1387:
1929:
1936:
1657:
1172:
2063:
1509:
13007:
12967:
6436:
12987:
12977:
12997:
1863:
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
1862:
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
4364:
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
1883:
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
1542:
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
2045:
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
845:
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
2766:
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
1543:
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
1305:
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
1616:
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
972:
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.
5779:
2660:
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
1447:
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).
4512:
1029:
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.
1064:
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".
5763:
518:
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
1846:
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
4269:
1955:
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
1538:
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
5949:
2462:
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.
1333:
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
1994:
3071:
2834:
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
2746:
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
1244:
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
2012:
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
6032:
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.
4496:
5809:
4919:
The Arabic inheritance of science was overwhelmingly Greek, but Hindu influences ranked next. In 773, at Mansur's behest, translations were made of the
1152:
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
12990:
6459:
4258:
3242:
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 (
6239:
1224:
was perhaps the earliest documented use of a numeral representing zero in the Old World. Ptolemy used it many times in his
1121:
had no symbol for zero (μηδέν, pronounced 'midén'), and did not use a digit placeholder for it. According to mathematician
1665:
The oldest, firmly dated use of zero as a decimal figure, found on the Sambor
Inscription. The number "605" is written in
5408:
1268:
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.
3359:
2835:
1057:
were used as a zero symbol for these Long Count dates, the earliest of which (on Stela 2 at Chiapa de Corzo,
17:
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
5092:
3219:
Zero derives from Hindu sunya – meaning void, emptiness – via Arabic sifr, Latin cephirum, Italian zevero.
3165:
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
1943:
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:
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.
3933:
12937:
12772:
5510:
5119:
4905:
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
4191:
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
2574:
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
834:
531:
31:
4703:
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:
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8317:
8312:
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8292:
8287:
8282:
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8272:
8259:
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8234:
8229:
8224:
8219:
8214:
8201:
8196:
8191:
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8176:
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8080:
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7969:
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7871:
7866:
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7848:
7843:
7838:
7833:
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7823:
7818:
7813:
7808:
7795:
7790:
7785:
7780:
7775:
7770:
7765:
7760:
7755:
7750:
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7690:
7685:
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7675:
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7657:
7652:
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7632:
7627:
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7604:
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7569:
7564:
7559:
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7541:
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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:
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6638:
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6631:
6628:
6626:
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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:
18:Zero function
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
1489:meters
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
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
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
4236:ISBN
4216:2024
4203:ISBN
4160:2024
4101:ISBN
4028:ISBN
3976:ISBN
3946:2020
3913:ISBN
3882:2020
3839:ISBN
3805:2020
3762:2022
3749:ISBN
3728:2022
3717:OCLC
3707:ISBN
3683:2016
3652:2024
3626:2024
3602:2024
3589:ISBN
3566:2015
3521:2024
3484:2021
3454:2021
3373:2019
3340:2013
3232:ISBN
3208:ISBN
3161:2012
3131:2016
3118:ISBN
3098:sifr
3090:zero
3086:zéro
2932:and
2892:and
2890:APIs
2862:and
2846:and
2821:true
2798:byte
2791:Java
2787:JDBC
2756:LISP
2750:and
2730:true
2654:zero
2567:and
2565:even
2528:(or
2520:and
2211:= 0.
2143:ring
2119:even
1818:The
1808:ṣifr
1761:The
1581:The
1554:Jain
1552:, a
1546:The
1402:The
1365:Bede
1289:20″
1240:and
1232:and
1117:The
1045:The
994:Kish
873:duck
859:love
802:ṣifr
780:zero
747:zero
728:Sifr
716:ṣifr
711:ṣifr
705:via
690:zero
681:zéro
676:zero
668:and
623:and
555:zero
544:Maya
526:and
499:has
461:zero
416:Thai
390:零, 〇
362:Urdu
78:For
67:For
34:and
12497:999
12492:998
12487:997
12482:996
12477:995
12472:994
12467:993
12462:992
12457:991
12452:990
12439:989
12434:988
12429:987
12424:986
12419:985
12414:984
12409:983
12404:982
12399:981
12394:980
12381:979
12376:978
12371:977
12366:976
12361:975
12356:974
12351:973
12346:972
12341:971
12336:970
12323:969
12318:968
12313:967
12308:966
12303:965
12298:964
12293:963
12288:962
12283:961
12278:960
12265:959
12260:958
12255:957
12250:956
12245:955
12240:954
12235:953
12230:952
12225:951
12220:950
12207:949
12202:948
12197:947
12192:946
12187:945
12182:944
12177:943
12172:942
12167:941
12162:940
12149:939
12144:938
12139:937
12134:936
12129:935
12124:934
12119:933
12114:932
12109:931
12104:930
12091:929
12086:928
12081:927
12076:926
12071:925
12066:924
12061:923
12056:922
12051:921
12046:920
12033:919
12028:918
12023:917
12018:916
12013:915
12008:914
12003:913
11998:912
11993:911
11988:910
11975:909
11970:908
11965:907
11960:906
11955:905
11950:904
11945:903
11940:902
11935:901
11930:900
11900:899
11895:898
11890:897
11885:896
11880:895
11875:894
11870:893
11865:892
11860:891
11855:890
11842:889
11837:888
11832:887
11827:886
11822:885
11817:884
11812:883
11807:882
11802:881
11797:880
11784:879
11779:878
11774:877
11769:876
11764:875
11759:874
11754:873
11749:872
11744:871
11739:870
11726:869
11721:868
11716:867
11711:866
11706:865
11701:864
11696:863
11691:862
11686:861
11681:860
11668:859
11663:858
11658:857
11653:856
11648:855
11643:854
11638:853
11633:852
11628:851
11623:850
11610:849
11605:848
11600:847
11595:846
11590:845
11585:844
11580:843
11575:842
11570:841
11565:840
11552:839
11547:838
11542:837
11537:836
11532:835
11527:834
11522:833
11517:832
11512:831
11507:830
11494:829
11489:828
11484:827
11479:826
11474:825
11469:824
11464:823
11459:822
11454:821
11449:820
11436:819
11431:818
11426:817
11421:816
11416:815
11411:814
11406:813
11401:812
11396:811
11391:810
11378:809
11373:808
11368:807
11363:806
11358:805
11353:804
11348:803
11343:802
11338:801
11333:800
11303:799
11298:798
11293:797
11288:796
11283:795
11278:794
11273:793
11268:792
11263:791
11258:790
11245:789
11240:788
11235:787
11230:786
11225:785
11220:784
11215:783
11210:782
11205:781
11200:780
11187:779
11182:778
11177:777
11172:776
11167:775
11162:774
11157:773
11152:772
11147:771
11142:770
11129:769
11124:768
11119:767
11114:766
11109:765
11104:764
11099:763
11094:762
11089:761
11084:760
11071:759
11066:758
11061:757
11056:756
11051:755
11046:754
11041:753
11036:752
11031:751
11026:750
11013:749
11008:748
11003:747
10998:746
10993:745
10988:744
10983:743
10978:742
10973:741
10968:740
10955:739
10950:738
10945:737
10940:736
10935:735
10930:734
10925:733
10920:732
10915:731
10910:730
10897:729
10892:728
10887:727
10882:726
10877:725
10872:724
10867:723
10862:722
10857:721
10852:720
10839:719
10834:718
10829:717
10824:716
10819:715
10814:714
10809:713
10804:712
10799:711
10794:710
10781:709
10776:708
10771:707
10766:706
10761:705
10756:704
10751:703
10746:702
10741:701
10736:700
10706:699
10701:698
10696:697
10691:696
10686:695
10681:694
10676:693
10671:692
10666:691
10661:690
10648:689
10643:688
10638:687
10633:686
10628:685
10623:684
10618:683
10613:682
10608:681
10603:680
10590:679
10585:678
10580:677
10575:676
10570:675
10565:674
10560:673
10555:672
10550:671
10545:670
10532:669
10527:668
10522:667
10517:666
10512:665
10507:664
10502:663
10497:662
10492:661
10487:660
10474:659
10469:658
10464:657
10459:656
10454:655
10449:654
10444:653
10439:652
10434:651
10429:650
10416:649
10411:648
10406:647
10401:646
10396:645
10391:644
10386:643
10381:642
10376:641
10371:640
10358:639
10353:638
10348:637
10343:636
10338:635
10333:634
10328:633
10323:632
10318:631
10313:630
10300:629
10295:628
10290:627
10285:626
10280:625
10275:624
10270:623
10265:622
10260:621
10255:620
10242:619
10237:618
10232:617
10227:616
10222:615
10217:614
10212:613
10207:612
10202:611
10197:610
10184:609
10179:608
10174:607
10169:606
10164:605
10159:604
10154:603
10149:602
10144:601
10139:600
10109:599
10104:598
10099:597
10094:596
10089:595
10084:594
10079:593
10074:592
10069:591
10064:590
10051:589
10046:588
10041:587
10036:586
10031:585
10026:584
10021:583
10016:582
10011:581
10006:580
9993:579
9988:578
9983:577
9978:576
9973:575
9968:574
9963:573
9958:572
9953:571
9948:570
9935:569
9930:568
9925:567
9920:566
9915:565
9910:564
9905:563
9900:562
9895:561
9890:560
9877:559
9872:558
9867:557
9862:556
9857:555
9852:554
9847:553
9842:552
9837:551
9832:550
9819:549
9814:548
9809:547
9804:546
9799:545
9794:544
9789:543
9784:542
9779:541
9774:540
9761:539
9756:538
9751:537
9746:536
9741:535
9736:534
9731:533
9726:532
9721:531
9716:530
9703:529
9698:528
9693:527
9688:526
9683:525
9678:524
9673:523
9668:522
9663:521
9658:520
9645:519
9640:518
9635:517
9630:516
9625:515
9620:514
9615:513
9610:512
9605:511
9600:510
9587:509
9582:508
9577:507
9572:506
9567:505
9562:504
9557:503
9552:502
9547:501
9542:500
9512:499
9507:498
9502:497
9497:496
9492:495
9487:494
9482:493
9477:492
9472:491
9467:490
9454:489
9449:488
9444:487
9439:486
9434:485
9429:484
9424:483
9419:482
9414:481
9409:480
9396:479
9391:478
9386:477
9381:476
9376:475
9371:474
9366:473
9361:472
9356:471
9351:470
9338:469
9333:468
9328:467
9323:466
9318:465
9313:464
9308:463
9303:462
9298:461
9293:460
9280:459
9275:458
9270:457
9265:456
9260:455
9255:454
9250:453
9245:452
9240:451
9235:450
9222:449
9217:448
9212:447
9207:446
9202:445
9197:444
9192:443
9187:442
9182:441
9177:440
9164:439
9159:438
9154:437
9149:436
9144:435
9139:434
9134:433
9129:432
9124:431
9119:430
9106:429
9101:428
9096:427
9091:426
9086:425
9081:424
9076:423
9071:422
9066:421
9061:420
9048:419
9043:418
9038:417
9033:416
9028:415
9023:414
9018:413
9013:412
9008:411
9003:410
8990:409
8985:408
8980:407
8975:406
8970:405
8965:404
8960:403
8955:402
8950:401
8945:400
8915:399
8910:398
8905:397
8900:396
8895:395
8890:394
8885:393
8880:392
8875:391
8870:390
8857:389
8852:388
8847:387
8842:386
8837:385
8832:384
8827:383
8822:382
8817:381
8812:380
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