1183:
1152:
841:
over a field. The irreducible fraction for a given element is unique up to multiplication of denominator and numerator by the same invertible element. In the case of the rational numbers this means that any number has two irreducible fractions, related by a change of sign of both numerator and
548:
Which method is faster "by hand" depends on the fraction and the ease with which common factors are spotted. In case a denominator and numerator remain that are too large to ensure they are coprime by inspection, a greatest common divisor computation is needed anyway to ensure the fraction is
466:
543:
837:: any element of such a field can be written as a fraction in which denominator and numerator are coprime, by dividing both by their greatest common divisor. This applies notably to
395:
A fraction that is reducible can be reduced by dividing both the numerator and denominator by a common factor. It can be fully reduced to lowest terms if both are divided by their
842:
denominator; this ambiguity can be removed by requiring the denominator to be positive. In the case of rational functions the denominator could similarly be required to be a
407:
can be used. The
Euclidean algorithm is commonly preferred because it allows one to reduce fractions with numerators and denominators too large to be easily factored.
1128:
471:
In the first step both numbers were divided by 10, which is a factor common to both 120 and 90. In the second step, they were divided by 3. The final result,
1006:
858:, an erroneous arithmetic procedure that produces the correct irreducible fraction by cancelling digits of the original unreduced form.
1121:
418:
1068:
1041:
1014:
983:
918:
594:
1335:
1340:
1114:
975:
893:
888:
834:
508:
940:, College text books, Sandhurst. Royal Military College, vol. 1, Longman, Brown, Green, and Longmans, p. 75
1294:
861:
687:
The fact that any rational number has a unique representation as an irreducible fraction is utilized in various
698:
could be represented as a ratio of integers, then it would have in particular the fully reduced representation
1282:
491:
396:
99:
1164:
855:
818:
47:
821:, so the premise that the square root of two has a representation as the ratio of two integers is false.
404:
400:
1240:
1225:
830:
1255:
1089:
1064:
1058:
1037:
1031:
1010:
979:
969:
914:
908:
883:
838:
111:
1000:
1299:
843:
688:
122:
can be represented as an irreducible fraction with positive denominator in exactly one way.
1289:
1250:
119:
267:
212:
1329:
1314:
1235:
1092:
1265:
1260:
1230:
1063:, Graduate Texts in Mathematics, vol. 242, Springer, Lemma 9.2, p. 183,
487:, is an irreducible fraction because 4 and 3 have no common factors other than 1.
58:
than 1 (and −1, when negative numbers are considered). In other words, a fraction
490:
The original fraction could have also been reduced in a single step by using the
115:
103:
647:, and so both sides of the latter must share the same prime factorization, yet
561:
representation as an irreducible fraction with a positive denominator (however
1309:
1304:
965:
938:
Elements of
Arithmetic and Algebra: For the Use of the Royal Military College
817:
is greater than 1), the latter is a ratio of two smaller integers. This is a
1151:
1097:
910:
The Legacy of Niels Henrik Abel: The Abel
Bicentennial, Oslo, June 3-8, 2002
17:
1172:
1137:
1272:
1245:
1168:
1106:
87:
55:
51:
691:
and of other irrational numbers. For example, one proof notes that if
593:
although both are irreducible). Uniqueness is a consequence of the
1141:
461:{\displaystyle {\frac {120}{90}}={\frac {12}{9}}={\frac {4}{3}}}
1110:
153:
is irreducible if and only if there is no other equal fraction
971:
Integers, Fractions, and
Arithmetic: A Guide for Teachers
864:, the approximation of real numbers by rational numbers.
114:
such that the numerator and the denominator are coprime
829:
The notion of irreducible fraction generalizes to the
655:
share no prime factors so the set of prime factors of
511:
421:
968:(2012), "9.1. Reducing a fraction to lowest terms",
399:. In order to find the greatest common divisor, the
1190:
1157:
974:, MSRI mathematical circles library, vol. 10,
689:
proofs of the irrationality of the square root of 2
328:are all irreducible fractions. On the other hand,
537:
460:
125:An equivalent definition is sometimes useful: if
1005:, Mathematical Association of America Textbooks,
538:{\displaystyle {\frac {120}{90}}={\frac {4}{3}}}
1122:
8:
659:(with multiplicity) is a subset of those of
907:Laudal, Olav Arnfinn; Piene, Ragni (2004),
344:is reducible since it is equal in value to
50:in which the numerator and denominator are
1129:
1115:
1107:
726:are the smallest possible; but given that
525:
512:
510:
448:
435:
422:
420:
931:
929:
874:
951:
7:
1007:Mathematical Association of America
782:(since cross-multiplying this with
999:Cuoco, Al; Rotman, Joseph (2013),
802:shows that they are equal). Since
25:
1181:
1150:
1057:Grillet, Pierre Antoine (2007),
133:are integers, then the fraction
494:of 90 and 120, which is 30. As
376:is less than the numerator of
78:is irreducible if and only if
1:
976:American Mathematical Society
557:Every rational number has a
889:Encyclopedia of Mathematics
835:unique factorization domain
1357:
1036:, CRC Press, p. 183,
595:unique prime factorization
54:that have no other common
1221:
1179:
1148:
1030:Garrett, Paul B. (2007),
913:, Springer, p. 155,
882:Stepanov, S. A. (2001) ,
862:Diophantine approximation
671:and by the same argument
27:Fully simplified fraction
663:and vice versa, meaning
36:fraction in lowest terms
1336:Fractions (mathematics)
1002:Learning Modern Algebra
936:Scott, William (1844),
492:greatest common divisor
397:greatest common divisor
360:, and the numerator of
100:greatest common divisor
856:Anomalous cancellation
549:actually irreducible.
539:
462:
1341:Elementary arithmetic
540:
463:
978:, pp. 131–134,
839:rational expressions
509:
419:
110:" may also refer to
108:irreducible fraction
32:irreducible fraction
597:of integers, since
405:prime factorization
401:Euclidean algorithm
1158:Division and ratio
1093:"Reduced Fraction"
1090:Weisstein, Eric W.
966:Sally, Paul J. Jr.
964:Sally, Judith D.;
831:field of fractions
535:
458:
197:| < |
181:| < |
112:rational fractions
1323:
1322:
533:
520:
456:
443:
430:
219:. (Two fractions
211:| means the
16:(Redirected from
1348:
1300:Musical interval
1213:
1212:
1210:
1209:
1206:
1203:
1185:
1184:
1154:
1131:
1124:
1117:
1108:
1103:
1102:
1075:
1073:
1060:Abstract Algebra
1054:
1048:
1046:
1033:Abstract Algebra
1027:
1021:
1019:
996:
990:
988:
961:
955:
949:
943:
941:
933:
924:
923:
903:
897:
896:
879:
844:monic polynomial
816:
815:
806: >
801:
799:
798:
793:
790:
781:
779:
778:
769:
766:
752:
751:
745:
743:
742:
737:
734:
717:
715:
714:
709:
706:
697:
696:
638:
637:
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629:
626:
617:
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614:
609:
606:
592:
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589:
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583:
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541:
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534:
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467:
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459:
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423:
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372:
369:
366:
359:
357:
356:
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343:
341:
340:
337:
334:
327:
325:
324:
321:
318:
311:
309:
308:
305:
302:
295:
293:
292:
289:
286:
258:
256:
255:
250:
247:
238:
236:
235:
230:
227:
210:
204:
202:
196:
188:
186:
180:
172:
170:
169:
164:
161:
152:
150:
149:
144:
141:
102:of 1. In higher
77:
75:
74:
69:
66:
44:reduced fraction
21:
1356:
1355:
1351:
1350:
1349:
1347:
1346:
1345:
1326:
1325:
1324:
1319:
1290:Just intonation
1217:
1207:
1204:
1201:
1200:
1198:
1197:
1186:
1182:
1177:
1155:
1144:
1135:
1088:
1087:
1084:
1079:
1078:
1071:
1056:
1055:
1051:
1044:
1029:
1028:
1024:
1017:
998:
997:
993:
986:
963:
962:
958:
950:
946:
935:
934:
927:
921:
906:
904:
900:
881:
880:
876:
871:
852:
827:
813:
811:
794:
791:
786:
785:
783:
770:
767:
757:
756:
754:
749:
747:
738:
735:
730:
729:
727:
710:
707:
702:
701:
699:
694:
692:
685:
630:
627:
622:
621:
619:
610:
607:
602:
601:
599:
598:
587:
584:
581:
580:
578:
571:
568:
565:
564:
562:
555:
507:
506:
499:
495:
481:
478:
475:
474:
472:
417:
416:
413:
386:
383:
380:
379:
377:
370:
367:
364:
363:
361:
354:
351:
348:
347:
345:
338:
335:
332:
331:
329:
322:
319:
316:
315:
313:
306:
303:
300:
299:
297:
290:
287:
284:
283:
281:
251:
248:
243:
242:
240:
231:
228:
223:
222:
220:
206:
198:
192:
190:
182:
176:
174:
165:
162:
157:
156:
154:
145:
142:
137:
136:
134:
120:rational number
70:
67:
62:
61:
59:
28:
23:
22:
15:
12:
11:
5:
1354:
1352:
1344:
1343:
1338:
1328:
1327:
1321:
1320:
1318:
1317:
1312:
1307:
1302:
1297:
1292:
1287:
1286:
1285:
1275:
1270:
1269:
1268:
1258:
1253:
1248:
1243:
1238:
1233:
1228:
1222:
1219:
1218:
1216:
1215:
1194:
1192:
1188:
1187:
1180:
1178:
1176:
1175:
1161:
1159:
1156:
1149:
1146:
1145:
1136:
1134:
1133:
1126:
1119:
1111:
1105:
1104:
1083:
1082:External links
1080:
1077:
1076:
1069:
1049:
1042:
1022:
1015:
1009:, p. 33,
991:
984:
956:
944:
925:
919:
898:
873:
872:
870:
867:
866:
865:
859:
851:
848:
826:
825:Generalization
823:
684:
681:
554:
551:
546:
545:
532:
529:
524:
519:
516:
469:
468:
455:
452:
447:
442:
439:
434:
429:
426:
412:
409:
268:if and only if
213:absolute value
205:, where |
90:, that is, if
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
1353:
1342:
1339:
1337:
1334:
1333:
1331:
1316:
1313:
1311:
1308:
1306:
1303:
1301:
1298:
1296:
1293:
1291:
1288:
1284:
1281:
1280:
1279:
1276:
1274:
1271:
1267:
1264:
1263:
1262:
1259:
1257:
1254:
1252:
1249:
1247:
1244:
1242:
1239:
1237:
1234:
1232:
1229:
1227:
1224:
1223:
1220:
1196:
1195:
1193:
1189:
1174:
1170:
1166:
1163:
1162:
1160:
1153:
1147:
1143:
1139:
1132:
1127:
1125:
1120:
1118:
1113:
1112:
1109:
1100:
1099:
1094:
1091:
1086:
1085:
1081:
1072:
1070:9780387715681
1066:
1062:
1061:
1053:
1050:
1045:
1043:9781584886907
1039:
1035:
1034:
1026:
1023:
1018:
1016:9781939512017
1012:
1008:
1004:
1003:
995:
992:
987:
985:9780821887981
981:
977:
973:
972:
967:
960:
957:
954:, p. 74.
953:
948:
945:
939:
932:
930:
926:
922:
920:9783540438267
916:
912:
911:
902:
899:
895:
891:
890:
885:
878:
875:
868:
863:
860:
857:
854:
853:
849:
847:
845:
840:
836:
832:
824:
822:
820:
819:contradiction
809:
805:
797:
789:
777:
773:
765:
761:
741:
733:
725:
721:
713:
705:
690:
682:
680:
678:
675: =
674:
670:
667: =
666:
662:
658:
654:
650:
646:
643: =
642:
633:
625:
613:
605:
596:
560:
552:
550:
530:
527:
522:
517:
514:
505:
504:
503:
493:
488:
453:
450:
445:
440:
437:
432:
427:
424:
415:
414:
410:
408:
406:
402:
398:
393:
280:For example,
278:
276:
273: =
272:
269:
266:
262:
254:
246:
234:
226:
218:
214:
209:
201:
195:
185:
179:
168:
160:
148:
140:
132:
128:
123:
121:
117:
113:
109:
105:
101:
97:
93:
89:
85:
81:
73:
65:
57:
53:
49:
45:
41:
40:simplest form
37:
33:
19:
1277:
1096:
1059:
1052:
1032:
1025:
1001:
994:
970:
959:
952:Scott (1844)
947:
937:
909:
901:
887:
877:
828:
807:
803:
795:
787:
775:
771:
763:
759:
739:
731:
723:
719:
711:
703:
686:
683:Applications
676:
672:
668:
664:
660:
656:
652:
648:
644:
640:
631:
623:
611:
603:
558:
556:
547:
496:120 ÷ 30 = 4
489:
470:
394:
279:
274:
270:
264:
260:
252:
244:
232:
224:
216:
207:
199:
193:
183:
177:
166:
158:
146:
138:
130:
126:
124:
107:
95:
91:
83:
79:
71:
63:
43:
39:
35:
31:
29:
18:Lowest terms
1278:Irreducible
1208:Denominator
502:, one gets
500:90 ÷ 30 = 3
116:polynomials
104:mathematics
1330:Categories
1310:Percentage
1305:Paper size
1214:= Quotient
905:E.g., see
884:"Fraction"
869:References
753:, so does
553:Uniqueness
265:equivalent
173:such that
1283:Reduction
1241:Continued
1226:Algebraic
1202:Numerator
1138:Fractions
1098:MathWorld
894:EMS Press
810:(because
1256:Egyptian
1191:Fraction
1173:Quotient
1165:Dividend
850:See also
639:implies
411:Examples
118:. Every
56:divisors
52:integers
48:fraction
1273:Integer
1246:Decimal
1211:
1199:
1169:Divisor
833:of any
812:√
800:
784:
780:
755:
748:√
746:equals
744:
728:
716:
700:
693:√
636:
620:
616:
600:
591:
579:
575:
563:
498:, and
485:
473:
390:
378:
374:
362:
358:
346:
342:
330:
326:
314:
310:
298:
294:
282:
257:
241:
237:
221:
171:
155:
151:
135:
98:have a
88:coprime
76:
60:
46:) is a
1266:Silver
1261:Golden
1251:Dyadic
1236:Binary
1231:Aspect
1142:ratios
1067:
1040:
1013:
982:
917:
718:where
559:unique
312:, and
203:|
191:|
187:|
175:|
261:equal
1315:Unit
1140:and
1065:ISBN
1038:ISBN
1011:ISBN
980:ISBN
915:ISBN
722:and
651:and
317:−101
259:are
239:and
129:and
94:and
86:are
82:and
34:(or
1295:LCD
515:120
425:120
403:or
323:100
277:.)
263:or
215:of
189:or
106:, "
42:or
30:An
1332::
1171:=
1167:÷
1095:.
928:^
892:,
886:,
846:.
774:−
762:−
679:.
645:bc
641:ad
618:=
588:−3
582:−2
577:=
518:90
438:12
428:90
392:.
296:,
275:bc
271:ad
38:,
1205:/
1130:e
1123:t
1116:v
1101:.
1074:.
1047:.
1020:.
989:.
942:.
814:2
808:b
804:a
796:b
792:/
788:a
776:b
772:a
768:/
764:a
760:b
758:2
750:2
740:b
736:/
732:a
724:b
720:a
712:b
708:/
704:a
695:2
677:d
673:b
669:c
665:a
661:c
657:a
653:b
649:a
632:d
628:/
624:c
612:b
608:/
604:a
585:/
572:3
569:/
566:2
531:3
528:4
523:=
482:3
479:/
476:4
454:3
451:4
446:=
441:9
433:=
387:4
384:/
381:2
371:2
368:/
365:1
355:2
352:/
349:1
339:4
336:/
333:2
320:/
307:6
304:/
301:5
291:4
288:/
285:1
253:d
249:/
245:c
233:b
229:/
225:a
217:a
208:a
200:b
194:d
184:a
178:c
167:d
163:/
159:c
147:b
143:/
139:a
131:b
127:a
96:b
92:a
84:b
80:a
72:b
68:/
64:a
20:)
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