3150:, which is opposite of the convention described above. The Unicode Standard states that the orientation of the Unicode characters follows Song dynasty convention, which differs from Han dynasty practice which represented digits as vertical lines, and tens as horizontal lines. Zero should be represented by U+3007 (〇, ideographic number zero) and the negative sign should be represented by U+20E5 (combining reverse solidus overlay). As these were recently added to the character set and since they are included in the SMP, font support may still be limited.
1267:
1215:
1239:
1259:
728:
1251:
3035:
3499:
1954年，考古学家在位于长江中游地区的湖南省长沙左家公山发掘了一座距今两千一百多年的战国时期楚墓。墓中出土了一个竹筒，里面盛放着戥秤、砝码、毛笔以及四十余根长度约为十二厘米的竹片。这是最早发现也是最古的算筹实物。In
English-In 1954, archaeologists discovered a tomb of the Chu Dynasty from the Warring States Period, which is more than 2,100 years old, at Zuojiagong Mountain in Changsha, Hunan Province, in
1522:
In Japan, mathematicians put counting rods on a counting board, a sheet of cloth with grids, and used only vertical forms relying on the grids. An 18th-century
Japanese mathematics book has a checker counting board diagram, with the order of magnitude symbols "千百十一分厘毛" (thousand, hundred, ten, unit,
1377:
The value of a number depends on its physical position on the counting board. A 9 at the rightmost position on the board stands for 9. Moving the batch of rods representing 9 to the left one position (i.e., to the tens place) gives 9 or 90. Shifting left again to the third position (to the hundreds
1278:
rod represents five. To avoid confusion, vertical and horizontal forms are alternately used. Generally, vertical rod numbers are used for the position for the units, hundreds, ten thousands, etc., while horizontal rod numbers are used for the tens, thousands, hundred thousands etc. It is written in
3104:
Before the introduction of a written zero, a space was used to indicate no units, and the rotation of the character in the subsequent unit column, by 90°, adopted, to help reduce the ambiguity in record values calculated on the rods. For example 107 (𝍠 𝍧) and 17 (𝍩𝍧) would be distinguished by
2018:
Rod numerals are a positional numeral system made from shapes of counting rods. Positive numbers are written as they are and the negative numbers are written with a slant bar at the last digit. The vertical bar in the horizontal forms 6–9 are drawn shorter to have the same character height.
1300:, which was mainly composed in the first century CE, stated "(when using subtraction) subtract same signed numbers, add different signed numbers, subtract a positive number from zero to make a negative number, and subtract a negative number from zero to make a positive number". Later, a
3105:
rotation, though multiple zero units could lead to ambiguity, eg. 1007 (𝍩 𝍧) , and 10007 (𝍠 𝍧). Once written zero came into play, the rod numerals had become independent, and their use indeed outlives the counting rods, after its replacement by
1514:
1500:
1486:
1378:
place) gives 9 or 900. Each time one shifts a number one position to the left, it is multiplied by 10. Each time one shifts a number one position to the right, it is divided by 10. This applies to single-digit numbers or multiple-digit numbers.
1507:
1493:
3500:
the middle reaches of the
Yangtze River. A bamboo tube was unearthed in the tomb, which contained a balance scale, weights, a brush, and more than 40 bamboo pieces about 12 centimeters long. This is the earliest and oldest abacus discovered.
3874:
1307:
This alternation of vertical and horizontal rod numeral form is very important to understanding written transcription of rod numerals on manuscripts correctly. For instance, in
Licheng suanjin, 81 was transcribed as
3902:
1330:; it is clear that the latter clearly had a blank zero on the "counting board" (i.e., floor or mat), even though on the written transcription, there was no blank. In the same manuscript, 405 was transcribed as
3878:
2030:
in 718, but some think it was created from the
Chinese text space filler "□", and others think that the Indians acquired it from China, because it resembles a Confucian philosophical symbol for "nothing".
3882:
3906:
3910:
1155:
A square lacquer box, dating from c. 168 BCE, containing a square chess board with the TLV patterns, chessmen, counting rods, and other items, was excavated in 1972, from
1444:
He treated the
Chinese order numbers as place value markers, and 七一八二四 became place value decimal number. He then wrote the rod numerals according to their place value:
1200:(finished 111 CE) recorded: "they calculate with bamboo, diameter one fen, length six cun, arranged into a hexagonal bundle of two hundred seventy one pieces".
1152:(206 BCE to 220 CE). On one of the wooden scripts was written: "当利二月定算𝍥". This is one of the earliest examples of using counting-rod numerals in writing.
3631:"How were Western written calculations introduced into China? — An analysis of the Tongwen suanzhi (Arithmetic Guidance in the Common Language, 1613)"
1297:
714:
3957:
434:
3057:
A fraction was expressed with rod numerals as two rod numerals one on top of another (without any other symbol, like the modern horizontal bar).
3595:
2718:
2703:
2613:
2585:
2538:
2510:
1218:
Toán trù 算籌 (counting rods) in a
Vietnamese mathematics textbook, Cửu chương lập thành toán pháp 九章立成算法 is shown at the bottom of the page.
2747:
2713:
2691:
2606:
2599:
2578:
2503:
2229:
2222:
2215:
3142:
for the horizontal digits 1–9 are U+1D360 to U+1D368 and those for the vertical digits 1–9 are U+1D369 to U+1D371. The former are called
1296:. Ancient Chinese clearly understood negative numbers and zero (leaving a blank space for it), though they had no symbol for the latter.
2958:
2940:
2757:
2740:
2421:
2414:
2407:
2400:
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2379:
2372:
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2339:
2332:
2325:
2318:
2311:
2304:
2297:
2290:
1951:
1944:
1934:
1915:
1908:
1702:
1695:
1688:
1681:
1674:
1667:
1660:
1653:
1631:
1624:
1617:
1610:
1603:
1596:
1589:
1582:
3851:
2978:
2934:
2887:
2882:
2877:
2806:
2801:
2796:
2791:
2784:
2779:
2774:
2769:
2730:
2696:
2681:
2664:
2659:
2647:
2642:
2592:
2571:
2564:
2531:
2524:
2517:
2496:
2489:
2358:
2283:
2208:
2201:
2194:
2187:
2180:
2154:
2147:
2140:
2133:
2126:
2119:
2112:
2105:
2002:
1995:
1988:
1976:
1966:
1922:
1886:
1879:
1872:
1865:
1858:
1851:
1844:
1837:
1815:
1808:
1801:
1794:
1787:
1780:
1773:
1766:
1646:
1575:
1366:
1358:
1352:
1347:
1341:
1333:
1327:
1311:
274:
3018:
2990:
2952:
2910:
2904:
2898:
2871:
2865:
2752:
2735:
2708:
2686:
2669:
2652:
2557:
2550:
2482:
2475:
2173:
2166:
2098:
2091:
1830:
1759:
1322:
1316:
3692:
65:
3512:
3585:
989:
763:
501:
3937:
777:
707:
289:
3482:
3932:
3718:
1083:(筭) are small bars, typically 3–14 cm (1" to 6") long, that were used by mathematicians for calculation in ancient
627:
3665:
3551:
637:
3947:
3942:
3927:
2038:
mathematicians changed digits for 4, 5, and 9 to reduce strokes. The new horizontal forms eventually transformed into
1385:
used hand-written
Chinese decimal orders 步十百千萬 as rod numeral place value, as evident from a facsimile from a page of
454:
3862:
1226:
flourished, counting rods were abandoned except in Japan, where rod numerals developed into a symbolic notation for
3098:
514:
1355:. In other words, transcribed rod numerals may not be positional, but on the counting board, they are positional.
1190:
of 771 BCE to 5th century BCE, mentions their use to make calculations to win wars before going into the battle;
610:
386:
1266:
1187:
1055:
700:
34:
1194:(died 531 BCE), writing in the Warring States period, said "a good calculator doesn't use counting rods". The
1148:
In 1973, archeologists unearthed a number of wood scripts from a tomb in Hubei dating from the period of the
690:
474:
78:
1214:
1207:(581 to 618 CE) mathematicians used triangular rods to represent positive numbers and rectangular rods for
381:
297:
3157:
3126:
1126:
1111:
492:
3749:
2831:
2819:
735:
587:
448:
441:
329:
1130:
3890:
3852:
https://web.archive.org/web/20010217175749/http://www.math.sfu.ca/histmath/China/Beginning/Rod.html
3410:
2823:
1386:
1243:
1103:
911:
669:
534:
485:
304:
236:
91:
52:
3807:
582:
342:
179:
174:
121:
3826:
3079:(筹算). Rod calculus can be used for a wide range of calculations, including finding the value of
1344:, with a blank space in between for obvious reasons, and could in no way be interpreted as "45"
3952:
3688:
3591:
3464:
888:
821:
664:
654:
642:
622:
577:
572:
508:
347:
319:
226:
159:
149:
136:
101:
96:
3430:
2827:
867:
844:
800:
567:
461:
221:
209:
154:
144:
111:
86:
3847:
For a look of the ancient counting rods, and further explanation, you can visit the sites
3516:
3135:
3110:
2039:
2023:
1293:
1289:
1208:
1167:
1107:
1092:
679:
649:
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547:
314:
282:
254:
231:
214:
73:
659:
1238:
3773:
3552:"BabelStone Blog : The Lost Game of Liubo Part 4 : Game Boards and Equipment"
2818:
developed the rod numerals into symbolic notation for algebra and drastically improved
2027:
1281:
1258:
674:
617:
597:
552:
164:
131:
116:
42:
3856:
3921:
3630:
3587:
Fleeting
Footsteps: Tracing the Conception of Arithmetic and Algebra in Ancient China
3425:
2815:
1275:
480:
376:
309:
184:
126:
106:
3509:
2717:
2702:
2612:
2584:
2537:
2509:
927:
3415:
3076:
3066:
2746:
2712:
2690:
2605:
2598:
2577:
2502:
2228:
2221:
2214:
2035:
874:
807:
632:
557:
3706:
3653:
3530:
1262:
A checker counting board diagram in an 18th-century
Japanese mathematics textbook
3788:
3420:
3162:
3086:
1204:
1196:
1163:
1149:
1134:
602:
467:
426:
416:
3453:"The Conceptual Origins of our Numeral System and the Symbolic Form of Algebra"
2957:
2939:
2756:
2739:
2420:
2413:
2406:
2399:
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2385:
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2371:
2364:
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2338:
2331:
2324:
2317:
2310:
2303:
2296:
2289:
1950:
1943:
1933:
1921:
1914:
1907:
1701:
1694:
1687:
1680:
1673:
1666:
1659:
1652:
1645:
1630:
1623:
1616:
1609:
1602:
1595:
1588:
1581:
1574:
1203:
At first, calculating rods were round in cross-section, but by the time of the
727:
3488:
3139:
3017:
2989:
2977:
2951:
2933:
2909:
2903:
2897:
2886:
2881:
2876:
2870:
2864:
2805:
2800:
2795:
2790:
2783:
2778:
2773:
2768:
2729:
2695:
2680:
2668:
2663:
2658:
2651:
2646:
2641:
2591:
2570:
2563:
2556:
2530:
2523:
2516:
2495:
2488:
2481:
2357:
2282:
2207:
2200:
2193:
2186:
2179:
2172:
2153:
2146:
2139:
2132:
2125:
2118:
2111:
2104:
2097:
2001:
1994:
1987:
1975:
1965:
1885:
1878:
1871:
1864:
1857:
1850:
1843:
1836:
1829:
1814:
1807:
1800:
1793:
1786:
1779:
1772:
1765:
1758:
1513:
1506:
1499:
1492:
1485:
1365:
1357:
1351:
1346:
1340:
1332:
1326:
1321:
1315:
1310:
739:
411:
169:
3468:
1122:
Chinese arithmeticians used counting rods well over two thousand years ago.
3114:
3090:
2751:
2734:
2707:
2685:
2549:
2474:
2165:
2090:
1301:
1250:
1156:
1084:
421:
3071:
The method for using counting rods for mathematical calculation was called
17:
3775:
Proposal to add Chinese counting rod numerals to Unicode and ISO/IEC 10646
3034:
1369:
is an exact image of the counting rod number 405 on a table top or floor.
3574:
Wu Wenjun ed, Grand Series of History of Chinese Mathematics, vol 1, p371
3094:
1382:
1138:
1030:
731:
3898:
3870:
3452:
3113:
is still in use for book-keeping and in herbal medicine prescription in
2022:
A circle (〇) is used for 0. Many historians think it was imported from
3131:
1227:
1175:
1171:
1088:
406:
391:
1166:-era (202 BCE to 9 CE) counting rods made of bones was unearthed from
3405:
3106:
1223:
1087:. They are placed either horizontally or vertically to represent any
948:
396:
3891:
Counting rods and go stones of a Japanese mathematician around 1872
3033:
1265:
1257:
1249:
1237:
1213:
1191:
1142:
1014:
964:
401:
363:
324:
3531:"The Siling (four cardinal animals) in Han Pictorial Art - vol 1"
1274:
Counting rods represent digits by the number of rods, and the
3134:
5.0 includes counting rod numerals in their own block in the
3310:
3300:
3280:
3260:
3255:
3235:
1270:
Counting rod numerals in grids in a Japanese mathematic book
3330:
3315:
3290:
3285:
3265:
3245:
3240:
3080:
2830:
was developed, and the rod numerals were used only for the
3340:
3335:
3295:
3275:
3270:
3250:
3230:
3225:
3721:, Number Words and Number Symbols, p 369, MIT Press, 1970
3519:– Copyright © 2010 – TutorVista.com, All rights reserved.
3325:
3320:
3736:
Zhongguo Shuxue Shi (The history of Chinese mathematics)
3685:
Sangi o koeta otoko (The man who exceeded counting rods)
3809:
The Unicode Standard, Version 15.0 – Electronic edition
3828:
The Unicode Standard, Version 5.0 – Electronic edition
3790:
The Unicode Standard, Version 5.0 – Electronic edition
1446:
1060:
893:
879:
826:
812:
994:
27:
Small bars used for calculating in ancient East Asia
3857:
http://mathforum.org/library/drmath/view/52557.html
3709:正負術曰: 同名相除，異名相益，正無入負之，負無入正之。其異名相除，同名相益，正無入正之，負無入負之。
2836:
2623:
2428:
2236:
2044:
2042:. Japanese continued to use the traditional forms.
1896:
1709:
1525:
1054:
1049:
1029:
1013:
1008:
988:
983:
963:
947:
942:
926:
910:
905:
887:
873:
866:
861:
843:
838:
820:
806:
799:
794:
776:
762:
757:
748:
3109:. One variation of horizontal rod numerals, the
932:
916:
3350:
1482:
1465:
1448:
1186:496 BCE), a military strategist at the end of
1035:
1019:
3345:
849:
782:
768:
708:
8:
3390:Grey areas indicate non-assigned code points
1304:stone was sometimes used to represent zero.
1174:. The use of counting rods must predate it;
1114:period (circa 475 BCE) to the 16th century.
969:
953:
1285:that "one is vertical, ten is horizontal".
1129:(5th century BCE to 221 BCE) were found in
1098:The written forms based on them are called
3152:
1046:
980:
858:
791:
715:
701:
58:
29:
1298:The Nine Chapters on the Mathematical Art
1125:In 1954, forty-odd counting rods of the
726:
3834:, Unicode, Inc., 2006, pp. 499–500
3815:, Unicode, Inc., 2022, pp. 869–870
3772:Christopher Cullen et John H. Jenkins,
3729:
3727:
3678:
3676:
3674:
3443:
41:
3163:Official Unicode Consortium code chart
745:
3457:Archive for History of Exact Sciences
7:
3584:Lam, Lay Yong; Ang, Tian Se (2004).
1110:for 1–9 and a blank for 0, from the
3529:WONG, Pui Yin Marianne (Mar 2006).
3138:(SMP) from U+1D360 to U+1D37F. The
3796:, Unicode, Inc., 2006, p. 558
1254:Japanese counting board with grids
25:
2430:Positive numbers (Southern Song)
3136:Supplementary Multilingual Plane
3016:
2988:
2976:
2956:
2950:
2938:
2932:
2908:
2902:
2896:
2885:
2880:
2875:
2869:
2863:
2804:
2799:
2794:
2789:
2782:
2777:
2772:
2767:
2755:
2750:
2745:
2738:
2733:
2728:
2716:
2711:
2706:
2701:
2694:
2689:
2684:
2679:
2667:
2662:
2657:
2650:
2645:
2640:
2611:
2604:
2597:
2590:
2583:
2576:
2569:
2562:
2555:
2548:
2536:
2529:
2522:
2515:
2508:
2501:
2494:
2487:
2480:
2473:
2419:
2412:
2405:
2398:
2391:
2384:
2377:
2370:
2363:
2356:
2344:
2337:
2330:
2323:
2316:
2309:
2302:
2295:
2288:
2281:
2227:
2220:
2213:
2206:
2199:
2192:
2185:
2178:
2171:
2164:
2152:
2145:
2138:
2131:
2124:
2117:
2110:
2103:
2096:
2089:
2000:
1993:
1986:
1974:
1964:
1949:
1942:
1932:
1920:
1913:
1906:
1884:
1877:
1870:
1863:
1856:
1849:
1842:
1835:
1828:
1813:
1806:
1799:
1792:
1785:
1778:
1771:
1764:
1757:
1700:
1693:
1686:
1679:
1672:
1665:
1658:
1651:
1644:
1629:
1622:
1615:
1608:
1601:
1594:
1587:
1580:
1573:
1512:
1505:
1498:
1491:
1484:
1364:
1356:
1350:
1345:
1339:
1331:
1325:
1320:
1314:
1309:
3958:Science and technology in China
3656:孫子算經: 先識其位，一從十橫，百立千僵，千十相望，萬百相當。
2238:Negative numbers (traditional)
2046:Positive numbers (traditional)
1523:tenth, hundredth, thousandth).
3367:
3365:
3363:
3361:
3359:
3357:
3355:
1159:M3, Changsha, Hunan Province.
1061:
995:
933:
894:
880:
850:
827:
813:
783:
769:
1:
1319:, and 108 was transcribed as
1242:Rod numeral place value from
1183:
1179:
3117:in some parts of the world.
1527:Zero and positive integers
1389:. He arranged 七萬一千八百二十四 as
1381:Song dynasty mathematician
742:in 1303, using rod numerals
3974:
3738:, Beijing: Kexue Chubanshe
3382:As of Unicode version 16.0
3371:
3155:
3146:and the latter are called
3124:
3099:system of linear equations
3064:
2828:Chinese numeral characters
1711:Zero and negative numbers
435:Non-standard radices/bases
3668:, 夢溪筆談: 如算法用赤籌、黑籌，以別正負之數。
2824:positional numeral system
2822:. After his period, the
1292:and black rods represent
1104:positional numeral system
1074:
1045:
1036:
1020:
979:
970:
954:
857:
790:
753:
3683:Wáng, Qīngxiáng (1999),
1188:Spring and Autumn period
839:Alternative Chinese name
764:Traditional Chinese
3901: to English:
3873: to English:
778:Simplified Chinese
691:List of numeral systems
3863:Counting rods in China
3734:Qian, Baocong (1964),
3687:, Tokyo: Tōyō Shoten,
3451:Lay-Yong, Lam (1986).
3054:
1271:
1263:
1255:
1247:
1219:
917:
895:su̯än⁵¹ t͡sz̩²¹⁴⁻²¹⁽⁴⁾
743:
3158:Counting Rod Numerals
3127:Counting Rod Numerals
3037:
2034:In the 13th century,
1269:
1261:
1253:
1241:
1217:
1162:In 1976, a bundle of
1127:Warring States period
730:
66:Hindu–Arabic numerals
3938:Japanese mathematics
3590:. World Scientific.
3556:www.babelstone.co.uk
2832:plus and minus signs
2820:Japanese mathematics
990:Revised Romanization
588:Prehistoric counting
371:Common radices/bases
53:Place-value notation
3933:Chinese mathematics
3629:Zhu, Yiwen (2018).
3411:Chinese mathematics
2431:
2239:
2047:
1712:
1528:
1387:Yongle Encyclopedia
1288:Red rods represent
1244:Yongle Encyclopedia
1234:Using counting rods
912:Vietnamese alphabet
535:Sign-value notation
3948:Mathematical tools
3943:Korean mathematics
3928:Chinese inventions
3750:"Chinese numerals"
3707:Chinese Wikisource
3666:Chinese Wikisource
3654:Chinese Wikisource
3515:2011-01-26 at the
3510:Ancient China Math
3095:higher order roots
3055:
2429:
2237:
2045:
1710:
1526:
1272:
1264:
1256:
1248:
1220:
1102:. They are a true
744:
198:East Asian systems
3610:孫子: 夫未戰而廟算勝者，得算多也
3597:978-981-256-725-3
3396:
3395:
3027:
3026:
2812:
2811:
2619:
2618:
2427:
2426:
2235:
2234:
2011:
2010:
1892:
1891:
1708:
1707:
1520:
1519:
1078:
1077:
1070:
1069:
1004:
1003:
901:
900:
868:Standard Mandarin
834:
833:
828:su̯än⁵¹ ʈ͡ʂʰoʊ̯³⁵
801:Standard Mandarin
738:, as depicted by
725:
724:
524:
523:
16:(Redirected from
3965:
3914:
3900:
3896:
3886:
3872:
3868:
3836:
3835:
3833:
3823:
3817:
3816:
3814:
3804:
3798:
3797:
3795:
3785:
3779:
3770:
3764:
3763:
3761:
3760:
3746:
3740:
3739:
3731:
3722:
3716:
3710:
3704:
3698:
3697:
3680:
3669:
3663:
3657:
3651:
3645:
3644:
3642:
3641:
3626:
3620:
3617:
3611:
3608:
3602:
3601:
3581:
3575:
3572:
3566:
3565:
3563:
3562:
3548:
3542:
3541:
3535:
3526:
3520:
3507:
3501:
3498:
3497:
3496:
3487:, archived from
3479:
3473:
3472:
3448:
3431:Unicode numerals
3389:
3381:
3353:
3348:
3343:
3338:
3333:
3328:
3323:
3318:
3313:
3303:
3298:
3293:
3288:
3283:
3278:
3273:
3268:
3263:
3258:
3253:
3248:
3243:
3238:
3233:
3228:
3153:
3097:, and solving a
3083:
3053:
3051:
3050:
3047:
3044:
3020:
2992:
2980:
2960:
2954:
2942:
2936:
2912:
2906:
2900:
2889:
2884:
2879:
2873:
2867:
2837:
2808:
2803:
2798:
2793:
2786:
2781:
2776:
2771:
2759:
2754:
2749:
2742:
2737:
2732:
2720:
2715:
2710:
2705:
2698:
2693:
2688:
2683:
2671:
2666:
2661:
2654:
2649:
2644:
2624:
2615:
2608:
2601:
2594:
2587:
2580:
2573:
2566:
2559:
2552:
2540:
2533:
2526:
2519:
2512:
2505:
2498:
2491:
2484:
2477:
2432:
2423:
2416:
2409:
2402:
2395:
2388:
2381:
2374:
2367:
2360:
2348:
2341:
2334:
2327:
2320:
2313:
2306:
2299:
2292:
2285:
2240:
2231:
2224:
2217:
2210:
2203:
2196:
2189:
2182:
2175:
2168:
2156:
2149:
2142:
2135:
2128:
2121:
2114:
2107:
2100:
2093:
2048:
2004:
1997:
1990:
1978:
1968:
1953:
1946:
1936:
1924:
1917:
1910:
1897:
1888:
1881:
1874:
1867:
1860:
1853:
1846:
1839:
1832:
1817:
1810:
1803:
1796:
1789:
1782:
1775:
1768:
1761:
1713:
1704:
1697:
1690:
1683:
1676:
1669:
1662:
1655:
1648:
1633:
1626:
1619:
1612:
1605:
1598:
1591:
1584:
1577:
1529:
1516:
1509:
1502:
1495:
1488:
1447:
1368:
1363:
1360:
1354:
1349:
1343:
1338:
1335:
1329:
1324:
1318:
1313:
1294:negative numbers
1290:positive numbers
1209:negative numbers
1185:
1181:
1066:
1065:
1047:
1041:
1040:
1025:
1024:
1000:
999:
981:
975:
974:
959:
958:
938:
937:
922:
897:
896:
883:
882:
859:
853:
852:
830:
829:
816:
815:
792:
786:
785:
772:
771:
746:
717:
710:
703:
506:
490:
472:
462:balanced ternary
459:
446:
59:
30:
21:
3973:
3972:
3968:
3967:
3966:
3964:
3963:
3962:
3918:
3917:
3894:
3889:
3866:
3861:
3845:
3840:
3839:
3831:
3825:
3824:
3820:
3812:
3806:
3805:
3801:
3793:
3787:
3786:
3782:
3771:
3767:
3758:
3756:
3748:
3747:
3743:
3733:
3732:
3725:
3717:
3713:
3705:
3701:
3695:
3682:
3681:
3672:
3664:
3660:
3652:
3648:
3639:
3637:
3628:
3627:
3623:
3618:
3614:
3609:
3605:
3598:
3583:
3582:
3578:
3573:
3569:
3560:
3558:
3550:
3549:
3545:
3533:
3528:
3527:
3523:
3517:Wayback Machine
3508:
3504:
3494:
3492:
3481:
3480:
3476:
3450:
3449:
3445:
3440:
3435:
3401:
3386:
3378:
3351:
3346:
3341:
3336:
3331:
3326:
3321:
3316:
3311:
3301:
3296:
3291:
3286:
3281:
3276:
3271:
3266:
3261:
3256:
3251:
3246:
3241:
3236:
3231:
3226:
3161:
3129:
3123:
3111:Suzhou numerals
3081:
3073:rod calculation
3069:
3063:
3048:
3045:
3042:
3041:
3039:
3032:
3023:
2995:
2983:
2963:
2945:
2915:
2891:
2040:Suzhou numerals
2024:Indian numerals
2016:
1375:
1361:
1336:
1236:
1168:Qianyang County
1137:Grave No.15 in
1120:
1093:rational number
1063:
1038:
1022:
997:
972:
956:
935:
919:
906:Vietnamese name
721:
685:
684:
607:
593:Proto-cuneiform
538:
537:
526:
525:
520:
519:
504:
488:
470:
457:
444:
431:
367:
366:
354:
353:
334:
294:
279:
270:
269:
260:
259:
241:
200:
199:
190:
189:
141:
83:
69:
68:
56:
55:
43:Numeral systems
28:
23:
22:
15:
12:
11:
5:
3971:
3969:
3961:
3960:
3955:
3950:
3945:
3940:
3935:
3930:
3920:
3919:
3916:
3915:
3887:
3859:
3854:
3844:
3843:External links
3841:
3838:
3837:
3818:
3799:
3780:
3765:
3741:
3723:
3719:Karl Menninger
3711:
3699:
3693:
3670:
3658:
3646:
3621:
3612:
3603:
3596:
3576:
3567:
3543:
3521:
3502:
3474:
3463:(3): 183–195.
3442:
3441:
3439:
3436:
3434:
3433:
3428:
3423:
3418:
3413:
3408:
3402:
3400:
3397:
3394:
3393:
3392:
3391:
3383:
3369:
3368:
3366:
3364:
3362:
3360:
3358:
3356:
3354:
3349:
3344:
3339:
3334:
3329:
3324:
3319:
3314:
3309:
3305:
3304:
3299:
3294:
3289:
3284:
3279:
3274:
3269:
3264:
3259:
3254:
3249:
3244:
3239:
3234:
3229:
3224:
3220:
3219:
3216:
3213:
3210:
3207:
3204:
3201:
3198:
3195:
3192:
3189:
3186:
3183:
3180:
3177:
3174:
3171:
3167:
3166:
3125:Main article:
3122:
3119:
3065:Main article:
3062:
3059:
3031:
3028:
3025:
3024:
3013:
3011:
3008:
2997:
2996:
2986:
2984:
2974:
2972:
2965:
2964:
2948:
2946:
2930:
2928:
2917:
2916:
2894:
2892:
2861:
2859:
2848:
2847:
2844:
2841:
2810:
2809:
2787:
2765:
2761:
2760:
2743:
2726:
2722:
2721:
2699:
2677:
2673:
2672:
2655:
2638:
2634:
2633:
2632:Southern Song
2630:
2627:
2617:
2616:
2609:
2602:
2595:
2588:
2581:
2574:
2567:
2560:
2553:
2546:
2542:
2541:
2534:
2527:
2520:
2513:
2506:
2499:
2492:
2485:
2478:
2471:
2467:
2466:
2463:
2460:
2457:
2454:
2451:
2448:
2445:
2442:
2439:
2436:
2425:
2424:
2417:
2410:
2403:
2396:
2389:
2382:
2375:
2368:
2361:
2354:
2350:
2349:
2342:
2335:
2328:
2321:
2314:
2307:
2300:
2293:
2286:
2279:
2275:
2274:
2271:
2268:
2265:
2262:
2259:
2256:
2253:
2250:
2247:
2244:
2233:
2232:
2225:
2218:
2211:
2204:
2197:
2190:
2183:
2176:
2169:
2162:
2158:
2157:
2150:
2143:
2136:
2129:
2122:
2115:
2108:
2101:
2094:
2087:
2083:
2082:
2079:
2076:
2073:
2070:
2067:
2064:
2061:
2058:
2055:
2052:
2028:Gautama Siddha
2015:
2012:
2009:
2008:
2005:
1998:
1991:
1984:
1980:
1979:
1972:
1969:
1962:
1959:
1955:
1954:
1947:
1940:
1937:
1930:
1926:
1925:
1918:
1911:
1904:
1901:
1890:
1889:
1882:
1875:
1868:
1861:
1854:
1847:
1840:
1833:
1826:
1823:
1819:
1818:
1811:
1804:
1797:
1790:
1783:
1776:
1769:
1762:
1755:
1752:
1748:
1747:
1744:
1741:
1738:
1735:
1732:
1729:
1726:
1723:
1720:
1717:
1706:
1705:
1698:
1691:
1684:
1677:
1670:
1663:
1656:
1649:
1642:
1639:
1635:
1634:
1627:
1620:
1613:
1606:
1599:
1592:
1585:
1578:
1571:
1568:
1564:
1563:
1560:
1557:
1554:
1551:
1548:
1545:
1542:
1539:
1536:
1533:
1518:
1517:
1510:
1503:
1496:
1489:
1481:
1480:
1477:
1474:
1471:
1468:
1464:
1463:
1460:
1457:
1454:
1451:
1442:
1441:
1440:
1439:
1438:
1437:
1436:
1435:
1434:
1433:
1432:
1431:
1430:
1429:
1428:
1427:
1426:
1425:
1424:
1423:
1422:
1421:
1420:
1419:
1416:
1374:
1371:
1282:Sunzi Suanjing
1235:
1232:
1131:Zuǒjiāgōngshān
1119:
1116:
1112:Warring states
1076:
1075:
1072:
1071:
1068:
1067:
1058:
1052:
1051:
1050:Transcriptions
1043:
1042:
1033:
1027:
1026:
1017:
1011:
1010:
1006:
1005:
1002:
1001:
992:
986:
985:
984:Transcriptions
977:
976:
967:
961:
960:
951:
945:
944:
940:
939:
930:
924:
923:
914:
908:
907:
903:
902:
899:
898:
891:
885:
884:
877:
871:
870:
864:
863:
862:Transcriptions
855:
854:
847:
841:
840:
836:
835:
832:
831:
824:
818:
817:
810:
804:
803:
797:
796:
795:Transcriptions
788:
787:
780:
774:
773:
766:
760:
759:
755:
754:
751:
750:
723:
722:
720:
719:
712:
705:
697:
694:
693:
687:
686:
683:
682:
677:
672:
667:
662:
657:
652:
647:
646:
645:
640:
635:
625:
620:
614:
613:
606:
605:
600:
595:
590:
585:
580:
575:
570:
565:
560:
555:
550:
544:
543:
542:Non-alphabetic
539:
533:
532:
531:
528:
527:
522:
521:
518:
517:
512:
499:
483:
478:
465:
452:
438:
437:
430:
429:
424:
419:
414:
409:
404:
399:
394:
389:
384:
379:
373:
372:
368:
361:
360:
359:
356:
355:
352:
351:
345:
339:
338:
333:
332:
327:
322:
317:
312:
307:
301:
300:
298:Post-classical
293:
292:
286:
285:
278:
277:
271:
267:
266:
265:
262:
261:
258:
257:
252:
246:
245:
240:
239:
234:
229:
224:
219:
218:
217:
206:
205:
201:
197:
196:
195:
192:
191:
188:
187:
182:
177:
172:
167:
162:
157:
152:
147:
140:
139:
134:
129:
124:
119:
114:
109:
104:
99:
94:
89:
82:
81:
79:Eastern Arabic
76:
74:Western Arabic
70:
64:
63:
62:
57:
51:
50:
49:
46:
45:
39:
38:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3970:
3959:
3956:
3954:
3951:
3949:
3946:
3944:
3941:
3939:
3936:
3934:
3931:
3929:
3926:
3925:
3923:
3912:
3908:
3904:
3895:(in Japanese)
3892:
3888:
3884:
3880:
3876:
3864:
3860:
3858:
3855:
3853:
3850:
3849:
3848:
3842:
3830:
3829:
3822:
3819:
3811:
3810:
3803:
3800:
3792:
3791:
3784:
3781:
3777:
3776:
3769:
3766:
3755:
3754:Maths History
3751:
3745:
3742:
3737:
3730:
3728:
3724:
3720:
3715:
3712:
3708:
3703:
3700:
3696:
3694:4-88595-226-3
3690:
3686:
3679:
3677:
3675:
3671:
3667:
3662:
3659:
3655:
3650:
3647:
3636:
3632:
3625:
3622:
3616:
3613:
3607:
3604:
3599:
3593:
3589:
3588:
3580:
3577:
3571:
3568:
3557:
3553:
3547:
3544:
3539:
3532:
3525:
3522:
3518:
3514:
3511:
3506:
3503:
3491:on 2007-11-21
3490:
3486:
3485:
3478:
3475:
3470:
3466:
3462:
3458:
3454:
3447:
3444:
3437:
3432:
3429:
3427:
3426:Tian yuan shu
3424:
3422:
3419:
3417:
3414:
3412:
3409:
3407:
3404:
3403:
3398:
3388:
3384:
3380:
3376:
3375:
3374:
3370:
3307:
3306:
3222:
3221:
3217:
3214:
3211:
3208:
3205:
3202:
3199:
3196:
3193:
3190:
3187:
3184:
3181:
3178:
3175:
3172:
3169:
3168:
3164:
3160:
3159:
3154:
3151:
3149:
3145:
3141:
3137:
3133:
3128:
3120:
3118:
3116:
3112:
3108:
3102:
3100:
3096:
3092:
3088:
3084:
3078:
3074:
3068:
3060:
3058:
3036:
3029:
3022:
3019:
3012:
3009:
3007:
3003:
2999:
2998:
2994:
2991:
2985:
2982:
2979:
2973:
2971:
2967:
2966:
2962:
2959:
2953:
2947:
2944:
2941:
2935:
2929:
2927:
2923:
2919:
2918:
2914:
2911:
2905:
2899:
2893:
2890:
2888:
2883:
2878:
2872:
2866:
2860:
2857:
2853:
2850:
2849:
2845:
2842:
2839:
2838:
2835:
2833:
2829:
2825:
2821:
2817:
2816:Seki Takakazu
2807:
2802:
2797:
2792:
2788:
2785:
2780:
2775:
2770:
2766:
2763:
2762:
2758:
2753:
2748:
2744:
2741:
2736:
2731:
2727:
2724:
2723:
2719:
2714:
2709:
2704:
2700:
2697:
2692:
2687:
2682:
2678:
2675:
2674:
2670:
2665:
2660:
2656:
2653:
2648:
2643:
2639:
2636:
2635:
2631:
2628:
2626:
2625:
2622:
2614:
2610:
2607:
2603:
2600:
2596:
2593:
2589:
2586:
2582:
2579:
2575:
2572:
2568:
2565:
2561:
2558:
2554:
2551:
2547:
2544:
2543:
2539:
2535:
2532:
2528:
2525:
2521:
2518:
2514:
2511:
2507:
2504:
2500:
2497:
2493:
2490:
2486:
2483:
2479:
2476:
2472:
2469:
2468:
2464:
2461:
2458:
2455:
2452:
2449:
2446:
2443:
2440:
2437:
2434:
2433:
2422:
2418:
2415:
2411:
2408:
2404:
2401:
2397:
2394:
2390:
2387:
2383:
2380:
2376:
2373:
2369:
2366:
2362:
2359:
2355:
2352:
2351:
2347:
2343:
2340:
2336:
2333:
2329:
2326:
2322:
2319:
2315:
2312:
2308:
2305:
2301:
2298:
2294:
2291:
2287:
2284:
2280:
2277:
2276:
2272:
2269:
2266:
2263:
2260:
2257:
2254:
2251:
2248:
2245:
2242:
2241:
2230:
2226:
2223:
2219:
2216:
2212:
2209:
2205:
2202:
2198:
2195:
2191:
2188:
2184:
2181:
2177:
2174:
2170:
2167:
2163:
2160:
2159:
2155:
2151:
2148:
2144:
2141:
2137:
2134:
2130:
2127:
2123:
2120:
2116:
2113:
2109:
2106:
2102:
2099:
2095:
2092:
2088:
2085:
2084:
2080:
2077:
2074:
2071:
2068:
2065:
2062:
2059:
2056:
2053:
2050:
2049:
2043:
2041:
2037:
2036:Southern Song
2032:
2029:
2025:
2020:
2013:
2006:
2003:
1999:
1996:
1992:
1989:
1985:
1982:
1981:
1977:
1973:
1970:
1967:
1963:
1960:
1957:
1956:
1952:
1948:
1945:
1941:
1938:
1935:
1931:
1928:
1927:
1923:
1919:
1916:
1912:
1909:
1905:
1902:
1899:
1898:
1895:
1887:
1883:
1880:
1876:
1873:
1869:
1866:
1862:
1859:
1855:
1852:
1848:
1845:
1841:
1838:
1834:
1831:
1827:
1824:
1821:
1820:
1816:
1812:
1809:
1805:
1802:
1798:
1795:
1791:
1788:
1784:
1781:
1777:
1774:
1770:
1767:
1763:
1760:
1756:
1753:
1750:
1749:
1745:
1742:
1739:
1736:
1733:
1730:
1727:
1724:
1721:
1718:
1715:
1714:
1703:
1699:
1696:
1692:
1689:
1685:
1682:
1678:
1675:
1671:
1668:
1664:
1661:
1657:
1654:
1650:
1647:
1643:
1640:
1637:
1636:
1632:
1628:
1625:
1621:
1618:
1614:
1611:
1607:
1604:
1600:
1597:
1593:
1590:
1586:
1583:
1579:
1576:
1572:
1569:
1566:
1565:
1561:
1558:
1555:
1552:
1549:
1546:
1543:
1540:
1537:
1534:
1531:
1530:
1524:
1515:
1511:
1508:
1504:
1501:
1497:
1494:
1490:
1487:
1483:
1478:
1475:
1472:
1469:
1466:
1461:
1458:
1455:
1452:
1449:
1445:
1417:
1414:
1413:
1412:
1411:
1410:
1409:
1408:
1407:
1406:
1405:
1404:
1403:
1402:
1401:
1400:
1399:
1398:
1397:
1396:
1395:
1394:
1393:
1392:
1391:
1390:
1388:
1384:
1379:
1372:
1370:
1367:
1359:
1353:
1348:
1342:
1334:
1328:
1323:
1317:
1312:
1305:
1303:
1299:
1295:
1291:
1286:
1284:
1283:
1277:
1276:perpendicular
1268:
1260:
1252:
1245:
1240:
1233:
1231:
1229:
1225:
1216:
1212:
1210:
1206:
1201:
1199:
1198:
1193:
1189:
1177:
1173:
1169:
1165:
1160:
1158:
1153:
1151:
1146:
1144:
1140:
1136:
1132:
1128:
1123:
1117:
1115:
1113:
1109:
1105:
1101:
1096:
1094:
1090:
1086:
1082:
1081:Counting rods
1073:
1059:
1057:
1053:
1048:
1044:
1034:
1032:
1028:
1018:
1016:
1012:
1009:Japanese name
1007:
993:
991:
987:
982:
978:
968:
966:
962:
952:
950:
946:
941:
931:
929:
925:
921:
915:
913:
909:
904:
892:
890:
886:
878:
876:
872:
869:
865:
860:
856:
848:
846:
842:
837:
825:
823:
819:
811:
809:
805:
802:
798:
793:
789:
781:
779:
775:
767:
765:
761:
756:
752:
749:Counting rods
747:
741:
737:
733:
729:
718:
713:
711:
706:
704:
699:
698:
696:
695:
692:
689:
688:
681:
678:
676:
673:
671:
668:
666:
663:
661:
658:
656:
653:
651:
648:
644:
641:
639:
636:
634:
631:
630:
629:
628:Alphasyllabic
626:
624:
621:
619:
616:
615:
612:
609:
608:
604:
601:
599:
596:
594:
591:
589:
586:
584:
581:
579:
576:
574:
571:
569:
566:
564:
561:
559:
556:
554:
551:
549:
546:
545:
541:
540:
536:
530:
529:
516:
513:
510:
503:
500:
497:
496:
487:
484:
482:
479:
476:
469:
466:
463:
456:
453:
450:
443:
440:
439:
436:
433:
432:
428:
425:
423:
420:
418:
415:
413:
410:
408:
405:
403:
400:
398:
395:
393:
390:
388:
385:
383:
380:
378:
375:
374:
370:
369:
365:
358:
357:
349:
346:
344:
341:
340:
336:
335:
331:
328:
326:
323:
321:
318:
316:
313:
311:
308:
306:
303:
302:
299:
296:
295:
291:
288:
287:
284:
281:
280:
276:
273:
272:
268:Other systems
264:
263:
256:
253:
251:
250:Counting rods
248:
247:
243:
242:
238:
235:
233:
230:
228:
225:
223:
220:
216:
213:
212:
211:
208:
207:
203:
202:
194:
193:
186:
183:
181:
178:
176:
173:
171:
168:
166:
163:
161:
158:
156:
153:
151:
148:
146:
143:
142:
138:
135:
133:
130:
128:
125:
123:
120:
118:
115:
113:
110:
108:
105:
103:
100:
98:
95:
93:
90:
88:
85:
84:
80:
77:
75:
72:
71:
67:
61:
60:
54:
48:
47:
44:
40:
36:
32:
31:
19:
3867:(in Chinese)
3846:
3827:
3821:
3808:
3802:
3789:
3783:
3774:
3768:
3757:. Retrieved
3753:
3744:
3735:
3714:
3702:
3684:
3661:
3649:
3638:. Retrieved
3635:ResearchGate
3634:
3624:
3619:老子: 善數者不用籌策。
3615:
3606:
3586:
3579:
3570:
3559:. Retrieved
3555:
3546:
3537:
3524:
3505:
3493:, retrieved
3489:the original
3483:
3477:
3460:
3456:
3446:
3416:Rod calculus
3387:
3379:
3372:
3156:
3147:
3143:
3130:
3103:
3087:square roots
3077:rod calculus
3072:
3070:
3067:Rod calculus
3061:Rod calculus
3056:
3014:
3005:
3001:
2987:
2975:
2969:
2949:
2931:
2925:
2921:
2895:
2862:
2855:
2851:
2813:
2620:
2033:
2021:
2017:
2014:Rod numerals
1893:
1521:
1443:
1380:
1376:
1306:
1287:
1280:
1273:
1221:
1202:
1195:
1161:
1154:
1147:
1124:
1121:
1100:rod numerals
1099:
1097:
1080:
1079:
1056:Romanization
875:Hanyu Pinyin
808:Hanyu Pinyin
758:Chinese name
494:
455:Signed-digit
337:Contemporary
249:
204:Contemporary
3540:: 370, 416.
3421:Tally marks
3148:tens digits
3144:unit digits
3140:code points
2846:After Seki
2629:Traditional
2545:Horizontal
2353:Horizontal
2161:Horizontal
1822:Horizontal
1638:Horizontal
1373:Place value
1205:Sui dynasty
1197:Book of Han
1164:Western Han
1150:Han dynasty
943:Korean name
734:(Pascal's)
633:Akṣarapallī
603:Tally marks
502:Non-integer
18:Rod numeral
3922:Categories
3759:2024-04-28
3640:2020-08-05
3561:2020-08-05
3495:2007-12-16
3438:References
3115:Chinatowns
3091:cube roots
3085:, finding
2814:In Japan,
2621:Examples:
1894:Examples:
1222:After the
740:Zhu Shijie
670:Glagolitic
643:Kaṭapayādi
611:Alphabetic
515:Asymmetric
364:radix/base
305:Cistercian
290:Babylonian
237:Vietnamese
92:Devanagari
3899:Translate
3871:Translate
3484:中国独特的计算工具
3469:0003-9519
3352:𝍸
3347:𝍷
3342:𝍶
3337:𝍵
3332:𝍴
3327:𝍳
3322:𝍲
3317:𝍱
3312:𝍰
3302:𝍯
3297:𝍮
3292:𝍭
3287:𝍬
3282:𝍫
3277:𝍪
3272:𝍩
3267:𝍨
3262:𝍧
3257:𝍦
3252:𝍥
3247:𝍤
3242:𝍣
3237:𝍢
3232:𝍡
3227:𝍠
3038:Fraction
3030:Fractions
2470:Vertical
2278:Vertical
2086:Vertical
1751:Vertical
1567:Vertical
1157:Mawangdui
1085:East Asia
638:Āryabhaṭa
583:Kharosthi
475:factorial
442:Bijective
350:(Iñupiaq)
180:Sundanese
175:Mongolian
122:Malayalam
3953:Numerals
3513:Archived
3399:See also
3308:U+1D37x
3223:U+1D36x
1383:Jia Xian
1246:: 71,824
1139:Changsha
1031:Hiragana
996:san'gaji
920:toán trù
918:que tính
814:suànchóu
736:triangle
732:Yang Hui
665:Georgian
655:Cyrillic
623:Armenian
578:Etruscan
573:Egyptian
481:Negative
348:Kaktovik
343:Cherokee
320:Pentadic
244:Historic
227:Japanese
160:Javanese
150:Balinese
137:Dzongkha
102:Gurmukhi
97:Gujarati
35:a series
33:Part of
3909:,
3905:,
3897: (
3881:,
3877:,
3869: (
3132:Unicode
3121:Unicode
3052:
3040:
2840:Western
2435:
2243:
2051:
2007:
1971:
1961:
1939:
1903:
1825:
1754:
1716:
1641:
1570:
1532:
1362:
1337:
1228:algebra
1182:544 to
1172:Shaanxi
1133:(左家公山)
1118:History
1089:integer
928:Hán-Nôm
845:Chinese
568:Chuvash
486:Complex
283:Ancient
275:History
222:Hokkien
210:Chinese
155:Burmese
145:Tibetan
132:Kannada
112:Sinhala
87:Bengali
3911:Yandex
3903:Google
3893:
3883:Yandex
3875:Google
3865:
3778:, 2004
3691:
3594:
3467:
3406:Abacus
3170:
3165:(PDF)
3107:abacus
2858:+ 246
2826:using
2764:−6720
1983:−6720
1224:abacus
1108:digits
1064:sanchū
998:sanmok
949:Hangul
881:suànzǐ
680:Hebrew
650:Coptic
563:Brahmi
548:Aegean
505:
489:
471:
458:
445:
315:Muisca
255:Tangut
232:Korean
215:Suzhou
127:Telugu
3832:(PDF)
3813:(PDF)
3794:(PDF)
3534:(PDF)
3373:Notes
3093:, or
2725:−407
2676:5089
1958:−407
1929:5089
1418:萬千百十步
1415:七一八二四
1192:Laozi
1176:Sunzi
1143:Hunan
1106:with
1062:sangi
1039:さんちゅう
1015:Kanji
965:Hanja
675:Greek
660:Geʽez
618:Abjad
598:Roman
558:Aztec
553:Attic
468:Mixed
325:Quipu
310:Mayan
165:Khmer
117:Tamil
3907:Bing
3879:Bing
3689:ISBN
3592:ISBN
3538:SOAS
3465:ISSN
3010:N/A
2843:Seki
2637:231
1900:231
330:Rumi
185:Thai
107:Odia
3075:or
2993:七甲乙
2924:− 6
2913:二四六
2834:.
2273:−9
2270:−8
2267:−7
2264:−6
2261:−5
2258:−4
2255:−3
2252:−2
2249:−1
2026:by
1746:−9
1743:−8
1740:−7
1737:−6
1734:−5
1731:−4
1728:−3
1725:−2
1722:−1
1170:in
1135:Chu
1091:or
1037:さんぎ
955:산가지
934:𣠗併
889:IPA
822:IPA
362:By
170:Lao
3924::
3752:.
3726:^
3673:^
3633:.
3554:.
3536:.
3461:36
3459:.
3455:.
3385:2.
3377:1.
3218:F
3101:.
3089:,
3021:八甲
3004:/
2981:甲乙
2970:xy
2961:六乙
2955:五甲
2854:+
2465:9
2462:8
2459:7
2456:6
2453:5
2450:4
2447:3
2444:2
2441:1
2438:0
2246:0
2081:9
2078:8
2075:7
2072:6
2069:5
2066:4
2063:3
2060:2
2057:1
2054:0
1719:0
1562:9
1559:8
1556:7
1553:6
1550:5
1547:4
1544:3
1541:2
1538:1
1535:0
1479:步
1476:十
1473:百
1470:千
1467:萬
1462:四
1459:二
1456:八
1453:一
1450:七
1302:go
1230:.
1211:.
1184:c.
1180:c.
1145:.
1141:,
1095:.
1023:算籌
1021:算木
973:算木
971:算-
957:산목
936:算籌
851:算子
784:算筹
770:算籌
427:60
422:20
417:16
412:12
407:10
37:on
3913:)
3885:)
3762:.
3643:.
3600:.
3564:.
3471:.
3215:E
3212:D
3209:C
3206:B
3203:A
3200:9
3197:8
3194:7
3191:6
3188:5
3185:4
3182:3
3179:2
3176:1
3173:0
3082:π
3049:7
3046:/
3043:1
3015:乙
3006:y
3002:x
3000:8
2968:7
2943:乙
2937:甲
2926:y
2922:x
2920:5
2907:乙
2901:甲
2874:乙
2868:甲
2856:y
2852:x
1178:(
716:e
709:t
702:v
511:)
509:φ
507:(
498:)
495:i
493:2
491:(
477:)
473:(
464:)
460:(
451:)
449:1
447:(
402:8
397:6
392:5
387:4
382:3
377:2
20:)
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