1277:
115:
25:
1092:
In any subset S of the real numbers which has an
Infimum and a Supremum, the absolute difference between any two numbers in S is less or equal then the absolute difference of the Infimum and Supremum of
884:
When it is desirable to avoid the absolute value function – for example because it is expensive to compute, or because its derivative is not continuous – it can sometimes be eliminated by the identity
1006:
662:
1077:
335:
860:
942:
732:
700:
386:
364:
568:
434:
482:
772:
247:
1206:
798:
508:
299:
279:
211:
191:
156:
136:
93:
1251:
35:
65:
1318:
1156:
72:
79:
1201:
1196:
61:
50:
947:
587:
1014:
308:
1311:
1149:
1211:
1191:
803:
889:
742:
is not non-negative or commutative, but it does obey the second and fourth properties above, since
705:
673:
254:
369:
347:
572:
515:
397:
86:
1347:
1337:
1304:
1292:
1241:
1142:
445:
46:
1236:
745:
216:
8:
866:
777:
666:
487:
1120:
1103:
1080:
284:
264:
196:
176:
141:
121:
1342:
1221:
1117:
874:
870:
1231:
342:
42:
1288:
1216:
437:
250:
1331:
1226:
578:
878:
338:
1165:
865:
The absolute difference is used to define other quantities including the
739:
670:); in the case of the absolute difference, equality holds if and only if
171:
1256:
1186:
114:
1246:
1125:
258:
159:
1276:
24:
1176:
302:
1284:
1134:
1115:
1017:
950:
892:
806:
780:
748:
708:
676:
590:
518:
490:
448:
400:
372:
350:
311:
287:
267:
219:
199:
179:
144:
124:
1071:
1000:
936:
854:
792:
766:
726:
694:
656:
562:
502:
476:
428:
380:
358:
329:
293:
273:
241:
205:
185:
150:
130:
1329:
391:As with any metric, the metric properties hold:
118:Showing the absolute difference of real numbers
366:and their completion, the set of real numbers
1312:
1150:
51:introducing citations to additional sources
1319:
1305:
1157:
1143:
374:
352:
1086:
113:
41:Relevant discussion may be found on the
1330:
1001:{\displaystyle (x-y)^{2}<(z-w)^{2}}
1138:
1116:
657:{\displaystyle |x-z|\leq |x-y|+|y-z|}
1271:
1207:Decidability of first-order theories
261:between the points corresponding to
158:as the distance between them on the
18:
1072:{\displaystyle |x-y|^{2}=(x-y)^{2}}
330:{\displaystyle 1\leq p\leq \infty }
257:. It describes the distance on the
16:Absolute value of (x - y), a metric
13:
324:
14:
1359:
1275:
1164:
34:relies largely or entirely on a
23:
855:{\displaystyle x-z=(x-y)+(y-z)}
1060:
1047:
1034:
1019:
989:
976:
964:
951:
937:{\displaystyle |x-y|<|z-w|}
930:
916:
908:
894:
849:
837:
831:
819:
650:
636:
628:
614:
606:
592:
556:
542:
534:
520:
464:
450:
416:
402:
301:. It is a special case of the
235:
221:
1:
1109:
727:{\displaystyle x\geq y\geq z}
695:{\displaystyle x\leq y\leq z}
484: if and only if
1291:. You can help Knowledge by
381:{\displaystyle \mathbb {R} }
359:{\displaystyle \mathbb {Q} }
7:
1097:
563:{\displaystyle |x-y|=|y-x|}
429:{\displaystyle |x-y|\geq 0}
10:
1364:
1270:
1083:on the nonnegative reals.
1212:Extended real number line
1172:
341:used for both the set of
477:{\displaystyle |x-y|=0}
440:is always non-negative.
1287:-related article is a
1073:
1002:
938:
856:
794:
768:
728:
696:
658:
564:
504:
478:
430:
382:
360:
331:
295:
275:
243:
207:
187:
163:
152:
132:
1252:Tarski axiomatization
1242:Real coordinate space
1192:Cantor–Dedekind axiom
1121:"Absolute Difference"
1087:Additional Properties
1074:
1003:
939:
869:, the L norm used in
857:
795:
769:
767:{\displaystyle x-y=0}
729:
697:
664: (
659:
565:
505:
479:
431:
383:
361:
332:
296:
276:
244:
242:{\displaystyle |x-y|}
208:
188:
153:
133:
117:
62:"Absolute difference"
1237:Rational zeta series
1015:
948:
890:
804:
778:
746:
738:By contrast, simple
706:
674:
588:
570: (
516:
488:
446:
398:
370:
348:
337:and is the standard
309:
285:
265:
217:
197:
177:
142:
122:
47:improve this article
1182:Absolute difference
1011:This follows since
867:relative difference
793:{\displaystyle x=y}
667:triangle inequality
503:{\displaystyle x=y}
168:absolute difference
1118:Weisstein, Eric W.
1104:Absolute deviation
1069:
998:
934:
875:graceful labelings
852:
790:
764:
724:
692:
654:
560:
500:
474:
426:
378:
356:
327:
291:
271:
239:
203:
183:
164:
148:
128:
1300:
1299:
1265:
1264:
1222:Irrational number
294:{\displaystyle y}
274:{\displaystyle x}
206:{\displaystyle y}
186:{\displaystyle x}
151:{\displaystyle y}
131:{\displaystyle x}
112:
111:
97:
1355:
1321:
1314:
1307:
1279:
1272:
1159:
1152:
1145:
1136:
1135:
1131:
1130:
1079:and squaring is
1078:
1076:
1075:
1070:
1068:
1067:
1043:
1042:
1037:
1022:
1007:
1005:
1004:
999:
997:
996:
972:
971:
943:
941:
940:
935:
933:
919:
911:
897:
871:taxicab geometry
861:
859:
858:
853:
799:
797:
796:
791:
773:
771:
770:
765:
733:
731:
730:
725:
701:
699:
698:
693:
663:
661:
660:
655:
653:
639:
631:
617:
609:
595:
569:
567:
566:
561:
559:
545:
537:
523:
509:
507:
506:
501:
483:
481:
480:
475:
467:
453:
435:
433:
432:
427:
419:
405:
387:
385:
384:
379:
377:
365:
363:
362:
357:
355:
343:rational numbers
336:
334:
333:
328:
300:
298:
297:
292:
280:
278:
277:
272:
248:
246:
245:
240:
238:
224:
212:
210:
209:
204:
192:
190:
189:
184:
157:
155:
154:
149:
137:
135:
134:
129:
107:
104:
98:
96:
55:
27:
19:
1363:
1362:
1358:
1357:
1356:
1354:
1353:
1352:
1328:
1327:
1326:
1325:
1268:
1266:
1261:
1232:Rational number
1168:
1163:
1112:
1100:
1089:
1063:
1059:
1038:
1033:
1032:
1018:
1016:
1013:
1012:
1009:
992:
988:
967:
963:
949:
946:
945:
944:if and only if
929:
915:
907:
893:
891:
888:
887:
805:
802:
801:
779:
776:
775:
774:if and only if
747:
744:
743:
707:
704:
703:
675:
672:
671:
649:
635:
627:
613:
605:
591:
589:
586:
585:
555:
541:
533:
519:
517:
514:
513:
489:
486:
485:
463:
449:
447:
444:
443:
415:
401:
399:
396:
395:
373:
371:
368:
367:
351:
349:
346:
345:
310:
307:
306:
286:
283:
282:
266:
263:
262:
234:
220:
218:
215:
214:
198:
195:
194:
178:
175:
174:
143:
140:
139:
123:
120:
119:
108:
102:
99:
56:
54:
40:
28:
17:
12:
11:
5:
1361:
1351:
1350:
1345:
1340:
1324:
1323:
1316:
1309:
1301:
1298:
1297:
1280:
1263:
1262:
1260:
1259:
1254:
1249:
1244:
1239:
1234:
1229:
1224:
1219:
1217:Gregory number
1214:
1209:
1204:
1199:
1194:
1189:
1184:
1179:
1173:
1170:
1169:
1162:
1161:
1154:
1147:
1139:
1133:
1132:
1111:
1108:
1107:
1106:
1099:
1096:
1095:
1094:
1088:
1085:
1066:
1062:
1058:
1055:
1052:
1049:
1046:
1041:
1036:
1031:
1028:
1025:
1021:
995:
991:
987:
984:
981:
978:
975:
970:
966:
962:
959:
956:
953:
932:
928:
925:
922:
918:
914:
910:
906:
903:
900:
896:
886:
851:
848:
845:
842:
839:
836:
833:
830:
827:
824:
821:
818:
815:
812:
809:
789:
786:
783:
763:
760:
757:
754:
751:
736:
735:
723:
720:
717:
714:
711:
691:
688:
685:
682:
679:
652:
648:
645:
642:
638:
634:
630:
626:
623:
620:
616:
612:
608:
604:
601:
598:
594:
583:
558:
554:
551:
548:
544:
540:
536:
532:
529:
526:
522:
511:
499:
496:
493:
473:
470:
466:
462:
459:
456:
452:
441:
438:absolute value
425:
422:
418:
414:
411:
408:
404:
376:
354:
326:
323:
320:
317:
314:
290:
270:
251:absolute value
237:
233:
230:
227:
223:
202:
182:
147:
127:
110:
109:
45:. Please help
31:
29:
22:
15:
9:
6:
4:
3:
2:
1360:
1349:
1348:Algebra stubs
1346:
1344:
1341:
1339:
1336:
1335:
1333:
1322:
1317:
1315:
1310:
1308:
1303:
1302:
1296:
1294:
1290:
1286:
1281:
1278:
1274:
1273:
1269:
1258:
1255:
1253:
1250:
1248:
1245:
1243:
1240:
1238:
1235:
1233:
1230:
1228:
1227:Normal number
1225:
1223:
1220:
1218:
1215:
1213:
1210:
1208:
1205:
1203:
1200:
1198:
1195:
1193:
1190:
1188:
1185:
1183:
1180:
1178:
1175:
1174:
1171:
1167:
1160:
1155:
1153:
1148:
1146:
1141:
1140:
1137:
1128:
1127:
1122:
1119:
1114:
1113:
1105:
1102:
1101:
1091:
1090:
1084:
1082:
1064:
1056:
1053:
1050:
1044:
1039:
1029:
1026:
1023:
993:
985:
982:
979:
973:
968:
960:
957:
954:
926:
923:
920:
912:
904:
901:
898:
885:
882:
880:
876:
872:
868:
863:
846:
843:
840:
834:
828:
825:
822:
816:
813:
810:
807:
787:
784:
781:
761:
758:
755:
752:
749:
741:
721:
718:
715:
712:
709:
689:
686:
683:
680:
677:
669:
668:
646:
643:
640:
632:
624:
621:
618:
610:
602:
599:
596:
584:
581:
580:
579:commutativity
575:
574:
552:
549:
546:
538:
530:
527:
524:
512:
497:
494:
491:
471:
468:
460:
457:
454:
442:
439:
423:
420:
412:
409:
406:
394:
393:
392:
389:
344:
340:
321:
318:
315:
312:
304:
288:
268:
260:
256:
252:
231:
228:
225:
200:
180:
173:
169:
161:
145:
125:
116:
106:
95:
92:
88:
85:
81:
78:
74:
71:
67:
64: –
63:
59:
58:Find sources:
52:
48:
44:
38:
37:
36:single source
32:This article
30:
26:
21:
20:
1338:Real numbers
1293:expanding it
1282:
1267:
1202:Construction
1197:Completeness
1181:
1166:Real numbers
1124:
1010:
883:
879:graph theory
864:
737:
665:
577:
571:
390:
213:is given by
172:real numbers
167:
165:
100:
90:
83:
76:
69:
57:
33:
740:subtraction
1332:Categories
1257:Vitali set
1187:Cantor set
1110:References
303:L distance
255:difference
73:newspapers
1247:Real line
1126:MathWorld
1081:monotonic
1054:−
1027:−
983:−
958:−
924:−
902:−
844:−
826:−
811:−
753:−
719:≥
713:≥
687:≤
681:≤
644:−
622:−
611:≤
600:−
550:−
528:−
458:−
421:≥
410:−
325:∞
322:≤
316:≤
259:real line
253:of their
229:−
160:real line
43:talk page
1343:Distance
1177:0.999...
1098:See also
573:symmetry
436:, since
305:for all
103:May 2024
1285:algebra
170:of two
87:scholar
873:, and
800:, and
339:metric
249:, the
89:
82:
75:
68:
60:
1283:This
94:JSTOR
80:books
1289:stub
974:<
913:<
281:and
193:and
166:The
138:and
66:news
877:in
702:or
576:or
49:by
1334::
1123:.
1093:S.
881:.
862:.
582:).
388:.
1320:e
1313:t
1306:v
1295:.
1158:e
1151:t
1144:v
1129:.
1065:2
1061:)
1057:y
1051:x
1048:(
1045:=
1040:2
1035:|
1030:y
1024:x
1020:|
1008:.
994:2
990:)
986:w
980:z
977:(
969:2
965:)
961:y
955:x
952:(
931:|
927:w
921:z
917:|
909:|
905:y
899:x
895:|
850:)
847:z
841:y
838:(
835:+
832:)
829:y
823:x
820:(
817:=
814:z
808:x
788:y
785:=
782:x
762:0
759:=
756:y
750:x
734:.
722:z
716:y
710:x
690:z
684:y
678:x
651:|
647:z
641:y
637:|
633:+
629:|
625:y
619:x
615:|
607:|
603:z
597:x
593:|
557:|
553:x
547:y
543:|
539:=
535:|
531:y
525:x
521:|
510:.
498:y
495:=
492:x
472:0
469:=
465:|
461:y
455:x
451:|
424:0
417:|
413:y
407:x
403:|
375:R
353:Q
319:p
313:1
289:y
269:x
236:|
232:y
226:x
222:|
201:y
181:x
162:.
146:y
126:x
105:)
101:(
91:·
84:·
77:·
70:·
53:.
39:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
