2804:
2472:
47:
2799:{\displaystyle {\begin{array}{l|rrrrrrr}{\text{Position}}&3&2&1&0&-1&-2&\cdots \\\hline {\text{Weight}}&b^{3}&b^{2}&b^{1}&b^{0}&b^{-1}&b^{-2}&\cdots \\{\text{Digit}}&a_{3}&a_{2}&a_{1}&a_{0}&c_{1}&c_{2}&\cdots \\\hline {\text{Decimal example weight}}&1000&100&10&1&0.1&0.01&\cdots \\{\text{Decimal example digit}}&4&3&2&7&0&0&\cdots \end{array}}}
808:
953:
3628:
3789:
1150:
1641:
The positional decimal system is presently universally used in human writing. The base 1000 is also used (albeit not universally), by grouping the digits and considering a sequence of three decimal digits as a single digit. This is the meaning of the common notation 1,000,234,567 used for very
1563:
More useful still are systems which employ special abbreviations for repetitions of symbols; for example, using the first nine letters of the alphabet for these abbreviations, with A standing for "one occurrence", B "two occurrences", and so on, one could then write C+ D/ for the number 304 (the
1572:
and other East Asian numerals based on
Chinese. The number system of the English language is of this type ("three hundred four"), as are those of other spoken languages, regardless of what written systems they have adopted. However, many languages use mixtures of bases, and other features, for
1141:
had not yet been widely accepted. Instead of a zero sometimes the digits were marked with dots to indicate their significance, or a space was used as a placeholder. The first widely acknowledged use of zero was in 876. The original numerals were very similar to the modern ones, even down to the
1488:
introduced the symbol for zero. The system slowly spread to other surrounding regions like Arabia due to their commercial and military activities with India. Middle-Eastern mathematicians extended the system to include negative powers of 10 (fractions), as recorded in a treatise by Syrian
3070:, one aspect of which is the representation of a sequence of non-negative integers of arbitrary size in the form of a sequence without delimiters, of "digits" from a collection of 36: a–z and 0–9, representing 0–25 and 26–35 respectively. There are also so-called threshold values (
2162:
1637:
Arithmetic is much easier in positional systems than in the earlier additive ones; furthermore, additive systems need a large number of different symbols for the different powers of 10; a positional system needs only ten different symbols (assuming that it uses base 10).
1699:
numerals (1, 10, 100, 1000, 10000 ...), respectively. The sign-value systems use only the geometric numerals and the positional systems use only the arithmetic numerals. A sign-value system does not need arithmetic numerals because they are made by repetition (except for the
1539:
The unary notation can be abbreviated by introducing different symbols for certain new values. Very commonly, these values are powers of 10; so for instance, if / stands for one, − for ten and + for 100, then the number 304 can be compactly represented as
1634:. Zero, which is not needed in the other systems, is of crucial importance here, in order to be able to "skip" a power. The Hindu–Arabic numeral system, which originated in India and is now used throughout the world, is a positional base 10 system.
1173:). They began to enter common use in the 15th century. By the end of the 20th century virtually all non-computerized calculations in the world were done with Arabic numerals, which have replaced native numeral systems in most cultures.
1688:). The command signals for different notes in the birdsong emanate from different points in the HVC. This coding works as space coding which is an efficient strategy for biological circuits due to its inherent simplicity and robustness.
1791:
natural numbers including zero are used. To generate the rest of the numerals, the position of the symbol in the figure is used. The symbol in the last position has its own value, and as it moves to the left its value is multiplied by
1958:
3653:
Fiete, I. R.; Seung, H. S. (2007). "Neural network models of birdsong production, learning, and coding". In Squire, L.; Albright, T.; Bloom, F.; Gage, F.; Spitzer, N. New
Encyclopedia of Neuroscience.
2477:
2464:
1934:
If a text (such as this one) discusses multiple bases, and if ambiguity exists, the base (itself represented in base 10) is added in subscript to the right of the number, like this: number
1626:
used by the system. In base 10, ten different digits 0, ..., 9 are used and the position of a digit is used to signify the power of ten that the digit is to be multiplied with, as in
3061:
3255:(i.e. 2), then the second-digit range is a–b (i.e. 0–1) with the second digit being most significant, while the range is c–9 (i.e. 2–35) in the presence of a third digit. Generally, for any
3364:
2254:
3170:
means that it is the most-significant digit, hence in the string this is the end of the number, and the next symbol (if present) is the least-significant digit of the next number.
1497:, who also wrote the earliest treatise on Arabic numerals. The Hindu–Arabic numeral system then spread to Europe due to merchants trading, and the digits used in Europe are called
2383:
3114:
1227:
mathematicians, are a decimal positional system able to represent not only zero but also negative numbers. Counting rods themselves predate the Hindu–Arabic numeral system. The
3249:
2328:
2978:
3304:
3181:(i.e. 0) marks the end of the number (it has just one digit), so in numbers of more than one digit, first-digit range is only b–9 (i.e. 1–35), therefore the weight
3168:
3141:
1189:(base 20), so it has twenty digits. The Mayas used a shell symbol to represent zero. Numerals were written vertically, with the ones place at the bottom. The
3699:(Alfred Louis Kroeber) (1876–1960), Handbook of the Indians of California, Bulletin 78 of the Bureau of American Ethnology of the Smithsonian Institution (1919)
1941:
By using a dot to divide the digits into two groups, one can also write fractions in the positional system. For example, the base 2 numeral 10.11 denotes
3445:
The flexibility in choosing threshold values allows optimization for number of digits depending on the frequency of occurrence of numbers of various sizes.
1666:
794:
1734:
between the set of all such digit-strings and the set of non-negative integers, avoiding the non-uniqueness caused by leading zeros. Bijective base-
2828:). An irrational number stays aperiodic (with an infinite number of non-repeating digits) in all integral bases. Thus, for example in base 2,
3472:
514:
3771:
3749:
3723:
3638:
68:
1684:
production. The nucleus in the brain of the songbirds that plays a part in both the learning and the production of bird song is the HVC (
1524:
represent one such system still in common use. The unary system is only useful for small numbers, although it plays an important role in
1441:
1434:
1427:
1420:
1413:
1406:
1399:
1392:
1385:
3477:
2157:{\displaystyle (a_{n}a_{n-1}\cdots a_{1}a_{0}.c_{1}c_{2}c_{3}\cdots )_{b}=\sum _{k=0}^{n}a_{k}b^{k}+\sum _{k=1}^{\infty }c_{k}b^{-k}.}
1378:
1337:
1330:
1323:
1316:
1309:
1302:
1295:
1288:
347:
1281:
1274:
3674:
90:
830:
The same sequence of symbols may represent different numbers in different numeral systems. For example, "11" represents the number
861:
The number the numeral represents is called its value. Not all number systems can represent the same set of numbers; for example,
1704:), and a positional system does not need geometric numerals because they are made by position. However, the spoken language uses
1469:
1205:
1137:. This system was established by the 7th century in India, but was not yet in its modern form because the use of the digit
1134:
138:
3685:
3819:
3563:
581:
3487:
787:
362:
3586:
2392:
1525:
707:
1661:) are commonly used. For very large integers, bases 2 or 2 (grouping binary digits by 32 or 64, the length of the
717:
3715:
2983:
2908:
534:
1691:
The numerals used when writing numbers with digits or symbols can be divided into two types that might be called the
61:
55:
3793:
1658:
1490:
594:
3309:
3741:
2194:
1130:
690:
459:
72:
3809:
3733:
780:
107:
3462:
1553:
1455:
1158:
770:
554:
151:
3550:
3502:
3467:
2337:
899:
454:
370:
3073:
3507:
3492:
1646:
1465:
820:
572:
35:
919:, but that name is ambiguous, as it could refer to different systems of numbers, such as the system of
3202:
2282:
3712:
Archaic
Bookkeeping: Early Writing and Techniques of Economic Administration in the Ancient Near East
3449:
2936:
2817:; this does not depend on the base. A number that terminates in one base may repeat in another (thus
2268:
1716:
1654:
1557:
1505:
1481:
851:
667:
528:
521:
402:
3522:
3512:
1758:
1692:
1606:
1549:
749:
614:
565:
377:
309:
164:
125:
3690:
3482:
3270:
2861:, above the common digits is a convention used to represent repeating rational expansions. Thus:
2257:
1696:
1598:
1529:
662:
415:
252:
247:
194:
1536:, expresses arbitrary-sized numbers by using unary to indicate the length of a binary numeral.
807:
3767:
3759:
3745:
3719:
3670:
3634:
3497:
1685:
1194:
744:
734:
722:
702:
657:
652:
588:
420:
392:
299:
232:
222:
209:
174:
169:
1185:
is unclear, but it is possible that it is older than the Hindu–Arabic system. The system was
883:
Give every number represented a unique representation (or at least a standard representation)
3814:
3567:
1569:
1533:
1228:
647:
541:
294:
282:
227:
217:
184:
159:
3146:
3119:
1208:
except for the symbols used to represent digits. The use of these digits is less common in
30:
This article is about expressing numbers with symbols. For different kinds of numbers, see
2814:
2176:
1645:
In computers, the main numeral systems are based on the positional system in base 2 (
1622:
1498:
943:
877:
824:
759:
729:
672:
642:
627:
387:
355:
327:
304:
287:
146:
3143:(in a given position in the number) that is lower than its corresponding threshold value
739:
3667:
The
Universal History of Numbers : From Prehistory to the Invention of the Computer
3611:
3263: + 1)-th digit is the weight of the previous one times (36 − threshold of the
2810:
1938:. Unless specified by context, numbers without subscript are considered to be decimal.
1772:
1701:
1677:
1509:
924:
903:
862:
754:
697:
677:
632:
505:
237:
204:
189:
3452:, where the zeros correspond to separators of numbers with digits which are non-zero.
952:
3803:
3527:
2894:
1743:
1610:, also known as place-value notation. The positional systems are classified by their
1216:
1201:
1182:
928:
915:
843:
560:
449:
382:
322:
257:
199:
179:
31:
17:
3696:
3680:
2886:
1673:
1662:
1650:
1494:
712:
637:
2334:≥ 0. For example, to describe the weight 1000 then four digits are needed because
2279:. In the positional system, the number of digits required to describe it is only
2926:
2264:
1521:
1485:
1168:
920:
855:
682:
547:
499:
489:
1440:
1433:
1426:
1419:
1412:
1405:
1398:
1391:
1384:
2930:
1377:
1336:
1329:
1322:
1315:
1308:
1301:
1294:
1287:
1280:
891:
484:
242:
2184:
1731:
1473:
1273:
1186:
1162:
494:
1653:, 0 and 1. Positional systems obtained by grouping binary digits by three (
3788:
2466:(in positions 1, 10, 100,... only for simplicity in the decimal example).
3067:
2920:
2848:
1681:
1209:
907:
3369:
So we have the following sequence of the numbers with at most 3 digits:
3517:
2893:
numerals whose expansion to the left never stops; these are called the
1800:
1461:
887:
873:
835:
479:
464:
1516:
is chosen, for example, then the number seven would be represented by
1730:), and zero being represented by an empty string. This establishes a
1597:). In English, one could say "four score less one", as in the famous
1212:
than it once was, but they are still used alongside Arabic numerals.
469:
838:
numeral system (today, the most common system globally), the number
3251:. Suppose the threshold values for the second and third digits are
1512:
is represented by a corresponding number of symbols. If the symbol
1924:
in descending order. The digits are natural numbers between 0 and
1776:
1616:
1477:
1224:
1220:
1190:
1148:
1143:
806:
474:
436:
397:
1601:
representing "87 years ago" as "four score and seven years ago".
934:, etc. Such systems are, however, not the topic of this article.
1138:
1834:
is the base, one writes a number in the numeral system of base
1149:
2907:
in which digits may be positive or negative; this is called a
947:
40:
3448:
The case with all threshold values equal to 1 corresponds to
1493:
in 952–953, and the decimal point notation was introduced by
3545:
3543:
3430:
each represent 35; yet the representation is unique because
3116:) which are fixed for every position in the number. A digit
3173:
For example, if the threshold value for the first digit is
2830:
846:
numeral system (used in modern computers), and the number
3710:
Hans J. Nissen; Peter
Damerow; Robert K. Englund (1993).
3706:, Fitzroy Dearborn Publishers, London and Chicago, 1997.
964:
819:
is a writing system for expressing numbers; that is, a
1564:
number of these abbreviations is sometimes called the
1468:
are credited with developing the integer version, the
3764:
Africa counts: number and pattern in
African cultures
3312:
3273:
3205:
3149:
3122:
3076:
2986:
2939:
2840:
can be written as the aperiodic 11.001001000011111...
2475:
2395:
2340:
2285:
2197:
1961:
1787:
basic symbols (or digits) corresponding to the first
3610:
David Eugene Smith; Louis
Charles Karpinski (1911).
1711:In some areas of computer science, a modified base
3358:
3298:
3267:-th digit). So the weight of the second symbol is
3243:
3162:
3135:
3108:
3055:
2972:
2903:It is also possible to define a variation of base
2809:A number has a terminating or repeating expansion
2798:
2459:{\displaystyle \log _{b}k+1=\log _{b}\log _{b}w+1}
2458:
2377:
2322:
2248:
2156:
1568:of the system). This system is used when writing
1197:, so their system could not represent fractions.
3693:. pp. 194–213, "Positional Number Systems".
3056:{\displaystyle a_{0}+a_{1}b_{1}+a_{2}b_{1}b_{2}}
1695:numerals (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) and the
1161:were accepted in European mathematical circles (
3422:-based numeral system, there are numbers like 9
2869: or 321.3217878787878... = 321.321
1676:system is employed. Unary numerals used in the
823:for representing numbers of a given set, using
1749:. Bijective base 1 is the same as unary.
1460:The most commonly used system of numerals is
1177:Other historical numeral systems using digits
788:
8:
3630:Design of an Efficient Multiplier using DBNS
3359:{\displaystyle 35(36-t_{1})=35\cdot 34=1190}
2256:. The highest used position is close to the
1484:in the 5th century and a century later
910:of digits, beginning with a non-zero digit.
872:Represent a useful set of numbers (e.g. all
811:Numbers written in different numeral systems
2249:{\displaystyle k=\log _{b}w=\log _{b}b^{k}}
1166:
2179:of the corresponding digits. The position
1548:without any need for zero. This is called
983:
795:
781:
131:
102:
3329:
3311:
3284:
3272:
3235:
3210:
3204:
3154:
3148:
3127:
3121:
3094:
3081:
3075:
3047:
3037:
3027:
3014:
3004:
2991:
2985:
2964:
2954:
2944:
2938:
2752:
2708:
2693:
2681:
2669:
2657:
2645:
2633:
2622:
2604:
2589:
2577:
2565:
2553:
2541:
2530:
2480:
2476:
2474:
2438:
2425:
2400:
2394:
2345:
2339:
2302:
2284:
2240:
2227:
2208:
2196:
2142:
2132:
2122:
2111:
2098:
2088:
2078:
2067:
2054:
2041:
2031:
2021:
2008:
1998:
1979:
1969:
1960:
1803:system (base 10), the numeral 4327 means
827:or other symbols in a consistent manner.
91:Learn how and when to remove this message
34:. For expressing numbers with words, see
3553:. January 2001. Retrieved on 2007-02-20.
3306:. And the weight of the third symbol is
3188:is 35 instead of 36. More generally, if
1742:-adic notation, not to be confused with
1620:, which is the number of symbols called
1233:
54:This article includes a list of general
3539:
3473:Non-standard positional numeral systems
3442:would terminate each of these numbers.
1501:, as they learned them from the Arabs.
1215:The rod numerals, the written forms of
114:
3704:Encyclopedia of Indo-European Culture
3549:O'Connor, J. J. and Robertson, E. F.
1153:The digits of the Maya numeral system
913:Numeral systems are sometimes called
7:
2915:Generalized variable-length integers
2378:{\displaystyle \log _{10}1000+1=3+1}
906:a unique representation as a finite
3199:-th digit, it is easy to show that
3109:{\displaystyle t_{0},t_{1},\ldots }
2385:. The number of digits required to
1708:arithmetic and geometric numerals.
1672:In certain biological systems, the
1504:The simplest numeral system is the
3587:"How Arabic Numbers Were Invented"
3478:History of ancient numeral systems
2123:
1888:and writing the enumerated digits
1715:positional system is used, called
865:cannot represent the number zero.
60:it lacks sufficient corresponding
25:
1560:was a modification of this idea.
3787:
3244:{\displaystyle b_{n+1}=36-t_{n}}
2323:{\displaystyle k+1=\log _{b}w+1}
1948:In general, numbers in the base
1665:) are used, as, for example, in
1439:
1432:
1425:
1418:
1411:
1404:
1397:
1390:
1383:
1376:
1335:
1328:
1321:
1314:
1307:
1300:
1293:
1286:
1279:
1272:
1193:had no equivalent of the modern
951:
868:Ideally, a numeral system will:
45:
3686:The Art of Computer Programming
2973:{\displaystyle a_{0}a_{1}a_{2}}
27:Notation for expressing numbers
3335:
3316:
2865:14/11 = 1.272727272727... = 1.
2051:
1962:
1719:, with digits 1, 2, ...,
1231:are variants of rod numerals.
944:Numerical digit § History
1:
3488:List of numeral system topics
1838:by expressing it in the form
3438:are not allowed – the first
2187:of the corresponding weight
1943:1×2 + 0×2 + 1×2 + 1×2 = 2.75
1775:greater than 1 known as the
1753:Positional systems in detail
1583:pedwar ar bymtheg a thrigain
1532:, which is commonly used in
1526:theoretical computer science
3716:University of Chicago Press
3299:{\displaystyle 36-t_{0}=35}
2909:signed-digit representation
1470:Hindu–Arabic numeral system
1206:Hindu–Arabic numeral system
1135:Hindu–Arabic numeral system
3836:
3702:J.P. Mallory; D.Q. Adams,
2918:
1756:
1738:numeration is also called
1659:hexadecimal numeral system
1556:was of this type, and the
1453:
1146:used to represent digits.
941:
515:Non-standard radices/bases
29:
3742:University of Texas Press
3734:Schmandt-Besserat, Denise
3613:The Hindu–Arabic numerals
3195:is the threshold for the
1573:instance 79 in French is
1131:positional numeral system
894:structure of the numbers.
3766:. Chicago Review Press.
2925:More general is using a
1952:system are of the form:
1628:304 = 3×100 + 0×10 + 4×1
1589:) or (somewhat archaic)
1235:Rod numerals (vertical)
1133:is considered to be the
3468:Computer number formats
3463:List of numeral systems
2929:notation (here written
1587:4 + (5 + 10) + (3 × 20)
1554:Egyptian numeral system
1491:Abu'l-Hasan al-Uqlidisi
1456:List of numeral systems
1159:Western Arabic numerals
1129:The first true written
898:For example, the usual
771:List of numeral systems
75:more precise citations.
3738:How Writing Came About
3503:Residue numeral system
3360:
3300:
3245:
3164:
3137:
3110:
3057:
2974:
2889:, one can define base-
2800:
2710:Decimal example weight
2460:
2379:
2324:
2250:
2158:
2127:
2083:
1544:and the number 123 as
1167:
1154:
900:decimal representation
812:
3820:Mathematical notation
3508:Long and short scales
3361:
3301:
3259:, the weight of the (
3246:
3165:
3163:{\displaystyle t_{i}}
3138:
3136:{\displaystyle a_{i}}
3111:
3058:
2975:
2801:
2754:Decimal example digit
2461:
2387:describe the position
2380:
2325:
2273:describing the weight
2251:
2159:
2107:
2063:
1767:numeral system (with
1763:In a positional base
1647:binary numeral system
1591:pedwar ugain namyn un
1466:Indian mathematicians
1181:The exact age of the
1157:By the 13th century,
1152:
821:mathematical notation
810:
139:Hindu–Arabic numerals
36:Numeral (linguistics)
18:Number representation
3796:at Wikimedia Commons
3689:. Volume 2, 3rd Ed.
3591:www.theclassroom.com
3450:bijective numeration
3310:
3271:
3203:
3147:
3120:
3074:
2984:
2937:
2823:= 0.0100110011001...
2473:
2393:
2338:
2283:
2269:unary numeral system
2195:
1959:
1799:For example, in the
1717:bijective numeration
1655:octal numeral system
1558:Roman numeral system
1506:unary numeral system
1482:place-value notation
1450:Main numeral systems
1204:is identical to the
902:gives every nonzero
852:unary numeral system
668:Prehistoric counting
444:Common radices/bases
126:Place-value notation
3616:. Ginn and Company.
3523:Numerical cognition
3513:Scientific notation
1759:Positional notation
1550:sign-value notation
1236:
1202:Thai numeral system
615:Sign-value notation
3760:Zaslavsky, Claudia
3627:Chowdhury, Arnab.
3483:History of numbers
3356:
3296:
3241:
3160:
3133:
3106:
3053:
2970:
2796:
2794:
2456:
2375:
2320:
2258:order of magnitude
2246:
2154:
1632:3×10 + 0×10 + 4×10
1630:or more precisely
1604:More elegant is a
1599:Gettysburg Address
1581:) and in Welsh is
1530:Elias gamma coding
1234:
1155:
963:. You can help by
836:decimal or base-10
813:
271:East Asian systems
3792:Media related to
3773:978-1-55652-350-2
3751:978-0-292-77704-0
3725:978-0-226-58659-5
3640:978-93-83006-18-2
3633:. GIAP Journals.
3585:Bradley, Jeremy.
3566:(February 2007).
3498:Repeating decimal
3418:Unlike a regular
2755:
2711:
2625:
2533:
2483:
1686:high vocal center
1607:positional system
1575:soixante dix-neuf
1508:, in which every
1447:
1446:
1195:decimal separator
1165:used them in his
1126:
1125:
981:
980:
805:
804:
604:
603:
101:
100:
93:
16:(Redirected from
3827:
3791:
3777:
3755:
3729:
3654:
3651:
3645:
3644:
3624:
3618:
3617:
3607:
3601:
3600:
3598:
3597:
3582:
3576:
3575:
3568:"All for Nought"
3560:
3554:
3547:
3365:
3363:
3362:
3357:
3334:
3333:
3305:
3303:
3302:
3297:
3289:
3288:
3250:
3248:
3247:
3242:
3240:
3239:
3221:
3220:
3169:
3167:
3166:
3161:
3159:
3158:
3142:
3140:
3139:
3134:
3132:
3131:
3115:
3113:
3112:
3107:
3099:
3098:
3086:
3085:
3066:This is used in
3062:
3060:
3059:
3054:
3052:
3051:
3042:
3041:
3032:
3031:
3019:
3018:
3009:
3008:
2996:
2995:
2979:
2977:
2976:
2971:
2969:
2968:
2959:
2958:
2949:
2948:
2872:
2868:
2856:
2839:
2833:
2827:
2805:
2803:
2802:
2797:
2795:
2756:
2753:
2712:
2709:
2698:
2697:
2686:
2685:
2674:
2673:
2662:
2661:
2650:
2649:
2638:
2637:
2626:
2623:
2612:
2611:
2597:
2596:
2582:
2581:
2570:
2569:
2558:
2557:
2546:
2545:
2534:
2531:
2484:
2481:
2465:
2463:
2462:
2457:
2443:
2442:
2430:
2429:
2405:
2404:
2384:
2382:
2381:
2376:
2350:
2349:
2329:
2327:
2326:
2321:
2307:
2306:
2275:would have been
2267:required in the
2255:
2253:
2252:
2247:
2245:
2244:
2232:
2231:
2213:
2212:
2163:
2161:
2160:
2155:
2150:
2149:
2137:
2136:
2126:
2121:
2103:
2102:
2093:
2092:
2082:
2077:
2059:
2058:
2046:
2045:
2036:
2035:
2026:
2025:
2013:
2012:
2003:
2002:
1990:
1989:
1974:
1973:
1944:
1930:
1923:
1887:
1826:
1822:
1783:of the system),
1729:
1680:responsible for
1633:
1629:
1596:
1588:
1580:
1570:Chinese numerals
1547:
1543:
1534:data compression
1519:
1515:
1443:
1436:
1429:
1422:
1415:
1408:
1401:
1394:
1387:
1380:
1339:
1332:
1325:
1318:
1311:
1304:
1297:
1290:
1283:
1276:
1237:
1172:
984:
976:
973:
955:
948:
927:, the system of
923:, the system of
878:rational numbers
844:binary or base-2
797:
790:
783:
586:
570:
552:
542:balanced ternary
539:
526:
132:
103:
96:
89:
85:
82:
76:
71:this article by
62:inline citations
49:
48:
41:
21:
3835:
3834:
3830:
3829:
3828:
3826:
3825:
3824:
3810:Numeral systems
3800:
3799:
3794:Numeral systems
3784:
3774:
3758:
3752:
3732:
3726:
3709:
3669:, Wiley, 1999.
3665:Georges Ifrah.
3662:
3657:
3652:
3648:
3641:
3626:
3625:
3621:
3609:
3608:
3604:
3595:
3593:
3584:
3583:
3579:
3562:
3561:
3557:
3551:Arabic Numerals
3548:
3541:
3537:
3532:
3458:
3411:(1261), ..., 99
3325:
3308:
3307:
3280:
3269:
3268:
3231:
3206:
3201:
3200:
3193:
3187:
3150:
3145:
3144:
3123:
3118:
3117:
3090:
3077:
3072:
3071:
3043:
3033:
3023:
3010:
3000:
2987:
2982:
2981:
2960:
2950:
2940:
2935:
2934:
2923:
2917:
2870:
2866:
2852:
2843:
2838:
2831:
2829:
2826:
2822:
2818:
2793:
2792:
2787:
2782:
2777:
2772:
2767:
2762:
2757:
2749:
2748:
2743:
2738:
2733:
2728:
2723:
2718:
2713:
2705:
2704:
2699:
2689:
2687:
2677:
2675:
2665:
2663:
2653:
2651:
2641:
2639:
2629:
2627:
2619:
2618:
2613:
2600:
2598:
2585:
2583:
2573:
2571:
2561:
2559:
2549:
2547:
2537:
2535:
2527:
2526:
2521:
2513:
2505:
2500:
2495:
2490:
2485:
2471:
2470:
2434:
2421:
2396:
2391:
2390:
2341:
2336:
2335:
2298:
2281:
2280:
2260:of the number.
2236:
2223:
2204:
2193:
2192:
2138:
2128:
2094:
2084:
2050:
2037:
2027:
2017:
2004:
1994:
1975:
1965:
1957:
1956:
1942:
1937:
1925:
1922:
1915:
1906:
1897:
1889:
1883:
1873:
1860:
1847:
1839:
1830:In general, if
1824:
1804:
1761:
1755:
1724:
1678:neural circuits
1642:large numbers.
1631:
1627:
1594:
1586:
1578:
1545:
1541:
1517:
1513:
1499:Arabic numerals
1458:
1452:
1229:Suzhou numerals
1179:
1127:
1022:Eastern Arabic
987:Western Arabic
977:
971:
968:
961:needs expansion
946:
940:
925:complex numbers
801:
765:
764:
687:
673:Proto-cuneiform
618:
617:
606:
605:
600:
599:
584:
568:
550:
537:
524:
511:
440:
439:
427:
426:
407:
367:
352:
343:
342:
333:
332:
314:
273:
272:
263:
262:
214:
156:
142:
141:
129:
128:
116:Numeral systems
97:
86:
80:
77:
67:Please help to
66:
50:
46:
39:
28:
23:
22:
15:
12:
11:
5:
3833:
3831:
3823:
3822:
3817:
3812:
3802:
3801:
3798:
3797:
3783:
3782:External links
3780:
3779:
3778:
3772:
3756:
3750:
3730:
3724:
3707:
3700:
3694:
3691:Addison–Wesley
3678:
3661:
3658:
3656:
3655:
3646:
3639:
3619:
3602:
3577:
3572:Feature Column
3564:Bill Casselman
3555:
3538:
3536:
3533:
3531:
3530:
3525:
3520:
3515:
3510:
3505:
3500:
3495:
3490:
3485:
3480:
3475:
3470:
3465:
3459:
3457:
3454:
3355:
3352:
3349:
3346:
3343:
3340:
3337:
3332:
3328:
3324:
3321:
3318:
3315:
3295:
3292:
3287:
3283:
3279:
3276:
3238:
3234:
3230:
3227:
3224:
3219:
3216:
3213:
3209:
3191:
3185:
3177:(i.e. 1) then
3157:
3153:
3130:
3126:
3105:
3102:
3097:
3093:
3089:
3084:
3080:
3050:
3046:
3040:
3036:
3030:
3026:
3022:
3017:
3013:
3007:
3003:
2999:
2994:
2990:
2967:
2963:
2957:
2953:
2947:
2943:
2919:Main article:
2916:
2913:
2875:
2874:
2841:
2836:
2835:= 3.1415926...
2824:
2820:
2811:if and only if
2807:
2806:
2791:
2788:
2786:
2783:
2781:
2778:
2776:
2773:
2771:
2768:
2766:
2763:
2761:
2758:
2751:
2750:
2747:
2744:
2742:
2739:
2737:
2734:
2732:
2729:
2727:
2724:
2722:
2719:
2717:
2714:
2707:
2706:
2703:
2700:
2696:
2692:
2688:
2684:
2680:
2676:
2672:
2668:
2664:
2660:
2656:
2652:
2648:
2644:
2640:
2636:
2632:
2628:
2621:
2620:
2617:
2614:
2610:
2607:
2603:
2599:
2595:
2592:
2588:
2584:
2580:
2576:
2572:
2568:
2564:
2560:
2556:
2552:
2548:
2544:
2540:
2536:
2529:
2528:
2525:
2522:
2520:
2517:
2514:
2512:
2509:
2506:
2504:
2501:
2499:
2496:
2494:
2491:
2489:
2486:
2479:
2478:
2455:
2452:
2449:
2446:
2441:
2437:
2433:
2428:
2424:
2420:
2417:
2414:
2411:
2408:
2403:
2399:
2374:
2371:
2368:
2365:
2362:
2359:
2356:
2353:
2348:
2344:
2319:
2316:
2313:
2310:
2305:
2301:
2297:
2294:
2291:
2288:
2263:The number of
2243:
2239:
2235:
2230:
2226:
2222:
2219:
2216:
2211:
2207:
2203:
2200:
2165:
2164:
2153:
2148:
2145:
2141:
2135:
2131:
2125:
2120:
2117:
2114:
2110:
2106:
2101:
2097:
2091:
2087:
2081:
2076:
2073:
2070:
2066:
2062:
2057:
2053:
2049:
2044:
2040:
2034:
2030:
2024:
2020:
2016:
2011:
2007:
2001:
1997:
1993:
1988:
1985:
1982:
1978:
1972:
1968:
1964:
1935:
1920:
1910:
1901:
1893:
1881:
1868:
1855:
1843:
1823:, noting that
1773:natural number
1754:
1751:
1552:. The ancient
1510:natural number
1489:mathematician
1480:developed the
1454:Main article:
1451:
1448:
1445:
1444:
1437:
1430:
1423:
1416:
1409:
1402:
1395:
1388:
1381:
1373:
1372:
1369:
1366:
1363:
1360:
1357:
1354:
1351:
1348:
1345:
1341:
1340:
1333:
1326:
1319:
1312:
1305:
1298:
1291:
1284:
1277:
1269:
1268:
1265:
1262:
1259:
1256:
1253:
1250:
1247:
1244:
1241:
1178:
1175:
1124:
1123:
1120:
1117:
1114:
1111:
1108:
1105:
1102:
1099:
1096:
1093:
1089:
1088:
1085:
1082:
1079:
1076:
1073:
1070:
1067:
1064:
1061:
1058:
1054:
1053:
1050:
1047:
1044:
1041:
1038:
1035:
1032:
1029:
1026:
1023:
1019:
1018:
1015:
1012:
1009:
1006:
1003:
1000:
997:
994:
991:
988:
982:
979:
978:
958:
956:
939:
936:
916:number systems
904:natural number
896:
895:
884:
881:
863:Roman numerals
817:numeral system
803:
802:
800:
799:
792:
785:
777:
774:
773:
767:
766:
763:
762:
757:
752:
747:
742:
737:
732:
727:
726:
725:
720:
715:
705:
700:
694:
693:
686:
685:
680:
675:
670:
665:
660:
655:
650:
645:
640:
635:
630:
624:
623:
622:Non-alphabetic
619:
613:
612:
611:
608:
607:
602:
601:
598:
597:
592:
579:
563:
558:
545:
532:
518:
517:
510:
509:
502:
497:
492:
487:
482:
477:
472:
467:
462:
457:
452:
446:
445:
441:
434:
433:
432:
429:
428:
425:
424:
418:
412:
411:
406:
405:
400:
395:
390:
385:
380:
374:
373:
371:Post-classical
366:
365:
359:
358:
351:
350:
344:
340:
339:
338:
335:
334:
331:
330:
325:
319:
318:
313:
312:
307:
302:
297:
292:
291:
290:
279:
278:
274:
270:
269:
268:
265:
264:
261:
260:
255:
250:
245:
240:
235:
230:
225:
220:
213:
212:
207:
202:
197:
192:
187:
182:
177:
172:
167:
162:
155:
154:
152:Eastern Arabic
149:
147:Western Arabic
143:
137:
136:
135:
130:
124:
123:
122:
119:
118:
112:
111:
99:
98:
53:
51:
44:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3832:
3821:
3818:
3816:
3813:
3811:
3808:
3807:
3805:
3795:
3790:
3786:
3785:
3781:
3775:
3769:
3765:
3761:
3757:
3753:
3747:
3743:
3739:
3735:
3731:
3727:
3721:
3717:
3713:
3708:
3705:
3701:
3698:
3695:
3692:
3688:
3687:
3682:
3679:
3676:
3675:0-471-37568-3
3672:
3668:
3664:
3663:
3659:
3650:
3647:
3642:
3636:
3632:
3631:
3623:
3620:
3615:
3614:
3606:
3603:
3592:
3588:
3581:
3578:
3573:
3569:
3565:
3559:
3556:
3552:
3546:
3544:
3540:
3534:
3529:
3528:Number system
3526:
3524:
3521:
3519:
3516:
3514:
3511:
3509:
3506:
3504:
3501:
3499:
3496:
3494:
3491:
3489:
3486:
3484:
3481:
3479:
3476:
3474:
3471:
3469:
3466:
3464:
3461:
3460:
3455:
3453:
3451:
3446:
3443:
3441:
3437:
3433:
3429:
3425:
3421:
3416:
3414:
3410:
3406:
3403:(71), ..., 99
3402:
3398:
3394:
3390:
3386:
3382:
3378:
3374:
3370:
3367:
3353:
3350:
3347:
3344:
3341:
3338:
3330:
3326:
3322:
3319:
3313:
3293:
3290:
3285:
3281:
3277:
3274:
3266:
3262:
3258:
3254:
3236:
3232:
3228:
3225:
3222:
3217:
3214:
3211:
3207:
3198:
3194:
3184:
3180:
3176:
3171:
3155:
3151:
3128:
3124:
3103:
3100:
3095:
3091:
3087:
3082:
3078:
3069:
3064:
3048:
3044:
3038:
3034:
3028:
3024:
3020:
3015:
3011:
3005:
3001:
2997:
2992:
2988:
2965:
2961:
2955:
2951:
2945:
2941:
2932:
2931:little-endian
2928:
2922:
2914:
2912:
2910:
2906:
2901:
2899:
2898:-adic numbers
2897:
2892:
2888:
2884:
2880:
2864:
2863:
2862:
2860:
2855:
2850:
2845:
2834:
2816:
2812:
2789:
2784:
2779:
2774:
2769:
2764:
2759:
2745:
2740:
2735:
2730:
2725:
2720:
2715:
2701:
2694:
2690:
2682:
2678:
2670:
2666:
2658:
2654:
2646:
2642:
2634:
2630:
2615:
2608:
2605:
2601:
2593:
2590:
2586:
2578:
2574:
2566:
2562:
2554:
2550:
2542:
2538:
2523:
2518:
2515:
2510:
2507:
2502:
2497:
2492:
2487:
2469:
2468:
2467:
2453:
2450:
2447:
2444:
2439:
2435:
2431:
2426:
2422:
2418:
2415:
2412:
2409:
2406:
2401:
2397:
2388:
2372:
2369:
2366:
2363:
2360:
2357:
2354:
2351:
2346:
2342:
2333:
2317:
2314:
2311:
2308:
2303:
2299:
2295:
2292:
2289:
2286:
2278:
2274:
2270:
2266:
2261:
2259:
2241:
2237:
2233:
2228:
2224:
2220:
2217:
2214:
2209:
2205:
2201:
2198:
2190:
2186:
2182:
2178:
2174:
2170:
2151:
2146:
2143:
2139:
2133:
2129:
2118:
2115:
2112:
2108:
2104:
2099:
2095:
2089:
2085:
2079:
2074:
2071:
2068:
2064:
2060:
2055:
2047:
2042:
2038:
2032:
2028:
2022:
2018:
2014:
2009:
2005:
1999:
1995:
1991:
1986:
1983:
1980:
1976:
1970:
1966:
1955:
1954:
1953:
1951:
1946:
1939:
1932:
1931:, inclusive.
1928:
1919:
1913:
1909:
1904:
1900:
1896:
1892:
1886:
1880:
1876:
1871:
1867:
1863:
1858:
1854:
1850:
1846:
1842:
1837:
1833:
1828:
1820:
1816:
1812:
1808:
1802:
1797:
1795:
1790:
1786:
1782:
1778:
1774:
1770:
1766:
1760:
1752:
1750:
1748:
1747:-adic numbers
1746:
1741:
1737:
1733:
1727:
1722:
1718:
1714:
1709:
1707:
1703:
1698:
1694:
1689:
1687:
1683:
1679:
1675:
1670:
1668:
1664:
1660:
1656:
1652:
1651:binary digits
1648:
1643:
1639:
1635:
1625:
1624:
1619:
1618:
1613:
1609:
1608:
1602:
1600:
1592:
1584:
1576:
1571:
1567:
1561:
1559:
1555:
1551:
1537:
1535:
1531:
1527:
1523:
1511:
1507:
1502:
1500:
1496:
1492:
1487:
1483:
1479:
1475:
1471:
1467:
1463:
1457:
1449:
1442:
1438:
1435:
1431:
1428:
1424:
1421:
1417:
1414:
1410:
1407:
1403:
1400:
1396:
1393:
1389:
1386:
1382:
1379:
1375:
1374:
1370:
1367:
1364:
1361:
1358:
1355:
1352:
1349:
1346:
1343:
1342:
1338:
1334:
1331:
1327:
1324:
1320:
1317:
1313:
1310:
1306:
1303:
1299:
1296:
1292:
1289:
1285:
1282:
1278:
1275:
1271:
1270:
1266:
1263:
1260:
1257:
1254:
1251:
1248:
1245:
1242:
1239:
1238:
1232:
1230:
1226:
1222:
1219:once used by
1218:
1217:counting rods
1213:
1211:
1207:
1203:
1198:
1196:
1192:
1188:
1184:
1183:Maya numerals
1176:
1174:
1171:
1170:
1164:
1160:
1151:
1147:
1145:
1140:
1136:
1132:
1121:
1118:
1115:
1112:
1109:
1106:
1103:
1100:
1097:
1094:
1091:
1090:
1086:
1083:
1080:
1077:
1074:
1071:
1068:
1065:
1062:
1059:
1056:
1055:
1051:
1048:
1045:
1042:
1039:
1036:
1033:
1030:
1027:
1024:
1021:
1020:
1016:
1013:
1010:
1007:
1004:
1001:
998:
995:
992:
989:
986:
985:
975:
966:
962:
959:This section
957:
954:
950:
949:
945:
937:
935:
933:
932:-adic numbers
931:
926:
922:
918:
917:
911:
909:
905:
901:
893:
889:
885:
882:
879:
875:
871:
870:
869:
866:
864:
859:
857:
853:
849:
845:
841:
837:
833:
828:
826:
822:
818:
809:
798:
793:
791:
786:
784:
779:
778:
776:
775:
772:
769:
768:
761:
758:
756:
753:
751:
748:
746:
743:
741:
738:
736:
733:
731:
728:
724:
721:
719:
716:
714:
711:
710:
709:
708:Alphasyllabic
706:
704:
701:
699:
696:
695:
692:
689:
688:
684:
681:
679:
676:
674:
671:
669:
666:
664:
661:
659:
656:
654:
651:
649:
646:
644:
641:
639:
636:
634:
631:
629:
626:
625:
621:
620:
616:
610:
609:
596:
593:
590:
583:
580:
577:
576:
567:
564:
562:
559:
556:
549:
546:
543:
536:
533:
530:
523:
520:
519:
516:
513:
512:
507:
503:
501:
498:
496:
493:
491:
488:
486:
483:
481:
478:
476:
473:
471:
468:
466:
463:
461:
458:
456:
453:
451:
448:
447:
443:
442:
438:
431:
430:
422:
419:
417:
414:
413:
409:
408:
404:
401:
399:
396:
394:
391:
389:
386:
384:
381:
379:
376:
375:
372:
369:
368:
364:
361:
360:
357:
354:
353:
349:
346:
345:
341:Other systems
337:
336:
329:
326:
324:
323:Counting rods
321:
320:
316:
315:
311:
308:
306:
303:
301:
298:
296:
293:
289:
286:
285:
284:
281:
280:
276:
275:
267:
266:
259:
256:
254:
251:
249:
246:
244:
241:
239:
236:
234:
231:
229:
226:
224:
221:
219:
216:
215:
211:
208:
206:
203:
201:
198:
196:
193:
191:
188:
186:
183:
181:
178:
176:
173:
171:
168:
166:
163:
161:
158:
157:
153:
150:
148:
145:
144:
140:
134:
133:
127:
121:
120:
117:
113:
109:
105:
104:
95:
92:
84:
74:
70:
64:
63:
57:
52:
43:
42:
37:
33:
32:Number system
19:
3763:
3737:
3711:
3703:
3697:A.L. Kroeber
3684:
3666:
3649:
3629:
3622:
3612:
3605:
3594:. Retrieved
3590:
3580:
3571:
3558:
3493:Number names
3447:
3444:
3439:
3435:
3431:
3427:
3426:where 9 and
3423:
3419:
3417:
3412:
3408:
3404:
3400:
3396:
3395:(37), ..., 9
3392:
3388:
3384:
3380:
3376:
3372:
3371:
3368:
3264:
3260:
3256:
3252:
3196:
3189:
3182:
3178:
3174:
3172:
3065:
2924:
2904:
2902:
2895:
2890:
2887:prime number
2882:
2878:
2876:
2858:
2853:
2846:
2808:
2386:
2331:
2276:
2272:
2262:
2188:
2180:
2172:
2168:
2167:The numbers
2166:
1949:
1947:
1940:
1933:
1926:
1917:
1911:
1907:
1902:
1898:
1894:
1890:
1884:
1878:
1874:
1869:
1865:
1861:
1856:
1852:
1848:
1844:
1840:
1835:
1831:
1829:
1818:
1814:
1810:
1806:
1798:
1793:
1788:
1784:
1780:
1768:
1764:
1762:
1744:
1739:
1735:
1725:
1720:
1712:
1710:
1705:
1702:Ionic system
1690:
1674:unary coding
1671:
1663:machine word
1649:), with two
1644:
1640:
1636:
1621:
1615:
1611:
1605:
1603:
1590:
1582:
1574:
1565:
1562:
1538:
1503:
1495:Sind ibn Ali
1459:
1214:
1199:
1180:
1156:
1128:
969:
965:adding to it
960:
929:
921:real numbers
914:
912:
897:
886:Reflect the
867:
860:
847:
839:
831:
829:
816:
814:
574:
535:Signed-digit
410:Contemporary
277:Contemporary
115:
87:
81:January 2011
78:
59:
3383:(2), ..., 9
2927:mixed radix
2857:, or dots,
2265:tally marks
1657:) or four (
1579:60 + 10 + 9
1522:Tally marks
1486:Brahmagupta
1169:Liber Abaci
1092:Devanagari
713:Akṣarapallī
683:Tally marks
582:Non-integer
73:introducing
3804:Categories
3596:2020-07-22
3535:References
2849:overscores
2191:, that is
1757:See also:
1693:arithmetic
1595:4 × 20 − 1
1478:Kusumapura
942:See also:
892:arithmetic
750:Glagolitic
723:Kaṭapayādi
691:Alphabetic
595:Asymmetric
437:radix/base
378:Cistercian
363:Babylonian
310:Vietnamese
165:Devanagari
56:references
3345:⋅
3323:−
3278:−
3229:−
3104:…
2790:⋯
2746:⋯
2702:⋯
2616:⋯
2606:−
2591:−
2524:⋯
2516:−
2508:−
2445:
2432:
2407:
2352:
2309:
2234:
2215:
2185:logarithm
2144:−
2124:∞
2109:∑
2065:∑
2048:⋯
1992:⋯
1984:−
1732:bijection
1697:geometric
1546:+ − − ///
1474:Aryabhata
1187:vigesimal
1163:Fibonacci
972:July 2024
888:algebraic
858:scores).
854:(used in
718:Āryabhaṭa
663:Kharosthi
555:factorial
522:Bijective
423:(Iñupiaq)
253:Sundanese
248:Mongolian
195:Malayalam
3762:(1999).
3736:(1996).
3681:D. Knuth
3456:See also
3415:(2450).
3407:(1260),
3068:Punycode
2921:Punycode
2847:Putting
2815:rational
2482:Position
2175:are the
1877:+ ... +
1817:×10) + (
1813:×10) + (
1809:×10) + (
1682:birdsong
1542:+++ ////
1225:Japanese
1210:Thailand
1057:Persian
908:sequence
874:integers
856:tallying
745:Georgian
735:Cyrillic
703:Armenian
658:Etruscan
653:Egyptian
561:Negative
421:Kaktovik
416:Cherokee
393:Pentadic
317:Historic
300:Japanese
233:Javanese
223:Balinese
210:Dzongkha
175:Gurmukhi
170:Gujarati
108:a series
106:Part of
3815:Numbers
3660:Sources
3518:-yllion
3063:, etc.
2933:) like
2183:is the
2177:weights
1801:decimal
1518:///////
1462:decimal
1221:Chinese
938:History
850:in the
842:in the
834:in the
648:Chuvash
566:Complex
356:Ancient
348:History
295:Hokkien
283:Chinese
228:Burmese
218:Tibetan
205:Kannada
185:Sinhala
160:Bengali
69:improve
3770:
3748:
3722:
3673:
3637:
3574:. AMS.
3399:(70),
3391:(36),
3387:(35),
2813:it is
2532:Weight
2330:, for
1825:10 = 1
1623:digits
1144:glyphs
832:eleven
825:digits
760:Hebrew
730:Coptic
643:Brahmi
628:Aegean
585:
569:
551:
538:
525:
388:Muisca
328:Tangut
305:Korean
288:Suzhou
200:Telugu
58:, but
3379:(1),
3375:(0),
2885:is a
2624:Digit
1777:radix
1617:radix
1191:Mayas
876:, or
840:three
755:Greek
740:Geʽez
698:Abjad
678:Roman
638:Aztec
633:Attic
548:Mixed
506:table
398:Quipu
383:Mayan
238:Khmer
190:Tamil
3768:ISBN
3746:ISBN
3720:ISBN
3671:ISBN
3635:ISBN
3434:and
3354:1190
2980:for
2741:0.01
2716:1000
2355:1000
2271:for
2171:and
1936:base
1916:...
1821:×10)
1781:base
1706:both
1612:base
1566:base
1223:and
1200:The
1139:zero
890:and
403:Rumi
258:Thai
180:Odia
3436:aca
3409:bcb
3401:bca
2877:If
2819:0.3
2736:0.1
2721:100
2436:log
2423:log
2398:log
2389:is
2343:log
2300:log
2225:log
2206:log
1929:− 1
1914:− 2
1905:− 1
1872:− 2
1859:− 1
1779:or
1728:≥ 1
1667:GMP
1614:or
1476:of
1371:–9
1368:–8
1365:–7
1362:–6
1359:–5
1356:–4
1353:–3
1350:–2
1347:–1
1344:–0
967:.
848:two
435:By
243:Lao
3806::
3744:.
3740:.
3718:.
3714:.
3683:.
3589:.
3570:.
3542:^
3432:ac
3393:cb
3389:bb
3381:ca
3377:ba
3366:.
3348:34
3342:35
3320:36
3314:35
3294:35
3275:36
3226:36
2911:.
2900:.
2881:=
2871:78
2867:27
2851:,
2844:.
2837:10
2821:10
2726:10
2347:10
1945:.
1864:+
1851:+
1827:.
1796:.
1771:a
1669:.
1528:.
1520:.
1472:.
1464:.
1267:9
1264:8
1261:7
1258:6
1255:5
1252:4
1249:3
1246:2
1243:1
1240:0
1122:९
1087:۹
1052:٩
1017:9
815:A
500:60
495:20
490:16
485:12
480:10
110:on
3776:.
3754:.
3728:.
3677:.
3643:.
3599:.
3440:a
3428:b
3424:b
3420:n
3413:b
3405:a
3397:b
3385:a
3373:a
3351:=
3339:=
3336:)
3331:1
3327:t
3317:(
3291:=
3286:0
3282:t
3265:n
3261:n
3257:n
3253:c
3237:n
3233:t
3223:=
3218:1
3215:+
3212:n
3208:b
3197:n
3192:n
3190:t
3186:1
3183:b
3179:a
3175:b
3156:i
3152:t
3129:i
3125:a
3101:,
3096:1
3092:t
3088:,
3083:0
3079:t
3049:2
3045:b
3039:1
3035:b
3029:2
3025:a
3021:+
3016:1
3012:b
3006:1
3002:a
2998:+
2993:0
2989:a
2966:2
2962:a
2956:1
2952:a
2946:0
2942:a
2905:b
2896:p
2891:p
2883:p
2879:b
2873:.
2859:ṅ
2854:n
2842:2
2832:π
2825:2
2785:0
2780:0
2775:7
2770:2
2765:3
2760:4
2731:1
2695:2
2691:c
2683:1
2679:c
2671:0
2667:a
2659:1
2655:a
2647:2
2643:a
2635:3
2631:a
2609:2
2602:b
2594:1
2587:b
2579:0
2575:b
2567:1
2563:b
2555:2
2551:b
2543:3
2539:b
2519:2
2511:1
2503:0
2498:1
2493:2
2488:3
2454:1
2451:+
2448:w
2440:b
2427:b
2419:=
2416:1
2413:+
2410:k
2402:b
2373:1
2370:+
2367:3
2364:=
2361:1
2358:+
2332:k
2318:1
2315:+
2312:w
2304:b
2296:=
2293:1
2290:+
2287:k
2277:w
2242:k
2238:b
2229:b
2221:=
2218:w
2210:b
2202:=
2199:k
2189:w
2181:k
2173:b
2169:b
2152:.
2147:k
2140:b
2134:k
2130:c
2119:1
2116:=
2113:k
2105:+
2100:k
2096:b
2090:k
2086:a
2080:n
2075:0
2072:=
2069:k
2061:=
2056:b
2052:)
2043:3
2039:c
2033:2
2029:c
2023:1
2019:c
2015:.
2010:0
2006:a
2000:1
1996:a
1987:1
1981:n
1977:a
1971:n
1967:a
1963:(
1950:b
1927:b
1921:0
1918:a
1912:n
1908:a
1903:n
1899:a
1895:n
1891:a
1885:b
1882:0
1879:a
1875:b
1870:n
1866:a
1862:b
1857:n
1853:a
1849:b
1845:n
1841:a
1836:b
1832:b
1819:7
1815:2
1811:3
1807:4
1805:(
1794:b
1789:b
1785:b
1769:b
1765:b
1745:p
1740:k
1736:k
1726:k
1723:(
1721:k
1713:k
1593:(
1585:(
1577:(
1514:/
1119:८
1116:७
1113:६
1110:५
1107:४
1104:३
1101:२
1098:१
1095:०
1084:۸
1081:۷
1078:۶
1075:۵
1072:۴
1069:۳
1066:۲
1063:۱
1060:۰
1049:٨
1046:٧
1043:٦
1040:٥
1037:٤
1034:٣
1031:٢
1028:١
1025:٠
1014:8
1011:7
1008:6
1005:5
1002:4
999:3
996:2
993:1
990:0
974:)
970:(
930:p
880:)
796:e
789:t
782:v
591:)
589:φ
587:(
578:)
575:i
573:2
571:(
557:)
553:(
544:)
540:(
531:)
529:1
527:(
508:)
504:(
475:8
470:6
465:5
460:4
455:3
450:2
94:)
88:(
83:)
79:(
65:.
38:.
20:)
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