general-purpose, block structured, procedural, imperative computer programming language developed in 1972 by Dennis Ritchie at the Bell Telephone Laboratories for use with the Unix operating system. ...

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. Modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic because of its use in the 24-hour clock system) is a system of arithmetic for integers, where numbers wrap around after they reach a certain value — the modulus. ... In mathematics and especially in abstract algebra, a congruence relation or simply congruence is an equivalence relation that is compatible with some algebraic operation(s). ...

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: 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 repr