0 - Knowledge

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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

1637: 1022: 1038: 1646: 5063: 2433: 1879:'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 4365:

the possible meters for any value "n". The answer is (2) = 128, as expected, but instead of seven doublings, the process (explained by the sutra) required only three doublings and two squarings – a handy time saver where "n" is large. Pingala's use of a zero symbol as a marker seems to be the first known explicit reference to zero

1426:(425–468 AD), to multiply or divide a number by 10, 100, 1000, or 10000, all one needs to do, with rods on the counting board, is to move them forwards, or back, by 1, 2, 3, or 4 places. The rods gave the decimal representation of a number, with an empty space denoting zero. The counting rod system is a 3028:

No long count date actually using the number 0 has been found before the 3rd century AD, but since the long count system would make no sense without some placeholder, and since Mesoamerican glyphs do not typically leave empty spaces, these earlier dates are taken as indirect evidence that the concept

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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

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In the Chandah-sutra of Pingala, dating perhaps the third or second century BC, Pingala's use of a zero symbol as a marker seems to be the first known explicit reference to zero. ... In the Chandah-sutra of Pingala, dating perhaps the third or second century BC, there are five questions concerning

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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

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bark fragments from different centuries forming the manuscript came to be packaged together. If the writing on the oldest birch bark fragments is as old as those fragments, it represents South Asia's oldest recorded use of a zero symbol. However, it is possible that the writing dates instead to the

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for representing numbers), the digit 0 plays the role of a placeholder, indicating that certain powers of the base do not contribute. For example, the decimal number 205 is the sum of two hundreds and five ones, with the 0 digit indicating that no tens are added. The digit plays the same role in

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201 may be pronounced "two oh one", and the year 1907 is often pronounced "nineteen oh seven". The presence of other digits, indicating that the string contains only numbers, avoids confusion with the letter O. For this reason, systems that include strings with both letters and numbers (such as

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introduced completely flexible basing for array subscripts (allowing any positive, negative, or zero integer as base for array subscripts), and most subsequent programming languages adopted one or other of these positions. For example, the elements of an array are numbered starting from 0 in

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time period of the youngest fragments, AD 885–993. The latter dating has been argued to be more consistent with the sophisticated use of zero within the document, as portions of it appear to show zero being employed as a number in its own right, rather than only as a positional placeholder.

2004:, and in some airline reservation systems. One variation uses a short vertical bar instead of the dot. 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). A further distinction is made in 1145:. Greeks seemed to have a philosophical opposition to using zero as a number. Other scholars give the Greek partial adoption of the Babylonian zero a later date, with neuroscientist Andreas Nieder giving a date of after 400 BC and mathematician Robert Kaplan dating it after the 5822:

In regard to services, sending a null value as an argument in a remote service call means that no data is sent. Because the receiving parameter is nullable, the receiving function creates a new, uninitialized value for the missing data then passes it to the requested service

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20″ was the sum of the radii of the Sun's and Moon's discs. Ptolemy's symbol was a placeholder as well as a number used by two continuous mathematical functions, one within another, so it meant zero, not none. Over time, Ptolemy's zero tended to increase in size and lose the

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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

3195:. Annual Report of the Board of Regents of the Smithsonian Institution; Harvard University Archives. "Sifr occurs in the meaning of "empty" even in the pre-Islamic time. ... Arabic sifr in the meaning of zero is a translation of the corresponding India sunya." 4923:– Indian astronomical treatises dating as far back as 425 BC; these versions may have the vehicle through which the "Arabic" numerals and the zero were brought from India into Islam. In 813, al-Khwarizmi used the Hindu numerals in his astronomical tables. 2026:

The concept of zero plays multiple roles in mathematics: as a digit, it is an important part of positional notation for representing numbers, while it also plays an important role as a number in its own right in many algebraic settings.

1784:, using Hindu numerals; and about 825, he published a book synthesizing Greek and Hindu knowledge and also contained his own contribution to mathematics including an explanation of the use of zero. This book was later translated into 2643:

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, for an

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hieroglyph was being used as a symbol for zero. The same symbol was also 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 line.

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can be defined for some physical systems, but negative-temperature systems are not actually colder.) This is in contrast to temperatures on the Celsius scale, for example, where zero is arbitrarily defined to be at the

1875:. 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 813:

Depending on the context, there may be different words used for the number zero, or the concept of zero. For the simple notion of lacking, the words "nothing" and "none" are often used. The British English words

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for zero. The origin of this symbol is unknown; it may have been borrowed from Indian sources or produced by modifying a square symbol. Chinese authors had been familiar with the idea of negative numbers by the

815: 1996:) is often used to distinguish the number from the letter (mostly in computing, navigation and in the military, for example). The digit 0 with a dot in the center seems to have originated as an option on 2054:

of other real numbers (indicating whether any tenths, hundredths, thousandths, etc., are present) and in bases other than 10 (for example, in binary, where it indicates which powers of 2 are omitted).

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but without an explicit exponent, and so only distinguished implicitly from context. The zero-like placeholder mark was only ever used in between digits, but never alone or at the end of a number.

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Since the eight earliest Long Count dates appear outside the Maya homeland, it is generally believed that the use of zero in the Americas predated the Maya and was possibly the invention of the

2133:: it is not prime because prime numbers are greater than 1 by definition, and it is not composite because it cannot be expressed as the product of two smaller natural numbers. (However, the 5030: 4467: 5972: 2838:, zero has two distinct representations, a positive one grouped with the positive numbers and a negative one grouped with the negatives. This kind of dual representation is known as 5168:

Bemer, R. W. (1967). "Towards standards for handwritten zero and oh: much ado about nothing (and a letter), or a partial dossier on distinguishing between handwritten zero and oh".

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corresponding to the place containing a 0 does not contribute to the total. For example, "205" in decimal means two hundreds, no tens, and five ones. The same principle applies in

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is credited with reintroducing the lost teachings into Catholic Europe. For this reason, the numerals came to be known in Europe as "Arabic numerals". The Italian mathematician

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originally made no distinction in shape between O and 0; some models did not even have a separate key for the digit 0. The distinction came into prominence on modern character

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results for three samples from the manuscript, indicating that they came from three different centuries: from AD 224–383, AD 680–779, and AD 885–993. It is not known how the

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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.

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word 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

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In mathematics, there is no "positive zero" or "negative zero" distinct from zero; both −0 and +0 represent exactly the same number. However, in some computer hardware

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indexing. The most common practice throughout human history has been to start counting at one, and this is the practice in early classic programming languages such as

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of immersion at first and last contact. Digits varied continuously from 0 to 12 to 0 as the Moon passed over the Sun (a triangular pulse), where twelve digits was the

2936:, recognition that some animals display awareness of the concept of zero leads to the conclusion that the capability for numerical abstraction arose early in the 4975:, remarked that if, in a calculation, no number appears in the place of tens, a little circle should be used "to keep the rows". This circle the Mosloems called 3557: 3152: 2113:

The number 0 can be regarded as neither positive nor negative or, alternatively, both positive and negative and is usually displayed as the central number in a

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by slitting open the digit 0 on the upper right side. In some systems either the letter O or the numeral 0, or both, are excluded from use, to avoid confusion.

3475: 1792:. This title means "al-Khwarizmi on the Numerals of the Indians". The word "Algoritmi" was the translator's Latinization of Al-Khwarizmi's name, and the word " 79: 68: 1806:, in 976, stated that if no number appears in the place of tens in a calculation, a little circle should be used "to keep the rows". This circle was called 1738:

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

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BC came 1 January AD 1. ... If you object to that no-year-zero scheme, then don't use it: use the astronomer's counting scheme, with negative year numbers.

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The Arabic inheritance of science was overwhelmingly Greek, but Hindu influences ranked next. In 773, at Mansur's behest, translations were made of the

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Greeks seemed unsure about the status of zero as a number. Some of them asked themselves, "How can not being be?", leading to philosophical and, by the

3580: 3364: 5020: 4645: 3640: 5713: 4459: 3796: 1756: 1910:, in 1488. The practice of calculating on paper using Hindu–Arabic numerals only gradually displaced calculation by abacus and recording with 5982: 2815:. 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 6371: 1892:

From the 13th century, manuals on calculation (adding, multiplying, extracting roots, etc.) became common in Europe where they were called

1781: 4825: 2452:: 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 5249: 3084:"figure which stands for naught in the Arabic notation," also "the absence of all quantity considered as quantity," c. 1600, from French 1141:). However, after using the Babylonian placeholder zero for astronomical calculations they would typically convert the numbers back into 6449: 6214: 5340: 3937: 1487:, in the form of short and long syllables (the latter equal in length to two short syllables), to identify the possible valid Sanskrit 853:

Slang words for zero include "zip", "zilch", "nada", and "scratch". In the context of sports, "nil" is sometimes used, especially in

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The idea of sunya and place numbers was transmitted to the Arabs who translated sunya or "leave a space" into their language as sifr.

1457: 535: 4414: 3445: 5476: 4542: 12690: 5736: 4130: 2704:, that is, using an "alphabet" that contains only two symbols, usually chosen to be "0" and "1". Binary coding is convenient for 6444: 916: 6381: 5941: 3614: 3323: 2009: 1439: 4049: 4002: 3870: 3079: 1310:, sometimes depicted as a large elongated 0-like omicron "Ο" or as omicron with overline "ō" instead of a dot with overline. 964:

hieroglyph to indicate cases where the amount of a foodstuff received was exactly equal to the amount disbursed. Egyptologist

6327: 6289: 6208: 6184: 6165: 6141: 6087: 6063: 6013: 5773: 5746: 5689: 5662: 5599: 5574: 5462: 5416: 5385: 5334: 5226: 5057: 5009: 4946: 4912: 4678: 4390: 4357: 4303: 4239: 4206: 4104: 4031: 3979: 3842: 3752: 3710: 3592: 3257: 3235: 3211: 3121: 5898: 5878: 5858: 5838: 5232: 1068:. Many of the earliest Long Count dates were found within the Olmec heartland, although the Olmec civilization ended by the 1019:

in that it did not explicitly specify the magnitude of the leading sexagesimal digit, so that for example the lone digit 1 (

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was perhaps the earliest documented use of a numeral representing zero in the Old World. Ptolemy used it many times in his

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had no symbol for zero (μηδέν, pronounced 'midén'), and did not use a digit placeholder for it. According to mathematician

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The oldest, firmly dated use of zero as a decimal figure, found on the Sambor Inscription. The number "605" is written in

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positional numeral system, while the combination meant a zero angle. Minutes of immersion was also a continuous function

59: 6496: 5679: 5652: 4996:. Sources and Studies in the History of Mathematics and Physical Sciences. Translated by Sigler, Laurence E. Springer. 3666: 1526:. A symbol for zero, a large dot likely to be the precursor of the still-current hollow symbol, is used throughout the 1001: 3547: 3138: 6349: 6308: 6270: 6119: 5924: 5520: 4871: 3916: 2889: 669: 550: 5298: 4718: 3467: 6155: 2148:

The following are some basic rules for dealing with the number 0. These rules apply for any real or complex number

1444: 1046: 2673:, the zero level is arbitrarily set at a reference value—for example, at a value for the threshold of hearing. In 2713: 2459: 1819: 1803: 1724: 1016: 2563:. As a function from the real numbers to the real numbers, the zero function is the only function that is both 13036: 12721: 4813: 2863: 1604: 5801: 1704:

in them, some of them possibly dated to the 6th century, but their date or authenticity may be open to doubt.

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were used as a zero symbol for these Long Count dates, the earliest of which (on Stela 2 at Chiapa de Corzo,

6366: 767:), who grew up in North Africa and is credited with introducing the decimal system to Europe, used the term 12980: 12767: 4576: 1049:

developed in south-central Mexico and Central America required the use of zero as a placeholder within its

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Zero derives from Hindu sunya – meaning void, emptiness – via Arabic sifr, Latin cephirum, Italian zevero.

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French zéro (1515 in Hatzfeld & Darmesteter) or its source Italian zero, for *zefiro, < Arabic çifr

12762: 12747: 12683: 4343: 4125: 3354: 2974: 2790: 2371:". That does not mean that the limit sought is necessarily undefined; rather, it means that the limit of 1146: 1850:

or Leonardo of Pisa was instrumental in bringing the system into European mathematics in 1202, stating:

1773:'s behest, translations were made of many ancient treatises including Greek, Roman, Indian, and others. 13046: 6404: 4637: 3966:. Sources and Studies in the History of Mathematics and Physical Sciences. Springer. pp. 232–235. 3908: 1739: 3509: 1378:, 0 was identified before the idea of negative things (i.e., quantities less than zero) was accepted. 12942: 12901: 12780: 4148: 3142: 2855: 2801: 2645: 2298: 425: 5705: 3788: 2402: 1645: 12786: 6394: 4994:

Fibonacci's Liber Abaci: A Translation into Modern English of Leonardo Pisano's Book of Calculation

4135: 3673:. The Whitney and Betty Macmillan Center for International and Area Studies at Yale. Archived from 1241: 2440:

The role of 0 as the smallest counting number can be generalized or extended in various ways. In

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Today, the numerical digit 0 is usually written as a circle or ellipse. Traditionally, many print

13006: 12729: 12713: 6194: 3148: 2721: 1126: 500: 345: 55: 49: 6005: 5998: 4535:"Carbon dating finds Bakhshali manuscript contains oldest recorded origins of the symbol 'zero'" 4379:"Sanskrit Prosody and Important Literary and Geographical Names in the Ancient History of India" 1969: 1221: 12970: 12790: 12739: 12676: 6489: 6464: 5377: 4817: 4096: 2264: 2214: 2051: 1903: 1375: 1207: 977: 665: 280: 17: 13010: 5246: 4378: 4038:. The pages in this edition have numbers six less than the same pages in the original edition. 3834: 3742: 3292: 3187: 12947: 12876: 5402: 5039: 4728: 4668: 4349: 2999: 2768: 2490: 2260: 1914:. In the 16th century, Hindu–Arabic numerals became the predominant numerals used in Europe. 1484: 1109:

positional system. Zero is represented by the absence of a knot in the appropriate position.

1026: 471:. Adding 0 to any number leaves that number unchanged. In mathematical terminology, 0 is the 35: 6198: 5324: 4324:

In these matrices we find negative numbers, which appear here for the first time in history.

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The Story of Civilization, Volume 4, The Age of Faith: Constantine to Dante – A.D. 325–1300

3306: 3204: 2759: 2709: 2708:, where "0" and "1" can stand for the absence or presence of electrical current in a wire. 2657: 2118: 2107: 1692: 1608:(7th century), which states the sum of zero with itself as zero, and incorrectly describes 1527: 1488: 1422: 1334: 847: 519: 492: 13000: 4785: 2333:, which may be obtained in an attempt to determine the limit of an expression of the form 1175:

Example of the early Greek symbol for zero (lower right corner) from a 2nd-century papyrus

1125:, the ancient Greeks did begin to adopt the Babylonian placeholder zero for their work in 8: 13031: 12906: 12815: 12810: 12804: 12796: 12757: 5361: 5025: 4701: 4623: 4421: 4246:

zero was regarded as a number in India ... whereas the Chinese employed a vacant position

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Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures

4121: 3437: 2859: 2851: 2847: 2842:, with the latter form sometimes called negative zero. These representations include the 2705: 2606: 2364: 2075: 2036: 1956: 1935: 1907: 1835: 1569: 1523: 1427: 1341:. This use was repeated in 525 in an equivalent table, that was translated via the Latin 1161: 1005: 943: 879:, a shortening of "duck's egg". "Goose egg" is another general slang term used for zero. 12996: 6390: 5123: 3995: 2811:

In databases, it is possible for a field not to have a value. It is then said to have a

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The number 0 is also used in several other ways within various branches of mathematics:

12952: 12896: 12800: 12717: 6386: 6053: 5452: 5371: 5185: 5135: 4772: 4764: 4534: 4089: 3830: 3818: 3403: 3386:

Lumpkin, Beatrice (2002). "Mathematics Used in Egyptian Construction and Bookkeeping".

2994: 2873:, in computing terminology, is the date and time associated with a zero timestamp. The 2617: 2580: 2368: 2142: 2021: 1535: 1087:" used for many depictions of the "zero" numeral, it is assumed not to have influenced 984:

positional numeral system. The lack of a positional value (or zero) was indicated by a

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Algebra, with Arithmetic and Mensuration from the Sanscrit of Brahmegupta and Bháscara

1712: 1678: 1656: 13041: 12866: 6482: 6417: 6345: 6323: 6304: 6285: 6266: 6235: 6204: 6180: 6161: 6137: 6115: 6093: 6083: 6059: 6019: 6009: 5920: 5769: 5742: 5685: 5658: 5595: 5570: 5516: 5468: 5458: 5412: 5381: 5330: 5222: 5139: 5053: 5005: 4942: 4908: 4877: 4867: 4776: 4744: 4707: 4674: 4572: 4504: 4386: 4353: 4299: 4235: 4202: 4100: 4027: 3975: 3912: 3838: 3823: 3748: 3716: 3706: 3588: 3407: 3231: 3207: 3117: 2933: 2870: 2743: 2725: 2682: 2678: 2537: 2509: 2047: 1899: 1843: 1512:

Bakhshali manuscript, with the numeral "zero" represented by a black dot (AD 224–383)

1359:, meaning "nothing", was used. These medieval zeros were used by all future medieval 1348: 947: 931: 738: 543: 472: 83: 6109: 6077: 3331: 1352: 12871: 12856: 6258: 5177: 5127: 5045: 4997: 4756: 4588: 4577:"The Bakhshālī Manuscript: A Response to the Bodleian Library's Radiocarbon Dating" 4198: 4194: 4053: 3967: 3905: 3859: 3395: 2893: 2843: 2689: 2505: 2497: 2240: 2172: 2130: 2122: 2099: 2062: 2042: 1923: 1855: 1609: 1531: 1508: 1480: 1245: 826: 685: 638: 604: 568: 496: 480: 395: 385: 353: 226: 216: 6301:

The Universal History of Numbers: From Prehistory to the Invention of the Computer

5189: 1053:(base-20) positional numeral system. Many different glyphs, including the partial 825:

It is often called "oh" in the context of reading out a string of digits, such as

12885: 12861: 12776: 5977: 5902: 5895: 5882: 5875: 5862: 5855: 5842: 5835: 5253: 5216: 4936: 4859: 4460:"Nothing matters: How India's invention of zero helped create modern mathematics" 4229: 4186: 4019: 3700: 3265: 3111: 2908: 2685: 2623: 2486: 2091: 1025:) might represent any of 1, 60, 3600 = 60, etc., similar to the significand of a 876: 854: 507: 488: 373: 357: 349: 244: 175: 162: 101: 3971: 2900:

typically use zero to indicate success and non-zero values to indicate specific

2267:(no real number multiplied by 0 produces 1), a consequence of the previous rule. 1574: 1501: 12922: 12841: 12725: 6376: 6105: 5506: 5448: 5131: 3039: 2878: 2517: 2478: 2466: 2270: 2200: 2103: 2083: 2079: 1911: 1777: 1720: 1666: 1404: 1338: 1142: 1118: 872: 830: 405: 257: 39: 6097: 6023: 5472: 5001: 4760: 4711: 3720: 2496:

The role of 0 as additive identity generalizes beyond elementary algebra. In

2001: 1530:, a practical manual on arithmetic for merchants. In 2017, researchers at the 13025: 12881: 12733: 12699: 12653: 12567: 12562: 12557: 12552: 12547: 12542: 12537: 12532: 12527: 12516: 6337: 6151: 6129: 6049: 5498: 4568: 4508: 4492: 3959: 3696: 3004: 2955: 2816: 2662: 2653: 2631: 2564: 2482: 2416: 2134: 1417: 1391: 1233: 1229: 1122: 1076: 1012:, a punctuation symbol (two slanted wedges) was repurposed as a placeholder. 965: 858: 838: 523: 435: 415: 361: 267: 6400: 5320: 4026:(Red Sea Press ed.). Red Sea Press. pp. 25, 53, 93, 183, Plate I. 3674: 3615:"Expedition Magazine | Time, Kingship, and the Maya Universe Maya Calendars" 3307:"Collins - Free online dictionary, thesaurus and reference materials - nill" 2785:

There can be confusion between 0- and 1-based indexing; for example, Java's

1363:. The initial "N" was used as a zero symbol in a table of Roman numerals by 12927: 12851: 12751: 12496: 12491: 12486: 12481: 12476: 12471: 12466: 12461: 12456: 12451: 12438: 12433: 12428: 12423: 12418: 12413: 12408: 12403: 12398: 12393: 12380: 12375: 12370: 12365: 12360: 12355: 12350: 12345: 12340: 12335: 12322: 12317: 12312: 12307: 12302: 12297: 12292: 12287: 12282: 12277: 12264: 12259: 12254: 12249: 12244: 12239: 12234: 12229: 12224: 12219: 12206: 12201: 12196: 12191: 12186: 12181: 12176: 12171: 12166: 12161: 12148: 12143: 12138: 12133: 12128: 12123: 12118: 12113: 12108: 12103: 12090: 12085: 12080: 12075: 12070: 12065: 12060: 12055: 12050: 12045: 12032: 12027: 12022: 12017: 12012: 12007: 12002: 11997: 11992: 11987: 11974: 11969: 11964: 11959: 11954: 11949: 11944: 11939: 11934: 11929: 11918: 11899: 11894: 11889: 11884: 11879: 11874: 11869: 11864: 11859: 11854: 11841: 11836: 11831: 11826: 11821: 11816: 11811: 11806: 11801: 11796: 11783: 11778: 11773: 11768: 11763: 11758: 11753: 11748: 11743: 11738: 11725: 11720: 11715: 11710: 11705: 11700: 11695: 11690: 11685: 11680: 11667: 11662: 11657: 11652: 11647: 11642: 11637: 11632: 11627: 11622: 11609: 11604: 11599: 11594: 11589: 11584: 11579: 11574: 11569: 11564: 11551: 11546: 11541: 11536: 11531: 11526: 11521: 11516: 11511: 11506: 11493: 11488: 11483: 11478: 11473: 11468: 11463: 11458: 11453: 11448: 11435: 11430: 11425: 11420: 11415: 11410: 11405: 11400: 11395: 11390: 11377: 11372: 11367: 11362: 11357: 11352: 11347: 11342: 11337: 11332: 11321: 11302: 11297: 11292: 11287: 11282: 11277: 11272: 11267: 11262: 11257: 11244: 11239: 11234: 11229: 11224: 11219: 11214: 11209: 11204: 11199: 11186: 11181: 11176: 11171: 11166: 11161: 11156: 11151: 11146: 11141: 11128: 11123: 11118: 11113: 11108: 11103: 11098: 11093: 11088: 11083: 11070: 11065: 11060: 11055: 11050: 11045: 11040: 11035: 11030: 11025: 11012: 11007: 11002: 10997: 10992: 10987: 10982: 10977: 10972: 10967: 10954: 10949: 10944: 10939: 10934: 10929: 10924: 10919: 10914: 10909: 10896: 10891: 10886: 10881: 10876: 10871: 10866: 10861: 10856: 10851: 10838: 10833: 10828: 10823: 10818: 10813: 10808: 10803: 10798: 10793: 10780: 10775: 10770: 10765: 10760: 10755: 10750: 10745: 10740: 10735: 10724: 10705: 10700: 10695: 10690: 10685: 10680: 10675: 10670: 10665: 10660: 10647: 10642: 10637: 10632: 10627: 10622: 10617: 10612: 10607: 10602: 10589: 10584: 10579: 10574: 10569: 10564: 10559: 10554: 10549: 10544: 10531: 10526: 10521: 10516: 10511: 10506: 10501: 10496: 10491: 10486: 10473: 10468: 10463: 10458: 10453: 10448: 10443: 10438: 10433: 10428: 10415: 10410: 10405: 10400: 10395: 10390: 10385: 10380: 10375: 10370: 10357: 10352: 10347: 10342: 10337: 10332: 10327: 10322: 10317: 10312: 10299: 10294: 10289: 10284: 10279: 10274: 10269: 10264: 10259: 10254: 10241: 10236: 10231: 10226: 10221: 10216: 10211: 10206: 10201: 10196: 10183: 10178: 10173: 10168: 10163: 10158: 10153: 10148: 10143: 10138: 10127: 10108: 10103: 10098: 10093: 10088: 10083: 10078: 10073: 10068: 10063: 10050: 10045: 10040: 10035: 10030: 10025: 10020: 10015: 10010: 10005: 9992: 9987: 9982: 9977: 9972: 9967: 9962: 9957: 9952: 9947: 9934: 9929: 9924: 9919: 9914: 9909: 9904: 9899: 9894: 9889: 9876: 9871: 9866: 9861: 9856: 9851: 9846: 9841: 9836: 9831: 9818: 9813: 9808: 9803: 9798: 9793: 9788: 9783: 9778: 9773: 9760: 9755: 9750: 9745: 9740: 9735: 9730: 9725: 9720: 9715: 9702: 9697: 9692: 9687: 9682: 9677: 9672: 9667: 9662: 9657: 9644: 9639: 9634: 9629: 9624: 9619: 9614: 9609: 9604: 9599: 9586: 9581: 9576: 9571: 9566: 9561: 9556: 9551: 9546: 9541: 9530: 9511: 9506: 9501: 9496: 9491: 9486: 9481: 9476: 9471: 9466: 9453: 9448: 9443: 9438: 9433: 9428: 9423: 9418: 9413: 9408: 9395: 9390: 9385: 9380: 9375: 9370: 9365: 9360: 9355: 9350: 9337: 9332: 9327: 9322: 9317: 9312: 9307: 9302: 9297: 9292: 9279: 9274: 9269: 9264: 9259: 9254: 9249: 9244: 9239: 9234: 9221: 9216: 9211: 9206: 9201: 9196: 9191: 9186: 9181: 9176: 9163: 9158: 9153: 9148: 9143: 9138: 9133: 9128: 9123: 9118: 9105: 9100: 9095: 9090: 9085: 9080: 9075: 9070: 9065: 9060: 9047: 9042: 9037: 9032: 9027: 9022: 9017: 9012: 9007: 9002: 8989: 8984: 8979: 8974: 8969: 8964: 8959: 8954: 8949: 8944: 8933: 8914: 8909: 8904: 8899: 8894: 8889: 8884: 8879: 8874: 8869: 8856: 8851: 8846: 8841: 8836: 8831: 8826: 8821: 8816: 8811: 8798: 8793: 8788: 8783: 8778: 8773: 8768: 8763: 8758: 8753: 8740: 8735: 8730: 8725: 8720: 8715: 8710: 8705: 8700: 8695: 8682: 8677: 8672: 8667: 8662: 8657: 8652: 8647: 8642: 8637: 8624: 8619: 8614: 8609: 8604: 8599: 8594: 8589: 8584: 8579: 8566: 8561: 8556: 8551: 8546: 8541: 8536: 8531: 8526: 8521: 8508: 8503: 8498: 8493: 8488: 8483: 8478: 8473: 8468: 8463: 8450: 8445: 8440: 8435: 8430: 8425: 8420: 8415: 8410: 8405: 8392: 8387: 8382: 8377: 8372: 8367: 8362: 8357: 8352: 8347: 8336: 8317: 8312: 8307: 8302: 8297: 8292: 8287: 8282: 8277: 8272: 8259: 8254: 8249: 8244: 8239: 8234: 8229: 8224: 8219: 8214: 8201: 8196: 8191: 8186: 8181: 8176: 8171: 8166: 8161: 8156: 8143: 8138: 8133: 8128: 8123: 8118: 8113: 8108: 8103: 8098: 8085: 8080: 8075: 8070: 8065: 8060: 8055: 8050: 8045: 8040: 8027: 8022: 8017: 8012: 8007: 8002: 7997: 7992: 7987: 7982: 7969: 7964: 7959: 7954: 7949: 7944: 7939: 7934: 7929: 7924: 7911: 7906: 7901: 7896: 7891: 7886: 7881: 7876: 7871: 7866: 7853: 7848: 7843: 7838: 7833: 7828: 7823: 7818: 7813: 7808: 7795: 7790: 7785: 7780: 7775: 7770: 7765: 7760: 7755: 7750: 7739: 7720: 7715: 7710: 7705: 7700: 7695: 7690: 7685: 7680: 7675: 7662: 7657: 7652: 7647: 7642: 7637: 7632: 7627: 7622: 7617: 7604: 7599: 7594: 7589: 7584: 7579: 7574: 7569: 7564: 7559: 7546: 7541: 7536: 7531: 7526: 7521: 7516: 7511: 7506: 7501: 7488: 7483: 7478: 7473: 7468: 7463: 7458: 7453: 7448: 7443: 7430: 7425: 7420: 7415: 7410: 7405: 7400: 7395: 7390: 7385: 7372: 7367: 7362: 7357: 7352: 7347: 7342: 7337: 7332: 7327: 7314: 7309: 7304: 7299: 7294: 7289: 7284: 7279: 7274: 7269: 7256: 7251: 7246: 7241: 7236: 7231: 7226: 7221: 7216: 7211: 7198: 7193: 7188: 7183: 7178: 7173: 7168: 7163: 7158: 6073: 5502: 4497:"Much ado about nothing: ancient Indian text contains earliest zero symbol" 3552: 3192: 3009: 2958: 2805: 2717: 2568: 2521: 2513: 2501: 2474: 2126: 1963: 1880: 1433:

Zero was not treated as a number at that time, but as a "vacant position".

1237: 1165: 1156:

period, religious arguments about the nature and existence of zero and the

993: 515: 208: 5215:

Bunt, Lucas Nicolaas Hendrik; Jones, Phillip S.; Bedient, Jack D. (1976).

5181: 5049: 4881: 2724:. 0 plays various important roles in high-level languages. For example, a 589:. In contexts where at least one adjacent digit distinguishes it from the 12648: 7123: 7118: 7113: 7108: 7103: 7098: 7093: 7088: 7083: 7078: 7065: 7060: 7055: 7050: 7045: 7040: 7035: 7030: 7025: 7020: 7007: 7002: 6997: 6992: 6987: 6982: 6977: 6972: 6967: 6962: 6949: 6944: 6939: 6934: 6929: 6924: 6919: 6914: 6909: 6904: 6891: 6886: 6881: 6876: 6871: 6866: 6861: 6856: 6851: 6846: 6833: 6828: 6823: 6818: 6813: 6808: 6803: 6798: 6793: 6788: 6775: 6770: 6765: 6760: 6755: 6750: 6745: 6740: 6735: 6730: 6717: 6712: 6707: 6702: 6697: 6692: 6687: 6682: 6677: 6672: 6659: 6654: 6649: 6644: 6639: 6634: 6629: 6624: 6619: 5738:

Understanding and Using C Pointers: Core Techniques for Memory Management

5560: 5290: 4900: 4564: 4339: 2897: 2839: 2701: 2627: 2610: 2445: 2367:

operator independently to both operands of the fraction, is a so-called "

2178: 2138: 2114: 2095: 2067: 1859: 1599: 1583: 1548: 1449: 1434: 1298: 1265: 1098: 989: 981: 939: 527: 484: 332: 205: 202: 199: 196: 193: 190: 187: 184: 154: 5564: 3046:, where 31 was written λα meaning 30+1, and 20 was written κ meaning 20. 786: 12932: 12891: 12743: 12643: 6420: 4642:

School of Mathematics and Statistics University of St Andrews, Scotland

3795:(Winter 2019 ed.). Metaphysics Research Lab, Stanford University. 3399: 2901: 2874: 2812: 2470: 2454: 2441: 2005: 2000:

displays and has continued with some modern computer typefaces such as

1952: 1868: 1823: 1492: 1443:

is the oldest surviving Chinese mathematical text using a round symbol

1386: 1188: 1054: 445: 319: 109: 72: 5919:. Prentice-Hall Software Series. Englewood Cliffs, NJ: Prentice Hall. 5836:"MySQL Cookbook: Solutions for Database Developers and Administrators" 4768: 2540:

with 0 as its only possible output value, that is, it is the function

2508:

for addition (if defined on the structure under consideration) and an

1906:

in the early 1200s and was one of the earliest scientific books to be

12638: 6425: 4175:, p. 12: "the ancient Chinese system is a place notation system" 2970: 2949: 2937: 2449: 2421: 2410: 1847: 1839: 1793: 1770: 1674: 1670: 1557: 1412:, and Japanese records dated from the 18th century, describe how the 1192: 1088: 1050: 842: 753: 539: 4706:. Translated by Henry Thomas Colebrooke. London: John Murray. 1817. 4593: 2512:

for multiplication (if defined). (Such elements may also be called

1800:" started to acquire a meaning of any arithmetic based on decimals. 1420:

system enabled one to perform decimal calculations. As noted in the

1171: 1105:

region to record accounting and other digital data, is encoded in a

4231:

The Nine Chapters on the Mathematical Art: Companion and Commentary

3899: 2763: 2620:, 0 is the smallest allowed value for the probability of any event. 2156: 1997: 1944: 1928: 1872: 1871:, I considered as almost a mistake in respect to the method of the 1864: 1797: 1728: 1716: 1682: 1565: 1496: 1360: 1330: 1326: 1314: 1307: 1216: 1153: 1084: 1080: 857:. Several sports have specific words for a score of zero, such as " 794: 791:

was already in existence (meaning "west wind" from Latin and Greek

468: 6435: 6028:

In the B.C./A.D. scheme there is no year zero. After 31 December 1

5373:

Discrete Mathematics: Proof Techniques and Mathematical Structures

5099:. Ginn and Company. pp. 134–136 – via Internet Archive. 4824:. University of British Columbia), American Mathematical Society. 1698:

There are numerous copper plate inscriptions, with the same small

819: 495:. Multiplying any number by 0 has the result 0, and consequently, 12633: 6505: 6134:

Beyond Infinity: An Expedition to the Outer Limits of Mathematics

5973:"Animals Count and Use Zero. How Far Does Their Number Sense Go?" 3895: 2929: 2882: 2831:. Any computation including a null value delivers a null result. 2747: 2674: 2666: 2087: 1707:

A stone tablet found in the ruins of a temple near Sambor on the

1561: 1469: 1184: 1138: 1106: 1058: 1015:

The Babylonian positional numeral system differed from the later

955: 935: 901: 511: 476: 167: 12668: 4749:

Bulletin of the School of Oriental Studies, University of London

4670:

The Britannica Guide to Numbers and Measurement (Math Explained)

2305:

is considered undefined in some contexts. For all positive real

2125:

of any other integer, rational, or real number. It is neither a

2082:

following 0 is 1 and no natural number precedes 0. The number 0

1072:, several centuries before the earliest known Long Count dates. 799:) and may have influenced the spelling when transcribing Arabic 12831: 12620: 12615: 12610: 12605: 12600: 12595: 12590: 12585: 12580: 2649: 1876: 1827: 1766: 1762: 1708: 1636: 1157: 1021: 862: 464: 293: 121: 5654:

C++ Without Fear: A Beginner's Guide That Makes You Feel Smart

5608: 5196: 3113:

Number Words and Number Symbols: A cultural history of numbers

2102:, including 0. When the real numbers are extended to form the 1168:

depend in large part on the uncertain interpretation of zero.

1129:

after 500 BC, representing it with the lowercase Greek letter

891: 4635: 3587:, Cambridge: Cambridge University Press, pp. 1005–1025, 3435: 3352: 2751: 2458:

to be the empty set. When this is done, the empty set is the

1831: 1785: 1735: 1539: 1322: 1102: 1094: 1065: 960: 641: 616: 574: 306: 254: 4563: 2804:

of characters ends. Also, 0 is a standard way to refer to a

1408:, of unknown date but estimated to be dated from the 1st to 1264:

0″ used the symbol as a placeholder in two positions of his

1037: 6474: 6000:

Marking Time: The epic quest to invent the perfect calendar

5457:. Vol. 1. Houston, Texas: OpenStax. pp. 454–459. 4228:

Shen Kanshen Crossley, John N.; Lun, Anthony W.-C. (1999).

2896:

that require applications to return an integer value as an

2885:

epoch begin the midnight before the first of January 1904.

2797: 2786: 2755: 2670: 2424:

0! evaluates to 1, as a special case of the empty product.

1553: 1364: 954:, a scribe recorded daily incomes and expenditures for the 607: 229: 5942:"Font Survey: 42 of the Best Monospaced Programming Fonts" 5270: 4298:. Translated by Wilson, Stephen S. Springer. p. 208. 4052:[Nineteen Year Cycle of Dionysius]. Archived from 3641:"Knots representing numbers: The mathematics of the Incas" 3548:"The golden age of Mayan civilisation – exhibition review" 3038:

Each place in Ptolemy's sexagesimal system was written in

2877:

begins the midnight before the first of January 1970. The

2432: 1822:(base 10) reached Western Europe in the 11th century, via 71:, "0#" redirects here. For the concept in set theory, see 7153: 7142: 6410: 2401:, if it exists, must be found by another method, such as 5896:"A Practical Guide to Computer Forensics Investigations" 5146: 4415:"Math for Poets and Drummers: The Mathematics of Rhythm" 3579:

Cyphers, Ann (2014), Renfrew, Colin; Bahn, Paul (eds.),

6614: 6530: 5678:

Oliveira, Suely; Stewart, David E. (7 September 2006).

4193:, Dordrecht: Springer Netherlands, pp. 1371–1378, 4091:

A History of Mathematics: From Mesopotamia to Modernity

3744:

A Brain for Numbers: The Biology of the Number Instinct

1595:"from place to place each is ten times the preceding". 181: 30:"Zero" and "Naught" redirect here. For other uses, see 4420:(slideshow). Saint Joseph's University. Archived from 1677:, was once part of a temple doorway, and was found in 732: 12846: 6600: 6593: 6586: 6579: 6572: 6565: 6558: 6551: 6544: 6265:. Berlin, Heidelberg, and New York: Springer-Verlag. 5762:

Wu, X.; Ichikawa, T.; Cercone, N. (25 October 1996).

5620: 5404:

From zero to infinity: what makes numbers interesting

4747:(1931). "A propos de l'origine des chiffres arabes". 4119: 3490: 3419: 3417: 2912: 2854:

binary form used in most modern computers), and most

2716:

that are more easily intelligible to humans than the

1972: 1948: 1248:

of the Sun. Minutes of immersion was tabulated from 0

590: 571: 151: 148: 145: 142: 139: 136: 133: 130: 127: 82:, ":0" redirects here. For the keyboard symbols, see 5541: 5497: 5041:

Primary Mathematics: Extending Knowledge in Practice

4840: 4601: 3768: 3527: 1951:

more rounded than the narrower, elliptical digit 0.

1893: 1727:). The number is the year of the inscription in the 1691:

A black dot is used as a decimal placeholder in the

1560:

surviving in a medieval Sanskrit translation of the

800: 726: 714: 644: 577: 5876:"Start Programming Using HTML, CSS, and JavaScript" 5529: 5428: 5258: 4971:, p. 241: "In 976, Muhammad ibn Ahmad, in his 4376: 2775:items the sequence of array indices runs from 0 to 1313:The earliest use of zero in the calculation of the 731:evolved to mean zero when it was used to translate 5997: 5765:Knowledge-Base Assisted Database Retrieval Systems 5681:Writing Scientific Software: A Guide to Good Style 5632: 4088: 3822: 3414: 2800:containing the value 0 serves to indicate where a 1988: 1776:In AD 813, astronomical tables were prepared by a 1750: 530:. The modern use of 0 in this manner derives from 5761: 5110:Pedersen, Olaf (1985). "In Quest of Sacrobosco". 3931: 1978: 1977: 1976: 1390:This is a depiction of zero expressed in Chinese 1000:), the scribe Bêl-bân-aplu used three hooks as a 13023: 5914: 5221:. Courier Dover Publications. pp. 254–255. 5214: 5086: 5084: 4666: 3116:. Courier Dover Publications. pp. 399–404. 2041:In positional number systems (such as the usual 5677: 4227: 4184: 4172: 3228:The Poetry of Physics and the Physics of Poetry 2736:and 0 is often the numerical representation of 1578:("void, empty") is also used to refer to zero. 1564:original, which is internally dated to AD 458 ( 6320:The Nothing That Is: A Natural History of Zero 5288: 5218:The historical roots of elementary mathematics 5090: 4673:. The Rosen Publishing Group. pp. 97–98. 4487: 4485: 3996:"Proposal to encode the Greek Zero in the UCS" 2630:, often denoted 0, and the related concept of 2175:(or neutral element) with respect to addition. 1917: 1516:The concept of zero as a written digit in the 1355:. When division produced zero as a remainder, 778: 694: 12684: 6490: 5081: 5029:. Vol. 11, no. 1. pp. 99–104. 3178: 3176: 2469:, corresponding to the empty set viewed as a 2084:may or may not be considered a natural number 1765:-language inheritance of science was largely 1301:sides), where d was the digit function and 31 784: 772: 745:) from India. The first known English use of 700: 688: 6234:(2nd ed.). Cambridge University Press. 5515:. Reading, MA: Addison-Wesley. p. 111. 5446: 4322:. New York: Dover Publications. pp. 32–33. " 3188:Oriental Elements of Culture in the Occident 2850:binary integer representations (but not the 2427: 1598:Rules governing the use of zero appeared in 866: 720: 679: 6232:Learning Scientific Programming with Python 4694: 4482: 3318: 3316: 2121:(that is, a multiple of 2), and is also an 2078:, and the largest nonpositive integer. The 1769:, followed by Hindu influences. In 773, at 1398:. An empty space is used to represent zero. 1342: 1329:document for the years 311 to 369, using a 792: 12986: 12976: 12691: 12677: 6497: 6483: 6415: 5970: 5684:. Cambridge University Press. p. 64. 5594:(3rd ed.). CRC Press. pp. 8–16. 4755:(2). Cambridge University Press: 323–328. 4439: 4334: 4332: 4082: 4080: 4018: 4012: 3860:"Consideration of the Greek symbol 'zero'" 3817: 3734: 3173: 2516:.) Examples include identity elements of 2414:) is 0, and the product of 0 numbers (the 1858:as state official in the customs house of 1757:History of the Hindu–Arabic numeral system 1695:, portions of which date from AD 224–993. 1669:(top), referring to the year it was made: 678:came into the English language via French 5931:Here: Sect.5.5 "Exit system call", p.114. 5329:. Springer Science & Business Media. 4895: 4893: 4891: 4866:. George Allen & Unwin. p. 165. 4858: 4592: 4385:. Motilal Banarsidass. pp. 648–649. 4383:The Student's Sanskrit-English Dictionary 4293: 4041: 3689: 3546:Mortaigne, Véronique (28 November 2014). 3545: 3109: 1813: 1458:The Nine Chapters on the Mathematical Art 1075:Although zero became an integral part of 1032: 522:that uses a base other than ten, such as 6282:The Olmecs: America's First Civilization 6257: 6193: 5650: 5626: 5163: 5161: 5109: 4734: 4667:William L. Hosch, ed. (15 August 2010). 4445: 4312: 4221: 3958: 3952: 3665:Wallin, Nils-Bertil (19 November 2002). 3313: 3198: 2591:in the domain of the function such that 2431: 2061: 1934: 1507: 1385: 1170: 1036: 976:By the middle of the 2nd millennium BC, 593:, the number is sometimes pronounced as 6342:Zero: The Biography of a Dangerous Idea 5589: 5093:"The spread of the numerals in Europe" 5091:Smith, D. E.; Karpinski, L. C. (1911). 4934: 4348:. Princeton University Press. pp. 4338: 4329: 4234:. Oxford University Press. p. 35. 4131:MacTutor History of Mathematics Archive 4086: 4077: 4047: 3894: 3888: 3851: 3793:The Stanford Encyclopedia of Philosophy 3786: 3702:Zero: The Biography of a Dangerous Idea 3578: 3385: 2665:of water. Measuring sound intensity in 1719:, includes the inscription of "605" in 850:) may exclude the use of the letter O. 777:in Italian, and was then contracted to 60:question marks, boxes, or other symbols 14: 13024: 6317: 6263:Elements of the History of Mathematics 6174: 6150: 6104: 6072: 5614: 5547: 5479:from the original on 23 September 2022 5369: 5202: 5037: 5033:from the original on 26 November 2023. 5021:"The Autobiography of Leonardo Pisano" 4991: 4968: 4899: 4888: 4783: 4724: 4575:; Wujastyk, Dominik (6 October 2017). 4545:from the original on 14 September 2017 4491: 4259:"Mathematics in the Near and Far East" 3774: 3740: 3664: 3496: 3448:from the original on 21 September 2021 3423: 3201:Mathematics: From the Birth of Numbers 3069: 2700:Modern computers store information in 1939:Oslo airport train station, Platform 0 1522:place value notation was developed in 1440:Mathematical Treatise in Nine Sections 1321:311, at the first entry in a table of 1112: 12672: 12513: 11916: 11319: 10722: 10125: 9528: 8931: 8334: 7737: 7140: 6516: 6478: 6416: 6336: 6298: 6279: 6250: 6128: 6048: 5995: 5971:Cepelewicz, Jordana (9 August 2021). 5812:from the original on 23 November 2021 5734: 5651:Overland, Brian (14 September 2004). 5559: 5535: 5434: 5276: 5264: 5167: 5158: 5152: 5018: 4938:Al-Khwarizmi: The Inventor of Algebra 4846: 4812: 4743: 4613: 4607: 4515:from the original on 20 November 2017 4457: 4451: 3695: 3560:from the original on 28 November 2014 3533: 3478:from the original on 7 September 2021 3367:from the original on 15 November 2019 3279: 3225: 2688:may possess and is the energy of the 2681:is the lowest possible energy that a 2634:, which generalize the zero function. 2057: 1731:, corresponding to a date of AD 683. 1673:(683 CE). The fragment, inscribed in 1101:and its predecessor societies in the 1097:, a knotted cord device, used in the 6367:Searching for the World's First Zero 6229: 5952:from the original on 24 January 2012 5638: 5590:Rex, Andrew; Finn, C. B. P. (2017). 5400: 5366:The integer 0 is even and is not odd 5319: 5112:Journal for the History of Astronomy 4828:from the original on 6 December 2015 4636:J J O'Connor; E F Robertson (2000). 4412: 4275:from the original on 4 November 2013 4060: 4008:from the original on 7 October 2022. 3876:from the original on 5 November 2020 3799:from the original on 10 January 2021 3741:Nieder, Andreas (19 November 2019). 3639:Leon, Manuel de (20 December 2022). 3638: 3436:J J O'Connor; E F Robertson (2000). 3353:J J O'Connor; E F Robertson (2000). 3273: 2911:to avoid confusion with the letter " 2465:Also in set theory, 0 is the lowest 1902:. One popular manual was written by 1788:in the 12th century under the title 887: 6217:from the original on 17 August 2019 5802:"Null values and the nullable type" 5491: 5409:Mathematical Association of America 5019:Grimm, Richard E. (February 1973). 4251: 4095:. Oxford University Press. p. 4072:Corpus Christianorum, Series Latina 3936:. MacTutor History of Mathematics. 3857: 2789:indexes parameters from 1 although 2695: 1394:, based on the example provided by 1236:. It represented the value of both 992:numerals. In a tablet unearthed at 721: 542:. It was independently used by the 534:that was transmitted to Europe via 24: 6391:Why numbering should start at zero 5782:from the original on 31 March 2022 5657:. Pearson Education. p. 132. 4458:Weiss, Ittay (20 September 2017). 3293:"Collins - Free online dictionary" 3100:"cipher," translation of Sanskrit 2819:. No longer is a condition either 2720:that are directly executed by the 2070:from −3 to 3, with 0 in the middle 1927: 1572:, including a zero. In this text, 25: 13058: 12698: 6360: 6004:. John Wiley & Sons. p. 5706:"ResultSet (Java Platform SE 8 )" 5343:from the original on 14 June 2021 5235:from the original on 23 June 2016 5066:from the original on 7 March 2021 4648:from the original on 11 July 2015 4470:from the original on 12 July 2018 4413:Hall, Rachel (15 February 2005). 4050:"Cyclus Decemnovennalis Dionysii" 3940:from the original on 7 April 2020 3262:Merriam Webster online Dictionary 3155:from the original on 7 March 2012 3029:of 0 already existed at the time. 2985:BC is numbered −1, and so forth. 2943: 2500:, 0 is commonly used to denote a 1854:After my father's appointment by 1723:(a set of numeral glyphs for the 670:Names for the number 0 in English 551:names for the number 0 in English 13005: 12995: 12985: 12975: 12966: 12965: 6434: 6157:Statistical Physics of Particles 6058:. New York: Palgrave Macmillan. 5989: 5964: 5934: 5423:zero neither prime nor composite 5301:from the original on 1 June 2013 4581:History of Science in South Asia 4320:A Concise History of Mathematics 4296:A History of Chinese Mathematics 4185:Eberhard-Bréard, Andrea (2008), 3962:(2010) . Alexander Jones (ed.). 2858:number representations (such as 2714:high-level programming languages 2006:falsification-hindering typeface 1898:after the Persian mathematician 1655: 1644: 1635: 1047:Mesoamerican Long Count calendar 1020: 693:, a contraction of the Venetian 637: 603: 567: 514:notation: it indicates that the 225: 6284:. London: Thames & Hudson. 6041: 5908: 5888: 5868: 5848: 5828: 5794: 5755: 5728: 5716:from the original on 9 May 2022 5698: 5671: 5644: 5583: 5553: 5440: 5394: 5355: 5313: 5282: 5208: 5103: 4982: 4962: 4928: 4852: 4806: 4660: 4629: 4557: 4527: 4406: 4370: 4294:Martzloff, Jean-Claude (2007). 4287: 4178: 4166: 4141: 4113: 4024:Ethiopic Astronomy and Computus 3988: 3932:O'Connor, J J; Robertson, E F. 3925: 3825:The Exact Sciences in Antiquity 3811: 3780: 3658: 3632: 3607: 3572: 3539: 3502: 3460: 3429: 3032: 3022: 2918: 2656:is the lowest possible value. ( 2460:von Neumann cardinal assignment 2436:The empty set has zero elements 1804:Muhammad ibn Ahmad al-Khwarizmi 1751:Transmission to Islamic culture 1734:The first known use of special 1195:, was using a symbol for zero ( 808: 713:. In pre-Islamic time the word 536:medieval Islamic mathematicians 12744:analytic theory of L-functions 12722:non-abelian class field theory 6179:. Princeton University Press. 6160:. Cambridge University Press. 5854:Arnold Robbins; Nelson Beebe. 4941:. The Rosen Publishing Group. 4199:10.1007/978-1-4020-4425-0_9453 3585:The Cambridge World Prehistory 3388:The Mathematical Intelligencer 3379: 3346: 3299: 3285: 3250: 3060: 2965:BC is the first year before AD 2793:itself uses 0-based indexing. 2728:stores a value that is either 2609:, 0 may be used to denote the 2015: 1888:... any number may be written. 1782:Muḥammad ibn Mūsā al-Khwārizmī 1745: 1593:sthānāt sthānaṁ daśaguṇaṁ syāt 783:in Venetian. The Italian word 13: 1: 4979:, "empty" whence our cipher." 4793:The Mon-Khmer Studies Journal 4173:Shen, Crossley & Lun 1999 3791:. In Zalta, Edward N. (ed.). 3360:mathshistory.st-andrews.ac.uk 3053: 2836:signed number representations 2754:. However, in the late 1950s 2477:(and especially its subfield 2074:The number 0 is the smallest 2030: 1834:, together with knowledge of 1588: 1505:explicitly to refer to zero. 1473: 841:, or years. For example, the 757: 12768:Transcendental number theory 6504: 6397:of a handwritten manuscript) 5376:. World Scientific. p. 4377:Vaman Shivaram Apte (1970). 3444:. University of St Andrews. 3363:. University of St Andrews. 3104:"empty place, desert, naught 2981:BC is numbered 0, the year 2 2363:as a result of applying the 1884:figures, and with the sign 0 1621: 1367:—or his colleagues—around AD 1228:(VI.8) for the magnitude of 1061:) has a date of 36 BC. 942:for the digits and were not 659: 510:, 0 plays a crucial role in 7: 12991:List of recreational topics 12763:Computational number theory 12748:probabilistic number theory 6322:. Oxford University Press. 5326:Number Theory for Beginners 4189:, in Selin, Helaine (ed.), 3972:10.1007/978-0-387-84826-6_7 2988: 2975:astronomical year numbering 2152:, unless otherwise stated. 2098:. All rational numbers are 1918:Symbols and representations 1820:Hindu–Arabic numeral system 1790:Algoritmi de numero Indorum 1725:Hindu–Arabic numeral system 801: 733: 727: 725:) had the meaning "empty". 715: 38:. For the music album, see 10: 13063: 6280:Diehl, Richard A. (2004). 6082:. New York: Pocket Books. 6076:(1978). "Nothing Counts". 5735:Reese, Richard M. (2013). 5566:Category Theory in Context 5370:Penner, Robert C. (1999). 5132:10.1177/002182868501600302 4048:Deckers, Michael (2003) . 3909:Princeton University Press 3747:. MIT Press. p. 286. 2947: 2923: 2771:, so that for an array of 2638: 2524:. Another example is the 2408:The sum of 0 numbers (the 2034: 2019: 1989:{\displaystyle 0\!\!\!{/}} 1921: 1754: 1742:, in India, dated AD 876. 1740:Chaturbhuj Temple, Gwalior 1079:, with a different, empty 908: 905:beautiful, pleasant, good 882: 752:The Italian mathematician 663: 77: 66: 29: 12961: 12943:Diophantine approximation 12915: 12902:Chinese remainder theorem 12824: 12706: 12629: 12576: 12523: 12514: 12509: 12447: 12389: 12331: 12273: 12215: 12157: 12099: 12041: 11983: 11925: 11912: 11850: 11792: 11734: 11676: 11618: 11560: 11502: 11444: 11386: 11328: 11315: 11253: 11195: 11137: 11079: 11021: 10963: 10905: 10847: 10789: 10731: 10718: 10656: 10598: 10540: 10482: 10424: 10366: 10308: 10250: 10192: 10134: 10121: 10059: 10001: 9943: 9885: 9827: 9769: 9711: 9653: 9595: 9537: 9524: 9462: 9404: 9346: 9288: 9230: 9172: 9114: 9056: 8998: 8940: 8927: 8865: 8807: 8749: 8691: 8633: 8575: 8517: 8459: 8401: 8343: 8330: 8268: 8210: 8152: 8094: 8036: 7978: 7920: 7862: 7804: 7746: 7733: 7671: 7613: 7555: 7497: 7439: 7381: 7323: 7265: 7207: 7149: 7136: 7074: 7016: 6958: 6900: 6842: 6784: 6726: 6668: 6610: 6525: 6512: 5917:Advanced UNIX Programming 5915:Marc J. Rochkind (1985). 5856:"Classic Shell Scripting" 5170:Communications of the ACM 5097:The Hindu–Arabic Numerals 5002:10.1007/978-1-4613-0079-3 4992:Sigler, Laurence (2003). 4786:"New Zeros and Old Khmer" 4761:10.1017/S0041977X00092806 3581:"The Olmec, 1800–400 bce" 3185:Smithsonian Institution. 3088:or directly from Italian 2866:floating-point formats). 2626:introduces the idea of a 2428:Other uses in mathematics 1838:and instruments like the 1297:(a triangular pulse with 742: 444: 434: 424: 414: 404: 394: 384: 372: 344: 331: 318: 305: 292: 279: 266: 253: 243: 215: 119: 96: 12787:Arithmetic combinatorics 11917: 11320: 10723: 10126: 9529: 8932: 8335: 7738: 7141: 6230:Hill, Christian (2020). 5901:24 February 2017 at the 5881:24 February 2017 at the 5861:24 February 2017 at the 5841:24 February 2017 at the 5617:, pp. 4–5, 103–104. 5401:Reid, Constance (1992). 5205:, pp. 156, 199–204. 4935:Brezina, Corona (2006). 4907:. Simon & Schuster. 4784:Diller, Anthony (1996). 4620:Aryabhatiya of Aryabhata 4318:Struik, Dirk J. (1987). 4136:University of St Andrews 3964:A Survey of the Almagest 3510:"Cyclical views of time" 3468:"Babylonian mathematics" 3110:Menninger, Karl (1992). 3070:Harper, Douglas (2011). 3015: 2907:Programmers often use a 2010:German car number plates 1947:made the capital letter 1867:, as well as the art of 1491:, a notation similar to 1464: 1396:A History of Mathematics 1381: 501:no meaning in arithmetic 12758:Geometric number theory 12714:Algebraic number theory 6450:Encyclopædia Britannica 6318:Kaplan, Robert (2000). 6299:Ifrah, Georges (2000). 3705:. Penguin. p. 39. 3149:Oxford University Press 2904:or warning conditions. 2742:0 also plays a role in 2722:central processing unit 2648:(typically measured in 996:(dating to as early as 822:" are also synonymous. 779: 695: 12877:Transcendental numbers 12791:additive number theory 12740:Analytic number theory 6517: 6465:Encyclopedia Americana 6453:(11th ed.). 1911. 6382:The History of Algebra 6177:Count Like an Egyptian 6175:Reimer, David (2014). 5996:Steel, Duncan (2000). 5592:Finn's Thermal Physics 5569:. Dover. p. 103. 5451:; et al. (2017). 5247:Extract of pp. 254–255 5038:Hansen, Alice (2008). 4864:A History of Astronomy 4187:"Mathematics in China" 4087:Hodgkin, Luke (2005). 3787:Huggett, Nick (2019). 3226:Logan, Robert (2010). 3199:Gullberg, Jan (1997). 3092:, from Medieval Latin 2437: 2265:multiplicative inverse 2110:of the complex plane. 2071: 2052:decimal representation 1990: 1940: 1932: 1904:Johannes de Sacrobosco 1894: 1890: 1814:Transmission to Europe 1619: 1612:in the following way: 1605:Brahmasputha Siddhanta 1513: 1479:or 2nd century BC), a 1399: 1343: 1208:mathematical astronomy 1176: 1147:conquests of Alexander 1042: 1033:Pre-Columbian Americas 978:Babylonian mathematics 867: 793: 785: 773: 701: 689: 680: 666:Names for the number 0 467:representing an empty 48:This article contains 13037:Elementary arithmetic 12948:Irrationality measure 12938:Diophantine equations 12781:Hodge–Arakelov theory 6439:Texts on Wikisource: 5279:, pp. 41, 48–53. 5182:10.1145/363534.363563 5050:10.4135/9781446276532 4638:"Aryabhata the Elder" 4541:. 14 September 2017. 4495:(13 September 2017). 3667:"The History of Zero" 3205:W.W. Norton & Co. 3000:Mathematical constant 2658:Negative temperatures 2491:partially ordered set 2435: 2065: 1991: 1938: 1931: 1852: 1614: 1568:380), uses a decimal 1511: 1389: 1361:calculators of Easter 1174: 1040: 1027:floating-point number 848:Canadian postal codes 656:have also been used. 520:place-value notations 239:, nought, naught, nil 36:Zero (disambiguation) 12907:Arithmetic functions 12773:Diophantine geometry 5808:. 12 December 2018. 5768:. World Scientific. 5512:Concrete Mathematics 4973:Keys of the Sciences 4345:Mathematics in India 4122:Robertson, Edmund F. 3911:. pp. 306–307. 3699:(1 September 2000). 3514:www.mexicolore.co.uk 3230:. World Scientific. 2760:zero-based numbering 2710:Computer programmers 2646:absolute temperature 2481:), 0 may denote the 1970: 1693:Bakhshali manuscript 1528:Bakhshali manuscript 1423:Xiahou Yang Suanjing 1335:Church of Alexandria 1214:, also known as the 1212:Syntaxis Mathematica 980:had a sophisticated 958:'s court, using the 816:"nought" or "naught" 619:terms for 0 include 493:algebraic structures 249:Zeroth, noughth, 0th 12953:Continued fractions 12816:Arithmetic dynamics 12811:Arithmetic topology 12805:P-adic Hodge theory 12797:Arithmetic geometry 12730:Iwasawa–Tate theory 6111:The Book of Nothing 5323:(6 December 2012). 5289:Weisstein, Eric W. 5252:10 May 2016 at the 5155:, pp. 588–590. 5124:1985JHA....16..175P 5026:Fibonacci Quarterly 4624:Walter Eugene Clark 4120:O'Connor, John J.; 4056:on 15 January 2019. 3934:"A history of Zero" 3619:Expedition Magazine 3472:The Open University 3355:"Egyptian numerals" 3268:on 6 December 2017. 2930:comparative zoology 2718:binary instructions 2706:digital electronics 2607:propositional logic 2263:, because 0 has no 2171:. That is, 0 is an 2076:nonnegative integer 2037:Positional notation 1844:Gerbert of Aurillac 1836:classical astronomy 1495:. Pingala used the 1428:positional notation 1325:as preserved in an 1113:Classical antiquity 1017:Hindu–Arabic system 968:suggested that the 871:, "the egg" – and " 835:credit card numbers 538:and popularized by 491:, as well as other 12897:Modular arithmetic 12867:Irrational numbers 12801:anabelian geometry 12718:class field theory 6418:Weisstein, Eric W. 6387:Edsger W. Dijkstra 6251:Historical studies 6203:. Evans Brothers. 6200:Digital Technology 5985:on 18 August 2021. 5948:. 18 August 2010. 5741:. O'Reilly Media. 4573:Montelle, Clemency 4567:; Keller, Agathe; 4427:on 22 January 2019 4149:"Chinese numerals" 4126:"Chinese numerals" 4066:C. W. Jones, ed., 3901:Ptolemy's Almagest 3858:Mercier, Raymond. 3831:Dover Publications 3789:"Zeno's Paradoxes" 3400:10.1007/BF03024613 3324:"'Aught' synonyms" 2995:Grammatical number 2973:. By contrast, in 2969:1; there is not a 2817:three-valued logic 2683:quantum mechanical 2618:probability theory 2581:zero of a function 2438: 2369:indeterminate form 2145:of the integers.) 2072: 2058:Elementary algebra 2022:Null (mathematics) 1986: 1941: 1933: 1826:, through Spanish 1629:Sambor Inscription 1570:place-value system 1536:radiocarbon dating 1514: 1400: 1317:occurred before AD 1177: 1043: 532:Indian mathematics 50:special characters 32:0 (disambiguation) 13047:Indian inventions 13019: 13018: 12916:Advanced concepts 12872:Algebraic numbers 12857:Composite numbers 12666: 12665: 12662: 12661: 12505: 12504: 11908: 11907: 11311: 11310: 10714: 10713: 10117: 10116: 9520: 9519: 8923: 8922: 8326: 8325: 7729: 7728: 7132: 7131: 6372:A History of Zero 6329:978-0-198-02945-8 6291:978-0-500-28503-9 6259:Bourbaki, Nicolas 6210:978-0-237-52725-9 6186:978-0-691-16012-2 6167:978-0-521-87342-0 6143:978-1-5416-4413-7 6089:978-0-671-82134-0 6079:Asimov on Numbers 6065:978-1-137-27984-2 6015:978-0-471-29827-4 5894:Darren R. Hayes. 5775:978-981-4501-75-0 5748:978-1-449-34455-9 5691:978-1-139-45862-7 5664:978-0-7686-8488-9 5601:978-1-4987-1887-5 5576:978-0-486-80903-8 5499:Graham, Ronald L. 5464:978-1-938168-02-4 5418:978-0-88385-505-8 5387:978-981-02-4088-2 5336:978-1-4612-9957-8 5228:978-0-486-13968-5 5059:978-0-85725-233-3 5011:978-1-4613-0079-3 4948:978-1-4042-0513-0 4914:978-0-9650007-5-8 4680:978-1-61530-108-9 4392:978-81-208-0045-8 4359:978-0-691-12067-6 4305:978-3-540-33783-6 4266:grmath4.phpnet.us 4241:978-0-19-853936-0 4208:978-1-4020-4425-0 4106:978-0-19-152383-0 4068:Opera Didascalica 4033:978-1-56902-440-9 3981:978-0-387-84825-9 3844:978-0-486-22332-2 3754:978-0-262-35432-5 3712:978-0-14-029647-1 3677:on 25 August 2016 3671:YaleGlobal online 3594:978-0-521-11993-1 3334:on 23 August 2014 3237:978-981-4295-92-5 3213:978-0-393-04002-9 3151:. December 2011. 3123:978-0-486-27096-8 2934:cognitive science 2894:operating systems 2762:for arrays while 2679:zero-point energy 2538:constant function 2510:absorbing element 2297:= 1, except that 2235:= 0, for nonzero 2100:algebraic numbers 2048:decimal fractions 1349:Dionysius Exiguus 1206:) in his work on 1091:numeral systems. 1041:Maya numeral zero 1006:Babylonian system 932:Egyptian numerals 928: 927: 924: 923: 920: 919: 888:Ancient Near East 827:telephone numbers 473:additive identity 454: 453: 426:Assamese, Bengali 115: 114: 84:List of emoticons 80:technical reasons 69:technical reasons 56:rendering support 16:(Redirected from 13054: 13009: 12999: 12989: 12988: 12979: 12978: 12969: 12968: 12862:Rational numbers 12693: 12686: 12679: 12670: 12669: 12511: 12510: 11914: 11913: 11317: 11316: 10720: 10719: 10123: 10122: 9526: 9525: 8929: 8928: 8332: 8331: 7735: 7734: 7138: 7137: 6514: 6513: 6499: 6492: 6485: 6476: 6475: 6469: 6454: 6438: 6431: 6430: 6355: 6333: 6314: 6295: 6276: 6245: 6241:978-1-10707541-2 6226: 6224: 6222: 6190: 6171: 6147: 6125: 6101: 6069: 6035: 6034: 6031: 6003: 5993: 5987: 5986: 5981:. Archived from 5968: 5962: 5961: 5959: 5957: 5938: 5932: 5930: 5912: 5906: 5892: 5886: 5872: 5866: 5852: 5846: 5832: 5826: 5825: 5819: 5817: 5798: 5792: 5791: 5789: 5787: 5759: 5753: 5752: 5732: 5726: 5725: 5723: 5721: 5702: 5696: 5695: 5675: 5669: 5668: 5648: 5642: 5636: 5630: 5624: 5618: 5612: 5606: 5605: 5587: 5581: 5580: 5557: 5551: 5545: 5539: 5533: 5527: 5526: 5503:Knuth, Donald E. 5495: 5489: 5488: 5486: 5484: 5444: 5438: 5432: 5426: 5425: 5407:(4th ed.). 5398: 5392: 5391: 5359: 5353: 5352: 5350: 5348: 5317: 5311: 5310: 5308: 5306: 5286: 5280: 5274: 5268: 5262: 5256: 5244: 5242: 5240: 5212: 5206: 5200: 5194: 5193: 5165: 5156: 5150: 5144: 5143: 5107: 5101: 5100: 5088: 5079: 5075: 5073: 5071: 5034: 5015: 4986: 4980: 4966: 4960: 4959: 4957: 4955: 4932: 4926: 4925: 4897: 4886: 4885: 4860:Pannekoek, Anton 4856: 4850: 4844: 4838: 4837: 4835: 4833: 4818:"All for Nought" 4810: 4804: 4800: 4790: 4780: 4738: 4732: 4722: 4716: 4715: 4698: 4692: 4691: 4689: 4687: 4664: 4658: 4657: 4655: 4653: 4633: 4627: 4622:, translated by 4617: 4611: 4605: 4599: 4598: 4596: 4561: 4555: 4554: 4552: 4550: 4539:Bodleian Library 4531: 4525: 4524: 4522: 4520: 4489: 4480: 4479: 4477: 4475: 4464:The Conversation 4455: 4449: 4443: 4437: 4436: 4434: 4432: 4426: 4419: 4410: 4404: 4403: 4401: 4399: 4374: 4368: 4367: 4336: 4327: 4316: 4310: 4309: 4291: 4285: 4284: 4282: 4280: 4274: 4263: 4255: 4249: 4248: 4225: 4219: 4218: 4217: 4215: 4182: 4176: 4170: 4164: 4163: 4161: 4159: 4145: 4139: 4138: 4124:(January 2004), 4117: 4111: 4110: 4094: 4084: 4075: 4064: 4058: 4057: 4045: 4039: 4037: 4020:Neugebauer, Otto 4016: 4010: 4009: 4007: 4001:. 31 July 2024. 4000: 3992: 3986: 3985: 3956: 3950: 3949: 3947: 3945: 3929: 3923: 3922: 3904:. Translated by 3892: 3886: 3885: 3883: 3881: 3875: 3864: 3855: 3849: 3848: 3828: 3819:Neugebauer, Otto 3815: 3809: 3808: 3806: 3804: 3784: 3778: 3772: 3766: 3765: 3763: 3761: 3738: 3732: 3731: 3729: 3727: 3693: 3687: 3686: 3684: 3682: 3662: 3656: 3655: 3653: 3651: 3636: 3630: 3629: 3627: 3625: 3611: 3605: 3604: 3603: 3601: 3576: 3570: 3569: 3567: 3565: 3543: 3537: 3531: 3525: 3524: 3522: 3520: 3506: 3500: 3494: 3488: 3487: 3485: 3483: 3464: 3458: 3457: 3455: 3453: 3433: 3427: 3421: 3412: 3411: 3383: 3377: 3376: 3374: 3372: 3350: 3344: 3343: 3341: 3339: 3330:. Archived from 3320: 3311: 3310: 3303: 3297: 3296: 3289: 3283: 3277: 3271: 3269: 3264:. Archived from 3254: 3248: 3244: 3222: 3180: 3171: 3167: 3162: 3160: 3134: 3132: 3130: 3106: 3082:on 3 July 2017. 3078:. Archived from 3064: 3047: 3045: 3036: 3030: 3026: 2984: 2980: 2968: 2964: 2852:two's complement 2848:ones' complement 2844:signed magnitude 2827:, but it can be 2781: 2726:Boolean variable 2696:Computer science 2601: 2562: 2558: 2554: 2543: 2535: 2506:identity element 2498:abstract algebra 2471:well-ordered set 2403:l'Hôpital's rule 2400: 2398: 2397: 2387: 2384: 2362: 2360: 2359: 2349: 2346: 2332: 2330: 2329: 2326: 2323: 2312: 2296: 2294: 2293: 2288: 2285: 2257: 2255: 2254: 2251: 2248: 2234: 2232: 2231: 2226: 2223: 2173:identity element 2131:composite number 2123:integer multiple 2106:, 0 becomes the 2043:decimal notation 1995: 1993: 1992: 1987: 1985: 1984: 1924:Symbols for zero 1897: 1887: 1703: 1702: 1659: 1648: 1639: 1610:division by zero 1590: 1532:Bodleian Library 1485:binary sequences 1481:Sanskrit prosody 1478: 1475: 1454: 1453:(2nd century AD) 1415: 1411: 1410:5th centuries AD 1370: 1346: 1320: 1304: 1296: 1295: 1294: 1288: 1284: 1282: 1281: 1278: 1275: 1263: 1259: 1255: 1251: 1246:angular diameter 1222:Hellenistic zero 1205: 1204: 1203: 1200: 1187:, influenced by 1182: 1071: 1024: 1011: 999: 953: 913: 912: 909: 892: 870: 831:street addresses 804: 798: 790: 782: 776: 766: 762: 759: 744: 736: 730: 724: 723: 718: 704: 699:form of Italian 698: 692: 683: 651: 650: 647: 646: 643: 627:. Historically, 614: 613: 610: 609: 584: 583: 580: 579: 576: 573: 497:division by zero 481:rational numbers 380: 368: 238: 236: 235: 232: 231: 98: 97: 94: 93: 21: 13062: 13061: 13057: 13056: 13055: 13053: 13052: 13051: 13022: 13021: 13020: 13015: 12957: 12923:Quadratic forms 12911: 12886:P-adic analysis 12842:Natural numbers 12820: 12777:Arakelov theory 12702: 12697: 12667: 12658: 12625: 12572: 12519: 12501: 12443: 12385: 12327: 12269: 12211: 12153: 12095: 12037: 11979: 11921: 11904: 11846: 11788: 11730: 11672: 11614: 11556: 11498: 11440: 11382: 11324: 11307: 11249: 11191: 11133: 11075: 11017: 10959: 10901: 10843: 10785: 10727: 10710: 10652: 10594: 10536: 10478: 10420: 10362: 10304: 10246: 10188: 10130: 10113: 10055: 9997: 9939: 9881: 9823: 9765: 9707: 9649: 9591: 9533: 9516: 9458: 9400: 9342: 9284: 9226: 9168: 9110: 9052: 8994: 8936: 8919: 8861: 8803: 8745: 8687: 8629: 8571: 8513: 8455: 8397: 8339: 8322: 8264: 8206: 8148: 8090: 8032: 7974: 7916: 7858: 7800: 7742: 7725: 7667: 7609: 7551: 7493: 7435: 7377: 7319: 7261: 7203: 7145: 7128: 7070: 7012: 6954: 6896: 6838: 6780: 6722: 6664: 6606: 6521: 6508: 6503: 6457: 6442: 6363: 6358: 6352: 6344:. Penguin USA. 6330: 6311: 6292: 6273: 6253: 6248: 6242: 6220: 6218: 6211: 6195:Woodford, Chris 6187: 6168: 6144: 6136:. Basic Books. 6122: 6106:Barrow, John D. 6090: 6066: 6044: 6039: 6038: 6029: 6016: 5994: 5990: 5978:Quanta Magazine 5969: 5965: 5955: 5953: 5946:codeproject.com 5940: 5939: 5935: 5927: 5913: 5909: 5903:Wayback Machine 5893: 5889: 5885:. 2015. p. 160. 5883:Wayback Machine 5873: 5869: 5863:Wayback Machine 5853: 5849: 5843:Wayback Machine 5833: 5829: 5815: 5813: 5800: 5799: 5795: 5785: 5783: 5776: 5760: 5756: 5749: 5733: 5729: 5719: 5717: 5710:docs.oracle.com 5704: 5703: 5699: 5692: 5676: 5672: 5665: 5649: 5645: 5637: 5633: 5625: 5621: 5613: 5609: 5602: 5588: 5584: 5577: 5558: 5554: 5546: 5542: 5534: 5530: 5523: 5507:Patashnik, Oren 5496: 5492: 5482: 5480: 5465: 5449:Strang, Gilbert 5447:Herman, Edwin; 5445: 5441: 5433: 5429: 5419: 5399: 5395: 5388: 5360: 5356: 5346: 5344: 5337: 5318: 5314: 5304: 5302: 5287: 5283: 5275: 5271: 5263: 5259: 5254:Wayback Machine 5238: 5236: 5229: 5213: 5209: 5201: 5197: 5166: 5159: 5151: 5147: 5108: 5104: 5089: 5082: 5078: 5069: 5067: 5060: 5012: 4987: 4983: 4967: 4963: 4953: 4951: 4949: 4933: 4929: 4917:. p. 241: 4915: 4898: 4889: 4874: 4857: 4853: 4845: 4841: 4831: 4829: 4814:Casselman, Bill 4811: 4807: 4803: 4788: 4739: 4735: 4723: 4719: 4700: 4699: 4695: 4685: 4683: 4681: 4665: 4661: 4651: 4649: 4634: 4630: 4618: 4614: 4606: 4602: 4594:10.18732/H2XT07 4562: 4558: 4548: 4546: 4533: 4532: 4528: 4518: 4516: 4490: 4483: 4473: 4471: 4456: 4452: 4444: 4440: 4430: 4428: 4424: 4417: 4411: 4407: 4397: 4395: 4393: 4375: 4371: 4360: 4337: 4330: 4317: 4313: 4306: 4292: 4288: 4278: 4276: 4272: 4268:. p. 262. 4261: 4257: 4256: 4252: 4242: 4226: 4222: 4213: 4211: 4209: 4183: 4179: 4171: 4167: 4157: 4155: 4147: 4146: 4142: 4118: 4114: 4107: 4085: 4078: 4070:, vol. 123C in 4065: 4061: 4046: 4042: 4034: 4017: 4013: 4005: 3998: 3994: 3993: 3989: 3982: 3957: 3953: 3943: 3941: 3930: 3926: 3919: 3893: 3889: 3879: 3877: 3873: 3862: 3856: 3852: 3845: 3816: 3812: 3802: 3800: 3785: 3781: 3773: 3769: 3759: 3757: 3755: 3739: 3735: 3725: 3723: 3713: 3694: 3690: 3680: 3678: 3663: 3659: 3649: 3647: 3645:EL PAÍS English 3637: 3633: 3623: 3621: 3613: 3612: 3608: 3599: 3597: 3595: 3577: 3573: 3563: 3561: 3544: 3540: 3532: 3528: 3518: 3516: 3508: 3507: 3503: 3495: 3491: 3481: 3479: 3466: 3465: 3461: 3451: 3449: 3434: 3430: 3422: 3415: 3384: 3380: 3370: 3368: 3351: 3347: 3337: 3335: 3322: 3321: 3314: 3305: 3304: 3300: 3291: 3290: 3286: 3278: 3274: 3256: 3255: 3251: 3247: 3238: 3214: 3181: 3174: 3170: 3158: 3156: 3137: 3128: 3126: 3124: 3065: 3061: 3056: 3051: 3050: 3043: 3037: 3033: 3027: 3023: 3018: 2991: 2982: 2978: 2966: 2962: 2952: 2946: 2926: 2921: 2776: 2698: 2692:of the system. 2686:physical system 2641: 2624:Category theory 2592: 2560: 2556: 2545: 2541: 2536:. This is the 2533: 2518:additive groups 2504:, which is the 2430: 2388: 2385: 2375: 2374: 2372: 2350: 2347: 2337: 2336: 2334: 2327: 2324: 2321: 2320: 2318: 2317:The expression 2310: 2289: 2286: 2281: 2280: 2278: 2252: 2249: 2244: 2243: 2241: 2227: 2224: 2221: 2220: 2218: 2104:complex numbers 2092:rational number 2086:, but it is an 2060: 2039: 2033: 2024: 2018: 1980: 1979: 1971: 1968: 1967: 1926: 1920: 1885: 1816: 1780:mathematician, 1759: 1753: 1748: 1713:Kratié Province 1700: 1699: 1689: 1688: 1687: 1686: 1679:Kratié province 1662: 1661: 1660: 1651: 1650: 1649: 1641: 1640: 1631: 1630: 1624: 1476: 1467: 1452: 1413: 1409: 1384: 1368: 1318: 1302: 1292: 1290: 1286: 1279: 1276: 1273: 1272: 1270: 1269: 1261: 1257: 1253: 1249: 1201: 1198: 1197: 1196: 1180: 1115: 1069: 1035: 1009: 997: 951: 950:written around 904: 896: 890: 885: 855:British English 811: 764: 760: 672: 664:Main articles: 662: 640: 636: 615:). Informal or 606: 602: 570: 566: 508:numerical digit 489:complex numbers 378: 366: 340: 327: 314: 301: 288: 275: 228: 224: 222: 211: 173: 172: 163:List of numbers 157: 92: 87: 76: 65: 64: 63: 54:Without proper 43: 28: 23: 22: 15: 12: 11: 5: 13060: 13050: 13049: 13044: 13039: 13034: 13017: 13016: 13014: 13013: 13003: 12993: 12983: 12981:List of topics 12973: 12962: 12959: 12958: 12956: 12955: 12950: 12945: 12940: 12935: 12930: 12925: 12919: 12917: 12913: 12912: 12910: 12909: 12904: 12899: 12894: 12889: 12882:P-adic numbers 12879: 12874: 12869: 12864: 12859: 12854: 12849: 12844: 12839: 12834: 12828: 12826: 12822: 12821: 12819: 12818: 12813: 12808: 12794: 12784: 12770: 12765: 12760: 12755: 12737: 12726:Iwasawa theory 12710: 12708: 12704: 12703: 12696: 12695: 12688: 12681: 12673: 12664: 12663: 12660: 12659: 12657: 12656: 12651: 12646: 12641: 12636: 12630: 12627: 12626: 12624: 12623: 12618: 12613: 12608: 12603: 12598: 12593: 12588: 12583: 12577: 12574: 12573: 12571: 12570: 12565: 12560: 12555: 12550: 12545: 12540: 12535: 12530: 12524: 12521: 12520: 12507: 12506: 12503: 12502: 12500: 12499: 12494: 12489: 12484: 12479: 12474: 12469: 12464: 12459: 12454: 12448: 12445: 12444: 12442: 12441: 12436: 12431: 12426: 12421: 12416: 12411: 12406: 12401: 12396: 12390: 12387: 12386: 12384: 12383: 12378: 12373: 12368: 12363: 12358: 12353: 12348: 12343: 12338: 12332: 12329: 12328: 12326: 12325: 12320: 12315: 12310: 12305: 12300: 12295: 12290: 12285: 12280: 12274: 12271: 12270: 12268: 12267: 12262: 12257: 12252: 12247: 12242: 12237: 12232: 12227: 12222: 12216: 12213: 12212: 12210: 12209: 12204: 12199: 12194: 12189: 12184: 12179: 12174: 12169: 12164: 12158: 12155: 12154: 12152: 12151: 12146: 12141: 12136: 12131: 12126: 12121: 12116: 12111: 12106: 12100: 12097: 12096: 12094: 12093: 12088: 12083: 12078: 12073: 12068: 12063: 12058: 12053: 12048: 12042: 12039: 12038: 12036: 12035: 12030: 12025: 12020: 12015: 12010: 12005: 12000: 11995: 11990: 11984: 11981: 11980: 11978: 11977: 11972: 11967: 11962: 11957: 11952: 11947: 11942: 11937: 11932: 11926: 11923: 11922: 11910: 11909: 11906: 11905: 11903: 11902: 11897: 11892: 11887: 11882: 11877: 11872: 11867: 11862: 11857: 11851: 11848: 11847: 11845: 11844: 11839: 11834: 11829: 11824: 11819: 11814: 11809: 11804: 11799: 11793: 11790: 11789: 11787: 11786: 11781: 11776: 11771: 11766: 11761: 11756: 11751: 11746: 11741: 11735: 11732: 11731: 11729: 11728: 11723: 11718: 11713: 11708: 11703: 11698: 11693: 11688: 11683: 11677: 11674: 11673: 11671: 11670: 11665: 11660: 11655: 11650: 11645: 11640: 11635: 11630: 11625: 11619: 11616: 11615: 11613: 11612: 11607: 11602: 11597: 11592: 11587: 11582: 11577: 11572: 11567: 11561: 11558: 11557: 11555: 11554: 11549: 11544: 11539: 11534: 11529: 11524: 11519: 11514: 11509: 11503: 11500: 11499: 11497: 11496: 11491: 11486: 11481: 11476: 11471: 11466: 11461: 11456: 11451: 11445: 11442: 11441: 11439: 11438: 11433: 11428: 11423: 11418: 11413: 11408: 11403: 11398: 11393: 11387: 11384: 11383: 11381: 11380: 11375: 11370: 11365: 11360: 11355: 11350: 11345: 11340: 11335: 11329: 11326: 11325: 11313: 11312: 11309: 11308: 11306: 11305: 11300: 11295: 11290: 11285: 11280: 11275: 11270: 11265: 11260: 11254: 11251: 11250: 11248: 11247: 11242: 11237: 11232: 11227: 11222: 11217: 11212: 11207: 11202: 11196: 11193: 11192: 11190: 11189: 11184: 11179: 11174: 11169: 11164: 11159: 11154: 11149: 11144: 11138: 11135: 11134: 11132: 11131: 11126: 11121: 11116: 11111: 11106: 11101: 11096: 11091: 11086: 11080: 11077: 11076: 11074: 11073: 11068: 11063: 11058: 11053: 11048: 11043: 11038: 11033: 11028: 11022: 11019: 11018: 11016: 11015: 11010: 11005: 11000: 10995: 10990: 10985: 10980: 10975: 10970: 10964: 10961: 10960: 10958: 10957: 10952: 10947: 10942: 10937: 10932: 10927: 10922: 10917: 10912: 10906: 10903: 10902: 10900: 10899: 10894: 10889: 10884: 10879: 10874: 10869: 10864: 10859: 10854: 10848: 10845: 10844: 10842: 10841: 10836: 10831: 10826: 10821: 10816: 10811: 10806: 10801: 10796: 10790: 10787: 10786: 10784: 10783: 10778: 10773: 10768: 10763: 10758: 10753: 10748: 10743: 10738: 10732: 10729: 10728: 10716: 10715: 10712: 10711: 10709: 10708: 10703: 10698: 10693: 10688: 10683: 10678: 10673: 10668: 10663: 10657: 10654: 10653: 10651: 10650: 10645: 10640: 10635: 10630: 10625: 10620: 10615: 10610: 10605: 10599: 10596: 10595: 10593: 10592: 10587: 10582: 10577: 10572: 10567: 10562: 10557: 10552: 10547: 10541: 10538: 10537: 10535: 10534: 10529: 10524: 10519: 10514: 10509: 10504: 10499: 10494: 10489: 10483: 10480: 10479: 10477: 10476: 10471: 10466: 10461: 10456: 10451: 10446: 10441: 10436: 10431: 10425: 10422: 10421: 10419: 10418: 10413: 10408: 10403: 10398: 10393: 10388: 10383: 10378: 10373: 10367: 10364: 10363: 10361: 10360: 10355: 10350: 10345: 10340: 10335: 10330: 10325: 10320: 10315: 10309: 10306: 10305: 10303: 10302: 10297: 10292: 10287: 10282: 10277: 10272: 10267: 10262: 10257: 10251: 10248: 10247: 10245: 10244: 10239: 10234: 10229: 10224: 10219: 10214: 10209: 10204: 10199: 10193: 10190: 10189: 10187: 10186: 10181: 10176: 10171: 10166: 10161: 10156: 10151: 10146: 10141: 10135: 10132: 10131: 10119: 10118: 10115: 10114: 10112: 10111: 10106: 10101: 10096: 10091: 10086: 10081: 10076: 10071: 10066: 10060: 10057: 10056: 10054: 10053: 10048: 10043: 10038: 10033: 10028: 10023: 10018: 10013: 10008: 10002: 9999: 9998: 9996: 9995: 9990: 9985: 9980: 9975: 9970: 9965: 9960: 9955: 9950: 9944: 9941: 9940: 9938: 9937: 9932: 9927: 9922: 9917: 9912: 9907: 9902: 9897: 9892: 9886: 9883: 9882: 9880: 9879: 9874: 9869: 9864: 9859: 9854: 9849: 9844: 9839: 9834: 9828: 9825: 9824: 9822: 9821: 9816: 9811: 9806: 9801: 9796: 9791: 9786: 9781: 9776: 9770: 9767: 9766: 9764: 9763: 9758: 9753: 9748: 9743: 9738: 9733: 9728: 9723: 9718: 9712: 9709: 9708: 9706: 9705: 9700: 9695: 9690: 9685: 9680: 9675: 9670: 9665: 9660: 9654: 9651: 9650: 9648: 9647: 9642: 9637: 9632: 9627: 9622: 9617: 9612: 9607: 9602: 9596: 9593: 9592: 9590: 9589: 9584: 9579: 9574: 9569: 9564: 9559: 9554: 9549: 9544: 9538: 9535: 9534: 9522: 9521: 9518: 9517: 9515: 9514: 9509: 9504: 9499: 9494: 9489: 9484: 9479: 9474: 9469: 9463: 9460: 9459: 9457: 9456: 9451: 9446: 9441: 9436: 9431: 9426: 9421: 9416: 9411: 9405: 9402: 9401: 9399: 9398: 9393: 9388: 9383: 9378: 9373: 9368: 9363: 9358: 9353: 9347: 9344: 9343: 9341: 9340: 9335: 9330: 9325: 9320: 9315: 9310: 9305: 9300: 9295: 9289: 9286: 9285: 9283: 9282: 9277: 9272: 9267: 9262: 9257: 9252: 9247: 9242: 9237: 9231: 9228: 9227: 9225: 9224: 9219: 9214: 9209: 9204: 9199: 9194: 9189: 9184: 9179: 9173: 9170: 9169: 9167: 9166: 9161: 9156: 9151: 9146: 9141: 9136: 9131: 9126: 9121: 9115: 9112: 9111: 9109: 9108: 9103: 9098: 9093: 9088: 9083: 9078: 9073: 9068: 9063: 9057: 9054: 9053: 9051: 9050: 9045: 9040: 9035: 9030: 9025: 9020: 9015: 9010: 9005: 8999: 8996: 8995: 8993: 8992: 8987: 8982: 8977: 8972: 8967: 8962: 8957: 8952: 8947: 8941: 8938: 8937: 8925: 8924: 8921: 8920: 8918: 8917: 8912: 8907: 8902: 8897: 8892: 8887: 8882: 8877: 8872: 8866: 8863: 8862: 8860: 8859: 8854: 8849: 8844: 8839: 8834: 8829: 8824: 8819: 8814: 8808: 8805: 8804: 8802: 8801: 8796: 8791: 8786: 8781: 8776: 8771: 8766: 8761: 8756: 8750: 8747: 8746: 8744: 8743: 8738: 8733: 8728: 8723: 8718: 8713: 8708: 8703: 8698: 8692: 8689: 8688: 8686: 8685: 8680: 8675: 8670: 8665: 8660: 8655: 8650: 8645: 8640: 8634: 8631: 8630: 8628: 8627: 8622: 8617: 8612: 8607: 8602: 8597: 8592: 8587: 8582: 8576: 8573: 8572: 8570: 8569: 8564: 8559: 8554: 8549: 8544: 8539: 8534: 8529: 8524: 8518: 8515: 8514: 8512: 8511: 8506: 8501: 8496: 8491: 8486: 8481: 8476: 8471: 8466: 8460: 8457: 8456: 8454: 8453: 8448: 8443: 8438: 8433: 8428: 8423: 8418: 8413: 8408: 8402: 8399: 8398: 8396: 8395: 8390: 8385: 8380: 8375: 8370: 8365: 8360: 8355: 8350: 8344: 8341: 8340: 8328: 8327: 8324: 8323: 8321: 8320: 8315: 8310: 8305: 8300: 8295: 8290: 8285: 8280: 8275: 8269: 8266: 8265: 8263: 8262: 8257: 8252: 8247: 8242: 8237: 8232: 8227: 8222: 8217: 8211: 8208: 8207: 8205: 8204: 8199: 8194: 8189: 8184: 8179: 8174: 8169: 8164: 8159: 8153: 8150: 8149: 8147: 8146: 8141: 8136: 8131: 8126: 8121: 8116: 8111: 8106: 8101: 8095: 8092: 8091: 8089: 8088: 8083: 8078: 8073: 8068: 8063: 8058: 8053: 8048: 8043: 8037: 8034: 8033: 8031: 8030: 8025: 8020: 8015: 8010: 8005: 8000: 7995: 7990: 7985: 7979: 7976: 7975: 7973: 7972: 7967: 7962: 7957: 7952: 7947: 7942: 7937: 7932: 7927: 7921: 7918: 7917: 7915: 7914: 7909: 7904: 7899: 7894: 7889: 7884: 7879: 7874: 7869: 7863: 7860: 7859: 7857: 7856: 7851: 7846: 7841: 7836: 7831: 7826: 7821: 7816: 7811: 7805: 7802: 7801: 7799: 7798: 7793: 7788: 7783: 7778: 7773: 7768: 7763: 7758: 7753: 7747: 7744: 7743: 7731: 7730: 7727: 7726: 7724: 7723: 7718: 7713: 7708: 7703: 7698: 7693: 7688: 7683: 7678: 7672: 7669: 7668: 7666: 7665: 7660: 7655: 7650: 7645: 7640: 7635: 7630: 7625: 7620: 7614: 7611: 7610: 7608: 7607: 7602: 7597: 7592: 7587: 7582: 7577: 7572: 7567: 7562: 7556: 7553: 7552: 7550: 7549: 7544: 7539: 7534: 7529: 7524: 7519: 7514: 7509: 7504: 7498: 7495: 7494: 7492: 7491: 7486: 7481: 7476: 7471: 7466: 7461: 7456: 7451: 7446: 7440: 7437: 7436: 7434: 7433: 7428: 7423: 7418: 7413: 7408: 7403: 7398: 7393: 7388: 7382: 7379: 7378: 7376: 7375: 7370: 7365: 7360: 7355: 7350: 7345: 7340: 7335: 7330: 7324: 7321: 7320: 7318: 7317: 7312: 7307: 7302: 7297: 7292: 7287: 7282: 7277: 7272: 7266: 7263: 7262: 7260: 7259: 7254: 7249: 7244: 7239: 7234: 7229: 7224: 7219: 7214: 7208: 7205: 7204: 7202: 7201: 7196: 7191: 7186: 7181: 7176: 7171: 7166: 7161: 7156: 7150: 7147: 7146: 7134: 7133: 7130: 7129: 7127: 7126: 7121: 7116: 7111: 7106: 7101: 7096: 7091: 7086: 7081: 7075: 7072: 7071: 7069: 7068: 7063: 7058: 7053: 7048: 7043: 7038: 7033: 7028: 7023: 7017: 7014: 7013: 7011: 7010: 7005: 7000: 6995: 6990: 6985: 6980: 6975: 6970: 6965: 6959: 6956: 6955: 6953: 6952: 6947: 6942: 6937: 6932: 6927: 6922: 6917: 6912: 6907: 6901: 6898: 6897: 6895: 6894: 6889: 6884: 6879: 6874: 6869: 6864: 6859: 6854: 6849: 6843: 6840: 6839: 6837: 6836: 6831: 6826: 6821: 6816: 6811: 6806: 6801: 6796: 6791: 6785: 6782: 6781: 6779: 6778: 6773: 6768: 6763: 6758: 6753: 6748: 6743: 6738: 6733: 6727: 6724: 6723: 6721: 6720: 6715: 6710: 6705: 6700: 6695: 6690: 6685: 6680: 6675: 6669: 6666: 6665: 6663: 6662: 6657: 6652: 6647: 6642: 6637: 6632: 6627: 6622: 6617: 6611: 6608: 6607: 6605: 6604: 6597: 6590: 6583: 6576: 6569: 6562: 6555: 6548: 6541: 6534: 6526: 6523: 6522: 6510: 6509: 6502: 6501: 6494: 6487: 6479: 6473: 6472: 6471: 6470: 6455: 6432: 6413: 6398: 6384: 6379: 6374: 6369: 6362: 6361:External links 6359: 6357: 6356: 6350: 6338:Seife, Charles 6334: 6328: 6315: 6309: 6296: 6290: 6277: 6271: 6254: 6252: 6249: 6247: 6246: 6240: 6227: 6209: 6191: 6185: 6172: 6166: 6152:Kardar, Mehran 6148: 6142: 6130:Cheng, Eugenia 6126: 6120: 6102: 6088: 6070: 6064: 6050:Aczel, Amir D. 6045: 6043: 6040: 6037: 6036: 6014: 5988: 5963: 5933: 5925: 5907: 5905:. 2014. p. 399 5887: 5874:Iztok Fajfar. 5867: 5865:. 2005. p. 274 5847: 5827: 5793: 5774: 5754: 5747: 5727: 5697: 5690: 5670: 5663: 5643: 5631: 5619: 5607: 5600: 5582: 5575: 5552: 5540: 5528: 5521: 5490: 5463: 5439: 5427: 5417: 5411:. p. 23. 5393: 5386: 5354: 5335: 5312: 5281: 5269: 5257: 5227: 5207: 5195: 5176:(8): 513–518. 5157: 5145: 5118:(3): 175–221. 5102: 5080: 5077: 5076: 5058: 5035: 5016: 5010: 4988: 4981: 4961: 4947: 4927: 4913: 4887: 4872: 4851: 4849:, p. 400. 4839: 4805: 4802: 4801: 4781: 4740: 4733: 4717: 4693: 4679: 4659: 4628: 4612: 4610:, p. 416. 4600: 4587:(1): 134–150. 4569:Hayashi, Takao 4556: 4526: 4493:Devlin, Hannah 4481: 4450: 4438: 4405: 4391: 4369: 4358: 4328: 4311: 4304: 4286: 4250: 4240: 4220: 4207: 4177: 4165: 4140: 4112: 4105: 4076: 4059: 4040: 4032: 4011: 3987: 3980: 3960:Pedersen, Olaf 3951: 3924: 3917: 3887: 3867:Home of Kairos 3850: 3843: 3837:–14, plate 2. 3829:(2 ed.). 3810: 3779: 3767: 3753: 3733: 3711: 3697:Seife, Charles 3688: 3657: 3631: 3606: 3593: 3571: 3538: 3536:, p. 186. 3526: 3501: 3499:, p. 172. 3489: 3459: 3428: 3413: 3378: 3345: 3312: 3298: 3284: 3272: 3249: 3246: 3245: 3240:. p. 38: 3236: 3223: 3216:. p. 26: 3212: 3196: 3182: 3172: 3169: 3168: 3135: 3122: 3107: 3096:, from Arabic 3066: 3058: 3057: 3055: 3052: 3049: 3048: 3040:Greek numerals 3031: 3020: 3019: 3017: 3014: 3013: 3012: 3007: 3002: 2997: 2990: 2987: 2948:Main article: 2945: 2944:Dating systems 2942: 2925: 2922: 2920: 2917: 2879:Classic Mac OS 2856:floating-point 2712:typically use 2697: 2694: 2663:freezing point 2640: 2637: 2636: 2635: 2632:zero morphisms 2621: 2614: 2603: 2532:) on a domain 2479:lattice theory 2467:ordinal number 2429: 2426: 2315: 2314: 2271:Exponentiation 2268: 2212: 2201:Multiplication 2198: 2176: 2090:, and hence a 2080:natural number 2059: 2056: 2035:Main article: 2032: 2029: 2017: 2014: 1983: 1975: 1922:Main article: 1919: 1916: 1912:Roman numerals 1815: 1812: 1752: 1749: 1747: 1744: 1721:Khmer numerals 1667:Khmer numerals 1664: 1663: 1654: 1653: 1652: 1643: 1642: 1634: 1633: 1632: 1628: 1627: 1626: 1625: 1623: 1620: 1483:scholar, used 1466: 1463: 1414:4th century BC 1405:Sūnzĭ Suànjīng 1383: 1380: 1353:Roman numerals 1339:Medieval Greek 1234:lunar eclipses 1143:Greek numerals 1119:ancient Greeks 1114: 1111: 1070:4th century BC 1034: 1031: 926: 925: 922: 921: 918: 917: 906: 898: 889: 886: 884: 881: 865:– from French 810: 807: 771:. This became 661: 658: 452: 451: 448: 442: 441: 438: 432: 431: 428: 422: 421: 418: 412: 411: 408: 402: 401: 398: 392: 391: 388: 382: 381: 376: 374:Hindu numerals 370: 369: 364: 342: 341: 338: 335: 329: 328: 325: 322: 316: 315: 312: 309: 303: 302: 299: 296: 290: 289: 286: 283: 277: 276: 273: 270: 264: 263: 260: 251: 250: 247: 241: 240: 219: 213: 212: 174: 171: 170: 165: 159: 158: 120: 117: 116: 113: 112: 107: 104: 91:Natural number 90: 58:, you may see 46: 45: 44: 40:Naught (album) 26: 9: 6: 4: 3: 2: 13059: 13048: 13045: 13043: 13040: 13038: 13035: 13033: 13030: 13029: 13027: 13012: 13008: 13004: 13002: 12998: 12994: 12992: 12984: 12982: 12974: 12972: 12964: 12963: 12960: 12954: 12951: 12949: 12946: 12944: 12941: 12939: 12936: 12934: 12931: 12929: 12928:Modular forms 12926: 12924: 12921: 12920: 12918: 12914: 12908: 12905: 12903: 12900: 12898: 12895: 12893: 12890: 12887: 12883: 12880: 12878: 12875: 12873: 12870: 12868: 12865: 12863: 12860: 12858: 12855: 12853: 12852:Prime numbers 12850: 12848: 12845: 12843: 12840: 12838: 12835: 12833: 12830: 12829: 12827: 12823: 12817: 12814: 12812: 12809: 12806: 12802: 12798: 12795: 12792: 12788: 12785: 12782: 12778: 12774: 12771: 12769: 12766: 12764: 12761: 12759: 12756: 12753: 12749: 12745: 12741: 12738: 12735: 12734:Kummer theory 12731: 12727: 12723: 12719: 12715: 12712: 12711: 12709: 12705: 12701: 12700:Number theory 12694: 12689: 12687: 12682: 12680: 12675: 12674: 12671: 12655: 12654:1,000,000,000 12652: 12650: 12647: 12645: 12642: 12640: 12637: 12635: 12632: 12631: 12628: 12622: 12619: 12617: 12614: 12612: 12609: 12607: 12604: 12602: 12599: 12597: 12594: 12592: 12589: 12587: 12584: 12582: 12579: 12578: 12575: 12569: 12566: 12564: 12561: 12559: 12556: 12554: 12551: 12549: 12546: 12544: 12541: 12539: 12536: 12534: 12531: 12529: 12526: 12525: 12522: 12518: 12512: 12508: 12498: 12495: 12493: 12490: 12488: 12485: 12483: 12480: 12478: 12475: 12473: 12470: 12468: 12465: 12463: 12460: 12458: 12455: 12453: 12450: 12449: 12446: 12440: 12437: 12435: 12432: 12430: 12427: 12425: 12422: 12420: 12417: 12415: 12412: 12410: 12407: 12405: 12402: 12400: 12397: 12395: 12392: 12391: 12388: 12382: 12379: 12377: 12374: 12372: 12369: 12367: 12364: 12362: 12359: 12357: 12354: 12352: 12349: 12347: 12344: 12342: 12339: 12337: 12334: 12333: 12330: 12324: 12321: 12319: 12316: 12314: 12311: 12309: 12306: 12304: 12301: 12299: 12296: 12294: 12291: 12289: 12286: 12284: 12281: 12279: 12276: 12275: 12272: 12266: 12263: 12261: 12258: 12256: 12253: 12251: 12248: 12246: 12243: 12241: 12238: 12236: 12233: 12231: 12228: 12226: 12223: 12221: 12218: 12217: 12214: 12208: 12205: 12203: 12200: 12198: 12195: 12193: 12190: 12188: 12185: 12183: 12180: 12178: 12175: 12173: 12170: 12168: 12165: 12163: 12160: 12159: 12156: 12150: 12147: 12145: 12142: 12140: 12137: 12135: 12132: 12130: 12127: 12125: 12122: 12120: 12117: 12115: 12112: 12110: 12107: 12105: 12102: 12101: 12098: 12092: 12089: 12087: 12084: 12082: 12079: 12077: 12074: 12072: 12069: 12067: 12064: 12062: 12059: 12057: 12054: 12052: 12049: 12047: 12044: 12043: 12040: 12034: 12031: 12029: 12026: 12024: 12021: 12019: 12016: 12014: 12011: 12009: 12006: 12004: 12001: 11999: 11996: 11994: 11991: 11989: 11986: 11985: 11982: 11976: 11973: 11971: 11968: 11966: 11963: 11961: 11958: 11956: 11953: 11951: 11948: 11946: 11943: 11941: 11938: 11936: 11933: 11931: 11928: 11927: 11924: 11920: 11915: 11911: 11901: 11898: 11896: 11893: 11891: 11888: 11886: 11883: 11881: 11878: 11876: 11873: 11871: 11868: 11866: 11863: 11861: 11858: 11856: 11853: 11852: 11849: 11843: 11840: 11838: 11835: 11833: 11830: 11828: 11825: 11823: 11820: 11818: 11815: 11813: 11810: 11808: 11805: 11803: 11800: 11798: 11795: 11794: 11791: 11785: 11782: 11780: 11777: 11775: 11772: 11770: 11767: 11765: 11762: 11760: 11757: 11755: 11752: 11750: 11747: 11745: 11742: 11740: 11737: 11736: 11733: 11727: 11724: 11722: 11719: 11717: 11714: 11712: 11709: 11707: 11704: 11702: 11699: 11697: 11694: 11692: 11689: 11687: 11684: 11682: 11679: 11678: 11675: 11669: 11666: 11664: 11661: 11659: 11656: 11654: 11651: 11649: 11646: 11644: 11641: 11639: 11636: 11634: 11631: 11629: 11626: 11624: 11621: 11620: 11617: 11611: 11608: 11606: 11603: 11601: 11598: 11596: 11593: 11591: 11588: 11586: 11583: 11581: 11578: 11576: 11573: 11571: 11568: 11566: 11563: 11562: 11559: 11553: 11550: 11548: 11545: 11543: 11540: 11538: 11535: 11533: 11530: 11528: 11525: 11523: 11520: 11518: 11515: 11513: 11510: 11508: 11505: 11504: 11501: 11495: 11492: 11490: 11487: 11485: 11482: 11480: 11477: 11475: 11472: 11470: 11467: 11465: 11462: 11460: 11457: 11455: 11452: 11450: 11447: 11446: 11443: 11437: 11434: 11432: 11429: 11427: 11424: 11422: 11419: 11417: 11414: 11412: 11409: 11407: 11404: 11402: 11399: 11397: 11394: 11392: 11389: 11388: 11385: 11379: 11376: 11374: 11371: 11369: 11366: 11364: 11361: 11359: 11356: 11354: 11351: 11349: 11346: 11344: 11341: 11339: 11336: 11334: 11331: 11330: 11327: 11323: 11318: 11314: 11304: 11301: 11299: 11296: 11294: 11291: 11289: 11286: 11284: 11281: 11279: 11276: 11274: 11271: 11269: 11266: 11264: 11261: 11259: 11256: 11255: 11252: 11246: 11243: 11241: 11238: 11236: 11233: 11231: 11228: 11226: 11223: 11221: 11218: 11216: 11213: 11211: 11208: 11206: 11203: 11201: 11198: 11197: 11194: 11188: 11185: 11183: 11180: 11178: 11175: 11173: 11170: 11168: 11165: 11163: 11160: 11158: 11155: 11153: 11150: 11148: 11145: 11143: 11140: 11139: 11136: 11130: 11127: 11125: 11122: 11120: 11117: 11115: 11112: 11110: 11107: 11105: 11102: 11100: 11097: 11095: 11092: 11090: 11087: 11085: 11082: 11081: 11078: 11072: 11069: 11067: 11064: 11062: 11059: 11057: 11054: 11052: 11049: 11047: 11044: 11042: 11039: 11037: 11034: 11032: 11029: 11027: 11024: 11023: 11020: 11014: 11011: 11009: 11006: 11004: 11001: 10999: 10996: 10994: 10991: 10989: 10986: 10984: 10981: 10979: 10976: 10974: 10971: 10969: 10966: 10965: 10962: 10956: 10953: 10951: 10948: 10946: 10943: 10941: 10938: 10936: 10933: 10931: 10928: 10926: 10923: 10921: 10918: 10916: 10913: 10911: 10908: 10907: 10904: 10898: 10895: 10893: 10890: 10888: 10885: 10883: 10880: 10878: 10875: 10873: 10870: 10868: 10865: 10863: 10860: 10858: 10855: 10853: 10850: 10849: 10846: 10840: 10837: 10835: 10832: 10830: 10827: 10825: 10822: 10820: 10817: 10815: 10812: 10810: 10807: 10805: 10802: 10800: 10797: 10795: 10792: 10791: 10788: 10782: 10779: 10777: 10774: 10772: 10769: 10767: 10764: 10762: 10759: 10757: 10754: 10752: 10749: 10747: 10744: 10742: 10739: 10737: 10734: 10733: 10730: 10726: 10721: 10717: 10707: 10704: 10702: 10699: 10697: 10694: 10692: 10689: 10687: 10684: 10682: 10679: 10677: 10674: 10672: 10669: 10667: 10664: 10662: 10659: 10658: 10655: 10649: 10646: 10644: 10641: 10639: 10636: 10634: 10631: 10629: 10626: 10624: 10621: 10619: 10616: 10614: 10611: 10609: 10606: 10604: 10601: 10600: 10597: 10591: 10588: 10586: 10583: 10581: 10578: 10576: 10573: 10571: 10568: 10566: 10563: 10561: 10558: 10556: 10553: 10551: 10548: 10546: 10543: 10542: 10539: 10533: 10530: 10528: 10525: 10523: 10520: 10518: 10515: 10513: 10510: 10508: 10505: 10503: 10500: 10498: 10495: 10493: 10490: 10488: 10485: 10484: 10481: 10475: 10472: 10470: 10467: 10465: 10462: 10460: 10457: 10455: 10452: 10450: 10447: 10445: 10442: 10440: 10437: 10435: 10432: 10430: 10427: 10426: 10423: 10417: 10414: 10412: 10409: 10407: 10404: 10402: 10399: 10397: 10394: 10392: 10389: 10387: 10384: 10382: 10379: 10377: 10374: 10372: 10369: 10368: 10365: 10359: 10356: 10354: 10351: 10349: 10346: 10344: 10341: 10339: 10336: 10334: 10331: 10329: 10326: 10324: 10321: 10319: 10316: 10314: 10311: 10310: 10307: 10301: 10298: 10296: 10293: 10291: 10288: 10286: 10283: 10281: 10278: 10276: 10273: 10271: 10268: 10266: 10263: 10261: 10258: 10256: 10253: 10252: 10249: 10243: 10240: 10238: 10235: 10233: 10230: 10228: 10225: 10223: 10220: 10218: 10215: 10213: 10210: 10208: 10205: 10203: 10200: 10198: 10195: 10194: 10191: 10185: 10182: 10180: 10177: 10175: 10172: 10170: 10167: 10165: 10162: 10160: 10157: 10155: 10152: 10150: 10147: 10145: 10142: 10140: 10137: 10136: 10133: 10129: 10124: 10120: 10110: 10107: 10105: 10102: 10100: 10097: 10095: 10092: 10090: 10087: 10085: 10082: 10080: 10077: 10075: 10072: 10070: 10067: 10065: 10062: 10061: 10058: 10052: 10049: 10047: 10044: 10042: 10039: 10037: 10034: 10032: 10029: 10027: 10024: 10022: 10019: 10017: 10014: 10012: 10009: 10007: 10004: 10003: 10000: 9994: 9991: 9989: 9986: 9984: 9981: 9979: 9976: 9974: 9971: 9969: 9966: 9964: 9961: 9959: 9956: 9954: 9951: 9949: 9946: 9945: 9942: 9936: 9933: 9931: 9928: 9926: 9923: 9921: 9918: 9916: 9913: 9911: 9908: 9906: 9903: 9901: 9898: 9896: 9893: 9891: 9888: 9887: 9884: 9878: 9875: 9873: 9870: 9868: 9865: 9863: 9860: 9858: 9855: 9853: 9850: 9848: 9845: 9843: 9840: 9838: 9835: 9833: 9830: 9829: 9826: 9820: 9817: 9815: 9812: 9810: 9807: 9805: 9802: 9800: 9797: 9795: 9792: 9790: 9787: 9785: 9782: 9780: 9777: 9775: 9772: 9771: 9768: 9762: 9759: 9757: 9754: 9752: 9749: 9747: 9744: 9742: 9739: 9737: 9734: 9732: 9729: 9727: 9724: 9722: 9719: 9717: 9714: 9713: 9710: 9704: 9701: 9699: 9696: 9694: 9691: 9689: 9686: 9684: 9681: 9679: 9676: 9674: 9671: 9669: 9666: 9664: 9661: 9659: 9656: 9655: 9652: 9646: 9643: 9641: 9638: 9636: 9633: 9631: 9628: 9626: 9623: 9621: 9618: 9616: 9613: 9611: 9608: 9606: 9603: 9601: 9598: 9597: 9594: 9588: 9585: 9583: 9580: 9578: 9575: 9573: 9570: 9568: 9565: 9563: 9560: 9558: 9555: 9553: 9550: 9548: 9545: 9543: 9540: 9539: 9536: 9532: 9527: 9523: 9513: 9510: 9508: 9505: 9503: 9500: 9498: 9495: 9493: 9490: 9488: 9485: 9483: 9480: 9478: 9475: 9473: 9470: 9468: 9465: 9464: 9461: 9455: 9452: 9450: 9447: 9445: 9442: 9440: 9437: 9435: 9432: 9430: 9427: 9425: 9422: 9420: 9417: 9415: 9412: 9410: 9407: 9406: 9403: 9397: 9394: 9392: 9389: 9387: 9384: 9382: 9379: 9377: 9374: 9372: 9369: 9367: 9364: 9362: 9359: 9357: 9354: 9352: 9349: 9348: 9345: 9339: 9336: 9334: 9331: 9329: 9326: 9324: 9321: 9319: 9316: 9314: 9311: 9309: 9306: 9304: 9301: 9299: 9296: 9294: 9291: 9290: 9287: 9281: 9278: 9276: 9273: 9271: 9268: 9266: 9263: 9261: 9258: 9256: 9253: 9251: 9248: 9246: 9243: 9241: 9238: 9236: 9233: 9232: 9229: 9223: 9220: 9218: 9215: 9213: 9210: 9208: 9205: 9203: 9200: 9198: 9195: 9193: 9190: 9188: 9185: 9183: 9180: 9178: 9175: 9174: 9171: 9165: 9162: 9160: 9157: 9155: 9152: 9150: 9147: 9145: 9142: 9140: 9137: 9135: 9132: 9130: 9127: 9125: 9122: 9120: 9117: 9116: 9113: 9107: 9104: 9102: 9099: 9097: 9094: 9092: 9089: 9087: 9084: 9082: 9079: 9077: 9074: 9072: 9069: 9067: 9064: 9062: 9059: 9058: 9055: 9049: 9046: 9044: 9041: 9039: 9036: 9034: 9031: 9029: 9026: 9024: 9021: 9019: 9016: 9014: 9011: 9009: 9006: 9004: 9001: 9000: 8997: 8991: 8988: 8986: 8983: 8981: 8978: 8976: 8973: 8971: 8968: 8966: 8963: 8961: 8958: 8956: 8953: 8951: 8948: 8946: 8943: 8942: 8939: 8935: 8930: 8926: 8916: 8913: 8911: 8908: 8906: 8903: 8901: 8898: 8896: 8893: 8891: 8888: 8886: 8883: 8881: 8878: 8876: 8873: 8871: 8868: 8867: 8864: 8858: 8855: 8853: 8850: 8848: 8845: 8843: 8840: 8838: 8835: 8833: 8830: 8828: 8825: 8823: 8820: 8818: 8815: 8813: 8810: 8809: 8806: 8800: 8797: 8795: 8792: 8790: 8787: 8785: 8782: 8780: 8777: 8775: 8772: 8770: 8767: 8765: 8762: 8760: 8757: 8755: 8752: 8751: 8748: 8742: 8739: 8737: 8734: 8732: 8729: 8727: 8724: 8722: 8719: 8717: 8714: 8712: 8709: 8707: 8704: 8702: 8699: 8697: 8694: 8693: 8690: 8684: 8681: 8679: 8676: 8674: 8671: 8669: 8666: 8664: 8661: 8659: 8656: 8654: 8651: 8649: 8646: 8644: 8641: 8639: 8636: 8635: 8632: 8626: 8623: 8621: 8618: 8616: 8613: 8611: 8608: 8606: 8603: 8601: 8598: 8596: 8593: 8591: 8588: 8586: 8583: 8581: 8578: 8577: 8574: 8568: 8565: 8563: 8560: 8558: 8555: 8553: 8550: 8548: 8545: 8543: 8540: 8538: 8535: 8533: 8530: 8528: 8525: 8523: 8520: 8519: 8516: 8510: 8507: 8505: 8502: 8500: 8497: 8495: 8492: 8490: 8487: 8485: 8482: 8480: 8477: 8475: 8472: 8470: 8467: 8465: 8462: 8461: 8458: 8452: 8449: 8447: 8444: 8442: 8439: 8437: 8434: 8432: 8429: 8427: 8424: 8422: 8419: 8417: 8414: 8412: 8409: 8407: 8404: 8403: 8400: 8394: 8391: 8389: 8386: 8384: 8381: 8379: 8376: 8374: 8371: 8369: 8366: 8364: 8361: 8359: 8356: 8354: 8351: 8349: 8346: 8345: 8342: 8338: 8333: 8329: 8319: 8316: 8314: 8311: 8309: 8306: 8304: 8301: 8299: 8296: 8294: 8291: 8289: 8286: 8284: 8281: 8279: 8276: 8274: 8271: 8270: 8267: 8261: 8258: 8256: 8253: 8251: 8248: 8246: 8243: 8241: 8238: 8236: 8233: 8231: 8228: 8226: 8223: 8221: 8218: 8216: 8213: 8212: 8209: 8203: 8200: 8198: 8195: 8193: 8190: 8188: 8185: 8183: 8180: 8178: 8175: 8173: 8170: 8168: 8165: 8163: 8160: 8158: 8155: 8154: 8151: 8145: 8142: 8140: 8137: 8135: 8132: 8130: 8127: 8125: 8122: 8120: 8117: 8115: 8112: 8110: 8107: 8105: 8102: 8100: 8097: 8096: 8093: 8087: 8084: 8082: 8079: 8077: 8074: 8072: 8069: 8067: 8064: 8062: 8059: 8057: 8054: 8052: 8049: 8047: 8044: 8042: 8039: 8038: 8035: 8029: 8026: 8024: 8021: 8019: 8016: 8014: 8011: 8009: 8006: 8004: 8001: 7999: 7996: 7994: 7991: 7989: 7986: 7984: 7981: 7980: 7977: 7971: 7968: 7966: 7963: 7961: 7958: 7956: 7953: 7951: 7948: 7946: 7943: 7941: 7938: 7936: 7933: 7931: 7928: 7926: 7923: 7922: 7919: 7913: 7910: 7908: 7905: 7903: 7900: 7898: 7895: 7893: 7890: 7888: 7885: 7883: 7880: 7878: 7875: 7873: 7870: 7868: 7865: 7864: 7861: 7855: 7852: 7850: 7847: 7845: 7842: 7840: 7837: 7835: 7832: 7830: 7827: 7825: 7822: 7820: 7817: 7815: 7812: 7810: 7807: 7806: 7803: 7797: 7794: 7792: 7789: 7787: 7784: 7782: 7779: 7777: 7774: 7772: 7769: 7767: 7764: 7762: 7759: 7757: 7754: 7752: 7749: 7748: 7745: 7741: 7736: 7732: 7722: 7719: 7717: 7714: 7712: 7709: 7707: 7704: 7702: 7699: 7697: 7694: 7692: 7689: 7687: 7684: 7682: 7679: 7677: 7674: 7673: 7670: 7664: 7661: 7659: 7656: 7654: 7651: 7649: 7646: 7644: 7641: 7639: 7636: 7634: 7631: 7629: 7626: 7624: 7621: 7619: 7616: 7615: 7612: 7606: 7603: 7601: 7598: 7596: 7593: 7591: 7588: 7586: 7583: 7581: 7578: 7576: 7573: 7571: 7568: 7566: 7563: 7561: 7558: 7557: 7554: 7548: 7545: 7543: 7540: 7538: 7535: 7533: 7530: 7528: 7525: 7523: 7520: 7518: 7515: 7513: 7510: 7508: 7505: 7503: 7500: 7499: 7496: 7490: 7487: 7485: 7482: 7480: 7477: 7475: 7472: 7470: 7467: 7465: 7462: 7460: 7457: 7455: 7452: 7450: 7447: 7445: 7442: 7441: 7438: 7432: 7429: 7427: 7424: 7422: 7419: 7417: 7414: 7412: 7409: 7407: 7404: 7402: 7399: 7397: 7394: 7392: 7389: 7387: 7384: 7383: 7380: 7374: 7371: 7369: 7366: 7364: 7361: 7359: 7356: 7354: 7351: 7349: 7346: 7344: 7341: 7339: 7336: 7334: 7331: 7329: 7326: 7325: 7322: 7316: 7313: 7311: 7308: 7306: 7303: 7301: 7298: 7296: 7293: 7291: 7288: 7286: 7283: 7281: 7278: 7276: 7273: 7271: 7268: 7267: 7264: 7258: 7255: 7253: 7250: 7248: 7245: 7243: 7240: 7238: 7235: 7233: 7230: 7228: 7225: 7223: 7220: 7218: 7215: 7213: 7210: 7209: 7206: 7200: 7197: 7195: 7192: 7190: 7187: 7185: 7182: 7180: 7177: 7175: 7172: 7170: 7167: 7165: 7162: 7160: 7157: 7155: 7152: 7151: 7148: 7144: 7139: 7135: 7125: 7122: 7120: 7117: 7115: 7112: 7110: 7107: 7105: 7102: 7100: 7097: 7095: 7092: 7090: 7087: 7085: 7082: 7080: 7077: 7076: 7073: 7067: 7064: 7062: 7059: 7057: 7054: 7052: 7049: 7047: 7044: 7042: 7039: 7037: 7034: 7032: 7029: 7027: 7024: 7022: 7019: 7018: 7015: 7009: 7006: 7004: 7001: 6999: 6996: 6994: 6991: 6989: 6986: 6984: 6981: 6979: 6976: 6974: 6971: 6969: 6966: 6964: 6961: 6960: 6957: 6951: 6948: 6946: 6943: 6941: 6938: 6936: 6933: 6931: 6928: 6926: 6923: 6921: 6918: 6916: 6913: 6911: 6908: 6906: 6903: 6902: 6899: 6893: 6890: 6888: 6885: 6883: 6880: 6878: 6875: 6873: 6870: 6868: 6865: 6863: 6860: 6858: 6855: 6853: 6850: 6848: 6845: 6844: 6841: 6835: 6832: 6830: 6827: 6825: 6822: 6820: 6817: 6815: 6812: 6810: 6807: 6805: 6802: 6800: 6797: 6795: 6792: 6790: 6787: 6786: 6783: 6777: 6774: 6772: 6769: 6767: 6764: 6762: 6759: 6757: 6754: 6752: 6749: 6747: 6744: 6742: 6739: 6737: 6734: 6732: 6729: 6728: 6725: 6719: 6716: 6714: 6711: 6709: 6706: 6704: 6701: 6699: 6696: 6694: 6691: 6689: 6686: 6684: 6681: 6679: 6676: 6674: 6671: 6670: 6667: 6661: 6658: 6656: 6653: 6651: 6648: 6646: 6643: 6641: 6638: 6636: 6633: 6631: 6628: 6626: 6623: 6621: 6618: 6616: 6613: 6612: 6609: 6602: 6598: 6595: 6591: 6588: 6584: 6581: 6577: 6574: 6570: 6567: 6563: 6560: 6556: 6553: 6549: 6546: 6542: 6539: 6535: 6532: 6528: 6527: 6524: 6520: 6515: 6511: 6507: 6500: 6495: 6493: 6488: 6486: 6481: 6480: 6477: 6467: 6466: 6461: 6456: 6452: 6451: 6446: 6441: 6440: 6437: 6433: 6428: 6427: 6422: 6419: 6414: 6412: 6408: 6407: 6402: 6399: 6396: 6392: 6388: 6385: 6383: 6380: 6378: 6375: 6373: 6370: 6368: 6365: 6364: 6353: 6351:0-14-029647-6 6347: 6343: 6339: 6335: 6331: 6325: 6321: 6316: 6312: 6310:0-471-39340-1 6306: 6302: 6297: 6293: 6287: 6283: 6278: 6274: 6272:3-540-64767-8 6268: 6264: 6260: 6256: 6255: 6243: 6237: 6233: 6228: 6216: 6212: 6206: 6202: 6201: 6196: 6192: 6188: 6182: 6178: 6173: 6169: 6163: 6159: 6158: 6153: 6149: 6145: 6139: 6135: 6131: 6127: 6123: 6121:0-09-928845-1 6117: 6113: 6112: 6107: 6103: 6099: 6095: 6091: 6085: 6081: 6080: 6075: 6074:Asimov, Isaac 6071: 6067: 6061: 6057: 6056: 6051: 6047: 6046: 6033: 6025: 6021: 6017: 6011: 6007: 6002: 6001: 5992: 5984: 5980: 5979: 5974: 5967: 5951: 5947: 5943: 5937: 5928: 5926:0-13-011818-4 5922: 5918: 5911: 5904: 5900: 5897: 5891: 5884: 5880: 5877: 5871: 5864: 5860: 5857: 5851: 5845:2014. p. 204. 5844: 5840: 5837: 5834:Paul DuBois. 5831: 5824: 5811: 5807: 5803: 5797: 5781: 5777: 5771: 5767: 5766: 5758: 5750: 5744: 5740: 5739: 5731: 5715: 5711: 5707: 5701: 5693: 5687: 5683: 5682: 5674: 5666: 5660: 5656: 5655: 5647: 5641:, p. 20. 5640: 5635: 5628: 5627:Woodford 2006 5623: 5616: 5611: 5603: 5597: 5593: 5586: 5578: 5572: 5568: 5567: 5562: 5556: 5550:, p. 35. 5549: 5544: 5538:, p. 60. 5537: 5532: 5524: 5522:0-201-14236-8 5518: 5514: 5513: 5508: 5504: 5500: 5494: 5478: 5474: 5470: 5466: 5460: 5456: 5455: 5450: 5443: 5437:, p. 47. 5436: 5431: 5424: 5420: 5414: 5410: 5406: 5405: 5397: 5389: 5383: 5379: 5375: 5374: 5367: 5363: 5358: 5342: 5338: 5332: 5328: 5327: 5322: 5316: 5300: 5296: 5292: 5285: 5278: 5273: 5267:, p. 32. 5266: 5261: 5255: 5251: 5248: 5234: 5230: 5224: 5220: 5219: 5211: 5204: 5199: 5191: 5187: 5183: 5179: 5175: 5171: 5164: 5162: 5154: 5149: 5141: 5137: 5133: 5129: 5125: 5121: 5117: 5113: 5106: 5098: 5094: 5087: 5085: 5065: 5061: 5055: 5051: 5047: 5043: 5042: 5036: 5032: 5028: 5027: 5022: 5017: 5013: 5007: 5003: 4999: 4995: 4990: 4989: 4985: 4978: 4974: 4970: 4965: 4950: 4944: 4940: 4939: 4931: 4924: 4922: 4916: 4910: 4906: 4902: 4896: 4894: 4892: 4883: 4879: 4875: 4873:9780045200023 4869: 4865: 4861: 4855: 4848: 4843: 4827: 4823: 4819: 4815: 4809: 4798: 4794: 4787: 4782: 4778: 4774: 4770: 4766: 4762: 4758: 4754: 4751:(in French). 4750: 4746: 4745:Cœdès, George 4742: 4741: 4737: 4730: 4726: 4721: 4713: 4709: 4705: 4704: 4697: 4682: 4676: 4672: 4671: 4663: 4647: 4643: 4639: 4632: 4625: 4621: 4616: 4609: 4604: 4595: 4590: 4586: 4582: 4578: 4574: 4570: 4566: 4560: 4544: 4540: 4536: 4530: 4514: 4510: 4506: 4502: 4498: 4494: 4488: 4486: 4469: 4465: 4461: 4454: 4447: 4446:Bourbaki 1998 4442: 4423: 4416: 4409: 4394: 4388: 4384: 4380: 4373: 4366: 4361: 4355: 4351: 4347: 4346: 4341: 4335: 4333: 4325: 4321: 4315: 4307: 4301: 4297: 4290: 4271: 4267: 4260: 4254: 4247: 4243: 4237: 4233: 4232: 4224: 4210: 4204: 4200: 4196: 4192: 4188: 4181: 4174: 4169: 4154: 4153:Maths History 4150: 4144: 4137: 4133: 4132: 4127: 4123: 4116: 4108: 4102: 4098: 4093: 4092: 4083: 4081: 4073: 4069: 4063: 4055: 4051: 4044: 4035: 4029: 4025: 4021: 4015: 4004: 3997: 3991: 3983: 3977: 3973: 3969: 3965: 3961: 3955: 3939: 3935: 3928: 3920: 3918:0-691-00260-6 3914: 3910: 3907: 3906:Toomer, G. J. 3903: 3902: 3897: 3891: 3872: 3868: 3861: 3854: 3846: 3840: 3836: 3832: 3827: 3826: 3820: 3814: 3798: 3794: 3790: 3783: 3777:, p. 17. 3776: 3771: 3756: 3750: 3746: 3745: 3737: 3722: 3718: 3714: 3708: 3704: 3703: 3698: 3692: 3676: 3672: 3668: 3661: 3646: 3642: 3635: 3620: 3616: 3610: 3596: 3590: 3586: 3582: 3575: 3559: 3555: 3554: 3549: 3542: 3535: 3530: 3515: 3511: 3505: 3498: 3493: 3477: 3473: 3469: 3463: 3447: 3443: 3442:Maths History 3439: 3432: 3425: 3420: 3418: 3409: 3405: 3401: 3397: 3393: 3389: 3382: 3366: 3362: 3361: 3356: 3349: 3333: 3329: 3328:Thesaurus.com 3325: 3319: 3317: 3308: 3302: 3294: 3288: 3282:, p. 589 3281: 3276: 3267: 3263: 3259: 3253: 3243: 3239: 3233: 3229: 3224: 3221: 3220: 3215: 3209: 3206: 3202: 3197: 3194: 3191:, p. 518, at 3190: 3189: 3184: 3183: 3179: 3177: 3166: 3154: 3150: 3146: 3144: 3140: 3136: 3125: 3119: 3115: 3114: 3108: 3105: 3103: 3099: 3095: 3091: 3087: 3081: 3077: 3073: 3068: 3067: 3063: 3059: 3041: 3035: 3025: 3021: 3011: 3008: 3006: 3005:Number theory 3003: 3001: 2998: 2996: 2993: 2992: 2986: 2976: 2972: 2960: 2957: 2951: 2941: 2939: 2935: 2931: 2916: 2914: 2910: 2905: 2903: 2899: 2895: 2891: 2886: 2884: 2880: 2876: 2872: 2867: 2865: 2861: 2857: 2853: 2849: 2845: 2841: 2837: 2832: 2830: 2826: 2822: 2818: 2814: 2809: 2807: 2803: 2799: 2794: 2792: 2788: 2783: 2779: 2774: 2770: 2765: 2761: 2757: 2753: 2749: 2745: 2740: 2739: 2735: 2731: 2727: 2723: 2719: 2715: 2711: 2707: 2703: 2693: 2691: 2687: 2684: 2680: 2676: 2672: 2668: 2664: 2659: 2655: 2651: 2647: 2633: 2629: 2625: 2622: 2619: 2615: 2612: 2608: 2604: 2599: 2595: 2590: 2586: 2583: 2582: 2577: 2576: 2575: 2572: 2570: 2566: 2552: 2548: 2539: 2531: 2527: 2526:zero function 2523: 2522:vector spaces 2519: 2515: 2514:zero elements 2511: 2507: 2503: 2499: 2494: 2492: 2488: 2484: 2483:least element 2480: 2476: 2472: 2468: 2463: 2461: 2457: 2456: 2451: 2447: 2443: 2434: 2425: 2423: 2419: 2418: 2417:empty product 2413: 2412: 2406: 2404: 2395: 2391: 2382: 2378: 2370: 2366: 2357: 2353: 2344: 2340: 2308: 2304: 2302: 2292: 2284: 2276: 2272: 2269: 2266: 2262: 2258: 2247: 2238: 2230: 2216: 2213: 2210: 2206: 2202: 2199: 2196: 2192: 2188: 2184: 2180: 2177: 2174: 2170: 2166: 2162: 2158: 2155: 2154: 2153: 2151: 2146: 2144: 2140: 2136: 2135:singleton set 2132: 2128: 2124: 2120: 2116: 2111: 2109: 2105: 2101: 2097: 2093: 2089: 2085: 2081: 2077: 2069: 2064: 2055: 2053: 2049: 2044: 2038: 2028: 2023: 2013: 2011: 2007: 2003: 1999: 1981: 1973: 1965: 1960: 1958: 1954: 1950: 1946: 1937: 1930: 1925: 1915: 1913: 1909: 1905: 1901: 1896: 1889: 1882: 1878: 1874: 1870: 1866: 1861: 1857: 1851: 1849: 1845: 1841: 1837: 1833: 1829: 1825: 1821: 1811: 1809: 1805: 1801: 1799: 1795: 1791: 1787: 1783: 1779: 1774: 1772: 1768: 1764: 1758: 1743: 1741: 1737: 1732: 1730: 1726: 1722: 1718: 1714: 1710: 1705: 1696: 1694: 1684: 1680: 1676: 1672: 1668: 1658: 1647: 1638: 1618: 1613: 1611: 1607: 1606: 1601: 1596: 1594: 1591:499), states 1586: 1585: 1579: 1577: 1576: 1571: 1567: 1563: 1559: 1555: 1551: 1550: 1544: 1541: 1537: 1533: 1529: 1525: 1521: 1520: 1510: 1506: 1504: 1503: 1498: 1494: 1490: 1486: 1482: 1471: 1462: 1460: 1459: 1455:, as seen in 1451: 1446: 1442: 1441: 1436: 1431: 1429: 1425: 1424: 1419: 1418:counting rods 1407: 1406: 1397: 1393: 1392:counting rods 1388: 1379: 1377: 1372: 1366: 1362: 1358: 1354: 1350: 1345: 1340: 1336: 1332: 1328: 1324: 1316: 1315:Julian Easter 1311: 1309: 1300: 1267: 1247: 1243: 1239: 1235: 1231: 1227: 1223: 1219: 1218: 1213: 1209: 1194: 1190: 1186: 1173: 1169: 1167: 1163: 1159: 1155: 1150: 1148: 1144: 1140: 1136: 1132: 1128: 1124: 1123:Charles Seife 1120: 1110: 1108: 1104: 1100: 1096: 1092: 1090: 1086: 1082: 1078: 1077:Maya numerals 1073: 1067: 1062: 1060: 1056: 1052: 1048: 1039: 1030: 1028: 1023: 1018: 1013: 1007: 1003: 995: 991: 987: 983: 979: 974: 971: 967: 966:Alan Gardiner 963: 962: 957: 949: 945: 941: 937: 933: 915: 914: 911: 910: 907: 903: 899: 894: 893: 880: 878: 874: 869: 864: 860: 856: 851: 849: 844: 840: 839:military time 836: 832: 828: 823: 821: 817: 806: 803: 797: 796: 789: 788: 781: 775: 770: 755: 750: 749:was in 1598. 748: 740: 735: 729: 717: 712: 708: 703: 697: 691: 687: 682: 677: 671: 667: 657: 655: 649: 634: 630: 626: 622: 618: 612: 600: 596: 592: 588: 582: 564: 560: 556: 552: 547: 545: 541: 537: 533: 529: 525: 521: 517: 513: 509: 504: 502: 498: 494: 490: 486: 482: 478: 474: 470: 466: 462: 458: 449: 447: 443: 439: 437: 436:Maya numerals 433: 429: 427: 423: 419: 417: 413: 409: 407: 403: 399: 397: 393: 389: 387: 383: 377: 375: 371: 365: 363: 359: 355: 351: 347: 343: 336: 334: 330: 323: 321: 317: 310: 308: 304: 297: 295: 291: 284: 282: 278: 271: 269: 265: 261: 259: 256: 252: 248: 246: 242: 234: 220: 218: 214: 210: 207: 204: 201: 198: 195: 192: 189: 186: 183: 180: 177: 169: 166: 164: 161: 160: 156: 153: 150: 147: 144: 141: 138: 135: 132: 129: 126: 123: 118: 111: 108: 105: 103: 100: 99: 95: 89: 85: 81: 74: 70: 61: 57: 53: 51: 41: 37: 33: 19: 12836: 12825:Key concepts 12752:sieve theory 6537: 6518: 6463: 6448: 6424: 6405: 6341: 6319: 6300: 6281: 6262: 6231: 6219:. Retrieved 6199: 6176: 6156: 6133: 6110: 6078: 6055:Finding Zero 6054: 6042:Bibliography 6027: 5999: 5991: 5983:the original 5976: 5966: 5954:. Retrieved 5945: 5936: 5916: 5910: 5890: 5870: 5850: 5830: 5821: 5814:. Retrieved 5805: 5796: 5784:. Retrieved 5764: 5757: 5737: 5730: 5718:. Retrieved 5709: 5700: 5680: 5673: 5653: 5646: 5634: 5629:, p. 9. 5622: 5610: 5591: 5585: 5565: 5561:Riehl, Emily 5555: 5543: 5531: 5511: 5493: 5481:. Retrieved 5453: 5442: 5430: 5422: 5403: 5396: 5372: 5365: 5357: 5345:. Retrieved 5325: 5315: 5303:. Retrieved 5294: 5284: 5272: 5260: 5237:. Retrieved 5217: 5210: 5198: 5173: 5169: 5148: 5115: 5111: 5105: 5096: 5068:. Retrieved 5040: 5024: 4993: 4984: 4976: 4972: 4964: 4954:26 September 4952:. Retrieved 4937: 4930: 4920: 4918: 4904: 4901:Durant, Will 4863: 4854: 4847:Ifrah (2000) 4842: 4830:. Retrieved 4821: 4808: 4796: 4792: 4752: 4748: 4736: 4720: 4702: 4696: 4686:26 September 4684:. Retrieved 4669: 4662: 4650:. Retrieved 4641: 4631: 4619: 4615: 4608:Ifrah (2000) 4603: 4584: 4580: 4565:Plofker, Kim 4559: 4547:. Retrieved 4538: 4529: 4519:14 September 4517:. Retrieved 4501:The Guardian 4500: 4472:. Retrieved 4463: 4453: 4448:, p. 46 4441: 4429:. Retrieved 4422:the original 4408: 4396:. Retrieved 4382: 4372: 4363: 4344: 4340:Plofker, Kim 4323: 4319: 4314: 4295: 4289: 4277:. Retrieved 4265: 4253: 4245: 4230: 4223: 4212:, retrieved 4190: 4180: 4168: 4156:. Retrieved 4152: 4143: 4129: 4115: 4090: 4071: 4067: 4062: 4054:the original 4043: 4023: 4014: 3990: 3963: 3954: 3942:. Retrieved 3927: 3900: 3890: 3878:. Retrieved 3866: 3853: 3824: 3813: 3801:. Retrieved 3792: 3782: 3770: 3758:. Retrieved 3743: 3736: 3724:. Retrieved 3701: 3691: 3679:. Retrieved 3675:the original 3670: 3660: 3648:. Retrieved 3644: 3634: 3622:. Retrieved 3618: 3609: 3598:, retrieved 3584: 3574: 3562:. Retrieved 3553:The Guardian 3551: 3541: 3534:Diehl (2004) 3529: 3517:. Retrieved 3513: 3504: 3492: 3480:. Retrieved 3471: 3462: 3450:. Retrieved 3441: 3431: 3394:(2): 20–25. 3391: 3387: 3381: 3369:. Retrieved 3358: 3348: 3336:. Retrieved 3332:the original 3327: 3301: 3287: 3275: 3266:the original 3261: 3252: 3241: 3227: 3218: 3217: 3200: 3193:Google Books 3186: 3164: 3157:. Retrieved 3141: 3127:. Retrieved 3112: 3101: 3097: 3093: 3089: 3085: 3083: 3080:the original 3075: 3062: 3034: 3024: 3010:Peano axioms 2977:, the year 1 2961:, the year 1 2959:calendar era 2953: 2940:of species. 2927: 2919:Other fields 2909:slashed zero 2906: 2887: 2868: 2833: 2829:undetermined 2828: 2824: 2820: 2810: 2806:null pointer 2795: 2784: 2777: 2772: 2741: 2737: 2733: 2729: 2699: 2690:ground state 2642: 2597: 2593: 2588: 2584: 2579: 2573: 2550: 2546: 2529: 2525: 2502:zero element 2495: 2475:order theory 2464: 2453: 2439: 2420:) is 1. The 2415: 2409: 2407: 2393: 2389: 2380: 2376: 2355: 2351: 2342: 2338: 2316: 2306: 2300: 2290: 2282: 2274: 2245: 2236: 2228: 2208: 2204: 2194: 2190: 2186: 2182: 2168: 2164: 2160: 2149: 2147: 2127:prime number 2112: 2073: 2040: 2025: 1964:slashed zero 1961: 1942: 1900:al-Khwārizmī 1891: 1881:Latin people 1856:his homeland 1853: 1817: 1807: 1802: 1789: 1775: 1760: 1733: 1706: 1697: 1690: 1671:605 Saka era 1615: 1603: 1597: 1592: 1582: 1580: 1573: 1547: 1545: 1518: 1517: 1515: 1500: 1468: 1456: 1438: 1432: 1421: 1403: 1401: 1395: 1373: 1356: 1351:, alongside 1347:("none") by 1312: 1225: 1215: 1211: 1178: 1166:Zeno of Elea 1151: 1134: 1130: 1116: 1093: 1074: 1063: 1044: 1014: 1004:in the same 985: 975: 969: 959: 938:. They used 929: 852: 824: 812: 809:Modern usage 768: 751: 746: 710: 706: 675: 673: 653: 632: 628: 624: 620: 598: 594: 586: 562: 558: 554: 548: 516:power of ten 505: 485:real numbers 460: 456: 455: 178: 124: 88: 47: 13011:Wikiversity 12933:L-functions 12649:100,000,000 6406:In Our Time 6114:. Vintage. 5816:23 November 5615:Kardar 2007 5548:Kardar 2007 5321:Weil, André 5203:Reimer 2014 4969:Durant 1950 4832:20 December 4725:Kaplan 2000 4431:20 December 3833:. pp. 3775:Kaplan 2000 3681:1 September 3497:Reimer 2014 3482:7 September 3452:7 September 3424:Kaplan 2000 3371:21 December 2898:exit status 2840:signed zero 2758:introduced 2628:zero object 2611:truth value 2587:is a point 2544:defined by 2446:cardinality 2444:, 0 is the 2179:Subtraction 2139:prime ideal 2115:number line 2096:real number 2068:number line 2050:and in the 2016:Mathematics 2008:as used on 2002:Andalé Mono 1953:Typewriters 1746:Middle Ages 1600:Brahmagupta 1584:Aryabhatiya 1549:Lokavibhāga 1450:Han dynasty 1435:Qín Jiǔsháo 1266:sexagesimal 1260:0″, where 0 1210:called the 1193:Babylonians 1099:Inca Empire 1085:shell shape 1002:placeholder 990:sexagesimal 948:one papyrus 940:hieroglyphs 900:heart with 765: 1250 761: 1170 528:hexadecimal 333:Hexadecimal 102:← −1 13032:0 (number) 13026:Categories 12892:Arithmetic 12644:10,000,000 6393:, EWD831 ( 6098:1105483009 6024:1135427740 5786:7 November 5536:Cheng 2017 5473:1022848630 5435:Cheng 2017 5277:Cheng 2017 5265:Cheng 2017 5153:Ifrah 2000 5070:7 November 4921:Siddhantas 4799:: 125–132. 4727:, p. 4712:1039515732 4549:25 October 3721:1005913932 3564:10 October 3519:20 January 3280:Ifrah 2000 3139:"zero, n." 3076:Etymonline 3054:References 2881:epoch and 2875:Unix epoch 2813:null value 2442:set theory 2207:· 0 = 0 · 2163:+ 0 = 0 + 2117:. Zero is 2031:As a digit 2020:See also: 1895:algorismus 1869:Pythagoras 1824:Al-Andalus 1755:See also: 1493:Morse code 1477: 3rd 1189:Hipparchus 1055:quatrefoil 944:positional 763: – c. 446:Morse code 320:Duodecimal 73:Zero sharp 12639:1,000,000 6426:MathWorld 6377:Zero Saga 6303:. Wiley. 5823:function. 5639:Hill 2020 5239:5 January 5140:118227787 4777:130482979 4509:0261-3077 4022:(2016) . 3898:(1998) . 3821:(1969) . 3624:13 August 3600:13 August 3408:120648746 3129:5 January 2971:year zero 2950:Year zero 2938:evolution 2864:IBM S/390 2808:in code. 2489:or other 2450:empty set 2422:factorial 2411:empty sum 2299:the case 2261:undefined 2137:{0} is a 1945:typefaces 1848:Fibonacci 1840:astrolabe 1794:Algorithm 1771:Al-Mansur 1675:Old Khmer 1622:Epigraphy 1558:cosmology 1534:reported 1162:paradoxes 1127:astronomy 1089:Old World 1051:vigesimal 843:area code 754:Fibonacci 684:from the 674:The word 660:Etymology 540:Fibonacci 450:_ _ _ _ _ 221:0, zero, 110:1 → 13042:Integers 13001:Wikibook 12971:Category 6506:Integers 6340:(2000). 6261:(1998). 6221:24 March 6215:Archived 6197:(2006). 6154:(2007). 6132:(2017). 6108:(2001). 6052:(2015). 5950:Archived 5899:Archived 5879:Archived 5859:Archived 5839:Archived 5810:Archived 5780:Archived 5714:Archived 5563:(2016). 5509:(1988). 5477:Archived 5454:Calculus 5341:Archived 5299:Archived 5250:Archived 5233:Archived 5064:Archived 5044:. SAGE. 5031:Archived 4903:(1950). 4862:(1961). 4826:Archived 4646:Archived 4543:Archived 4513:Archived 4468:Archived 4398:21 April 4342:(2009). 4270:Archived 4214:28 April 4158:28 April 4003:Archived 3944:28 March 3938:Archived 3880:28 March 3871:Archived 3803:9 August 3797:Archived 3760:30 April 3726:30 April 3558:Archived 3476:Archived 3474:. 2016. 3446:Archived 3365:Archived 3338:23 April 3153:Archived 3094:zephirum 2989:See also 2860:IEEE 754 2796:In C, a 2764:Algol 58 2667:decibels 2555:for all 2530:zero map 2215:Division 2189:and 0 − 2157:Addition 1998:IBM 3270 1957:displays 1865:algorism 1798:Algorism 1729:Saka era 1717:Cambodia 1683:Cambodia 1566:Saka era 1556:text on 1497:Sanskrit 1437:'s 1247 1430:system. 1416:Chinese 1376:cultures 1374:In most 1327:Ethiopic 1308:overline 1256:20″ to 0 1252:0″ to 31 1226:Almagest 1217:Almagest 1191:and the 1154:medieval 1107:base ten 1081:tortoise 988:between 934:were of 930:Ancient 795:Zephyrus 769:zephyrum 739:Sanskrit 719:(Arabic 591:letter O 553:include 477:integers 469:quantity 217:Cardinal 168:Integers 12832:Numbers 12634:100,000 6468:. 1920. 6409:at the 5956:22 July 5483:26 July 5364:B.2.2, 5347:6 April 5305:4 April 5295:Wolfram 5120:Bibcode 4822:ams.org 4474:12 July 3896:Ptolemy 3159:4 March 3102:sunya-m 3044:0 to 59 2954:In the 2924:Biology 2883:Palm OS 2748:Fortran 2675:physics 2650:kelvins 2639:Physics 2487:lattice 2455:defined 2448:of the 2399:⁠ 2373:⁠ 2361:⁠ 2335:⁠ 2331:⁠ 2319:⁠ 2295:⁠ 2279:⁠ 2256:⁠ 2242:⁠ 2233:⁠ 2219:⁠ 2141:in the 2088:integer 1908:printed 1828:Muslims 1778:Persian 1562:Prakrit 1519:decimal 1470:Pingala 1293:d(24−d) 1291:√ 1283:⁠ 1271:⁠ 1242:minutes 1220:. This 1185:Ptolemy 1139:omicron 1135:όμικρον 1083:-like " 1059:Chiapas 982:base 60 956:pharaoh 952:1770 BC 936:base 10 902:trachea 897: 883:History 877:cricket 818:, and " 686:Italian 652:), and 585:), and 549:Common 512:decimal 475:of the 463:) is a 396:Burmese 386:Chinese 354:Persian 350:Kurdish 281:Ternary 245:Ordinal 12707:Fields 12621:90,000 12616:80,000 12611:70,000 12606:60,000 12601:50,000 12596:40,000 12591:30,000 12586:20,000 12581:10,000 6603: 6599: 6596: 6592: 6589: 6585: 6582: 6578: 6575: 6571: 6568: 6564: 6561: 6557: 6554: 6550: 6547: 6543: 6540: 6536: 6533: 6529: 6348: 6326: 6307: 6288: 6269: 6238: 6207: 6183: 6164: 6140: 6118: 6096: 6086: 6062: 6030: 6022: 6012: 5923: 5772: 5745: 5688: 5661: 5598: 5573: 5519: 5471: 5461: 5415: 5384: 5333: 5291:"Zero" 5225: 5190:294510 5188: 5138: 5056: 5008: 4945: 4911: 4882:840043 4880: 4870: 4775: 4769:607661 4767: 4710: 4677: 4652:26 May 4507: 4389: 4356: 4302: 4279:7 June 4238: 4205: 4103: 4030: 3978: 3915: 3841: 3751: 3719: 3709: 3650:5 June 3591: 3438:"Zero" 3406: 3258:"Zero" 3234: 3210: 3145:Online 3120: 3072:"Zero" 2983: 2979: 2967: 2963: 2802:string 2738:false. 2734:false, 2702:binary 2677:, the 2613:false. 2473:. In 2239:. But 2185:− 0 = 2129:nor a 2108:origin 2094:and a 1886: 1877:Euclid 1873:Hindus 1830:, the 1796:" or " 1763:Arabic 1736:glyphs 1709:Mekong 1369: 1323:epacts 1319: 1299:convex 1238:digits 1181: 1160:. The 1158:vacuum 1103:Andean 1066:Olmecs 1010:300 BC 998:700 BC 863:tennis 787:zefiro 774:zefiro 707:ṣafira 702:zefiro 696:zevero 654:cipher 563:naught 559:nought 524:binary 487:, and 465:number 358:Sindhi 346:Arabic 294:Senary 268:Binary 262:nulli- 258:prefix 223:"oh" ( 27:Number 18:Nought 12847:Unity 5720:9 May 5368:, in 5362:Lemma 5186:S2CID 5136:S2CID 4789:(PDF) 4773:S2CID 4765:JSTOR 4729:68–75 4425:(PDF) 4418:(PDF) 4350:54–56 4273:(PDF) 4262:(PDF) 4006:(PDF) 3999:(PDF) 3874:(PDF) 3863:(PDF) 3404:S2CID 3042:from 3016:Notes 2902:error 2888:Many 2871:epoch 2825:false 2752:COBOL 2744:array 2671:phons 2600:) = 0 2553:) = 0 2485:of a 2311:0 = 0 1860:Bugia 1832:Moors 1786:Latin 1767:Greek 1617:zero. 1575:śūnya 1540:birch 1524:India 1502:śūnya 1499:word 1489:meter 1465:India 1382:China 1371:725. 1357:nihil 1344:nulla 1331:Ge'ez 1230:solar 1183:150, 1179:By AD 1095:Quipu 1008:. By 986:space 946:. In 875:" in 868:l'œuf 861:" in 743:शून्य 734:śūnya 633:aught 629:ought 621:zilch 617:slang 506:As a 406:Khmer 307:Octal 255:Latin 12568:9000 12563:8000 12558:7000 12553:6000 12548:5000 12543:4000 12538:3000 12533:2000 12528:1000 12517:1000 11919:900s 11322:800s 10725:700s 10128:600s 9531:500s 8934:400s 8337:300s 7740:200s 7143:100s 6460:Zero 6445:Zero 6401:Zero 6346:ISBN 6324:ISBN 6305:ISBN 6286:ISBN 6267:ISBN 6236:ISBN 6223:2016 6205:ISBN 6181:ISBN 6162:ISBN 6138:ISBN 6116:ISBN 6094:OCLC 6084:ISBN 6060:ISBN 6020:OCLC 6010:ISBN 5958:2021 5921:ISBN 5818:2021 5788:2020 5770:ISBN 5743:ISBN 5722:2022 5686:ISBN 5659:ISBN 5596:ISBN 5571:ISBN 5517:ISBN 5485:2022 5469:OCLC 5459:ISBN 5413:ISBN 5382:ISBN 5349:2021 5331:ISBN 5307:2018 5241:2016 5223:ISBN 5072:2020 5054:ISBN 5006:ISBN 4977:ṣifr 4956:2016 4943:ISBN 4909:ISBN 4878:OCLC 4868:ISBN 4834:2015 4708:OCLC 4688:2016 4675:ISBN 4654:2013 4551:2017 4521:2017 4505:ISSN 4476:2018 4433:2015 4400:2017 4387:ISBN 4354:ISBN 4300:ISBN 4281:2012 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