This article is about the number and digit 0. It is not to be confused with the letter O or the O distinguish used to represent affirmation. “ Naught ” redirects here. For the Stolen Babies album, see Naught ( album )
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natural number
0 ( zero ) is a count, [ 1 ] and the numeric finger used to represent that number in numerals. It fulfills a central function in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures. As a digit, 0 is used as a placeholder in position prize systems. Names for the phone number 0 in English include zero, nought ( UK ), naught ( US ; ), nil, or—in context where at least one adjacent digit distinguishes it from the letter “ O ” — oh or o ( ). Informal or slang terms for zero include zilch and zip. [ 2 ] Ought and aught ( ), [ 3 ] equally well as cipher, [ 4 ] have besides been used historically. [ 5 ] [ 6 ]
etymology
The discussion zero came into the english linguistic process via french zéro from the italian zero, a contraction of the venetian zevero form of italian zefiro via ṣafira or ṣifr. [ 7 ] In preIslamic time the discussion ṣifr ( Arabic صفر ) had the mean “ empty ”. [ 8 ] Sifr evolved to mean zero when it was used to translate śūnya ( Sanskrit : शून्य ) from India. [ 8 ] The first base known English consumption of zero was in 1598. [ 9 ] The italian mathematician Fibonacci ( c. 1170–1250 ), who grew up in North Africa and is credited with introducing the decimal system to Europe, used the condition zephyrum. This became zefiro in italian, and was then contracted to zero in Venetian. The italian give voice zefiro was already in being ( meaning “ west wind instrument ” from Latin and Greek zephyrus ) and may have influenced the spelling when transcribing Arabic ṣifr. [ 10 ]
modern usage
Depending on the context, there may be unlike words used for the issue zero, or the concept of zero. For the elementary notion of miss, the words “ nothing ” and “ none ” are often used. sometimes, the parole “ zero ” or “ nothing ” is used. It is frequently called “ ohio ” in the context of call numbers. Slang words for zero include “ travel rapidly ”, “ nothing ”, “ nothing ”, and “ boodle. ” [ 11 ] “ nothing ” is used for many sports in british English. several sports have specific words for a mark of zero, such as “ love “ in tennis – from french l’oeuf, “ the egg ” – and “ hedge “ in cricket, a shortening of “ duck ‘s egg ” ; “ goose testis ” is another general slang term used for zero. [ 11 ]
history
ancient Near East
nfr 
heart with trachea beautiful, pleasant, good 


Ancient egyptian numerals were of basis 10. [ 12 ] They used hieroglyph for the digits and were not positional. By 1770 BC, the Egyptians had a symbol for zero in accounting textbook. The symbol nfr, meaning beautiful, was besides used to indicate the base level in drawings of tombs and pyramids, and distances were measured relative to the base line as being above or below this production line. [ 13 ] By the middle of the 2nd millennium BC, the babylonian mathematics had a sophisticate sexagesimal positional numeral system. The miss of a positional value ( or zero ) was indicated by a space between sexagesimal numerals. In a pill unearthed at Kish ( dating to angstrom early as 700 BC ), the scribe Bêlbânaplu used three hooks as a proxy in the same Babylonian organization. [ 14 ] By 300 BC, a punctuation symbol ( two slanted wedges ) was coopted to serve as this placeholder. [ 15 ] [ 16 ] The babylonian proxy was not a true zero because it was not used alone, nor was it used at the end of a number. therefore 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 concluding sexagesimal placeholder. only context could differentiate them. [ citation needed ]
precolumbian america
The Mesoamerican Long Count calendar developed in southcentral Mexico and Central America required the use of zero as a proxy within its vigesimal ( base20 ) 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. [ a ] Since the eight earliest Long Count dates appear outside the Maya fatherland, [ 17 ] it is broadly believed that the practice of zero in the Americas predated the Maya and was possibly the invention of the Olmecs. [ 18 ] Many of the earliest long Count dates were found within the Olmec heartland, although the Olmec culture ended by the fourth century BC, several centuries before the earliest known Long Count dates .
Maya numeral zero. Although zero became an integral contribution of Maya numerals, with a different, empty tortoise like “ shell human body “ used for many depictions of the “ zero ” numeral, it is assumed not to have influenced Old World numeral systems. Quipu, a knotted cord device, used in the Inca Empire and its predecessor societies in the Andean region to record accounting and other digital data, is encoded in a base ten positional organization. Zero is represented by the absence of a knot in the appropriate position .
classical antiquity
The ancient Greeks had no symbol for zero ( μηδέν ), and did not use a finger proxy for it. [ 19 ] They seemed diffident about the condition of zero as a number. They asked themselves, “ How can nothing be something ? “, leading to philosophic and, by the chivalric menstruation, religious arguments about the nature and universe of zero and the void. The paradoxes of Zeno of Elea depend in big part on the unsealed rendition of zero. [ 20 ]
model of the early greek symbol for zero ( lower right corner ) from a 2ndcentury papyrus By AD 150, Ptolemy, influenced by Hipparchus and the Babylonians, was using a symbol for zero ( —° ) [ 21 ] [ 22 ] in his work on mathematical astronomy called the Syntaxis Mathematica, besides known as the Almagest. [ 23 ] This hellenic zero was possibly the earliest documented use of a numeral representing zero in the Old World. [ 24 ] Ptolemy used it many times in his Almagest ( VI.8 ) for the order of magnitude of solar and lunar eclipses. It represented the value of both digits and minutes of ingress at beginning and last contact. Digits varied endlessly from 0 to 12 to 0 as the Moon passed over the Sun ( a triangular pulse ), where twelve digits was the angular diameter of the Sun. Minutes of ingress was tabulated from 0′0″ to 31′20″ to 0′0″, where 0′0″ used the symbol as a placeholder in two positions of his sexagesimal positional numeral system, [ b ] while the combination meant a zero angle. Minutes of submersion was besides a continuous function 1/12 31′20″ √d ( 24−d ) ( a trilateral pulsate with convex sides ), where five hundred was the digit function and 31′20″ was the sum of the radius of the Sun ‘s and Moon ‘s phonograph record. [ 25 ] Ptolemy ‘s symbol was a placeholder equally well as a number used by two continuous mathematical functions, one within another, so it meant zero, not none. The earliest use of zero in the calculation of the julian Easter occurred before AD 311, at the first submission in a board of epacts as preserved in an Ethiopic document for the years AD 311 to 369, using a Ge’ez give voice for “ none ” ( english translation is “ 0 ” elsewhere ) alongside Ge’ez numerals ( based on Greek numerals ), which was translated from an equivalent table published by the Church of Alexandria in Medieval Greek. [ 26 ] This function was repeated in AD 525 in an equivalent table, that was translated via the Latin nulla or “ none ” by Dionysius Exiguus, aboard Roman numerals. [ 27 ] When division produced zero as a remainder, nihil, meaning “ nothing ”, was used. These medieval zeros were used by all future medieval calculators of Easter. The initial “ N ” was used as a zero symbol in a mesa of Roman numerals by Bede —or his colleagues around AD 725. [ 28 ]
China
A History of Mathematics. An empty space is used to represent zero.[29] This is a depiction of zero expressed in chinese count rods, based on the model provided by. An empty space is used to represent zero. The Sūnzĭ Suànjīng, of unknown date but estimated to be dated from the 1st to 5th centuries AD, and japanese records dated from the eighteenth century, describe how the c. fourth century BC Chinese counting rods organization enabled one to perform decimal calculations. As noted in Xiahou Yang ‘s Suanjing ( 425–468 AD ) that states that to multiply or divide a number by 10, 100, 1000, or 10000, all one needs to do, with rods on the counting circuit board, is to move them forwards, or back, by 1, 2, 3, or 4 places, [ 30 ] According to A History of Mathematics, the rods “ gave the decimal representation of a count, with an empty space denoting nothing. ” [ 29 ] The count rod system is considered a positional notation arrangement. [ 31 ] In AD 690, Empress Wǔ promulgated Zetian characters, one of which was “ 〇 ” ; primitively meaning ‘star ‘, it subsequently [ when? ] came to represent zero. Zero was not treated as a total at that time, but as a “ vacant side ”. [ 32 ] Qín Jiǔsháo ‘s 1247 Mathematical Treatise in Nine Sections is the oldest surviving chinese numerical textbook using a polish symbol for nothing. [ 33 ] chinese authors had been familiar with the idea of minus numbers by the Han Dynasty ( second hundred AD ), as seen in The Nine Chapters on the Mathematical Art. [ 34 ]
India
Pingala ( c. 3rd/2nd hundred BC [ 35 ] ), a Sanskrit prosody scholar, [ 36 ] used binary numbers in the form of short and retentive syllables ( the latter equal in duration to two inadequate syllables ), a note similar to Morse code. [ 37 ] Pingala used the Sanskrit word śūnya explicitly to refer to zero. [ 35 ] The concept of nothing as a written digit in the decimal place value notation was developed in India, presumably a early as during the Gupta period ( c. fifth hundred ), with the oldest unequivocal evidence dating to the seventh hundred. [ 38 ] A symbol for zero, a boastfully scatter likely to be the precursor of the stillcurrent hollow symbol, is used throughout the Bakhshali manuscript, a practical manual on arithmetical for merchants. [ 39 ] In 2017, three samples from the manuscript were shown by radiocarbon dating to come from three different centuries : from AD 224–383, AD 680–779, and AD 885–993, making it South Asia ‘s oldest recorded use of the zero symbol. It is not known how the birch bark fragments from different centuries forming the manuscript came to be packaged together. [ 40 ] [ 41 ] [ 42 ] The Lokavibhāga, a Jain text on cosmology surviving in a medieval Sanskrit translation of the Prakrit original, which is internally dated to AD 458 ( Saka earned run average 380 ), uses a decimal placevalue system, including a zero. In this text, śūnya ( “ void, empty ” ) is besides used to refer to zero. [ 43 ] The Aryabhatiya ( c. 500 ), states sthānāt sthānaṁ daśaguṇaṁ syāt “ from position to place each is ten times the predate. ” [ 44 ] [ 45 ] [ 46 ] Rules governing the use of zero appeared in Brahmagupta ‘s Brahmasputha Siddhanta ( seventh century ), which states the sum of zero with itself as zero, and falsely class by zero as : [ 47 ] [ 48 ]
A positive or damaging number when divided by zero is a fraction with the zero as denominator. Zero divided by a negative or plus number is either zero or is expressed as a divide with zero as numerator and the finite quantity as denominator. Zero divided by zero is zero .
epigraphy
The number 605 in Khmer numerals, from the Sambor inscription ( Saka earned run average 605 corresponds to AD 683 ). The earliest known material use of zero as a decimal fraction digit. There are numerous bull plate inscriptions, with the like little o in them, some of them possibly dated to the sixth century, but their date or authenticity may be open to doubt. [ 14 ] A stone pad found in the ruins of a temple near Sambor on the Mekong, Kratié Province, Cambodia, includes the inscription of “ 605 ” in Khmer numerals ( a fixed of numeral glyph for the Hindu–Arabic numeral system ). The number is the year of the dedication in the Saka era, corresponding to a date of AD 683. [ 49 ] The first base known use of particular glyph for the decimal digits that includes the beyond doubt appearance of a symbol for the digit zero, a little circle, appears on a stone inscription found at the Chaturbhuj Temple, Gwalior, in India, dated 876. [ 50 ] [ 51 ] Zero is besides used as a proxy in the Bakhshali manuscript, portions of which date from AD 224–383. [ 52 ]
Middle Ages
transmission to Islamic polish
The Arabic language inheritance of science was largely greek, [ 53 ] followed by Hindu influences. [ 54 ] In 773, at AlMansur ‘s behest, translations were made of many ancient treatises including Greek, Roman, Indian, and others. In AD 813, astronomic tables were prepared by a iranian mathematician, Muḥammad ibn Mūsā alKhwārizmī, using Hindu numerals ; [ 54 ] and about 825, he published a record synthesizing greek and Hindu cognition and besides contained his own contribution to mathematics including an explanation of the practice of zero. [ 55 ] This record was subsequently translated into Latin in the twelfth hundred under the title Algoritmi de numero Indorum. This style means “ alKhwarizmi on the Numerals of the Indians ”. The word “ Algoritmi ” was the translator ‘s Latinization of AlKhwarizmi ‘s name, and the news “ Algorithm ” or “ algorism ” started to acquire a intend of any arithmetical based on decimals. [ 54 ] Muhammad ibn Ahmad alKhwarizmi, in 976, stated that if no total appears in the place of tens in a calculation, a small circle should be used “ to keep the rows ”. This lap was called ṣifr. [ 56 ]
transmission to Europe
The Hindu–Arabic numeral organization ( base 10 ) reached Western Europe in the eleventh hundred, via AlAndalus, through spanish Muslims, the Moors, together with cognition of classical astronomy and instruments like the astrolabe ; Gerbert of Aurillac is credited with reintroducing the suffer teachings into Catholic Europe. For this argue, the numerals came to be known in Europe as “ Arabic numerals ”. The italian mathematician Fibonacci or Leonardo of Pisa was instrumental in bringing the system into european mathematics in 1202, stating :
After my beget ‘s appointment by his fatherland as submit official in the customs house of Bugia for the Pisan merchants who thronged to it, he took charge ; and in scene of its future utility 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 education in the art, to the nine digits of the Hindus, the cognition 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 vary methods ; and at these places thereafter, while on business. I pursued my study in astuteness and learned the banter of debate. But all this flush, and the algorism, vitamin a well as the art of Pythagoras, I considered as about a error in deference to the method acting of the Hindus ( Modus Indorum ). consequently, embracing more strictly that method acting of the Hindus, and taking hardandfast pains in its study, while adding certain things from my own agreement and insert besides certain things from the niceties of Euclid ‘s geometric art. I have striven to compose this book in its entirety a intelligibly as I could, dividing it into fifteen chapters. Almost everything which I have introduced I have displayed with claim proofread, in order that those further seeking this cognition, with its preeminent method, might be instructed, and far, in order that the Latin people might not be discovered to be without it, as they have been up to nowadays. If I have possibly omitted anything more or less proper or necessity, I beg indulgence, since there is no one who is blameless and absolutely 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 phone number may be written. [ 57 ] [ 58 ] [ 59 ]
here Leonardo of Pisa uses the phrase “ sign 0 ”, indicating it is like a signal to do operations like addition or multiplication. From the thirteenth century, manuals on calculation ( adding, multiplying, extracting roots, etc. ) became common in Europe where they were called algorismus after the iranian mathematician alKhwārizmī. The most popular was written by Johannes de Sacrobosco, about 1235 and was one of the earliest scientific books to be printed in 1488. Until the recently fifteenth century, Hindu–Arabic numerals seem to have predominated among mathematicians, while merchants preferred to use the Roman numerals. In the sixteenth hundred, they became normally used in Europe .
Mathematics
0 is the integer immediately preceding 1. Zero is an even number [ 60 ] because it is divisible by 2 with no remainder. 0 is neither incontrovertible nor negative, [ 61 ] or both plus and negative. [ 62 ] many definitions [ 63 ] include 0 as a natural act, in which subject it is the only natural number that is not positive. Zero is a number which quantifies a count or an come of nothing size. In most cultures, 0 was identified before the mind of negative things ( i.e., quantities less than zero ) was accepted. As a rate or a number, zero is not the lapp as the digit zero, used in numeral systems with positional notation. consecutive positions of digits have higher weights, so the digit zero is used inside a numeral to skip a position and give allow weights to the past and following digits. A zero finger is not constantly necessary in a positional number system ( e.g., the number 02 ). In some instances, a leading zero may be used to distinguish a total .
elementary algebra
The number 0 is the smallest nonnegative integer. The natural number following 0 is 1 and no natural issue precedes 0. The count 0 may or may not be considered a natural number, but it is an integer, and therefore a rational number and a real number issue ( arsenic well as an algebraic number and a complex total ). The issue 0 is neither cocksure nor negative, and is normally displayed as the cardinal number in a act channel. It is neither a prime number nor a composite number. It can not be prime because it has an space total of factors, and can not be complex because it can not be expressed as a product of prime numbers ( as 0 must constantly be one of the factors ). [ 64 ] Zero is, however, even ( i.e. a multiple of 2, a well as being a multiple of any early integer, rational number, or real numeral ). The succeed are some basic ( elementary ) rules for dealing with the number 0. These rules apply for any real or complex number x, unless otherwise stated.
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 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 (no real number multiplied by 0 produces 1), a consequence of the previous rule.
 Exponentiation: x0 =
x
/
x
= 1, except that the case x = 0 may be left undefined in some contexts. For all positive real x, 0 x = 0.
The saying 0/0, which may be obtained in an attempt to determine the terminus ad quem of an expression of the form f ( x ) / g ( x ) as a resultant role of applying the lim operator independently to both operands of the divide, is a alleged “ indeterminate mannequin “. That does not plainly 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 principle. The summarize of 0 numbers ( the empty sum ) is 0, and the product of 0 numbers ( the empty product ) is 1. The factorial 0 ! evaluates to 1, as a particular case of the empty product .
other branches of mathematics
related mathematical terms
 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. This is related to 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. The zero function is the only function that is both even and odd. 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 noninvertible square matrices is a zero map.
 Several branches of mathematics have zero elements, which generalize either the property 0 + x = x, or the property 0 · x = 0, or both.
Physics
The value zero plays a limited 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 randomly chosen. For case, for an absolute temperature ( as measured in kelvins ), zero is the lowest possible respect ( negative temperatures are defined, but negativetemperature systems are not actually colder ). This is in contrast to for example temperatures on the Celsius scale, where zero is randomly defined to be at the freezing point of water. Measuring sound intensity in decibels or phons, the zero flush is randomly set at a mention value—for exemplar, at a value for the brink of hearing. In physics, the zeropoint energy is the lowest possible energy that a quantum mechanical physical system may possess and is the department of energy of the grate state of the system .
chemistry
Zero has been proposed as the atomic count of the theoretical element tetraneutron. It has been shown that a cluster of four neutrons may be stable adequate to be considered an atom in its own veracious. This would create an chemical element with no protons and no care on its nucleus. ampere early on as 1926, Andreas von Antropoff coined the condition neutronium for a speculate shape of topic made up of neutrons with no protons, which he placed as the chemical element of atomic number zero at the mind of his new interpretation of the periodic postpone. It was subsequently placed as a noble flatulence in the middle of several corkscrew representations of the periodic system for classifying the chemical elements .
Computer science
The most coarse practice throughout human history has been to start counting at one, and this is the practice in early classic computer programming languages such as Fortran and COBOL. however, in the belated 1950s LISP introduced zerobased number for arrays while Algol 58 introduced wholly flexible freebase for array subscripts ( allowing any incontrovertible, negative, or zero integer as floor for array subscripts ), and most subsequent program languages adopted one or other of these positions. For exemplar, the elements of an align are numbered starting from 0 in C, so that for an array of n items the sequence of array indices runs from 0 to n −1. This permits an range chemical element ‘s placement to be calculated by adding the index directly to address of the array, whereas 1based languages precalculate the range ‘s basis savoirfaire to be the position one element before the first. [ citation needed ] There can be confusion between 0 and 1based indexing, for exemplar Java ‘s JDBC indexes parameters from 1 although Java itself uses 0based index. [ citation needed ] In databases, it is possible for a field not to have a value. It is then said to have a nothing value. [ 65 ] For numeral fields it is not the respect zero. For text fields this is not blank nor the empty string. The presence of nothing values leads to threevalued logic. No long is a stipulate either true or false, but it can be undetermined. Any calculation including a nothing value delivers a null solution. [ 66 ] A null pointer is a pointer in a calculator plan that does not point to any object or function. In C, the integer constant 0 is converted into the nothing pointer at compile time when it appears in a pointer context, and so 0 is a standard means to refer to the null pointer in code. however, the internal representation of the nothing pointer may be any moment form ( possibly different values for different data types ). [ citation needed ] In mathematics −0 = +0 = 0 ; both −0 and +0 represent precisely the same act, i.e., there is no “ positive zero ” or “ negative zero ” clearcut from zero. however, in some calculator hardware signed number representations, zero has two distinct representations, a positive one grouped with the plus numbers and a negative one group with the negatives ; this kind of dual representation is known as sign zero, with the latter form sometimes called negative zero. These representations include the signed magnitude and one ‘s complement binary integer representations ( but not the two ‘s complement binary form used in most modern computers ), and most float point total representations ( such as IEEE 754 and IBM S/390 floating steer formats ). In binary star, 0 represents the value for “ off ”, which means no electricity menstruate. Zero is the value of faithlessly in many program languages. The Unix epoch ( the date and prison term associated with a zero timestamp ) begins the midnight before the first of January 1970. [ 68 ] [ 69 ] [ 70 ] The Classic Mac OS era and Palm OS epoch ( the date and time associated with a zero timestamp ) begins the midnight before the first of January 1904. [ 71 ] many APIs and operating systems that require applications to return an integer value as an exit status typically use zero to indicate success and nonzero values to indicate particular error or warning conditions. Programmers much use a slashed zero to avoid confusion with the letter “ O “. [ 72 ]
early fields
Symbols and representations
The modern numerical finger 0 is normally written as a circle or ellipse. traditionally, many print typefaces made the capital letter O more round off than the minute, eggshaped digit 0. [ 74 ] Typewriters primitively made no eminence in condition between O and 0 ; some models did not even have a separate key for the digit 0. The distinction came into bulge on modern character displays. [ 74 ] A slash zero can be used to distinguish the number from the letter. The digit 0 with a dot in the center seems to have originated as an option on IBM 3270 displays and has continued with some modern calculator typefaces such as Andalé Mono, and in some airline reservation systems. One variation uses a abruptly vertical bar alternatively of the department of transportation. Some fonts designed for use with computers made one of the capitalO–digit0 match more round and the other more angular ( closer to a rectangle ). A far distinction is made in falsificationhindering font as used on german car number plates by slitting open the digit 0 on the amphetamine correct side. Sometimes the digit 0 is used either entirely, or not at all, to avoid confusion wholly .
class pronounce
In the BC calendar era, the class 1 BC is the first class earlier AD 1 ; there is not a year zero. By contrast, in astronomic year number, the class 1 BC is total 0, the year 2 BC is numbered −1, and therefore forth. [ 75 ]
See besides
Notes
 ^ No long count date actually using the number 0 has been found before the third hundred AD, but since the long consider organization would make no sense without some proxy, and since Mesoamerican glyphs do not typically leave empty spaces, these earlier dates are taken as collateral attest that the concept of 0 already existed at the clock time .
 ^ 0 to 59, where 31 was written λα meaning 30+1, and 20 was written κ meaning 20. Each place in Ptolemy ‘s sexagesimal system was written in greek numerals from, where 31 was written λα meaning 30+1, and 20 was written κ meaning 20 .
References
bibliography
historic studies
 Bourbaki, Nicolas (1998). Elements of the History of Mathematics. Berlin, Heidelberg, and New York: SpringerVerlag. ISBN 3540647678.
 Diehl, Richard A. (2004). The Olmecs: America’s First Civilization. London: Thames & Hudson.
 Ifrah, Georges (2000). The Universal History of Numbers: From Prehistory to the Invention of the Computer. Wiley. ISBN 0471393401.
 Kaplan, Robert (2000). The Nothing That Is: A Natural History of Zero. Oxford University Press.
 Seife, Charles (2000). Zero: The Biography of a Dangerous Idea. Penguin USA. ISBN 0140296476.
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