3691:
3350:
3357:
3095:
3925:
3686:{\displaystyle {\begin{aligned}\pm 8.123{\overline {4567}}&=\pm \left(8+{\frac {123}{10^{3}}}+{\frac {4567}{(10^{4}-1)\times 10^{3}}}\right)&{\text{from above}}\\&=\pm {\frac {8\times (10^{4}-1)\times 10^{3}+123\times (10^{4}-1)+4567}{(10^{4}-1)\times 10^{3}}}&{\text{common denominator}}\\&=\pm {\frac {81226444}{9999000}}&{\text{multiplying, and summing the numerator}}\\&=\pm {\frac {20306611}{2499750}}&{\text{reducing}}\\\end{aligned}}}
3345:{\displaystyle {\begin{aligned}0.000{\overline {4567}}&=4567\times 0.000{\overline {0001}}\\&=4567\times 0.{\overline {0001}}\times {\frac {1}{10^{3}}}\\&=4567\times {\frac {1}{9999}}\times {\frac {1}{10^{3}}}\\&={\frac {4567}{9999}}\times {\frac {1}{10^{3}}}\\&={\frac {4567}{(10^{4}-1)\times 10^{3}}}&{\text{The exponents are the number of non-repeating digits after the decimal point (3) and the number of repeating digits (4).}}\end{aligned}}}
43:
3737:
3920:{\displaystyle {\begin{aligned}\pm 8.1234&=\pm \left(8+{\frac {1234}{10^{4}}}\right)&\\&=\pm {\frac {8\times 10^{4}+1234}{10^{4}}}&{\text{common denominator}}\\&=\pm {\frac {81234}{10000}}&{\text{multiplying, and summing the numerator}}\\&=\pm {\frac {40617}{5000}}&{\text{reducing}}\\\end{aligned}}}
2670:
1861:
3055:
Every decimal representation of a rational number can be converted to a fraction by converting it into a sum of the integer, non-repeating, and repeating parts and then converting that sum to a single fraction with a common denominator.
2915:
595:
2489:
876:
1998:
2162:
247:
2985:(i.e. can alternatively be represented as a ratio of an integer and a positive integer). Also the converse is true: The decimal expansion of a rational number is either finite, or endlessly repeating.
1167:
341:
1221:
3742:
3362:
3100:
1096:
1405:(where the infinite sequences of trailing 0's or 9's, respectively, are represented by "..."). Conventionally, the decimal representation without trailing 9's is preferred. Moreover, in the
1750:
2714:
2067:
2314:
2778:
1263:
3090:
1623:
738:
3729:
2509:
1311:
485:
1713:
1584:
630:
381:
787:
407:
2248:
1656:
1500:
1020:
994:
2219:
1338:
914:
656:
2189:
1904:
1740:
1527:
1365:
949:
782:
670:. For having a one-to-one correspondence between nonnegative real numbers and decimal representations, decimal representations with a trailing infinite sequence of
465:
434:
2787:
1551:
1474:
1451:
1427:
1399:
2361:
1915:
164:
1476:
will avoid the problem of trailing 9's. For instance, the following algorithmic procedure will give the standard decimal representation: Given
1103:
2072:
107:
79:
60:
2940:
Some real numbers have decimal expansions that eventually get into loops, endlessly repeating a sequence of one or more digits:
2988:
Finite decimal representations can also be seen as a special case of infinite repeating decimal representations. For example,
86:
268:
1177:
3335:
The exponents are the number of non-repeating digits after the decimal point (3) and the number of repeating digits (4).
1025:
93:
4055:
4010:
126:
17:
2683:
2003:
2253:
75:
2719:
1401:
have two infinite decimal representations. For example, the number 1 may be equally represented by 1.000... as by
3974:
4079:
2665:{\displaystyle x={\frac {p}{2^{n}5^{m}}}={\frac {2^{m}5^{n}p}{2^{n+m}5^{n+m}}}={\frac {2^{m}5^{n}p}{10^{n+m}}}}
1226:
64:
3731:, although since that makes the repeating term zero the sum simplifies to two terms and a simpler conversion.
1912:); otherwise, it continues indefinitely to give an infinite sequence of decimal digits. It can be shown that
3062:
1589:
3698:
685:
1268:
599:
Every nonnegative real number has at least one such representation; it has two such representations (with
1856:{\displaystyle a_{0}+{\frac {a_{1}}{10}}+{\frac {a_{2}}{10^{2}}}+\cdots +{\frac {a_{k}}{10^{k}}}\leq x.}
1661:
100:
1556:
602:
353:
53:
30:
This article is about decimal expansion of real numbers. For finite decimal representation, see
879:
471:, the separator is also omitted, resulting in a finite sequence of digits, which represents a
386:
3950:
3050:
2224:
1628:
1479:
999:
973:
2910:{\displaystyle x=\sum _{i=0}^{n}10^{n-i}a_{i}/10^{n}=\sum _{i=0}^{n}{\frac {a_{i}}{10^{i}}}}
2198:
1316:
884:
635:
3940:
3010:, numbers that cannot be represented as a ratio of integers. Some well-known examples are:
2167:
1882:
1718:
1505:
1343:
927:
760:
479:
443:
412:
590:{\displaystyle r=\sum _{i=0}^{k}b_{i}10^{i}+\sum _{i=1}^{\infty }{\frac {a_{i}}{10^{i}}}.}
8:
2484:{\textstyle x=\sum _{i=0}^{n}{\frac {a_{i}}{10^{i}}}=\sum _{i=0}^{n}10^{n-i}a_{i}/10^{n}}
262:
3970:
3695:
If there are no repeating digits one assumes that there is a forever repeating 0, e.g.
3007:
1536:
1459:
1436:
1412:
1384:
3006:
Other real numbers have decimal expansions that never repeat. These are precisely the
2998:= 1.44 = 1.4400000...; the endlessly repeated sequence is the one-digit sequence "0".
1370:
4006:
3026:
2935:
254:
3014:
2192:
2982:
964:
871:{\displaystyle 0.a_{1}a_{2}\ldots =\sum _{i=1}^{\infty }{\frac {a_{i}}{10^{i}}},}
437:
4038:
3980:
659:
472:
4073:
1993:{\textstyle x=\sup _{k}\left\{\sum _{i=0}^{k}{\frac {a_{i}}{10^{i}}}\right\}}
158:
4047:
3994:
1430:
144:
4030:
4002:
963:
Any real number can be approximated to any desired degree of accuracy by
147:
42:
3945:
1402:
1376:
1371:
Non-uniqueness of decimal representation and notational conventions
154:
440:. If it is finite, the lacking digits are assumed to be 0. If all
347:, which are symbols representing integers in the range 0, ..., 9.
3935:
31:
2337:
is a rational number whose denominator is of the form 25, where
2157:{\displaystyle a_{1},a_{2},a_{3}\ldots \in \{0,1,2,\ldots ,9\},}
1456:
Certain procedures for constructing the decimal expansion of
1367:
has a finite decimal representation is easily established.)
242:{\displaystyle r=b_{k}b_{k-1}\ldots b_{0}.a_{1}a_{2}\ldots }
1429:, an infinite sequence of trailing 0's appearing after the
1162:{\displaystyle r_{n}\leq x<r_{n}+{\frac {1}{10^{n}}}.}
3001:
3035:
336:{\displaystyle b_{0},\ldots ,b_{k},a_{1},a_{2},\ldots }
3065:
2687:
2364:
1918:
1664:
1194:
757:
in the remainder of this article. The sequence of the
688:
3740:
3701:
3360:
3098:
2790:
2722:
2686:
2512:
2256:
2227:
2201:
2170:
2075:
2006:
1885:
1753:
1721:
1631:
1592:
1559:
1539:
1508:
1482:
1462:
1439:
1415:
1387:
1346:
1319:
1271:
1229:
1180:
1106:
1028:
1002:
976:
930:
887:
790:
763:
638:
605:
488:
446:
415:
389:
356:
271:
167:
2929:
1313:, and the result follows from dividing all sides by
1216:{\displaystyle r_{n}=\textstyle {\frac {p}{10^{n}}}}
666:, and the other has a trailing infinite sequence of
1433:is omitted, along with the decimal point itself if
67:. Unsourced material may be challenged and removed.
3919:
3723:
3685:
3344:
3084:
2909:
2772:
2708:
2664:
2483:
2329:The decimal expansion of non-negative real number
2308:
2242:
2213:
2183:
2156:
2061:
1992:
1898:
1855:
1734:
1707:
1650:
1617:
1578:
1545:
1521:
1494:
1468:
1445:
1421:
1393:
1359:
1332:
1305:
1257:
1215:
1161:
1091:{\displaystyle r_{n}=a_{0}.a_{1}a_{2}\cdots a_{n}}
1090:
1014:
988:
943:
908:
870:
776:
732:
650:
624:
589:
459:
428:
401:
375:
335:
241:
2333:will end in zeros (or in nines) if, and only if,
1742:inductively to be the largest integer such that:
4071:
2250:and denoting the resultant decimal expansion by
1926:
958:
161:traditionally written with a single separator:
677:
2981:Every time this happens the number is still a
2709:{\displaystyle \textstyle {\frac {p}{10^{k}}}}
2062:{\displaystyle x=a_{0}.a_{1}a_{2}a_{3}\cdots }
2309:{\displaystyle -a_{0}.a_{1}a_{2}a_{3}\cdots }
2773:{\displaystyle p=\sum _{i=0}^{n}10^{i}a_{i}}
2148:
2118:
1612:
1606:
1252:
1236:
27:Expression of numbers as sequences of digits
3987:
1658:the procedure terminates. Otherwise, for
478:The decimal representation represents the
3963:
3051:Fraction § Arithmetic with fractions
3044:
1258:{\displaystyle p=\lfloor 10^{n}x\rfloor }
127:Learn how and when to remove this message
3085:{\textstyle \pm 8.123{\overline {4567}}}
662:one has a trailing infinite sequence of
4029:
3979:. Vol. 1: Fundamental Algorithms.
1618:{\displaystyle a_{0}=\lfloor x\rfloor }
733:{\textstyle \sum _{i=0}^{k}b_{i}10^{i}}
436:—the digits after the dot—is generally
14:
4072:
4045:
3880:multiplying, and summing the numerator
3724:{\displaystyle 1.9=1.9{\overline {0}}}
3646:multiplying, and summing the numerator
1906:is found such that equality holds in (
1553:) to be the largest integer such that
3993:
3969:
3002:Non-repeating decimal representations
967:with finite decimal representations.
1744:
1306:{\displaystyle p\leq 10^{n}x<p+1}
65:adding citations to reliable sources
36:
3999:Principles of Mathematical Analysis
3092:to a fraction one notes the lemma:
2221:by applying the above procedure to
2195:. This construction is extended to
24:
4023:
3040: = 3.14159265358979323846...
3031: = 2.71828182845904523536...
2502:Conversely, if the denominator of
1879:The procedure terminates whenever
836:
555:
25:
4091:
2930:Repeating decimal representations
1708:{\textstyle (a_{i})_{i=0}^{k-1}}
41:
4058:from the original on 2018-07-16
3976:The Art of Computer Programming
1407:standard decimal representation
52:needs additional citations for
3593:
3574:
3563:
3544:
3519:
3500:
3447:
3428:
3354:Thus one converts as follows:
3312:
3293:
1679:
1665:
900:
888:
13:
1:
3956:
959:Finite decimal approximations
4046:Savard, John J. G. (2018) .
3716:
3376:
3166:
3137:
3111:
3077:
2354:If the decimal expansion of
2164:and the nonnegative integer
678:Integer and fractional parts
7:
3929:
3022:= 1.41421356237309504880...
2924:
2345:are non-negative integers.
2000:(conventionally written as
1908:
1869:
1579:{\displaystyle a_{0}\leq x}
625:{\displaystyle b_{k}\neq 0}
376:{\displaystyle b_{k}\neq 0}
10:
4096:
3048:
2933:
2495:, then the denominator of
1374:
1022:there is a finite decimal
29:
4048:"Decimal Representations"
2324:
1715:already found, we define
996:. Then for every integer
157:of symbols consisting of
3059:For example, to convert
2499:is of the form 10 = 25.
2319:
674:are sometimes excluded.
402:{\displaystyle k\geq 1.}
76:"Decimal representation"
2243:{\displaystyle -x>0}
1651:{\displaystyle x=a_{0}}
1495:{\displaystyle x\geq 0}
1015:{\displaystyle n\geq 1}
989:{\displaystyle x\geq 0}
784:represents the number
153:is its expression as a
3921:
3725:
3687:
3346:
3086:
3045:Conversion to fraction
2911:
2882:
2817:
2774:
2749:
2710:
2666:
2485:
2439:
2391:
2358:will end in zeros, or
2310:
2244:
2215:
2214:{\displaystyle x<0}
2185:
2158:
2063:
1994:
1960:
1900:
1857:
1736:
1709:
1652:
1619:
1580:
1547:
1523:
1496:
1470:
1447:
1423:
1395:
1361:
1334:
1333:{\displaystyle 10^{n}}
1307:
1259:
1217:
1163:
1092:
1016:
990:
945:
910:
909:{\displaystyle [0,1),}
872:
840:
778:
734:
709:
652:
651:{\displaystyle k>0}
626:
591:
559:
515:
461:
430:
403:
377:
337:
243:
141:decimal representation
4080:Mathematical notation
4035:Mathematical analysis
3922:
3726:
3688:
3347:
3087:
3049:Further information:
2912:
2862:
2797:
2775:
2729:
2711:
2667:
2486:
2419:
2371:
2311:
2245:
2216:
2186:
2184:{\displaystyle a_{0}}
2159:
2064:
1995:
1940:
1901:
1899:{\displaystyle a_{k}}
1858:
1737:
1735:{\displaystyle a_{k}}
1710:
1653:
1620:
1581:
1548:
1524:
1522:{\displaystyle a_{0}}
1497:
1471:
1448:
1424:
1396:
1362:
1360:{\displaystyle r_{n}}
1335:
1308:
1260:
1218:
1164:
1093:
1017:
991:
946:
944:{\displaystyle a_{i}}
911:
878:which belongs to the
873:
820:
779:
777:{\displaystyle a_{i}}
735:
689:
653:
627:
592:
539:
495:
462:
460:{\displaystyle a_{i}}
431:
429:{\displaystyle a_{i}}
404:
378:
338:
244:
3941:Series (mathematics)
3738:
3699:
3358:
3096:
3063:
2788:
2720:
2684:
2510:
2362:
2254:
2225:
2199:
2168:
2073:
2004:
1916:
1883:
1751:
1719:
1662:
1629:
1590:
1557:
1537:
1506:
1480:
1460:
1437:
1413:
1385:
1344:
1317:
1269:
1227:
1178:
1104:
1026:
1000:
974:
928:
885:
788:
761:
748:, and is denoted by
686:
636:
603:
486:
444:
413:
409:The sequence of the
387:
354:
269:
165:
61:improve this article
4037:(Second ed.).
3971:Knuth, Donald Ervin
2965:= 0.142857142857...
2921:will end in zeros.
2506:is of the form 25,
1704:
682:The natural number
263:nonnegative integer
3917:
3915:
3850:common denominator
3721:
3683:
3681:
3616:common denominator
3342:
3340:
3082:
3008:irrational numbers
2907:
2770:
2706:
2705:
2662:
2481:
2306:
2240:
2211:
2191:is represented in
2181:
2154:
2059:
1990:
1934:
1896:
1853:
1732:
1705:
1678:
1648:
1615:
1576:
1543:
1519:
1502:, we first define
1492:
1466:
1443:
1419:
1391:
1381:Some real numbers
1357:
1330:
1303:
1255:
1213:
1212:
1159:
1088:
1012:
986:
941:
916:and is called the
906:
868:
774:
730:
648:
622:
587:
457:
426:
399:
373:
333:
239:
3911:
3904:
3881:
3874:
3851:
3844:
3787:
3719:
3677:
3670:
3647:
3640:
3617:
3610:
3476:
3464:
3417:
3379:
3336:
3329:
3275:
3255:
3235:
3215:
3189:
3169:
3140:
3114:
3080:
2977:= 7.1243243243...
2936:Repeating decimal
2905:
2703:
2660:
2612:
2546:
2414:
1983:
1925:
1877:
1876:
1842:
1809:
1782:
1546:{\displaystyle x}
1469:{\displaystyle x}
1446:{\displaystyle x}
1422:{\displaystyle x}
1394:{\displaystyle x}
1340:. (The fact that
1210:
1154:
924:(except when all
863:
582:
255:decimal separator
137:
136:
129:
111:
18:Decimal expansion
16:(Redirected from
4087:
4066:
4064:
4063:
4042:
4017:
4016:
3991:
3985:
3984:
3967:
3926:
3924:
3923:
3918:
3916:
3912:
3909:
3905:
3897:
3886:
3882:
3879:
3875:
3867:
3856:
3852:
3849:
3845:
3843:
3842:
3833:
3826:
3825:
3809:
3798:
3795:
3793:
3789:
3788:
3786:
3785:
3773:
3730:
3728:
3727:
3722:
3720:
3712:
3692:
3690:
3689:
3684:
3682:
3678:
3675:
3671:
3663:
3652:
3648:
3645:
3641:
3633:
3622:
3618:
3615:
3611:
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3608:
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3586:
3585:
3572:
3556:
3555:
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3477:
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3330:
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3326:
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3304:
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3280:
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3236:
3234:
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3221:
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3208:
3194:
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3187:
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3162:
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3141:
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3115:
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2281:
2269:
2268:
2249:
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2220:
2218:
2217:
2212:
2193:decimal notation
2190:
2188:
2187:
2182:
2180:
2179:
2163:
2161:
2160:
2155:
2111:
2110:
2098:
2097:
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2045:
2044:
2035:
2034:
2022:
2021:
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1997:
1996:
1991:
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1981:
1972:
1971:
1962:
1959:
1954:
1933:
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1807:
1798:
1797:
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1741:
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1703:
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1286:
1264:
1262:
1261:
1256:
1248:
1247:
1222:
1220:
1219:
1214:
1211:
1209:
1208:
1196:
1190:
1189:
1168:
1166:
1165:
1160:
1155:
1153:
1152:
1140:
1135:
1134:
1116:
1115:
1097:
1095:
1094:
1089:
1087:
1086:
1074:
1073:
1064:
1063:
1051:
1050:
1038:
1037:
1021:
1019:
1018:
1013:
995:
993:
992:
987:
965:rational numbers
954:
950:
948:
947:
942:
940:
939:
923:
915:
913:
912:
907:
877:
875:
874:
869:
864:
862:
861:
852:
851:
842:
839:
834:
813:
812:
803:
802:
783:
781:
780:
775:
773:
772:
756:
747:
740:, is called the
739:
737:
736:
731:
729:
728:
719:
718:
708:
703:
673:
669:
665:
657:
655:
654:
649:
631:
629:
628:
623:
615:
614:
596:
594:
593:
588:
583:
581:
580:
571:
570:
561:
558:
553:
535:
534:
525:
524:
514:
509:
470:
466:
464:
463:
458:
456:
455:
435:
433:
432:
427:
425:
424:
408:
406:
405:
400:
382:
380:
379:
374:
366:
365:
342:
340:
339:
334:
326:
325:
313:
312:
300:
299:
281:
280:
260:
252:
248:
246:
245:
240:
235:
234:
225:
224:
212:
211:
199:
198:
183:
182:
152:
132:
125:
121:
118:
112:
110:
69:
45:
37:
21:
4095:
4094:
4090:
4089:
4088:
4086:
4085:
4084:
4070:
4069:
4061:
4059:
4026:
4024:Further reading
4021:
4020:
4013:
3992:
3988:
3968:
3964:
3959:
3932:
3914:
3913:
3908:
3906:
3896:
3884:
3883:
3878:
3876:
3866:
3854:
3853:
3848:
3846:
3838:
3834:
3821:
3817:
3810:
3808:
3796:
3794:
3781:
3777:
3772:
3765:
3761:
3751:
3741:
3739:
3736:
3735:
3711:
3700:
3697:
3696:
3680:
3679:
3674:
3672:
3662:
3650:
3649:
3644:
3642:
3632:
3620:
3619:
3614:
3612:
3603:
3599:
3581:
3577:
3573:
3551:
3547:
3529:
3525:
3507:
3503:
3493:
3491:
3479:
3478:
3473:
3471:
3457:
3453:
3435:
3431:
3427:
3422:
3411:
3407:
3402:
3395:
3391:
3381:
3371:
3361:
3359:
3356:
3355:
3339:
3338:
3333:
3331:
3322:
3318:
3300:
3296:
3292:
3287:
3278:
3277:
3269:
3265:
3260:
3247:
3238:
3237:
3229:
3225:
3220:
3207:
3192:
3191:
3183:
3179:
3174:
3161:
3143:
3142:
3132:
3116:
3106:
3099:
3097:
3094:
3093:
3072:
3064:
3061:
3060:
3053:
3047:
3017:
3015:
3004:
2994:
2990:
2989:
2983:rational number
2973:
2969:
2968:
2961:
2957:
2956:
2949:
2945:
2944:
2938:
2932:
2927:
2899:
2895:
2889:
2885:
2883:
2877:
2866:
2853:
2849:
2844:
2838:
2834:
2822:
2818:
2812:
2801:
2789:
2786:
2785:
2764:
2760:
2754:
2750:
2744:
2733:
2721:
2718:
2717:
2697:
2693:
2688:
2685:
2682:
2681:
2680:is of the form
2648:
2644:
2634:
2630:
2624:
2620:
2619:
2617:
2599:
2595:
2583:
2579:
2578:
2568:
2564:
2558:
2554:
2553:
2551:
2539:
2535:
2529:
2525:
2524:
2519:
2511:
2508:
2507:
2475:
2471:
2466:
2460:
2456:
2444:
2440:
2434:
2423:
2408:
2404:
2398:
2394:
2392:
2386:
2375:
2363:
2360:
2359:
2327:
2322:
2297:
2293:
2287:
2283:
2277:
2273:
2264:
2260:
2255:
2252:
2251:
2226:
2223:
2222:
2200:
2197:
2196:
2175:
2171:
2169:
2166:
2165:
2106:
2102:
2093:
2089:
2080:
2076:
2074:
2071:
2070:
2050:
2046:
2040:
2036:
2030:
2026:
2017:
2013:
2005:
2002:
2001:
1977:
1973:
1967:
1963:
1961:
1955:
1944:
1939:
1935:
1929:
1917:
1914:
1913:
1890:
1886:
1884:
1881:
1880:
1836:
1832:
1826:
1822:
1820:
1803:
1799:
1793:
1789:
1787:
1773:
1769:
1767:
1758:
1754:
1752:
1749:
1748:
1726:
1722:
1720:
1717:
1716:
1693:
1682:
1672:
1668:
1663:
1660:
1659:
1642:
1638:
1630:
1627:
1626:
1597:
1593:
1591:
1588:
1587:
1564:
1560:
1558:
1555:
1554:
1538:
1535:
1534:
1513:
1509:
1507:
1504:
1503:
1481:
1478:
1477:
1461:
1458:
1457:
1453:is an integer.
1438:
1435:
1434:
1414:
1411:
1410:
1386:
1383:
1382:
1379:
1373:
1351:
1347:
1345:
1342:
1341:
1324:
1320:
1318:
1315:
1314:
1282:
1278:
1270:
1267:
1266:
1243:
1239:
1228:
1225:
1224:
1204:
1200:
1195:
1185:
1181:
1179:
1176:
1175:
1148:
1144:
1139:
1130:
1126:
1111:
1107:
1105:
1102:
1101:
1082:
1078:
1069:
1065:
1059:
1055:
1046:
1042:
1033:
1029:
1027:
1024:
1023:
1001:
998:
997:
975:
972:
971:
961:
952:
935:
931:
929:
926:
925:
921:
918:fractional part
886:
883:
882:
857:
853:
847:
843:
841:
835:
824:
808:
804:
798:
794:
789:
786:
785:
768:
764:
762:
759:
758:
755:
749:
745:
724:
720:
714:
710:
704:
693:
687:
684:
683:
680:
671:
667:
663:
637:
634:
633:
610:
606:
604:
601:
600:
576:
572:
566:
562:
560:
554:
543:
530:
526:
520:
516:
510:
499:
487:
484:
483:
468:
451:
447:
445:
442:
441:
420:
416:
414:
411:
410:
388:
385:
384:
361:
357:
355:
352:
351:
321:
317:
308:
304:
295:
291:
276:
272:
270:
267:
266:
258:
250:
230:
226:
220:
216:
207:
203:
188:
184:
178:
174:
166:
163:
162:
150:
133:
122:
116:
113:
70:
68:
58:
46:
35:
28:
23:
22:
15:
12:
11:
5:
4093:
4083:
4082:
4068:
4067:
4043:
4039:Addison-Wesley
4025:
4022:
4019:
4018:
4011:
4005:. p. 11.
3986:
3981:Addison-Wesley
3961:
3960:
3958:
3955:
3954:
3953:
3948:
3943:
3938:
3931:
3928:
3907:
3903:
3900:
3895:
3892:
3889:
3887:
3885:
3877:
3873:
3870:
3865:
3862:
3859:
3857:
3855:
3847:
3841:
3837:
3832:
3829:
3824:
3820:
3816:
3813:
3807:
3804:
3801:
3799:
3797:
3792:
3784:
3780:
3776:
3771:
3768:
3764:
3760:
3757:
3754:
3752:
3750:
3747:
3744:
3743:
3718:
3715:
3710:
3707:
3704:
3673:
3669:
3666:
3661:
3658:
3655:
3653:
3651:
3643:
3639:
3636:
3631:
3628:
3625:
3623:
3621:
3613:
3606:
3602:
3598:
3595:
3592:
3589:
3584:
3580:
3576:
3571:
3568:
3565:
3562:
3559:
3554:
3550:
3546:
3543:
3540:
3537:
3532:
3528:
3524:
3521:
3518:
3515:
3510:
3506:
3502:
3499:
3496:
3490:
3487:
3484:
3482:
3480:
3472:
3469:
3460:
3456:
3452:
3449:
3446:
3443:
3438:
3434:
3430:
3426:
3421:
3414:
3410:
3406:
3401:
3398:
3394:
3390:
3387:
3384:
3382:
3378:
3375:
3370:
3367:
3364:
3363:
3332:
3325:
3321:
3317:
3314:
3311:
3308:
3303:
3299:
3295:
3291:
3286:
3283:
3281:
3279:
3272:
3268:
3264:
3259:
3254:
3251:
3246:
3243:
3241:
3239:
3232:
3228:
3224:
3219:
3214:
3211:
3206:
3203:
3200:
3197:
3195:
3193:
3186:
3182:
3178:
3173:
3168:
3165:
3160:
3157:
3154:
3151:
3148:
3146:
3144:
3139:
3136:
3131:
3128:
3125:
3122:
3119:
3117:
3113:
3110:
3105:
3102:
3101:
3079:
3076:
3071:
3068:
3046:
3043:
3042:
3041:
3032:
3023:
3003:
3000:
2979:
2978:
2966:
2954:
2934:Main article:
2931:
2928:
2926:
2923:
2902:
2898:
2892:
2888:
2880:
2875:
2872:
2869:
2865:
2861:
2856:
2852:
2847:
2841:
2837:
2831:
2828:
2825:
2821:
2815:
2810:
2807:
2804:
2800:
2796:
2793:
2767:
2763:
2757:
2753:
2747:
2742:
2739:
2736:
2732:
2728:
2725:
2700:
2696:
2692:
2657:
2654:
2651:
2647:
2642:
2637:
2633:
2627:
2623:
2616:
2608:
2605:
2602:
2598:
2592:
2589:
2586:
2582:
2576:
2571:
2567:
2561:
2557:
2550:
2542:
2538:
2532:
2528:
2523:
2518:
2515:
2478:
2474:
2469:
2463:
2459:
2453:
2450:
2447:
2443:
2437:
2432:
2429:
2426:
2422:
2418:
2411:
2407:
2401:
2397:
2389:
2384:
2381:
2378:
2374:
2370:
2367:
2326:
2323:
2321:
2318:
2305:
2300:
2296:
2290:
2286:
2280:
2276:
2272:
2267:
2263:
2259:
2239:
2236:
2233:
2230:
2210:
2207:
2204:
2178:
2174:
2153:
2150:
2147:
2144:
2141:
2138:
2135:
2132:
2129:
2126:
2123:
2120:
2117:
2114:
2109:
2105:
2101:
2096:
2092:
2088:
2083:
2079:
2058:
2053:
2049:
2043:
2039:
2033:
2029:
2025:
2020:
2016:
2012:
2009:
1988:
1980:
1976:
1970:
1966:
1958:
1953:
1950:
1947:
1943:
1938:
1932:
1928:
1924:
1921:
1893:
1889:
1875:
1874:
1865:
1863:
1852:
1849:
1846:
1839:
1835:
1829:
1825:
1819:
1816:
1813:
1806:
1802:
1796:
1792:
1786:
1781:
1776:
1772:
1766:
1761:
1757:
1729:
1725:
1702:
1699:
1696:
1691:
1688:
1685:
1681:
1675:
1671:
1667:
1645:
1641:
1637:
1634:
1614:
1611:
1608:
1605:
1600:
1596:
1575:
1572:
1567:
1563:
1542:
1516:
1512:
1491:
1488:
1485:
1465:
1442:
1418:
1390:
1375:Main article:
1372:
1369:
1354:
1350:
1327:
1323:
1302:
1299:
1296:
1293:
1290:
1285:
1281:
1277:
1274:
1254:
1251:
1246:
1242:
1238:
1235:
1232:
1207:
1203:
1199:
1193:
1188:
1184:
1158:
1151:
1147:
1143:
1138:
1133:
1129:
1125:
1122:
1119:
1114:
1110:
1085:
1081:
1077:
1072:
1068:
1062:
1058:
1054:
1049:
1045:
1041:
1036:
1032:
1011:
1008:
1005:
985:
982:
979:
960:
957:
938:
934:
905:
902:
899:
896:
893:
890:
867:
860:
856:
850:
846:
838:
833:
830:
827:
823:
819:
816:
811:
807:
801:
797:
793:
771:
767:
753:
727:
723:
717:
713:
707:
702:
699:
696:
692:
679:
676:
660:if and only if
647:
644:
641:
621:
618:
613:
609:
586:
579:
575:
569:
565:
557:
552:
549:
546:
542:
538:
533:
529:
523:
519:
513:
508:
505:
502:
498:
494:
491:
473:natural number
454:
450:
423:
419:
398:
395:
392:
372:
369:
364:
360:
332:
329:
324:
320:
316:
311:
307:
303:
298:
294:
290:
287:
284:
279:
275:
238:
233:
229:
223:
219:
215:
210:
206:
202:
197:
194:
191:
187:
181:
177:
173:
170:
159:decimal digits
135:
134:
49:
47:
40:
26:
9:
6:
4:
3:
2:
4092:
4081:
4078:
4077:
4075:
4057:
4053:
4049:
4044:
4040:
4036:
4032:
4028:
4027:
4014:
4012:0-07-054235-X
4008:
4004:
4000:
3996:
3995:Rudin, Walter
3990:
3983:. p. 21.
3982:
3978:
3977:
3972:
3966:
3962:
3952:
3949:
3947:
3944:
3942:
3939:
3937:
3934:
3933:
3927:
3901:
3898:
3893:
3890:
3888:
3871:
3868:
3863:
3860:
3858:
3839:
3835:
3830:
3827:
3822:
3818:
3814:
3811:
3805:
3802:
3800:
3790:
3782:
3778:
3774:
3769:
3766:
3762:
3758:
3755:
3753:
3748:
3745:
3734:For example:
3732:
3713:
3708:
3705:
3702:
3693:
3667:
3664:
3659:
3656:
3654:
3637:
3634:
3629:
3626:
3624:
3604:
3600:
3596:
3590:
3587:
3582:
3578:
3569:
3566:
3560:
3557:
3552:
3548:
3541:
3538:
3535:
3530:
3526:
3522:
3516:
3513:
3508:
3504:
3497:
3494:
3488:
3485:
3483:
3467:
3458:
3454:
3450:
3444:
3441:
3436:
3432:
3424:
3419:
3412:
3408:
3404:
3399:
3396:
3392:
3388:
3385:
3383:
3373:
3368:
3365:
3352:
3323:
3319:
3315:
3309:
3306:
3301:
3297:
3289:
3284:
3282:
3270:
3266:
3262:
3257:
3252:
3249:
3244:
3242:
3230:
3226:
3222:
3217:
3212:
3209:
3204:
3201:
3198:
3196:
3184:
3180:
3176:
3171:
3163:
3158:
3155:
3152:
3149:
3147:
3134:
3129:
3126:
3123:
3120:
3118:
3108:
3103:
3074:
3069:
3066:
3057:
3052:
3039:
3038:
3034:
3033:
3030:
3029:
3025:
3024:
3021:
3013:
3012:
3011:
3009:
2999:
2986:
2984:
2967:
2955:
2943:
2942:
2941:
2937:
2922:
2920:
2900:
2896:
2890:
2886:
2878:
2873:
2870:
2867:
2863:
2859:
2854:
2850:
2845:
2839:
2835:
2829:
2826:
2823:
2819:
2813:
2808:
2805:
2802:
2798:
2794:
2791:
2783:
2765:
2761:
2755:
2751:
2745:
2740:
2737:
2734:
2730:
2726:
2723:
2698:
2694:
2690:
2679:
2675:
2655:
2652:
2649:
2645:
2640:
2635:
2631:
2625:
2621:
2614:
2606:
2603:
2600:
2596:
2590:
2587:
2584:
2580:
2574:
2569:
2565:
2559:
2555:
2548:
2540:
2536:
2530:
2526:
2521:
2516:
2513:
2505:
2500:
2498:
2494:
2476:
2472:
2467:
2461:
2457:
2451:
2448:
2445:
2441:
2435:
2430:
2427:
2424:
2420:
2416:
2409:
2405:
2399:
2395:
2387:
2382:
2379:
2376:
2372:
2368:
2365:
2357:
2352:
2350:
2346:
2344:
2340:
2336:
2332:
2317:
2303:
2298:
2294:
2288:
2284:
2278:
2274:
2270:
2265:
2261:
2257:
2237:
2234:
2231:
2228:
2208:
2205:
2202:
2194:
2176:
2172:
2151:
2145:
2142:
2139:
2136:
2133:
2130:
2127:
2124:
2121:
2115:
2112:
2107:
2103:
2099:
2094:
2090:
2086:
2081:
2077:
2056:
2051:
2047:
2041:
2037:
2031:
2027:
2023:
2018:
2014:
2010:
2007:
1986:
1978:
1974:
1968:
1964:
1956:
1951:
1948:
1945:
1941:
1936:
1930:
1922:
1919:
1911:
1910:
1891:
1887:
1873:
1866:
1864:
1850:
1847:
1844:
1837:
1833:
1827:
1823:
1817:
1814:
1811:
1804:
1800:
1794:
1790:
1784:
1779:
1774:
1770:
1764:
1759:
1755:
1747:
1746:
1743:
1727:
1723:
1700:
1697:
1694:
1689:
1686:
1683:
1673:
1669:
1643:
1639:
1635:
1632:
1609:
1603:
1598:
1594:
1573:
1570:
1565:
1561:
1540:
1532:
1514:
1510:
1489:
1486:
1483:
1463:
1454:
1440:
1432:
1431:decimal point
1416:
1408:
1404:
1388:
1378:
1368:
1352:
1348:
1325:
1321:
1300:
1297:
1294:
1291:
1288:
1283:
1279:
1275:
1272:
1249:
1244:
1240:
1233:
1230:
1205:
1201:
1197:
1191:
1186:
1182:
1173:
1169:
1156:
1149:
1145:
1141:
1136:
1131:
1127:
1123:
1120:
1117:
1112:
1108:
1099:
1083:
1079:
1075:
1070:
1066:
1060:
1056:
1052:
1047:
1043:
1039:
1034:
1030:
1009:
1006:
1003:
983:
980:
977:
968:
966:
956:
951:are equal to
936:
932:
919:
903:
897:
894:
891:
881:
865:
858:
854:
848:
844:
831:
828:
825:
821:
817:
814:
809:
805:
799:
795:
791:
769:
765:
752:
743:
725:
721:
715:
711:
705:
700:
697:
694:
690:
675:
661:
645:
642:
639:
619:
616:
611:
607:
597:
584:
577:
573:
567:
563:
550:
547:
544:
540:
536:
531:
527:
521:
517:
511:
506:
503:
500:
496:
492:
489:
481:
476:
474:
452:
448:
439:
421:
417:
396:
393:
390:
370:
367:
362:
358:
348:
346:
330:
327:
322:
318:
314:
309:
305:
301:
296:
292:
288:
285:
282:
277:
273:
264:
256:
236:
231:
227:
221:
217:
213:
208:
204:
200:
195:
192:
189:
185:
179:
175:
171:
168:
160:
156:
149:
146:
142:
131:
128:
120:
109:
106:
102:
99:
95:
92:
88:
85:
81:
78: –
77:
73:
72:Find sources:
66:
62:
56:
55:
50:This article
48:
44:
39:
38:
33:
19:
4060:. Retrieved
4051:
4034:
4031:Apostol, Tom
4001:. New York:
3998:
3989:
3975:
3965:
3951:Simon Stevin
3733:
3694:
3353:
3058:
3054:
3036:
3027:
3005:
2987:
2980:
2953:= 0.33333...
2939:
2918:
2781:
2677:
2673:
2503:
2501:
2496:
2492:
2355:
2353:
2348:
2347:
2342:
2338:
2334:
2330:
2328:
1907:
1878:
1867:
1531:integer part
1530:
1455:
1406:
1380:
1171:
1170:
1100:
969:
962:
917:
750:
742:integer part
741:
681:
598:
480:infinite sum
477:
349:
344:
145:non-negative
140:
138:
123:
117:January 2022
114:
104:
97:
90:
83:
71:
59:Please help
54:verification
51:
4003:McGraw-Hill
1098:such that:
148:real number
4062:2018-07-16
3957:References
3475:from above
350:Commonly,
87:newspapers
4052:quadibloc
3894:±
3864:±
3815:×
3806:±
3759:±
3746:±
3717:¯
3660:±
3630:±
3597:×
3588:−
3558:−
3542:×
3523:×
3514:−
3498:×
3489:±
3451:×
3442:−
3389:±
3377:¯
3366:±
3316:×
3307:−
3258:×
3218:×
3205:×
3172:×
3167:¯
3156:×
3138:¯
3127:×
3112:¯
3078:¯
3067:±
2864:∑
2827:−
2799:∑
2780:for some
2731:∑
2672:for some
2491:for some
2449:−
2421:∑
2373:∑
2304:⋯
2258:−
2229:−
2140:…
2116:∈
2113:…
2069:), where
2057:⋯
1942:∑
1845:≤
1815:⋯
1698:−
1613:⌋
1607:⌊
1571:≤
1487:≥
1276:≤
1253:⌋
1237:⌊
1118:≤
1076:⋯
1007:≥
981:≥
837:∞
822:∑
815:…
691:∑
617:≠
556:∞
541:∑
497:∑
394:≥
368:≠
331:…
286:…
237:…
201:…
193:−
4074:Category
4056:Archived
4033:(1974).
3997:(1976).
3973:(1973).
3946:IEEE 754
3930:See also
3910:reducing
3676:reducing
3665:20306611
3635:81226444
2925:Infinite
2676:. While
1403:0.999...
1377:0.999...
1223:, where
880:interval
438:infinite
155:sequence
3936:Decimal
3668:2499750
3638:9999000
3016:√
2993:⁄
2972:⁄
2960:⁄
2948:⁄
1625:). If
1586:(i.e.,
1265:. Then
970:Assume
253:is the
101:scholar
32:Decimal
4009:
3749:8.1234
2325:Finite
1174:: Let
345:digits
265:, and
103:
96:
89:
82:
74:
3899:40617
3872:10000
3869:81234
3369:8.123
3130:0.000
3104:0.000
3070:8.123
2784:. By
2349:Proof
2320:Types
1529:(the
1172:Proof
261:is a
249:Here
143:of a
108:JSTOR
94:books
4007:ISBN
3902:5000
3831:1234
3775:1234
3570:4567
3425:4567
3374:4567
3290:4567
3253:9999
3250:4567
3213:9999
3202:4567
3164:0001
3153:4567
3135:0001
3124:4567
3109:4567
3075:4567
2970:1318
2341:and
2235:>
2206:<
1292:<
1124:<
643:>
467:are
343:are
80:news
3709:1.9
3703:1.9
3539:123
3405:123
2974:185
1927:sup
1533:of
1409:of
955:).
920:of
744:of
632:if
383:if
63:by
4076::
4054:.
4050:.
3836:10
3819:10
3779:10
3601:10
3579:10
3549:10
3527:10
3505:10
3455:10
3433:10
3409:10
3320:10
3298:10
3267:10
3227:10
3181:10
3159:0.
2995:25
2991:36
2917:,
2897:10
2851:10
2820:10
2752:10
2716:,
2695:10
2646:10
2473:10
2442:10
2406:10
2351::
2316:.
1975:10
1834:10
1801:10
1780:10
1322:10
1280:10
1241:10
1202:10
1146:10
855:10
792:0.
722:10
658:)
574:10
528:10
482::
475:.
397:1.
257:,
139:A
4065:.
4041:.
4015:.
3891:=
3861:=
3840:4
3828:+
3823:4
3812:8
3803:=
3791:)
3783:4
3770:+
3767:8
3763:(
3756:=
3714:0
3706:=
3657:=
3627:=
3605:3
3594:)
3591:1
3583:4
3575:(
3567:+
3564:)
3561:1
3553:4
3545:(
3536:+
3531:3
3520:)
3517:1
3509:4
3501:(
3495:8
3486:=
3468:)
3459:3
3448:)
3445:1
3437:4
3429:(
3420:+
3413:3
3400:+
3397:8
3393:(
3386:=
3324:3
3313:)
3310:1
3302:4
3294:(
3285:=
3271:3
3263:1
3245:=
3231:3
3223:1
3210:1
3199:=
3185:3
3177:1
3150:=
3121:=
3037:π
3028:e
3018:2
2962:7
2958:1
2950:3
2946:1
2919:x
2901:i
2891:i
2887:a
2879:n
2874:0
2871:=
2868:i
2860:=
2855:n
2846:/
2840:i
2836:a
2830:i
2824:n
2814:n
2809:0
2806:=
2803:i
2795:=
2792:x
2782:n
2766:i
2762:a
2756:i
2746:n
2741:0
2738:=
2735:i
2727:=
2724:p
2699:k
2691:p
2678:x
2674:p
2656:m
2653:+
2650:n
2641:p
2636:n
2632:5
2626:m
2622:2
2615:=
2607:m
2604:+
2601:n
2597:5
2591:m
2588:+
2585:n
2581:2
2575:p
2570:n
2566:5
2560:m
2556:2
2549:=
2541:m
2537:5
2531:n
2527:2
2522:p
2517:=
2514:x
2504:x
2497:x
2493:n
2477:n
2468:/
2462:i
2458:a
2452:i
2446:n
2436:n
2431:0
2428:=
2425:i
2417:=
2410:i
2400:i
2396:a
2388:n
2383:0
2380:=
2377:i
2369:=
2366:x
2356:x
2343:n
2339:m
2335:x
2331:x
2299:3
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2289:2
2285:a
2279:1
2275:a
2271:.
2266:0
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2238:0
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2209:0
2203:x
2177:0
2173:a
2152:,
2149:}
2146:9
2143:,
2137:,
2134:2
2131:,
2128:1
2125:,
2122:0
2119:{
2108:3
2104:a
2100:,
2095:2
2091:a
2087:,
2082:1
2078:a
2052:3
2048:a
2042:2
2038:a
2032:1
2028:a
2024:.
2019:0
2015:a
2011:=
2008:x
1987:}
1979:i
1969:i
1965:a
1957:k
1952:0
1949:=
1946:i
1937:{
1931:k
1923:=
1920:x
1909:*
1892:k
1888:a
1872:)
1870:*
1868:(
1851:.
1848:x
1838:k
1828:k
1824:a
1818:+
1812:+
1805:2
1795:2
1791:a
1785:+
1775:1
1771:a
1765:+
1760:0
1756:a
1728:k
1724:a
1701:1
1695:k
1690:0
1687:=
1684:i
1680:)
1674:i
1670:a
1666:(
1644:0
1640:a
1636:=
1633:x
1610:x
1604:=
1599:0
1595:a
1574:x
1566:0
1562:a
1541:x
1515:0
1511:a
1490:0
1484:x
1464:x
1441:x
1417:x
1389:x
1353:n
1349:r
1326:n
1301:1
1298:+
1295:p
1289:x
1284:n
1273:p
1250:x
1245:n
1234:=
1231:p
1206:n
1198:p
1192:=
1187:n
1183:r
1157:.
1150:n
1142:1
1137:+
1132:n
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1121:x
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1109:r
1084:n
1080:a
1071:2
1067:a
1061:1
1057:a
1053:.
1048:0
1044:a
1040:=
1035:n
1031:r
1010:1
1004:n
984:0
978:x
953:9
937:i
933:a
922:r
904:,
901:)
898:1
895:,
892:0
889:[
866:,
859:i
849:i
845:a
832:1
829:=
826:i
818:=
810:2
806:a
800:1
796:a
770:i
766:a
754:0
751:a
746:r
726:i
716:i
712:b
706:k
701:0
698:=
695:i
672:9
668:9
664:0
646:0
640:k
620:0
612:k
608:b
585:.
578:i
568:i
564:a
551:1
548:=
545:i
537:+
532:i
522:i
518:b
512:k
507:0
504:=
501:i
493:=
490:r
469:0
453:i
449:a
422:i
418:a
391:k
371:0
363:k
359:b
328:,
323:2
319:a
315:,
310:1
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302:,
297:k
293:b
289:,
283:,
278:0
274:b
259:k
251:.
232:2
228:a
222:1
218:a
214:.
209:0
205:b
196:1
190:k
186:b
180:k
176:b
172:=
169:r
151:r
130:)
124:(
119:)
115:(
105:·
98:·
91:·
84:·
57:.
34:.
20:)
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