72:. In some cases, an object or value that does not (originally) belong to a given sequence, but which could be naturally placed before its initial element, may be termed the zeroth element. There is no wide agreement regarding the correctness of using zero as an ordinal (nor regarding the use of the term
607:
Hence, the efficiency benefit at run time of zero-based indexing is not inherent, but is an artifact of the decision to represent an array with the address of its first element rather than the address of the fictitious zeroth element. However, the address of that fictitious element could very well be
1941:
quit after 1993. However, in 2014 the series moved to drivers carrying career-long personalised numbers, instead of team-allocated numbers, other than the defending champion still having the option to carry number 1. Therefore 0 is no longer used in this scenario. It is not clear if it is available
611:
Superficially, the fictitious element doesn't scale well to multidimensional arrays. Indexing multidimensional arrays from zero makes a naive (contiguous) conversion to a linear address space (systematically varying one index after the other) look simpler than when indexing from one. For instance,
1707:
In biology, an organism is said to have zero-order intentionality if it shows "no intention of anything at all". This would include a situation where the organism's genetically predetermined phenotype results in a fitness benefit to itself, because it did not "intend" to express its genes. In the
1628:
This situation can lead to some confusion in terminology. In a zero-based indexing scheme, the first element is "element number zero"; likewise, the twelfth element is "element number eleven". Therefore, an analogy from the ordinal numbers to the quantity of objects numbered appears; the highest
191:. Discussing possible designs of array ranges by enclosing them in a chained inequality, combining sharp and standard inequalities to four possibilities, demonstrating that to his conviction zero-based arrays are best represented by non-overlapping index ranges, which start at zero, alluding to
309:
on virtually all computer architectures, so this design detail in C makes compilation easier, at the cost of some human factors. In this context using "zeroth" as an ordinal is not strictly correct, but a widespread habit in this profession. Other programming languages, such as
1929:, when a defending world champion did not compete in the following season, the number 1 was not assigned to any driver, but one driver of the world champion team would carry the number 0, and the other, number 2. This did happen both in 1993 and 1994 with
1832:
In some universities, including Oxford and
Cambridge, "week 0" or occasionally "noughth week" refers to the week before the first week of lectures in a term. In Australia, some universities refer to this as "O week", which serves as a pun on
561:; that is, rather than using the address of the first array element, such a language would use the address of a fictitious element located immediately before the first actual element. The indexing expression for a 1-based index would then be
1781:
in buildings is considered as floor number 0 rather than as the "1st floor", the naming convention usually found in the United States of
America. This makes a consistent set with underground floors marked with negative numbers.
810:
2004:
are numbered starting from zero. This is for historical reasons: the list originally had only three freedoms, and when the fourth was added it was placed in the zeroth position as it was considered more basic.
378:, . Empty ranges, which often occur in algorithms, are tricky to express with a closed interval without resorting to obtuse conventions like . Because of this property, zero-based indexing potentially reduces
298:. In these three, sequence types (C arrays, Java arrays and lists, and Lisp lists and vectors) are indexed beginning with the zero subscript. Particularly in C, where arrays are closely tied to
427:. Because of this, many formulas in algorithms (such as that for calculating hash table indices) can be elegantly expressed in code using the modulo operation when array indices start at zero.
111:
zero times. Such usage corresponds to naming an element not properly belonging to the sequence but preceding it: the zeroth derivative is not really a derivative at all. However, just as the
1854:
starts basic training each
Wednesday, and the first week (of eight) is considered to begin with the following Sunday. The four days before that Sunday are often referred to as "zero week".
99:, and so forth. In some mathematical contexts, zero-based numbering can be used without confusion, when ordinal forms have well established meaning with an obvious candidate to come before
692:
359:. Proofs and other sorts of mathematical reasoning in computer science often begin with zero. For these reasons, in computer science it is not unusual to number from zero rather than one.
47: 0, rather than the index 1 as is typical in everyday non-mathematical or non-programming circumstances. Under zero-based numbering, the initial element is sometimes termed the
1528:
1864:
use 0 to denote the first (zeroth) hour of the day, consistent with using the 0 to denote the first (zeroth) minute of the hour and the first (zeroth) second of the minute. Also, the
1785:
While the ordinal of 0 mostly finds use in communities directly connected to mathematics, physics, and computer science, there are also instances in classical music. The composer
602:
490:
535:
442:
is the index of the desired element. To compute the address of the desired element, if the index numbers count from 1, the desired address is computed by this expression:
302:
arithmetic, this makes for a simpler implementation: the subscript refers to an offset from the starting position of an array, so the first element has an offset of zero.
1888:
were stolen, so he drew a whole new issue, which was published as issue 1. Later he re-inked his photocopies of the stolen artwork and published it as issue 0.
1461:
1297:
1083:
916:
1128:
1708:
similar sense, a computer may be considered from this perspective a zero-order intentional entity, as it does not "intend" to express the code of the programs it runs.
841:, which the invention of the zero made possible. In positional notation, tens, hundreds, thousands and all other digits start with zero, only units start at one.
430:
Pointer operations can also be expressed more elegantly on a zero-based index due to the underlying address/offset logic mentioned above. To illustrate, suppose
277:
1993:
wheel contains the number 0 as well as 1-36. It appears in green, so is classed as neither a "red" nor "black" number for betting purposes. The card game
1915:
195:
as with the real numbers. Dijkstra's criteria for preferring this convention are in detail that it represents empty sequences in a more natural way
397:(inclusive) less intuitive. Some authors prefer one-based indexing, as it corresponds more closely to how entities are indexed in other contexts.
1799:("doesn't count") on the score and a circle with a crossbar, intending it to mean "invalid". But posthumously, this work came to be known as
2424:
140:
1899:, but Brussels (being the capital city) was deemed deserving of a more basic number. Similarly the (unfinished) orbital motorway around
151:), designed arrays initiating at 0 as the natural position to start accessing the array contents in the language, since the value of a
541:
351:
Zero is the lowest unsigned integer value, one of the most fundamental types in programming and hardware design. In computer science,
170:
2399:
703:
176:, so the indirection optimization provided by these arrays was done at compile time. The optimization was nevertheless important.
1869:
319:
2428:
2055:
299:
152:
1942:
as a driver's chosen number, or whether they must be between 2 and 99, but it has not been used to date under this system.
1872:
use 0 to denote the hour immediately after midnight and noon in contrast to 12 used elsewhere, in order to avoid confusion
2228:
333:
547:
However, a language wishing to index arrays from 1 could adopt the convention that every array address is represented by
226:), and that with half-open "intervals" of naturals, the length of a sub-sequence equals the upper minus the lower bound (
2079:
1700:
323:
65:
1914:, especially in schemes where even numbers are one side of the street and odd numbers on the other. A case in point is
1759:
291:
2045:
2014:
1826:
1816:
697:
Organizing all arrays with 1-based indices (, , , ), and assuming an analogous arrangement of the elements, gives
638:
341:
337:
1806:
1800:
295:
2445:
1692:
287:
148:
2122:
1851:
1983:
1968:
375:
340:
allows setting the index origin to 0 or 1 during runtime programmatically. Some recent languages, such as
192:
2220:
2320:
1704:
was formulated after the first, second, and third laws, but considered more fundamental, thus its name.
2100:
1479:
448:
1684:
1645:
283:
88:
499:
is the size of each element. In contrast, if the index numbers count from 0, the expression becomes
2450:
1723:
838:
567:
505:
2265:
2216:
1972:
1743:
1711:
In biological or medical experiments, the first day of an experiment is often numbered as day 0.
401:
180:
173:
1101:
2395:
2051:
1767:
1665:
306:
91:
indices usually start at 0 in modern programming languages, so computer programmers might use
76:), as it creates ambiguity for all subsequent elements of the sequence when lacking context.
2257:
2047:
An
Introduction to Object-Oriented Programming in C++ with Applications in Computer Graphics
2019:
1954:
1834:
1822:
1429:
1265:
1051:
884:
405:
379:
84:
79:
Numbering sequences starting at 0 is quite common in mathematics notation, in particular in
1949:(in addition to the typical range of 1-99). The NFL voted to allow this from 2023 onwards.
2349:
1997:
has number cards running from 0 to 9 along with special cards, within each coloured suit.
1994:
1911:
1747:
383:
363:
1946:
1919:
1786:
1771:
435:
188:
44:
362:
If an array is used to represent a cycle, it is convenient to obtain the index with a
2439:
2001:
1934:
1892:
1873:
1865:
1857:
336:
allows the range of an array to be of any ordinal type (including enumerated types).
80:
2370:
2269:
2284:
1979:
1879:
1727:
1669:
345:
62:
2145:
1938:
1926:
1904:
1661:
1657:
187:
argued that arrays subscripts should start at zero as the latter being the most
2427:
prior to 1 November 2008 and incorporated under the "relicensing" terms of the
1644:
th element. For this reason, the first element is sometimes referred to as the
1930:
1715:
1680:
815:
to access the same element, which arguably looks more complicated. Of course,
352:
108:
83:, though programming languages for mathematics usually index from 1. In
2192:
1960:
1884:
1739:
1676:
356:
36:
2261:
2170:
1766:(where it coincides with the Gregorian year 1 BC), as well as in all
305:
Referencing memory by an address and an offset is represented directly in
17:
1990:
1900:
1861:
1763:
166:
40:
1895:
road in
Belgium is numbered R0. It was built after the ring road around
1896:
1719:
1688:
311:
318:, have array subscripts starting with one, because they were meant as
1838:
1837:". As a parallel, the introductory weeks at university educations in
1778:
374:
With zero-based numbering, a range can be expressed as the half-open
1793:
to be unworthy of including in the canon of his works, and he wrote
282:
This usage follows from design choices embedded in many influential
608:
the address of some other item in memory not related to the array.
278:
Comparison of programming languages (array) § Array dimensions
1952:
A chronological prequel of a series may be numbered as 0, such as
315:
2335:
Youtube title: Number grid / Counting / Early Math / Khan
Academy
1971:
number certain classes of rolling stock from zero, for example,
1874:
whether 12 a.m. and 12 p.m. represent noon or midnight
1755:
1751:
355:
is thus often used as the base case for many kinds of numerical
144:
69:
390:
is calculated in advance, making the use of counting from 0 to
1933:
carrying the number 0 in both seasons, as defending champion
322:, and as such they had to have a correspondence to the usual
2248:
Brown, Jim (December 1978). "In
Defense of Index Origin 0".
1687:
is defined, representing total mass in the case of physical
805:{\displaystyle r'=(z'-1)\cdot M\cdot N+(y'-1)\cdot M+(x'-0)}
2420:
1762:(where it coincides with the Julian year 1 BC) and in
632:
in zero-based indexing, i.e. , , , and , is calculated by
327:
49:
2371:"The Human Story – A new history of mankind's Evolution"
2350:"The Thinking Ape: Evolutionary Origins of Intelligence"
2309:
Programming
Microsoft® Visual C# 2005 by Donis Marshall.
2050:(2nd ed.). British Library: Springer. p. 391.
540:
This simpler expression is more efficient to compute at
348:, have adopted the same convention for the same reason.
1664:
are indexed by nonnegative integers, for example, the
628:
in the linear array to access a specific element with
1482:
1432:
1268:
1104:
1054:
887:
706:
641:
570:
508:
451:
400:
Another property of this convention is in the use of
2153:. Massachusetts Institute of Technology. p. 11.
1945:
Some team sports allow 0 to be chosen as a player's
1522:
1455:
1291:
1122:
1077:
910:
804:
686:
596:
529:
484:
107:of a function is the function itself, obtained by
404:as implemented in modern computers. Usually, the
2419:This article incorporates material taken from
2221:"Why numbering should start at zero (EWD 831)"
123:(or the original function itself) precede the
1842:
687:{\displaystyle r=z\cdot M\cdot N+y\cdot M+x.}
8:
1805:, even though it was actually written after
1794:
1648:element, in an attempt to avoid confusion.
165:in memory. BCPL was first compiled for the
2322:Math 1st Grade / Place Value / Number grid
2211:
2209:
2164:
2162:
2160:
1922:, whose address is 0 Garden Street.
1815:of Bruckner's, which is sometimes called
1481:
1431:
1267:
1103:
1053:
886:
705:
640:
612:when mapping the three-dimensional array
569:
507:
450:
158:used as an address accesses the position
2123:"Indexing Arrays, Matrices, and Vectors"
2002:Four Essential Freedoms of Free Software
1937:quit after 1992, and defending champion
1691:, or total probability, i.e. one, for a
1227:
846:
2036:
1615:The table content represents the index
1215:The table content represents the index
837:A simple and everyday-life example is
438:of the first element of an array, and
386:. On the other hand, the repeat count
27:Counting from "0" instead of "1" first
2101:"How to : Get Elements of Lists"
1986:, essentially making them four laws.
1982:eventually added a Zeroth Law to his
95:in situations where others might use
7:
2425:Free On-line Dictionary of Computing
193:open, half-open and closed intervals
1882:'s drawings for the first issue of
1758:. However, there is a year zero in
1742:does not exist in the widely used
1660:, many sequences of numbers or of
185:Why numbering should start at zero
39:in which the initial element of a
25:
2285:"Is Index Origin 0 a Hindrance?"
1750:. Under those systems, the year
1523:{\displaystyle (y'-1)\cdot M+x'}
485:{\displaystyle a+s\times (i-1),}
320:high-level programming languages
1860:and the international standard
2078:Steve Eddins and Loren Shure.
1808:Symphony No. 1 in C minor
1802:Symphony No. 0 in D minor
1500:
1483:
799:
782:
770:
753:
735:
718:
476:
464:
272:Usage in programming languages
1:
2394:. Holt McDougal. p. 35.
2229:University of Texas at Austin
597:{\displaystyle a'+s\times i.}
169:; the language introduced no
1701:zeroth law of thermodynamics
530:{\displaystyle a+s\times i.}
366:, which can result in zero.
68:corresponding to the number
2080:"Matrix Indexing in MATLAB"
1811:. There is an even earlier
1760:astronomical year numbering
1746:or in its predecessor, the
2467:
2015:Zeroth-order approximation
1123:{\displaystyle y\cdot M+x}
275:
2250:ACM SIGAPL APL Quote Quad
2147:The BCPL Reference Manual
1978:In the realm of fiction,
1614:
1588:
1562:
1541:
1424:
1403:
1377:
1351:
1313:
1214:
1188:
1162:
1141:
1046:
1025:
999:
973:
932:
211:than closed "intervals" (
147:language (a precursor of
53:element, rather than the
2144:Martin Richards (1967).
2044:M. Seed, Graham (1965).
1693:probability distribution
408:maps any integer modulo
2431:, version 1.3 or later.
2193:"The IBM 7094 and CTSS"
1910:Zero is sometimes used
1852:United States Air Force
1821:. The Russian composer
1777:In many countries, the
1640:, and it refers to the
2390:Towle, Albert (1989).
2225:E. W. Dijkstra Archive
2191:Tom Van Vleck (1995).
1984:Three Laws of Robotics
1969:Swiss Federal Railways
1843:
1795:
1524:
1457:
1456:{\displaystyle y'=y+1}
1293:
1292:{\displaystyle x'=x+1}
1124:
1079:
1078:{\displaystyle y=y'-1}
912:
911:{\displaystyle x=x'-1}
806:
688:
598:
531:
486:
412:to one of the numbers
183:in his pertinent note
2262:10.1145/586050.586053
2217:Dijkstra, Edsger Wybe
1955:Ring 0: Birthday
1903:in Hungary is called
1841:are generally called
1525:
1458:
1294:
1125:
1080:
913:
807:
689:
599:
532:
487:
328:invention of the zero
284:programming languages
1480:
1430:
1266:
1102:
1052:
885:
704:
639:
624:elements, the index
568:
506:
449:
370:Numerical properties
252:possible values for
131:Computer programming
33:Zero-based numbering
1912:in street addresses
1827:Symphony No. 0
1813:Symphony in F minor
1791:Symphony in D minor
1789:regarded his early
1233:
852:
839:positional notation
174:indirection lookups
119:, so also does the
2348:Byrne, Richard W.
1764:ISO 8601:2004
1744:Gregorian calendar
1520:
1453:
1289:
1228:
1120:
1075:
908:
847:
802:
684:
616:to a linear array
594:
527:
482:
402:modular arithmetic
326:which predate the
181:Edsger W. Dijkstra
103:; for instance, a
2171:"Citation Needed"
1724:population sample
1718:) is the initial
1714:Patient zero (or
1666:Bernoulli numbers
1623:
1622:
1223:
1222:
307:computer hardware
143:, creator of the
121:zeroth derivative
117:second derivative
105:zeroth derivative
16:(Redirected from
2458:
2406:
2405:
2387:
2381:
2380:
2378:
2377:
2366:
2360:
2359:
2357:
2356:
2345:
2339:
2337:
2332:
2330:
2316:
2310:
2307:
2301:
2300:
2298:
2296:
2280:
2274:
2273:
2245:
2239:
2238:
2236:
2235:
2213:
2204:
2203:
2201:
2199:
2188:
2182:
2181:
2179:
2177:
2166:
2155:
2154:
2152:
2141:
2135:
2134:
2132:
2130:
2119:
2113:
2112:
2110:
2108:
2097:
2091:
2090:
2088:
2086:
2075:
2069:
2068:
2066:
2064:
2041:
2020:Off-by-one error
1846:
1835:orientation week
1823:Alfred Schnittke
1798:
1643:
1639:
1633:objects will be
1632:
1618:
1531:
1529:
1527:
1526:
1521:
1519:
1493:
1464:
1462:
1460:
1459:
1454:
1440:
1343:
1331:
1325:
1319:
1300:
1298:
1296:
1295:
1290:
1276:
1245:
1240:
1234:
1218:
1131:
1129:
1127:
1126:
1121:
1086:
1084:
1082:
1081:
1076:
1068:
968:
962:
950:
944:
938:
919:
917:
915:
914:
909:
901:
864:
859:
853:
836:
832:
829:
825:
811:
809:
808:
803:
792:
763:
728:
714:
693:
691:
690:
685:
631:
627:
623:
619:
615:
603:
601:
600:
595:
578:
560:
536:
534:
533:
528:
498:
491:
489:
488:
483:
441:
433:
426:
419:
411:
396:
389:
384:fencepost errors
330:by a long time.
251:
239:
225:
210:
164:
125:first derivative
113:first derivative
85:computer science
43:is assigned the
21:
2466:
2465:
2461:
2460:
2459:
2457:
2456:
2455:
2446:Ordinal numbers
2436:
2435:
2434:
2415:
2410:
2409:
2402:
2389:
2388:
2384:
2375:
2373:
2369:Dunbar, Robin.
2368:
2367:
2363:
2354:
2352:
2347:
2346:
2342:
2328:
2326:
2318:
2317:
2313:
2308:
2304:
2294:
2292:
2282:
2281:
2277:
2247:
2246:
2242:
2233:
2231:
2219:(May 2, 2008).
2215:
2214:
2207:
2197:
2195:
2190:
2189:
2185:
2175:
2173:
2168:
2167:
2158:
2150:
2143:
2142:
2138:
2128:
2126:
2121:
2120:
2116:
2106:
2104:
2099:
2098:
2094:
2084:
2082:
2077:
2076:
2072:
2062:
2060:
2058:
2043:
2042:
2038:
2033:
2028:
2011:
1772:Hindu calendars
1754:is followed by
1748:Julian calendar
1736:
1730:investigation.
1728:epidemiological
1654:
1641:
1634:
1630:
1626:
1616:
1512:
1486:
1478:
1477:
1475:
1433:
1428:
1427:
1425:
1341:
1329:
1323:
1317:
1269:
1264:
1263:
1261:
1246:
1243:
1241:
1238:
1232:-based indices
1216:
1100:
1099:
1097:
1061:
1050:
1049:
1047:
966:
960:
948:
942:
936:
894:
883:
882:
880:
865:
862:
860:
857:
851:-based indices
834:
830:
827:
816:
785:
756:
721:
707:
702:
701:
637:
636:
629:
625:
621:
617:
613:
571:
566:
565:
548:
504:
503:
496:
447:
446:
439:
431:
421:
413:
409:
406:modulo function
391:
387:
372:
364:modulo function
324:ordinal numbers
280:
274:
268:all integers).
241:
227:
212:
196:
159:
141:Martin Richards
138:
133:
109:differentiating
28:
23:
22:
15:
12:
11:
5:
2464:
2462:
2454:
2453:
2448:
2438:
2437:
2433:
2432:
2416:
2414:
2411:
2408:
2407:
2400:
2392:Modern Biology
2382:
2361:
2340:
2325:. Khan Academy
2311:
2302:
2275:
2240:
2205:
2183:
2156:
2136:
2114:
2092:
2070:
2056:
2035:
2034:
2032:
2029:
2027:
2024:
2023:
2022:
2017:
2010:
2007:
1947:uniform number
1920:Harvard Square
1866:12-hour clocks
1858:24-hour clocks
1787:Anton Bruckner
1735:
1732:
1653:
1650:
1625:
1624:
1621:
1620:
1612:
1611:
1608:
1605:
1603:
1601:
1599:
1596:
1593:
1590:
1586:
1585:
1582:
1579:
1577:
1575:
1573:
1570:
1567:
1564:
1560:
1559:
1557:
1555:
1553:
1551:
1549:
1547:
1545:
1543:
1539:
1538:
1536:
1534:
1532:
1518:
1515:
1511:
1508:
1505:
1502:
1499:
1496:
1492:
1489:
1485:
1473:
1471:
1469:
1467:
1465:
1452:
1449:
1446:
1443:
1439:
1436:
1422:
1421:
1419:
1417:
1415:
1413:
1411:
1409:
1407:
1405:
1401:
1400:
1397:
1394:
1392:
1390:
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1382:
1379:
1375:
1374:
1371:
1368:
1366:
1364:
1362:
1359:
1356:
1353:
1349:
1348:
1345:
1339:
1337:
1335:
1333:
1327:
1321:
1315:
1311:
1310:
1307:
1304:
1301:
1288:
1285:
1282:
1279:
1275:
1272:
1259:
1256:
1253:
1250:
1247:
1242:
1237:
1224:
1221:
1220:
1212:
1211:
1208:
1205:
1203:
1201:
1199:
1196:
1193:
1190:
1186:
1185:
1182:
1179:
1177:
1175:
1173:
1170:
1167:
1164:
1160:
1159:
1157:
1155:
1153:
1151:
1149:
1147:
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1143:
1139:
1138:
1136:
1134:
1132:
1119:
1116:
1113:
1110:
1107:
1095:
1093:
1091:
1089:
1087:
1074:
1071:
1067:
1064:
1060:
1057:
1044:
1043:
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1039:
1037:
1035:
1033:
1031:
1029:
1027:
1023:
1022:
1019:
1016:
1014:
1012:
1010:
1007:
1004:
1001:
997:
996:
993:
990:
988:
986:
984:
981:
978:
975:
971:
970:
964:
958:
956:
954:
952:
946:
940:
934:
930:
929:
926:
923:
920:
907:
904:
900:
897:
893:
890:
878:
875:
872:
869:
866:
861:
856:
843:
813:
812:
801:
798:
795:
791:
788:
784:
781:
778:
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769:
766:
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520:
517:
514:
511:
493:
492:
481:
478:
475:
472:
469:
466:
463:
460:
457:
454:
436:memory address
414:0, 1, 2, ...,
371:
368:
273:
270:
189:natural number
137:
134:
132:
129:
66:ordinal number
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24:
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2218:
2212:
2210:
2206:
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2187:
2184:
2172:
2165:
2163:
2161:
2157:
2149:
2148:
2140:
2137:
2124:
2118:
2115:
2102:
2096:
2093:
2081:
2074:
2071:
2059:
2053:
2049:
2048:
2040:
2037:
2030:
2025:
2021:
2018:
2016:
2013:
2012:
2008:
2006:
2003:
1998:
1996:
1992:
1987:
1985:
1981:
1976:
1974:
1970:
1965:
1963:
1962:
1957:
1956:
1950:
1948:
1943:
1940:
1936:
1935:Nigel Mansell
1932:
1928:
1923:
1921:
1917:
1916:Christ Church
1913:
1908:
1906:
1902:
1898:
1894:
1893:Brussels ring
1889:
1887:
1886:
1881:
1877:
1875:
1871:
1867:
1863:
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1859:
1855:
1853:
1848:
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1840:
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1825:also wrote a
1824:
1820:
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1725:
1721:
1717:
1712:
1709:
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1703:
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1694:
1690:
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1682:
1678:
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1663:
1659:
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1391:
1389:
1386:
1383:
1380:
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1372:
1369:
1367:
1365:
1363:
1360:
1357:
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1350:
1346:
1340:
1338:
1336:
1334:
1328:
1322:
1316:
1312:
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1283:
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1273:
1270:
1260:
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1034:
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1002:
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741:
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669:
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289:
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271:
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267:
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234:
230:
223:
219:
215:
208:
204:
200:
194:
190:
186:
182:
177:
175:
172:
168:
167:IBM 7094
162:
157:
154:
150:
146:
142:
135:
130:
128:
126:
122:
118:
115:precedes the
114:
110:
106:
102:
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94:
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82:
81:combinatorics
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75:
71:
67:
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2385:
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2364:
2353:. Retrieved
2343:
2334:
2327:. Retrieved
2321:
2314:
2305:
2293:. Retrieved
2288:
2283:Hui, Roger.
2278:
2253:
2249:
2243:
2232:. Retrieved
2224:
2196:. Retrieved
2186:
2174:. Retrieved
2146:
2139:
2127:. Retrieved
2117:
2105:. Retrieved
2095:
2083:. Retrieved
2073:
2061:. Retrieved
2046:
2039:
1999:
1988:
1980:Isaac Asimov
1977:
1975:000 to 118.
1966:
1959:
1953:
1951:
1944:
1925:Formerly in
1924:
1909:
1890:
1883:
1880:Robert Crumb
1878:
1856:
1849:
1831:
1817:
1812:
1807:
1801:
1790:
1784:
1779:ground floor
1776:
1737:
1734:Other fields
1713:
1710:
1706:
1699:
1697:
1674:
1670:Bell numbers
1655:
1635:
1627:
1229:
848:
821:
817:
814:
696:
620:, both with
610:
606:
557:
553:
549:
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539:
494:
429:
422:
415:
399:
392:
373:
361:
350:
346:Visual Basic
332:
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286:, including
281:
265:
261:
257:
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243:
236:
232:
228:
221:
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120:
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35:is a way of
32:
31:
29:
2291:. JSoftware
2169:Mike Hoye.
2129:23 February
2125:. Maplesoft
2107:23 February
2085:23 February
2063:11 February
1989:A standard
1973:Re 460
1939:Alain Prost
1927:Formula One
1847:(zeroing).
1818:No. 00
1662:polynomials
1658:mathematics
224:− 1) ?
2451:0 (number)
2440:Categories
2376:2010-05-18
2355:2010-05-18
2319:Sal Khan.
2295:19 January
2234:2011-03-16
2198:28 January
2176:28 January
2057:1852334509
2026:References
1931:Damon Hill
1796:gilt nicht
1716:index case
1681:statistics
380:off-by-one
276:See also:
18:Zero-based
2103:. Wolfram
2031:Citations
1961:Zork Zero
1885:Zap Comix
1756:AD 1
1752:1 BC
1740:year zero
1677:mechanics
1629:index of
1504:⋅
1495:−
1109:⋅
1070:−
903:−
794:−
774:⋅
765:−
745:⋅
739:⋅
730:−
670:⋅
658:⋅
652:⋅
622:M ⋅ N ⋅ P
586:×
519:×
471:−
462:×
357:recursion
57:element;
37:numbering
2329:July 28,
2270:40187000
2256:(2): 7.
2009:See also
1991:roulette
1901:Budapest
1868:used in
1844:nollning
1768:Buddhist
1675:In both
1668:and the
1517:′
1491:′
1438:′
1274:′
1225:
1066:′
899:′
790:′
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726:′
712:′
576:′
542:run time
420:, where
376:interval
209: ?)
179:In 1982
171:run-time
41:sequence
2423:at the
2413:Sources
1897:Antwerp
1722:in the
1720:patient
1689:density
1652:Science
1530:
1476:
1463:
1426:
1299:
1262:
1130:
1098:
1085:
1048:
918:
881:
434:is the
312:Fortran
300:pointer
256:, with
153:pointer
2421:zeroth
2398:
2268:
2054:
1839:Sweden
1726:of an
1685:moment
1646:zeroth
826:since
495:where
334:Pascal
294:, and
240:gives
136:Origin
93:zeroth
74:zeroth
63:coined
59:zeroth
50:zeroth
2266:S2CID
2151:(PDF)
1870:Japan
630:L = A
316:COBOL
235:<
205:<
101:first
97:first
89:array
61:is a
55:first
45:index
2429:GFDL
2396:ISBN
2331:2018
2297:2015
2200:2014
2178:2014
2131:2021
2109:2021
2087:2021
2065:2020
2052:ISBN
2000:The
1967:The
1891:The
1850:The
1770:and
1738:The
1698:The
1679:and
1610:100
849:Zero
833:and
824:+ 1,
820:′ =
552:′ =
382:and
353:zero
344:and
296:Lisp
292:Java
145:BCPL
70:zero
2258:doi
1995:Uno
1958:or
1918:on
1656:In
1638:− 1
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1021:29
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425:≥ 1
418:− 1
395:− 1
342:Lua
338:APL
314:or
220:≤ (
163:+ 0
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