84:
74:
53:
22:
968:
The rule is on page 42. Maybe this is interesting: Descartes also makes the statement that you can determine the number of negative roots (called "false" by
Descartes) by counting unchanging signs between consecutive coefficients. Descartes does not say what to do with coefficients that are zero, but
911:
I don't know why you think that the complex roots version of the rule only applies to purely imaginary roots. This is not correct. Perhaps you were misled by the simple example in that section. I have replaced that example with a more general one where the complex roots are not purely imaginary. The
505:
How about being patent nonsense? It relies solely on "(X + i)(X - i) = X^2 + b" to work in all forms. It only detects 'purely imaginary' (not complex) numbers in that manner. (X + i)(X + i) instantly defeats it, I believe... with 1 positive sign change and 1 negative sign change.
1209:
article. The statements there are clearly not definitive and the most that can be concluded from the passages is that there is a difference of opinion on the matter. Your statement about
Descartes (above) I would place in the category of
1080:, "Descartes' rule of signs" has about 4060 results in Google scholar; "Chasles's theorem" has 224. Also, to me, "s's" implies that both s's should be pronounced separately, which is the incorrect pronounciation in this case. —
1172:). 2) Singular nouns that terminate with a silent "s/z/x" (as in some French names, an example of which is the name Descartes discussed here), almost uniformly take the " 's " in their possessive form (see
1102:
to be very convincing. While you can certainly find both forms in the literature, you will also find many clearly incorrect forms, but the majority, in my experience, have used the apostrophe only form.
876:
It only detects "purely imaginary pole pairs", now you COULD say If random(5) <= 5, a minimum of 0 "purely imaginary" roots exist, but that is a silly assertion... (But just as valid as this claim)
840:
721:
1244:
Singular nouns that terminate with a silent "s/z/x" (as in some French names, an example of which is the name
Descartes discussed here), almost uniformly take the " 's " in their possessive form
140:
1098:
requires the use of "s's"; it specifically mentions that there are two systems and that one should be consistent within an article. Under these circumstances, I find David's appeal to
609:
262:
361:
I have reversed the recent move to "Descartes's rule of signs". The usual possessive form of
Descartes is Descartes' - this is the standard followed on other sites such as
1275:
as above. There is no universal grammatical rule that requires the additional s. The omission has been taught in
British schools for the past sixty years at least.
341:
The proof is clearly lacking. I have neither the time nor the inclination to fix it. I will add words similar to "A rough outline or sketch of the proof follows."
176:
Actually, it was correct before "fixing". The factorization and roots discussed in the final lines refer to the original polynomial, not the sign-flipped version. --
1020:
969:
you have to insert an arbitrary sign for any zero to make the rules work. Another point: Descartes is talking only about polynomials with all real roots.
1399:
130:
1394:
1168:. Two main reasons: 1) Singular nouns, even the ones that end in s/z, take the " 's " as opposed to a single apostrophe in their possessive form (see
402:
This polynomial has two sign changes, meaning the original polynomial has 2 or 0 negative roots and this second polynomial has 2 or 0 positive roots.
319:
There seem to be a few neat expositions of various proofs floating around, I added the one from Cut the Knot since they cater for most audiences.
940:) complex roots. If you accept that the real roots version of the rule is correct, then the complex roots version is quite obviously true also.
264:
this polynomial has two roots with the value x=-1, and one with value x=1. So counting roots we have 2 negative roots and one positive root. --
106:
890:
517:
1008:
Subsequent comments should be made in a new section on the talk page. Editors desiring to contest the closing decision should consider a
976:
427:
475:
306:
I happen to know that the proof is rather long, so perhaps it would be apt to provide an external link to it, if anyone can find it.
97:
58:
513:-1.66489 + 0.00000i -0.62797 + 0.90355i -0.62797 - 0.90355i 1.16202 + 0.00000i 0.37941 + 0.53195i 0.37941 - 0.53195i
1307:
As I've mentioned in similar discussions elsewhere, the universality of this statement seems to be highly suspect (maybe in
736:
620:
912:
complex roots version of the rule is actually a very simple extension of the real roots version - if you know there are a
33:
1176:), especially since this " 's " is actually pronounced: "de-CART" for the name itself, "de-CARTz" for its possessive (
490:
True. I have updated that section of the article to add the condition that polynomials must not not have a root at 0.
1336:
1040:
999:
1085:
894:
521:
412:
21:
980:
431:
1156:
1066:
479:
390:
536:
850:) has a maximum of 3 negative real roots. The "complex roots" version of Descartes' Rule then says that f(
1180:, note that the spelling doesn't imply the "de-CART-siz" pronunciation as you incorrectly suggested e.g.
407:
Aren't the positive roots counted from the original polynomial and the negative roots from the second? --
1346:
1009:
441:
39:
346:
285:
200:
83:
1272:
1118:
1099:
1077:
972:
471:
269:
1228:
Not even sure how to interpret your last sentence. Which specific "statement" do you believe false?
1316:
1229:
1185:
1177:
1135:
1081:
1058:
1054:
945:
863:
495:
453:
408:
386:
213:
342:
281:
196:
105:
on
Knowledge. If you would like to participate, please visit the project page, where you can join
1320:
1251:
1233:
1219:
1189:
1152:
1108:
1062:
324:
278:
Wouldn't the very much more standard ``Roots are counted taking into account their multiplicities
168:
89:
1356:
I've reverted the recent move and falsification of links. Please abide by the above decision.
884:
This IS patent nonsense, it works under a select number of cases and does not work consistently.
73:
52:
371:
879:(x + i)(x - i)(x - 1) is another fun thing, which it fails to detect the imaginary pole pair.
1028:
615:
because it only applies to polynomials with real coefficients. As for your other example,
468:
This result is a bit off; it needs to take into account the fact that 0 could be a root.
265:
858:
of 6-3-3 = 0 complex roots, which is entirely consistent with it having 6 complex roots.
377:
1379:
1324:
1298:
1255:
1237:
1223:
1193:
1160:
1139:
1131:
1112:
1089:
1070:
1033:
984:
949:
941:
898:
867:
859:
525:
499:
491:
483:
457:
449:
435:
416:
382:
366:
350:
328:
313:
289:
273:
204:
187:
182:
171:
161:
507:
362:
1388:
1247:
1215:
1122:
1104:
1095:
1050:
338:
320:
158:
334:
310:
195:
What does «Multiple roots of the same value are counted separately» mean, exactly?
1357:
1276:
102:
1205:. I do not see how you arrive at these conclusions from what is written in the
1345:
Subsequent comments should be made in a new section on this talk page or in a
1206:
1173:
1169:
1025:
79:
889:
And what is more... you don't even list the conditions under which it works!
177:
1125:
in my opinion. Wolfram, Cut The Knot and
Purplemath all use the <s': -->
965:
1214:
as a simple google scholar search clearly shows its falsity.--
15:
303:
The example is all well and good, but it's not a proof.
1181:
1045:
835:{\displaystyle f(-x)=x^{6}-x^{5}-x^{4}+x^{3}-x^{2}-x-1}
731:) has a maximum of 3 positive real roots. And we have
716:{\displaystyle f(x)=x^{6}+x^{5}-x^{4}-x^{3}-x^{2}+x-1}
442:
Knowledge is an encyclopedic reference, not a textbook
1335:
The above discussion is preserved as an archive of a
739:
623:
539:
216:
101:, a collaborative effort to improve the coverage of
1012:. No further edits should be made to this section.
508:
http://mathworld.wolfram.com/DescartesSignRule.html
1349:. No further edits should be made to this section.
1151:per David Eppstein, Bill Cherowitzo and Gandalf61.
834:
715:
603:
256:
157:pay attention to the last example: it's rong... --
510:uses a general polynomial , which has roots of
444:. To see a proof you can follow the link under
1017:The result of the move request was: Not moved
8:
19:
1057:seems to insist. The move was reverted by
998:The following is a closed discussion of a
964:Here is a source for Descartes Geometrie:
426:The page lacks a proof; please put it in.
47:
814:
801:
788:
775:
762:
738:
695:
682:
669:
656:
643:
622:
577:
538:
233:
215:
357:Moved back to "Descartes' rule of signs"
49:
1243:
604:{\displaystyle (x+i)(x+i)=x^{2}+2ix-1}
966:http://www.gutenberg.org/ebooks/26400
369:, and in the titles of books such as
7:
95:This article is within the scope of
846:which also has 3 sign changes so f(
531:You can't apply Dsecartes' Rule to
367:Stanford Encyclopedia of Philosophy
38:It is of interest to the following
960:Original source + unchanging signs
356:
14:
1400:Mid-priority mathematics articles
115:Knowledge:WikiProject Mathematics
1395:Start-Class mathematics articles
924:real roots then there must be a
448:or consult an algebra textbook.
118:Template:WikiProject Mathematics
82:
72:
51:
20:
1049:– The rule is to use "s's" per
991:Requested move 31 December 2017
378:Descartes' Metaphysical Physics
135:This article has been rated as
752:
743:
727:this has 3 sign changes, so f(
633:
627:
567:
555:
552:
540:
257:{\displaystyle (x+1)^{2}(x-1)}
251:
239:
230:
217:
1:
1113:19:48, 31 December 2017 (UTC)
1090:06:43, 31 December 2017 (UTC)
1071:06:04, 31 December 2017 (UTC)
484:17:45, 22 February 2012 (UTC)
329:14:25, 20 February 2008 (UTC)
290:18:33, 15 February 2008 (UTC)
188:16:42, 13 February 2007 (UTC)
109:and see a list of open tasks.
417:22:56, 15 October 2010 (UTC)
391:16:53, 5 February 2008 (UTC)
351:11:17, 11 October 2023 (UTC)
314:20:24, 3 November 2007 (UTC)
274:00:46, 8 February 2008 (UTC)
205:00:31, 8 February 2008 (UTC)
172:12:37, 8 February 2007 (UTC)
167:fixed, thanks for spotting.
162:23:19, 7 February 2007 (UTC)
1325:02:19, 5 January 2018 (UTC)
1299:10:37, 4 January 2018 (UTC)
1256:04:57, 5 January 2018 (UTC)
1238:02:21, 5 January 2018 (UTC)
1224:04:34, 4 January 2018 (UTC)
1194:01:23, 4 January 2018 (UTC)
1161:19:10, 1 January 2018 (UTC)
1140:12:32, 1 January 2018 (UTC)
1034:19:37, 6 January 2018 (UTC)
950:10:28, 4 October 2013 (UTC)
899:20:19, 3 October 2013 (UTC)
868:07:38, 3 October 2013 (UTC)
526:06:08, 3 October 2013 (UTC)
500:07:45, 3 October 2013 (UTC)
1416:
1182:in your edit summary here
1046:Descartes's rule of signs
458:09:34, 22 July 2011 (UTC)
436:20:02, 21 July 2011 (UTC)
134:
67:
46:
1342:Please do not modify it.
1041:Descartes' rule of signs
1005:Please do not modify it.
985:17:56, 11 May 2014 (UTC)
141:project's priority scale
1380:07:53, 1 May 2018 (UTC)
98:WikiProject Mathematics
836:
717:
605:
258:
28:This article is rated
837:
718:
606:
259:
1311:schools, but not in
737:
621:
537:
214:
121:mathematics articles
1130:the proposed move.
1094:I do not see that
832:
713:
601:
516:again, "defeated".
254:
90:Mathematics portal
34:content assessment
1159:
1024:
1021:non-admin closure
975:comment added by
474:comment added by
155:
154:
151:
150:
147:
146:
1407:
1377:
1344:
1296:
1248:Bill Cherowitzo
1216:Bill Cherowitzo
1157:(talk)(contribs)
1155:
1105:Bill Cherowitzo
1048:
1018:
1007:
987:
841:
839:
838:
833:
819:
818:
806:
805:
793:
792:
780:
779:
767:
766:
722:
720:
719:
714:
700:
699:
687:
686:
674:
673:
661:
660:
648:
647:
610:
608:
607:
602:
582:
581:
486:
372:Descartes' Error
263:
261:
260:
255:
238:
237:
123:
122:
119:
116:
113:
92:
87:
86:
76:
69:
68:
63:
55:
48:
31:
25:
24:
16:
1415:
1414:
1410:
1409:
1408:
1406:
1405:
1404:
1385:
1384:
1358:
1353:
1340:
1277:
1044:
1031:
1003:
993:
970:
962:
810:
797:
784:
771:
758:
735:
734:
691:
678:
665:
652:
639:
619:
618:
573:
535:
534:
514:
469:
466:
424:
409:Patrick McLaren
398:
359:
301:
229:
212:
211:
186:
120:
117:
114:
111:
110:
88:
81:
61:
32:on Knowledge's
29:
12:
11:
5:
1413:
1411:
1403:
1402:
1397:
1387:
1386:
1383:
1382:
1352:
1351:
1337:requested move
1331:
1330:
1329:
1328:
1327:
1302:
1301:
1265:
1264:
1263:
1262:
1261:
1260:
1259:
1258:
1197:
1196:
1178:David Eppstein
1163:
1153:—Roman Spinner
1145:
1144:
1143:
1142:
1115:
1082:David Eppstein
1059:David Eppstein
1039:
1037:
1029:
1015:
1014:
1000:requested move
994:
992:
989:
961:
958:
957:
956:
955:
954:
953:
952:
904:
903:
902:
901:
891:173.168.30.119
887:
885:
882:
880:
877:
871:
870:
844:
843:
842:
831:
828:
825:
822:
817:
813:
809:
804:
800:
796:
791:
787:
783:
778:
774:
770:
765:
761:
757:
754:
751:
748:
745:
742:
725:
724:
723:
712:
709:
706:
703:
698:
694:
690:
685:
681:
677:
672:
668:
664:
659:
655:
651:
646:
642:
638:
635:
632:
629:
626:
613:
612:
611:
600:
597:
594:
591:
588:
585:
580:
576:
572:
569:
566:
563:
560:
557:
554:
551:
548:
545:
542:
518:173.168.30.119
512:
503:
502:
465:
462:
461:
460:
446:External links
423:
420:
397:
396:Second Example
394:
358:
355:
354:
353:
331:
300:
297:
296:
295:
294:
293:
253:
250:
247:
244:
241:
236:
232:
228:
225:
222:
219:
193:
192:
191:
190:
180:
153:
152:
149:
148:
145:
144:
133:
127:
126:
124:
107:the discussion
94:
93:
77:
65:
64:
56:
44:
43:
37:
26:
13:
10:
9:
6:
4:
3:
2:
1412:
1401:
1398:
1396:
1393:
1392:
1390:
1381:
1378:
1376:
1373:
1370:
1367:
1364:
1361:
1355:
1354:
1350:
1348:
1343:
1338:
1333:
1332:
1326:
1322:
1318:
1314:
1310:
1306:
1305:
1304:
1303:
1300:
1297:
1295:
1292:
1289:
1286:
1283:
1280:
1274:
1273:WP:COMMONNAME
1270:
1267:
1266:
1257:
1253:
1249:
1245:
1241:
1240:
1239:
1235:
1231:
1227:
1226:
1225:
1221:
1217:
1213:
1208:
1204:
1201:
1200:
1199:
1198:
1195:
1191:
1187:
1183:
1179:
1175:
1171:
1167:
1164:
1162:
1158:
1154:
1150:
1147:
1146:
1141:
1137:
1133:
1129:
1124:
1120:
1119:WP:COMMONNAME
1116:
1114:
1110:
1106:
1101:
1100:WP:COMMONNAME
1097:
1093:
1092:
1091:
1087:
1083:
1079:
1078:WP:COMMONNAME
1075:
1074:
1073:
1072:
1068:
1064:
1063:GeoffreyT2000
1060:
1056:
1052:
1047:
1042:
1036:
1035:
1032:
1027:
1022:
1013:
1011:
1006:
1001:
996:
995:
990:
988:
986:
982:
978:
977:93.220.47.168
974:
967:
959:
951:
947:
943:
939:
935:
931:
927:
923:
919:
915:
910:
909:
908:
907:
906:
905:
900:
896:
892:
888:
886:
883:
881:
878:
875:
874:
873:
872:
869:
865:
861:
857:
853:
849:
845:
829:
826:
823:
820:
815:
811:
807:
802:
798:
794:
789:
785:
781:
776:
772:
768:
763:
759:
755:
749:
746:
740:
733:
732:
730:
726:
710:
707:
704:
701:
696:
692:
688:
683:
679:
675:
670:
666:
662:
657:
653:
649:
644:
640:
636:
630:
624:
617:
616:
614:
598:
595:
592:
589:
586:
583:
578:
574:
570:
564:
561:
558:
549:
546:
543:
533:
532:
530:
529:
528:
527:
523:
519:
511:
509:
501:
497:
493:
489:
488:
487:
485:
481:
477:
473:
464:Complex Roots
463:
459:
455:
451:
447:
443:
440:
439:
438:
437:
433:
429:
428:76.181.64.113
421:
419:
418:
414:
410:
405:
403:
395:
393:
392:
388:
384:
380:
379:
374:
373:
368:
364:
352:
348:
344:
340:
337:Glooper and @
336:
332:
330:
326:
322:
318:
317:
316:
315:
312:
307:
304:
298:
292:
291:
287:
283:
277:
276:
275:
271:
267:
248:
245:
242:
234:
226:
223:
220:
209:
208:
207:
206:
202:
198:
189:
184:
179:
175:
174:
173:
170:
169:DavidMcKenzie
166:
165:
164:
163:
160:
142:
138:
132:
129:
128:
125:
108:
104:
100:
99:
91:
85:
80:
78:
75:
71:
70:
66:
60:
57:
54:
50:
45:
41:
35:
27:
23:
18:
17:
1374:
1371:
1368:
1365:
1362:
1359:
1341:
1334:
1312:
1308:
1293:
1290:
1287:
1284:
1281:
1278:
1268:
1211:
1202:
1165:
1148:
1127:
1038:
1016:
1004:
997:
971:— Preceding
963:
937:
933:
929:
925:
921:
917:
913:
855:
851:
847:
728:
515:
504:
476:71.174.58.74
470:— Preceding
467:
445:
425:
406:
401:
399:
376:
370:
360:
308:
305:
302:
279:
194:
156:
137:Mid-priority
136:
96:
62:Mid‑priority
40:WikiProjects
1347:move review
1242:This one. "
1010:move review
280:be better?
112:Mathematics
103:mathematics
59:Mathematics
30:Start-class
1389:Categories
1207:Apostrophe
1117:I agree -
266:Salix alba
1212:fake news
1132:Gandalf61
942:Gandalf61
860:Gandalf61
492:Gandalf61
450:Gandalf61
383:Gandalf61
363:MathWorld
1317:cherkash
1230:cherkash
1186:cherkash
1126:form. I
1055:Cherkash
973:unsigned
854:) has a
472:unsigned
365:and the
339:Chenxlee
321:Chenxlee
309:Thanks,
159:Arirossa
1203:Comment
1166:Support
1121:trumps
1061:twice.
926:minimum
914:maximum
856:minimum
422:Comment
343:Nickalh
335:Glooper
311:Glooper
282:Mariano
197:Mariano
139:on the
1269:Oppose
1149:Oppose
1128:oppose
1123:WP:MOS
1096:WP:MOS
1051:WP:MOS
36:scale.
1026:Ⓩⓟⓟⓘⓧ
299:Proof
210:Take
1321:talk
1309:some
1271:per
1252:talk
1246:".--
1234:talk
1220:talk
1190:talk
1174:here
1170:here
1136:talk
1109:talk
1086:talk
1076:Per
1067:talk
1030:Talk
981:talk
946:talk
895:talk
864:talk
522:talk
496:talk
480:talk
454:talk
432:talk
413:talk
387:talk
375:and
347:talk
325:talk
286:talk
270:talk
201:talk
183:talk
178:mglg
1339:.
1315:).
1313:all
1184:).
1053:as
932:- (
928:of
916:of
131:Mid
1391::
1323:)
1254:)
1236:)
1222:)
1192:)
1138:)
1111:)
1103:--
1088:)
1069:)
1043:→
1002:.
983:)
948:)
897:)
866:)
827:−
821:−
808:−
782:−
769:−
747:−
708:−
689:−
676:−
663:−
596:−
524:)
498:)
482:)
456:)
434:)
415:)
404:"
389:)
381:.
349:)
327:)
288:)
272:)
246:−
203:)
1375:s
1372:r
1369:i
1366:f
1363:b
1360:D
1319:(
1294:s
1291:r
1288:i
1285:f
1282:b
1279:D
1250:(
1232:(
1218:(
1188:(
1134:(
1107:(
1084:(
1065:(
1023:)
1019:(
979:(
944:(
938:q
936:+
934:p
930:n
922:q
920:+
918:p
893:(
862:(
852:x
848:x
830:1
824:x
816:2
812:x
803:3
799:x
795:+
790:4
786:x
777:5
773:x
764:6
760:x
756:=
753:)
750:x
744:(
741:f
729:x
711:1
705:x
702:+
697:2
693:x
684:3
680:x
671:4
667:x
658:5
654:x
650:+
645:6
641:x
637:=
634:)
631:x
628:(
625:f
599:1
593:x
590:i
587:2
584:+
579:2
575:x
571:=
568:)
565:i
562:+
559:x
556:(
553:)
550:i
547:+
544:x
541:(
520:(
494:(
478:(
452:(
430:(
411:(
400:"
385:(
345:(
333:@
323:(
284:(
268:(
252:)
249:1
243:x
240:(
235:2
231:)
227:1
224:+
221:x
218:(
199:(
185:)
181:(
143:.
42::
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
