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digits is the same as the integer part," but that seems like such a fundamental statement that there must be some sort of unstated axiom in play -- whatever the case, that should be either rewritten as I proposed or somebody should have some sort of explanation or reference to what is really meant by
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These functions are usually used with floating point numbers, an approximation of the real number system that has some important differences. Using them with real numbers gives some results that may be unexpected: floor(0.99999...) = 1 because 0.9 repeating is equal to 1. floor( max(0,1) ) = 0 using
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I think a common name for the "integral part" function is "truncation". Certainly in computer science it is called that. May want to mention it. Also it sure looks like does not mean this truncation function all the time, or even the majority of the time. Can't find any modern examples of it's use,
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Can someone please explain this? From what I can tell, rounding up to the nearest integer should be equivalent to ceiling, and similarly for rounding down equaling floor. The section referenced and the definitions of floor and ceiling are all incredibly technical and thus useless to anyone that has
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Both "smaller" and "lower" involve an implicit agreement between reader and author about how bigness (whether in the language of size or height) works for negative numbers. We could make explicit that the order relation in question is the usual total order on the real numbers, but I don't believe
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If a word/phrase is used to mean a certain thing, it takes that definition no matter the original meaning(see the words "awful" and "awesome" for example). On top of this, without the common definitions of the terms "round up" and "round down", the terms are,to my knowledge, meaningless in math.
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I saw that in a StackOverflow question when searching for whether means floor or round-toward-zero, but no reliabla reference. I agree it is possible this is in error, and in fact does not define the behavior for negative numbers. That would at least make it not conflict with the use of for
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does not normally results into 1 (some clever compilers may produce 1., by rearranging the order of operations). Here the floor function is specific because it is not continuous, which means that one cannot get a good approximation of the correct result by increasing the accuracy (size of the
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I am a bit baffled by the statement, "truncation to zero significant digits is the same as the integer part". In my mind, truncation of a value to zero significant digits is the same as introducing complete ambiguity in its magnitude. I was tempted to change the text to "truncation to zero
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I also agree, in lay use the term "round down" and "round up" are equivalent floor and ceiling for positive numbers. Perhaps they are technically incorrect by using the word "round" but not returning the closest integer, but the terms are certainly used this way right or
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I think it would be useful to say somewhere that the definition of "integer part" given in the introduction is only valid in some countries. In France for example, this is equal to the floor function : see the french page "partie entière" of wikipedia.
1894:{\displaystyle {\begin{aligned}\lfloor x\rfloor +\lfloor y\rfloor &\leq \lfloor x+y\rfloor \leq \lfloor x\rfloor +\lfloor y\rfloor +1,\\\lceil x\rceil +\lceil y\rceil -1&\leq \lceil x+y\rceil \leq \lceil x\rceil +\lceil y\rceil .\end{aligned}}}
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are, I think, ambiguous between the "floor" definition and the "truncate towards zero" definition. We can certainly rewrite the lead (with sources) to clarify all that. I don't have the time to do that right now, but I encourage you to do so yourself.
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Because of these two things, I feel adding them would increase the average person's understanding without degrading the expert's understanding, especially if the clarification that this would be a simplified version of the definition.
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on two points here: first, that the current first description is bizarrely technical, and second, that the floor and ceiling functions are particular rounding functions. In particular, there are lots of rounding rules (see
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I'm concerned that at least some people think -1 is "smaller" than -2, and the use of the word "smallest" in the opening paragraphs is incorrect. I tried changing it to "lowest" but that got reverted. Any opinions on this?
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Since this function and notation is very specific, is only mentioned briefly in the lead and the
Notation section, and, as far as I know, rarely used, I think this should be moved in its entirety back into the Applications
969:"floor(0.99999...) = 1 because 0.9 repeating is equal to 1." -- this is a fact about the rounding of 0.999..., not about the floor function; if floor(0.999...) returns 1, it means that the argument was actually 1.0.
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this will help the (in my estimation, very small) percentage of people who believe both that -1 is smaller than -2 and that -2 is lower than -1. But, I would also be happy to hear the views of other editors. --
2511:{\displaystyle \lfloor x+y\rfloor =\lfloor \lfloor x\rfloor +\{x\}+\lfloor y\rfloor +\{y\}\rfloor =\lfloor x\rfloor +\lfloor y\rfloor +\lfloor \{x\}+\{y\}\rfloor \leq \lfloor x\rfloor +\lfloor y\rfloor +1.}
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2043:{\displaystyle {\begin{aligned}\lfloor x\rfloor +\lfloor y\rfloor &=\lfloor x\rfloor +\lfloor y\rfloor +1,\\\lceil x\rceil +\lceil y\rceil -1&=\lceil x\rceil +\lceil y\rceil .\end{aligned}}}
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I wonder if this is strictly an issue of language -- in which case it doesn't belong in the
English WP -- or if different authors have different definitions, in which case we should mention that. --
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I have only seen the function described as "trunc(x)" or "int(x)". I would like to see some reference that anybody uses "" for truncation. I am worried that this is a wikipedia-made-up "fact".
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Apparently this notation (and names) was invented by the APL programming language designers in the 1960s. To do: Find a reliable source. Previous (and different) notations go back to Gauss.
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as better "filled color" (red in this case). The line is also filled, like a bolded line. The addiction of the "empty point" representation of the line function needs to be bold.
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So I think it would make sense to update the article's text to clarify that while each inequality need not be strict, they can't simultaneously be equal. That is, for any given
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We had a section on truncation in this article, but there's actually a full article on it. I added a mention to the lead with a link and removed the unnecessary section. --
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seemed to correct the verbal explanation, but it was still difficult to follow. I think the simple definition is intuitively obvious and needs no further explanation. —
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I agree that the current wording is problematic, but I haven't come up with a concise alternative that is clearer. Maybe just drop the "in words" part? --
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A quick literature search shows that definitions vary even in
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One of the two inequalities in each of these two "chains" must be strict, otherwise we would end up with the following contradictions:
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I agree that this seems dubious without a very good citation, but I also think it was removed again (perhaps, by me)? --
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related articles on
Knowledge. If you would like to participate, please visit the project page, where you can join
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It is rare in mathematics, but common in programming. The full article on it should have any necessary details. --
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The article says "the integer part is often represented as ". That is the statement I am unsure is true.
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The wording in the article doesn't say that is used for truncation. It just says that "the operation of
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until a consensus is reached, and readers of this page are welcome to contribute to the discussion.
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These functions are defined for real numbers, not only for floating point numbers, which all are
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cannot be simultaneously equal to both 1 and −1. This situation seems completely analogous. --
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The "Equivalences" section currently states that the following inequalities are always true:
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I agree with the IP who previously edited this, that it is ambiguous. For example, if
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Either relation in each "chain" can be an equality. Let's examine the floor case.
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Also, Reference 1 cites this wikipedia page. It is a circular reference.
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means round-towards-zero and not floor is a made-up "fact" by wikipedia.
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Maybe put a footnote? This doesn't seem that important to belabor. –
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means something completely different, such as the
Iverson bracket).—
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Knowledge:Redirects for discussion/Log/2022 May 19#Greatest integer
989:(edit conflict) I have removed this reference, and two others per
806:"Not quite rounding. See Floor and ceiling functions#Rounding."
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An editor has identified a potential problem with the redirect
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https://www.maplesoft.com/support/help/content/1939/image84.png
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clearly contradicts the present definition in our article. --
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the range notation for all numbers between 0 and 1 exclusive.
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Find pictures for the biographies of computer scientists (see
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Different definition of integer part in different countries
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I concur. The standard convention in mathematics is that
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generalizes this to a specified number of digits". --
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975:So I don't think any changes are called for. --
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810:not gotten a degree in mathematics. --
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1404:Ah, I think I see what you're saying.
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533:Tag all relevant articles in
273:and see a list of open tasks.
42:Put new text under old text.
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998:rational numbers
731:
715:
687:
679:
645:
644:
641:
638:
635:
614:
609:
608:
598:
591:
590:
585:
577:
570:
546:
540:
415:Computer science
344:Article requests
333:
326:
325:
313:
287:
286:
283:
280:
277:
276:Computer science
267:Computer science
256:
249:
248:
243:
239:Computer science
235:
228:
211:
202:
201:
194:
193:
185:
179:
178:
164:
95:Article policies
16:
2787:
2786:
2782:
2781:
2780:
2778:
2777:
2776:
2712:
2711:
2667:
2666:
2639:
2638:
2592:
2591:
2542:
2541:
2522:
2521:
2350:
2349:
2298:
2297:
2260:
2259:
2216:
2215:
2125:
2124:
2089:
2088:
2057:
2056:
2037:
2036:
2005:
1975:
1974:
1937:
1907:
1906:
1888:
1887:
1838:
1808:
1807:
1752:
1722:
1721:
1716:
1679:
1656:
1614:
1613:
1588:
1587:
1562:
1561:
1543:
1529:
1525:
1522:
1486:
1437:
1251:
1220:
1210:
1188:
1186:
1116:
1105:
1058:
1054:
1029:
1028:
1025:
1001:
945:
804:
743:
727:
716:
710:
692:
642:
639:
636:
633:
632:
610:
603:
583:
552:
549:
544:
538:
526:Project-related
521:
502:
483:
457:
438:
419:
400:
381:
357:
284:
281:
278:
275:
274:
241:
212:on Knowledge's
209:
199:
121:
116:
115:
114:
91:
61:
12:
11:
5:
2785:
2783:
2775:
2774:
2769:
2764:
2759:
2754:
2749:
2744:
2739:
2734:
2729:
2724:
2714:
2713:
2710:
2709:
2686:
2683:
2680:
2677:
2674:
2653:
2649:
2646:
2626:
2623:
2620:
2617:
2614:
2611:
2608:
2605:
2602:
2599:
2588:
2549:
2529:
2507:
2504:
2501:
2498:
2495:
2492:
2489:
2486:
2483:
2480:
2477:
2474:
2471:
2468:
2465:
2462:
2459:
2456:
2453:
2450:
2447:
2444:
2441:
2438:
2435:
2432:
2429:
2426:
2423:
2420:
2417:
2414:
2411:
2408:
2405:
2402:
2399:
2396:
2393:
2390:
2387:
2384:
2381:
2378:
2375:
2372:
2369:
2366:
2363:
2360:
2357:
2335:
2332:
2329:
2326:
2323:
2320:
2317:
2314:
2311:
2308:
2305:
2285:
2282:
2279:
2276:
2273:
2270:
2267:
2247:
2244:
2241:
2238:
2235:
2232:
2229:
2226:
2223:
2201:
2198:
2195:
2192:
2189:
2186:
2183:
2180:
2177:
2174:
2171:
2168:
2165:
2162:
2159:
2156:
2153:
2150:
2147:
2144:
2141:
2138:
2135:
2132:
2109:
2105:
2102:
2099:
2096:
2076:
2073:
2070:
2067:
2064:
2035:
2032:
2029:
2026:
2023:
2020:
2017:
2014:
2011:
2008:
2006:
2004:
2001:
1998:
1995:
1992:
1989:
1986:
1983:
1980:
1977:
1976:
1973:
1970:
1967:
1964:
1961:
1958:
1955:
1952:
1949:
1946:
1943:
1940:
1938:
1936:
1933:
1930:
1927:
1924:
1921:
1918:
1915:
1914:
1886:
1883:
1880:
1877:
1874:
1871:
1868:
1865:
1862:
1859:
1856:
1853:
1850:
1847:
1844:
1841:
1839:
1837:
1834:
1831:
1828:
1825:
1822:
1819:
1816:
1813:
1810:
1809:
1806:
1803:
1800:
1797:
1794:
1791:
1788:
1785:
1782:
1779:
1776:
1773:
1770:
1767:
1764:
1761:
1758:
1755:
1753:
1751:
1748:
1745:
1742:
1739:
1736:
1733:
1730:
1729:
1715:
1712:
1711:
1710:
1678:
1675:
1655:
1652:
1651:
1650:
1649:
1648:
1627:
1624:
1621:
1601:
1598:
1595:
1575:
1572:
1569:
1521:
1518:
1499:for discussion
1485:
1479:
1478:
1477:
1476:
1475:
1474:
1473:
1472:
1471:
1470:
1469:
1468:
1467:
1466:
1465:
1464:
1463:
1462:
1461:
1460:
1459:
1428:
1405:
1250:
1247:
1246:
1245:
1244:
1243:
1242:
1241:
1240:
1239:
1238:
1237:
1184:
1128:= –3.9, then |
1122:
1121:
1120:
1088:
1042:
1039:
1036:
1024:
1021:
1020:
1019:
1018:
1017:
994:
973:
970:
967:
944:
941:
940:
939:
938:
937:
907:
906:
905:
860:
859:
858:
855:
852:
849:
847:
844:
841:
803:
800:
799:
798:
742:
739:
734:
733:
721:
718:
717:
712:
708:
706:
703:
702:
694:
693:
688:
682:
675:
674:
671:
670:
667:
666:
655:
649:
648:
646:
629:the discussion
616:
615:
599:
587:
586:
578:
566:
565:
562:
561:
558:
557:
554:
553:
551:
550:
548:
547:
530:
522:
520:
519:
513:
503:
501:
500:
494:
484:
482:
481:
476:
468:
458:
456:
455:
449:
439:
437:
436:
430:
420:
418:
417:
411:
401:
399:
398:
392:
382:
380:
379:
374:
368:
358:
356:
355:
349:
337:
335:
334:
322:
321:
309:
308:
301:Mid-importance
297:
291:
290:
288:
271:the discussion
257:
245:
244:
242:Mid‑importance
236:
224:
223:
217:
195:
181:
180:
118:
117:
113:
112:
107:
102:
93:
92:
90:
89:
82:
77:
68:
62:
60:
59:
48:
39:
38:
35:
34:
28:
13:
10:
9:
6:
4:
3:
2:
2784:
2773:
2770:
2768:
2765:
2763:
2760:
2758:
2755:
2753:
2750:
2748:
2745:
2743:
2740:
2738:
2735:
2733:
2730:
2728:
2725:
2723:
2720:
2719:
2717:
2708:
2704:
2700:
2681:
2675:
2672:
2647:
2644:
2624:
2621:
2615:
2609:
2606:
2603:
2600:
2597:
2589:
2587:
2584:
2581:
2577:
2576:
2575:
2574:
2570:
2566:
2561:
2547:
2527:
2518:
2505:
2502:
2496:
2490:
2484:
2478:
2469:
2463:
2457:
2448:
2442:
2436:
2430:
2424:
2415:
2409:
2403:
2397:
2391:
2385:
2379:
2370:
2364:
2361:
2358:
2347:
2333:
2330:
2324:
2318:
2312:
2306:
2303:
2283:
2280:
2274:
2268:
2265:
2242:
2236:
2230:
2224:
2221:
2212:
2199:
2193:
2190:
2187:
2181:
2172:
2166:
2160:
2151:
2145:
2139:
2133:
2122:
2103:
2097:
2074:
2071:
2065:
2053:
2050:
2033:
2027:
2021:
2015:
2009:
2007:
2002:
1999:
1993:
1987:
1981:
1971:
1968:
1965:
1959:
1953:
1947:
1941:
1939:
1931:
1925:
1919:
1904:
1901:
1884:
1878:
1872:
1866:
1860:
1854:
1851:
1848:
1842:
1840:
1835:
1832:
1826:
1820:
1814:
1804:
1801:
1798:
1792:
1786:
1780:
1774:
1768:
1765:
1762:
1756:
1754:
1746:
1740:
1734:
1719:
1713:
1709:
1705:
1701:
1696:
1695:
1694:
1693:
1689:
1685:
1676:
1674:
1671:
1667:
1666:79.147.146.38
1663:
1653:
1647:
1644:
1641:
1622:
1596:
1570:
1559:
1558:
1557:
1553:
1549:
1541:
1540:
1539:
1538:
1535:
1519:
1517:
1516:
1512:
1508:
1504:
1500:
1496:
1491:
1484:
1480:
1457:
1453:
1449:
1445:
1441:
1434:
1429:
1427:
1423:
1419:
1414:
1410:
1406:
1403:
1402:
1401:
1397:
1393:
1389:
1388:
1387:
1383:
1379:
1375:
1371:
1370:
1369:
1365:
1361:
1357:
1356:
1355:
1351:
1347:
1343:
1342:
1341:
1337:
1333:
1328:
1327:
1326:
1322:
1318:
1314:
1313:
1312:
1308:
1304:
1299:
1298:
1297:
1293:
1289:
1285:
1284:
1283:
1279:
1275:
1271:
1270:
1269:
1268:
1264:
1260:
1255:
1248:
1236:
1232:
1228:
1214:
1209:
1208:
1207:
1203:
1199:
1194:
1185:
1183:
1179:
1175:
1171:
1167:
1166:
1165:
1161:
1157:
1153:
1152:
1151:
1147:
1143:
1139:
1135:
1131:
1127:
1123:
1114:
1113:
1109:
1104:
1103:
1102:
1098:
1094:
1089:
1087:
1083:
1079:
1075:
1074:
1073:
1072:
1068:
1064:
1037:
1023:Integral part
1022:
1016:
1012:
1008:
999:
995:
992:
988:
987:
986:
982:
978:
974:
971:
968:
965:
964:
963:
962:
958:
954:
949:
942:
936:
932:
928:
927:Bastian 51234
923:
922:
921:
917:
913:
908:
904:
900:
896:
892:
887:
886:Bastian 51234
883:
882:
881:
877:
873:
869:
865:
861:
856:
853:
850:
848:
845:
842:
839:
838:
836:
832:
828:
824:
823:
822:
821:
817:
813:
812:Bastian 51234
807:
801:
796:
792:
788:
784:
780:
774:
773:
772:
770:
766:
762:
758:
754:
748:
740:
738:
730:
725:
720:
719:
705:
704:
701:
700:
696:
695:
691:
686:
681:
680:
664:
660:
654:
651:
650:
647:
630:
626:
622:
621:
613:
607:
602:
600:
597:
593:
592:
588:
582:
579:
576:
572:
543:
536:
532:
531:
529:
527:
523:
518:
515:
514:
512:
510:
509:
504:
499:
496:
495:
493:
491:
490:
485:
480:
477:
474:
470:
469:
467:
465:
464:
459:
454:
451:
450:
448:
446:
445:
440:
435:
432:
431:
429:
427:
426:
421:
416:
413:
412:
410:
408:
407:
402:
397:
394:
393:
391:
389:
388:
383:
378:
375:
373:
370:
369:
367:
365:
364:
359:
354:
351:
350:
348:
346:
345:
340:
339:
336:
332:
328:
327:
324:
323:
319:
315:
314:
310:
306:
302:
296:
293:
292:
289:
272:
268:
264:
263:
258:
255:
251:
250:
246:
240:
237:
234:
230:
225:
221:
215:
207:
206:
196:
192:
187:
186:
177:
173:
170:
167:
163:
159:
155:
152:
149:
146:
143:
140:
137:
134:
131:
127:
124:
123:Find sources:
120:
119:
111:
110:Verifiability
108:
106:
103:
101:
98:
97:
96:
87:
83:
81:
78:
76:
72:
69:
67:
64:
63:
57:
53:
52:Learn to edit
49:
46:
41:
40:
37:
36:
32:
26:
22:
18:
17:
2562:
2519:
2348:
2213:
2123:
2054:
2051:
1905:
1902:
1720:
1717:
1680:
1660:— Preceding
1657:
1523:
1487:
1438:— Preceding
1432:
1412:
1408:
1259:89.95.99.135
1256:
1252:
1192:
1137:
1133:
1129:
1125:
1026:
953:Subcelestial
950:
946:
867:
863:
834:
830:
826:
808:
805:
777:— Preceding
751:— Preceding
744:
737:
723:
697:
689:
659:Mid-priority
658:
618:
584:Mid‑priority
525:
524:
508:Unreferenced
506:
505:
487:
486:
461:
460:
442:
441:
423:
422:
404:
403:
385:
384:
361:
360:
342:
341:
300:
260:
220:WikiProjects
203:
171:
165:
157:
150:
144:
138:
132:
122:
94:
19:This is the
1444:Dodecagon12
1005:mantissa).
634:Mathematics
625:mathematics
581:Mathematics
148:free images
31:not a forum
2716:Categories
1433:fractional
1374:truncation
2580:jacobolus
2565:Ilikefood
1407:Both the
1332:Anita5192
1330:section.—
1213:Anita5192
1198:Anita5192
1174:Anita5192
1170:This edit
1142:Anita5192
872:Anita5192
396:Computing
208:is rated
88:if needed
71:Be polite
21:talk page
2296:implies
1677:Smallest
1662:unsigned
1507:D.Lazard
1452:contribs
1440:unsigned
1418:Macrakis
1413:notation
1378:Macrakis
1346:Macrakis
1317:Macrakis
1288:Macrakis
1274:Macrakis
1156:Macrakis
1063:D.Lazard
1007:D.Lazard
977:Macrakis
891:Rounding
791:contribs
779:unsigned
765:contribs
753:unsigned
724:365 days
690:Archives
444:Maintain
387:Copyedit
56:get help
29:This is
27:article.
1684:Spitzak
1548:Spitzak
1534:Lambiam
1392:Spitzak
1360:Spitzak
1303:Spitzak
1093:Spitzak
991:WP:ELNO
912:Spitzak
831:greater
827:integer
783:Spitzak
661:on the
425:Infobox
363:Cleanup
303:on the
210:B-class
154:WP refs
142:scholar
2055:Since
910:wrong.
757:Krauss
406:Expand
216:scale.
126:Google
835:least
489:Stubs
463:Photo
320:with:
197:This
169:JSTOR
130:books
84:Seek
2703:talk
2637:for
2569:talk
2540:and
2331:<
2281:<
2087:and
1704:talk
1688:talk
1670:talk
1640:Emil
1586:and
1552:talk
1530:= −3
1526:= −2
1511:talk
1448:talk
1422:talk
1409:name
1396:talk
1382:talk
1364:talk
1350:talk
1336:talk
1321:talk
1307:talk
1292:talk
1278:talk
1263:talk
1231:talk
1227:DVdm
1202:talk
1193:Done
1178:talk
1160:talk
1146:talk
1108:DVdm
1097:talk
1082:talk
1067:talk
1011:talk
981:talk
957:talk
931:talk
916:talk
899:talk
876:talk
864:down
816:talk
787:talk
761:talk
162:FENS
136:news
73:and
2699:JBL
2673:sin
2607:sin
2583:(t)
1700:JBL
1078:JBL
895:JBL
653:Mid
295:Mid
176:TWL
2718::
2705:)
2676:
2648:∈
2622:≤
2610:
2604:≤
2598:−
2571:)
2506:1.
2500:⌋
2494:⌊
2488:⌋
2482:⌊
2479:≤
2476:⌋
2452:⌊
2446:⌋
2440:⌊
2434:⌋
2428:⌊
2422:⌋
2407:⌋
2401:⌊
2383:⌋
2377:⌊
2374:⌊
2368:⌋
2356:⌊
2307:≤
2269:≤
2234:⌋
2228:⌊
2197:⌋
2185:⌊
2182:≤
2179:⌋
2176:⌋
2170:⌊
2164:⌋
2158:⌊
2155:⌊
2149:⌋
2143:⌊
2137:⌋
2131:⌊
2104:∈
2101:⌋
2095:⌊
2072:≤
2069:⌋
2063:⌊
2031:⌉
2025:⌈
2019:⌉
2013:⌈
2000:−
1997:⌉
1991:⌈
1985:⌉
1979:⌈
1963:⌋
1957:⌊
1951:⌋
1945:⌊
1935:⌋
1929:⌊
1923:⌋
1917:⌊
1882:⌉
1876:⌈
1870:⌉
1864:⌈
1861:≤
1858:⌉
1846:⌈
1843:≤
1833:−
1830:⌉
1824:⌈
1818:⌉
1812:⌈
1796:⌋
1790:⌊
1784:⌋
1778:⌊
1775:≤
1772:⌋
1760:⌊
1757:≤
1750:⌋
1744:⌊
1738:⌋
1732:⌊
1706:)
1690:)
1643:J.
1600:⌋
1594:⌊
1554:)
1513:)
1454:)
1450:•
1424:)
1416:--
1398:)
1384:)
1366:)
1352:)
1338:)
1323:)
1309:)
1294:)
1280:)
1265:)
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