122:
36:
2967:
1944:
88:
1955:
and so the arrows have no effect on 0, otherwise 0 is used as a link to create a perpetual cycle). The pattern also works with multiples of 10, by starting at 1 and simply adding 0, giving you 10, then just apply every number in the pattern to the "tens" unit as you would normally do as usual to the "ones" unit.
2356:
in products less than ten. Chinese and
Japanese share a similar system of eighty-one short, easily memorable sentences taught to students to help them learn the multiplication table up to 9 × 9. In current usage, the sentences that express products less than ten include an additional particle in both
1954:
Figure 1 is used for multiples of 1, 3, 7, and 9. Figure 2 is used for the multiples of 2, 4, 6, and 8. These patterns can be used to memorize the multiples of any number from 0 to 10, except 5. As you would start on the number you are multiplying, when you multiply by 0, you stay on 0 (0 is external
3002:
that basic mathematics facts must be learned, though there is no consensus on whether rote memorization is the best method. In recent years, a number of nontraditional methods have been devised to help children learn multiplication facts, including video-game style apps and books that aim to teach
2990:(NCTM) developed new standards which were based on the belief that all students should learn higher-order thinking skills, which recommended reduced emphasis on the teaching of traditional methods that relied on rote memorization, such as multiplication tables. Widely adopted texts such as
73:
multiplication table was traditionally taught as an essential part of elementary arithmetic around the world, as it lays the foundation for arithmetic operations with base-ten numbers. Many educators believe it is necessary to memorize the table up to 9 × 9.
173:) the product of every number from 2 to 50 times and the rows were "a list of numbers starting with one thousand, descending by hundreds to one hundred, then descending by tens to ten, then by ones to one, and then the fractions down to 1/144."
189:
published a multiplication table up to 1000 × 1000, which allows numbers to be multiplied in pairs of digits at a time. Leslie also recommended that young pupils memorize the multiplication table up to 50 × 50.
1971:
After coming to the top of this column, start with the bottom of the next column, and travel in the same direction. The number is 8. So think of the next number after 21 that ends with 8, which is 28.
2995:
2966:
1809:
This form of writing the multiplication table in columns with complete number sentences is still used in some countries, such as Bosnia and
Herzegovina, instead of the modern grids above.
3175:
The
Philosophy of Arithmetic; Exhibiting a Progressive View of the Theory and Practice of Calculation, with Tables for the Multiplication of Numbers as Far as One Thousand
2999:
3109:
2998:
after its producer, Technical
Education Research Centers) omitted aids such as multiplication tables in early editions. NCTM made it clear in their 2006
2393:. In particular, the Japanese multiplication table uses non-standard pronunciations for numbers in some specific instances (such as the replacement of
136:(570–495 BC). It is also called the Table of Pythagoras in many languages (for example French, Italian and Russian), sometimes in English. The
3188:
2987:
2991:
1817:
There is a pattern in the multiplication table that can help people to memorize the table more easily. It uses the figures below:
2973:
2346:
suggest that the multiplication table may have been introduced to Japan through
Chinese mathematical treatises such as the
1947:
Cycles of the unit digit of multiples of integers ending in 1, 3, 7 and 9 (upper row), and 2, 4, 6 and 8 (lower row) on a
111:
3209:
3074:
2389:, because the sentences remind them to shift one column to the right when inputting a product that does not begin with a
1968:
The next number in the direction of the arrow is 1. So think of the next number after 14 that ends with 1, which is 21.
1965:
The next number in the direction of the arrow is 4. So think of the next number after 7 that ends with 4, which is 14.
3147:
193:
The illustration below shows a table up to 12 × 12, which is a size commonly used nowadays in
English-world schools.
3232:
39:
Multiplication table from 1 to 10 drawn to scale with the upper-right half labeled with prime factorisations
3237:
2334:
107:
about 4000 years ago. However, they used a base of 60. The oldest known tables using a base of 10 are the
150:
29:
2976:
A modern representation of the
Warring States decimal multiplication table used to calculate 12 × 34.5
739:, many schools use a smaller table as below. Some schools even remove the first column since 1 is the
35:
20:
3227:
3127:
740:
186:
3192:
158:
multiplication table is on a wax tablet dated to the 1st century AD and currently housed in the
166:
104:
3159:
David W. Maher and John F. Makowski. "Literary evidence for Roman arithmetic with fractions".
2957:
92:
3020:, an early computer that used tables stored in memory to perform addition and multiplication
121:
3121:
2960:(清華簡) collection is the world's earliest known example of a decimal multiplication table.
998:
of multiplication was based on memorization of columns in the table, arranged as follows.
8:
3140:
History of
Mathematics, Volume I: General Survey of the History of Elementary Mathematics
2318:
1997:
1993:
108:
3092:
3047:
2041:
2001:
56:
3143:
3096:
155:
145:
126:
132:
The multiplication table is sometimes attributed to the ancient Greek mathematician
3082:
2005:
1948:
63:
2390:
3038:
Trivett, John (1980), "The
Multiplication Table: To Be Memorized or Mastered!",
2343:
2953:
2347:
170:
159:
115:
96:
60:
3087:
28:"Times table" redirects here. For a table of departure and arrival times, see
3221:
995:
3210:
The 2,300-year-old matrix is the world's oldest decimal multiplication table
3012:
2350:, because their expression of the multiplication table share the character
2009:
1943:
736:
141:
137:
87:
44:
3051:
133:
1974:
Proceed in the same way until the last number, 3, corresponding to 63.
2947:
3142:. New York: Dover Publications (a reprint of the 1951 publication),
3070:
3017:
2322:
1980:
Then, start again with the 7. This time it will correspond to 77.
2386:
2382:
70:
2981:
2339:
2328:
2381:). This is useful for those who practice calculation with a
3110:
Wikisource:Page:Popular
Science Monthly Volume 26.djvu/467
1962:
Look at the 7 in the first picture and follow the arrow.
103:
The oldest known multiplication tables were used by the
2376:
169:
wrote a 98-column multiplication table which gave (in
3075:"Ancient times table hidden in Chinese bamboo strips"
2952:
A bundle of 21 bamboo slips dated 305 BC in the
2364:
1977:
Next, use the 0 at the bottom. It corresponds to 70.
3191:. Yomiuri Shimbun. December 4, 2010. Archived from
2948:Warring States decimal multiplication bamboo slips
2357:languages. In the case of modern Chinese, this is
3219:
1820:
1958:For example, to recall all the multiples of 7:
99:era decimal multiplication table of 305 BC
3003:times tables through character-based stories.
2370:
2358:
2351:
112:decimal multiplication table on bamboo strips
2982:Standards-based mathematics reform in the US
1992:Tables can also define binary operations on
114:dating to about 305 BC, during China's
2988:National Council of Teachers of Mathematics
2992:Investigations in Numbers, Data, and Space
2329:Chinese and Japanese multiplication tables
3086:
1812:
148:, included a multiplication table in his
1942:
120:
86:
34:
3037:
3220:
3172:
1987:
735:Because multiplication of integers is
3065:
3063:
3061:
3177:. Edinburgh: Abernethy & Walker.
3123:An Elementary Treatise on Arithmetic
2054:
3069:
2008:. In such contexts they are called
82:
13:
3058:
2405:The Japanese multiplication table
14:
3249:
2019:, addition and multiplication in
1796:12 × 12 = 144
1794:11 × 12 = 132
1792:10 × 12 = 120
1736:12 × 11 = 132
1734:11 × 11 = 121
1732:10 × 11 = 110
1676:12 × 10 = 120
1674:11 × 10 = 110
1672:10 × 10 = 100
19:For the Matthew Shipp album, see
2965:
1790:9 × 12 = 108
1614:12 × 9 = 108
144:(60–120 AD), a follower of
3040:For the Learning of Mathematics
2044:.) For example, the tables for
1785:8 × 12 = 96
1780:7 × 12 = 84
1775:6 × 12 = 72
1770:5 × 12 = 60
1765:4 × 12 = 48
1760:3 × 12 = 36
1755:2 × 12 = 24
1750:1 × 12 = 12
1730:9 × 11 = 99
1725:8 × 11 = 88
1720:7 × 11 = 77
1715:6 × 11 = 66
1710:5 × 11 = 55
1705:4 × 11 = 44
1700:3 × 11 = 33
1695:2 × 11 = 22
1690:1 × 11 = 11
1670:9 × 10 = 90
1665:8 × 10 = 80
1660:7 × 10 = 70
1655:6 × 10 = 60
1650:5 × 10 = 50
1645:4 × 10 = 40
1640:3 × 10 = 30
1635:2 × 10 = 20
1630:1 × 10 = 10
1612:11 × 9 = 99
1610:10 × 9 = 90
1554:12 × 8 = 96
1552:11 × 8 = 88
1550:10 × 8 = 80
1494:12 × 7 = 84
1492:11 × 7 = 77
1490:10 × 7 = 70
1434:12 × 6 = 72
1432:11 × 6 = 66
1430:10 × 6 = 60
1374:12 × 5 = 60
1372:11 × 5 = 55
1370:10 × 5 = 50
1312:12 × 4 = 48
1310:11 × 4 = 44
1308:10 × 4 = 40
1252:12 × 3 = 36
1250:11 × 3 = 33
1248:10 × 3 = 30
1192:12 × 2 = 24
1190:11 × 2 = 22
1188:10 × 2 = 20
1132:12 × 1 = 12
1130:11 × 1 = 11
1128:10 × 1 = 10
176:
154:, whereas the oldest surviving
3199:
3181:
3166:
3163:, 96/4 (October 2001), p. 383.
3153:
3132:
3114:
3103:
3031:
2028:, the ring of integers modulo
1745:0 × 12 = 0
1685:0 × 11 = 0
1625:0 × 10 = 0
1608:9 × 9 = 81
1603:8 × 9 = 72
1598:7 × 9 = 63
1593:6 × 9 = 54
1588:5 × 9 = 45
1583:4 × 9 = 36
1578:3 × 9 = 27
1573:2 × 9 = 18
1548:9 × 8 = 72
1543:8 × 8 = 64
1538:7 × 8 = 56
1533:6 × 8 = 48
1528:5 × 8 = 40
1523:4 × 8 = 32
1518:3 × 8 = 24
1513:2 × 8 = 16
1488:9 × 7 = 63
1483:8 × 7 = 56
1478:7 × 7 = 49
1473:6 × 7 = 42
1468:5 × 7 = 35
1463:4 × 7 = 28
1458:3 × 7 = 21
1453:2 × 7 = 14
1428:9 × 6 = 54
1423:8 × 6 = 48
1418:7 × 6 = 42
1413:6 × 6 = 36
1408:5 × 6 = 30
1403:4 × 6 = 24
1398:3 × 6 = 18
1393:2 × 6 = 12
1368:9 × 5 = 45
1363:8 × 5 = 40
1358:7 × 5 = 35
1353:6 × 5 = 30
1348:5 × 5 = 25
1343:4 × 5 = 20
1338:3 × 5 = 15
1333:2 × 5 = 10
1306:9 × 4 = 36
1301:8 × 4 = 32
1296:7 × 4 = 28
1291:6 × 4 = 24
1286:5 × 4 = 20
1281:4 × 4 = 16
1276:3 × 4 = 12
1246:9 × 3 = 27
1241:8 × 3 = 24
1236:7 × 3 = 21
1231:6 × 3 = 18
1226:5 × 3 = 15
1221:4 × 3 = 12
1186:9 × 2 = 18
1181:8 × 2 = 16
1176:7 × 2 = 14
1171:6 × 2 = 12
1166:5 × 2 = 10
1072:12 × 0 = 0
1070:11 × 0 = 0
1068:10 × 0 = 0
1:
3024:
1568:1 × 9 = 9
1563:0 × 9 = 0
1508:1 × 8 = 8
1503:0 × 8 = 0
1448:1 × 7 = 7
1443:0 × 7 = 0
1388:1 × 6 = 6
1383:0 × 6 = 0
1328:1 × 5 = 5
1323:0 × 5 = 0
1271:2 × 4 = 8
1266:1 × 4 = 4
1261:0 × 4 = 0
1216:3 × 3 = 9
1211:2 × 3 = 6
1206:1 × 3 = 3
1201:0 × 3 = 0
1161:4 × 2 = 8
1156:3 × 2 = 6
1151:2 × 2 = 4
1146:1 × 2 = 2
1141:0 × 2 = 0
1126:9 × 1 = 9
1121:8 × 1 = 8
1116:7 × 1 = 7
1111:6 × 1 = 6
1106:5 × 1 = 5
1101:4 × 1 = 4
1096:3 × 1 = 3
1091:2 × 1 = 2
1086:1 × 1 = 1
1081:0 × 1 = 0
1066:9 × 0 = 0
1061:8 × 0 = 0
1056:7 × 0 = 0
1051:6 × 0 = 0
1046:5 × 0 = 0
1041:4 × 0 = 0
1036:3 × 0 = 0
1031:2 × 0 = 0
1026:1 × 0 = 0
1021:0 × 0 = 0
51:(sometimes, less formally, a
2369:); and in Japanese, this is
2335:Chinese multiplication table
183:The Philosophy of Arithmetic
7:
3006:
2377:
2365:
10:
3254:
2332:
862:
835:
811:
790:
772:
757:
151:Introduction to Arithmetic
77:
30:Timetable (disambiguation)
27:
18:
3088:10.1038/nature.2014.14482
2371:
2359:
2352:
2015:For every natural number
1935:
1932:
1927:
1921:
1904:
1901:
1860:
1849:
1846:
1834:
1829:
1826:
1823:
125:"Table of Pythagoras" on
66:for an algebraic system.
3189:"「九九」は中国伝来...平城宮跡から木簡出土"
2317:For other examples, see
22:The Multiplication Table
3138:David E. Smith (1958),
741:multiplicative identity
1951:
1813:Patterns in the tables
167:Victorius of Aquitaine
129:
100:
40:
3233:Mathematics education
3173:Leslie, John (1820).
2958:Tsinghua Bamboo Slips
2877:hachi shichi gojūroku
2872:hachi roku shijūhachi
2824:shichi shichi shijūku
2794:shichi ichi ga shichi
2032:, is described by an
1946:
124:
93:Tsinghua Bamboo Slips
90:
38:
3195:on December 7, 2010.
2809:shichi shi nijūhachi
2665:shi shichi nijūhachi
49:multiplication table
3238:Mathematical tables
3161:Classical Philology
3073:(January 7, 2014).
2930:ku shichi rokujūsan
2847:hachi ichi ga hachi
2834:shichi ku rokujūsan
2819:shichi roku shijūni
2804:shichi san nijūichi
2771:roku shichi shijūni
2766:roku roku sanjūroku
2612:san shichi nijūichi
2406:
1988:In abstract algebra
1983:Continue like this.
2829:shichi ha gojūroku
2776:roku ha shijūhachi
2506:inshichi ga shichi
2404:
2042:Modular arithmetic
1952:
130:
101:
57:mathematical table
41:
16:Mathematical table
2994:(widely known as
2945:
2944:
2940:ku ku hachijūichi
2910:ku san nijūshichi
2862:hachi shi sanjūni
2857:hachi san nijūshi
2814:shichi go sanjūgo
2741:roku ichi ga roku
2718:go shichi sanjūgo
2622:san ku nijūshichi
2314:
2313:
2307:
2306:
2181:
2180:
2006:algebraic systems
1940:
1939:
1806:
1805:
991:
990:
732:
731:
181:In his 1820 book
146:Neopythagoreanism
59:used to define a
3245:
3212:
3203:
3197:
3196:
3185:
3179:
3178:
3170:
3164:
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3130:
3118:
3112:
3107:
3101:
3100:
3090:
3067:
3056:
3054:
3035:
2969:
2935:ku ha shichijūni
2915:ku shi sanjūroku
2887:hakku shichijūni
2756:roku shi nijūshi
2751:roku san jūhachi
2675:shi ku sanjūroku
2660:shi roku nijūshi
2607:saburoku jūhachi
2511:inhachi ga hachi
2407:
2403:
2380:
2374:
2373:
2368:
2362:
2361:
2355:
2354:
2185:
2184:
2059:
2058:
2055:
1949:telephone keypad
1821:
1789:
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1002:
1001:
994:The traditional
747:
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197:
196:
185:, mathematician
165:In 493 AD,
83:Pre-modern times
3253:
3252:
3248:
3247:
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3120:for example in
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3108:
3104:
3068:
3059:
3036:
3032:
3027:
3009:
2984:
2979:
2978:
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2975:
2970:
2950:
2925:ku roku gojūshi
2882:happa rokujūshi
2852:hachi ni jūroku
2799:shichi ni jūshi
2640:shi ni ga hachi
2635:shi ichi ga shi
2582:san ichi ga san
2564:ni hachi jūroku
2559:ni shichi jūshi
2544:ni shi ga hachi
2476:in'ichi ga ichi
2337:
2331:
2315:
2049:
2026:
1990:
1941:
1936:Figure 2: Even
1815:
1807:
1797:
1795:
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733:
179:
85:
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33:
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12:
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5:
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3241:
3240:
3235:
3230:
3228:Multiplication
3214:
3213:
3198:
3180:
3165:
3152:
3150:, pp. 58, 129.
3131:
3113:
3102:
3057:
3029:
3028:
3026:
3023:
3022:
3021:
3015:
3008:
3005:
2983:
2980:
2972:
2971:
2964:
2963:
2962:
2956:period in the
2954:Warring States
2949:
2946:
2943:
2942:
2937:
2932:
2927:
2922:
2917:
2912:
2907:
2902:
2897:
2890:
2889:
2884:
2879:
2874:
2869:
2867:hachi go shijū
2864:
2859:
2854:
2849:
2844:
2837:
2836:
2831:
2826:
2821:
2816:
2811:
2806:
2801:
2796:
2791:
2784:
2783:
2778:
2773:
2768:
2763:
2758:
2753:
2748:
2743:
2738:
2731:
2730:
2725:
2720:
2715:
2710:
2705:
2700:
2695:
2690:
2685:
2678:
2677:
2672:
2670:shi ha sanjūni
2667:
2662:
2657:
2652:
2650:shi shi jūroku
2647:
2642:
2637:
2632:
2625:
2624:
2619:
2614:
2609:
2604:
2599:
2594:
2589:
2587:san ni ga roku
2584:
2579:
2572:
2571:
2566:
2561:
2556:
2551:
2546:
2541:
2539:ni san ga roku
2536:
2531:
2526:
2519:
2518:
2513:
2508:
2503:
2501:inroku ga roku
2498:
2493:
2488:
2483:
2478:
2473:
2466:
2465:
2459:
2453:
2447:
2441:
2435:
2429:
2423:
2417:
2411:
2348:Sunzi Suanjing
2342:discovered at
2333:Main article:
2330:
2327:
2312:
2311:
2305:
2304:
2301:
2298:
2295:
2292:
2289:
2285:
2284:
2281:
2278:
2275:
2272:
2269:
2265:
2264:
2261:
2258:
2255:
2252:
2249:
2245:
2244:
2241:
2238:
2235:
2232:
2229:
2225:
2224:
2221:
2218:
2215:
2212:
2209:
2205:
2204:
2201:
2198:
2195:
2192:
2189:
2182:
2179:
2178:
2175:
2172:
2169:
2166:
2163:
2159:
2158:
2155:
2152:
2149:
2146:
2143:
2139:
2138:
2135:
2132:
2129:
2126:
2123:
2119:
2118:
2115:
2112:
2109:
2106:
2103:
2099:
2098:
2095:
2092:
2089:
2086:
2083:
2079:
2078:
2075:
2072:
2069:
2066:
2063:
2053:
2047:
2022:
1989:
1986:
1985:
1984:
1981:
1978:
1975:
1972:
1969:
1966:
1963:
1938:
1937:
1934:
1933:Figure 1: Odd
1930:
1929:
1926:
1923:
1920:
1918:
1915:
1913:
1910:
1907:
1906:
1903:
1899:
1898:
1895:
1893:
1890:
1887:
1884:
1880:
1879:
1877:
1875:
1873:
1870:
1867:
1863:
1862:
1859:
1856:
1854:
1851:
1848:
1845:
1842:
1839:
1836:
1832:
1831:
1828:
1825:
1819:
1814:
1811:
1804:
1803:
1801:
1799:
1739:
1679:
1618:
1617:
1557:
1497:
1437:
1377:
1316:
1315:
1255:
1195:
1135:
1075:
1014:
1013:
1011:
1009:
1007:
1005:
1000:
989:
988:
985:
982:
979:
976:
973:
970:
967:
964:
961:
957:
956:
953:
950:
947:
944:
941:
938:
935:
932:
929:
925:
924:
922:
919:
916:
913:
910:
907:
904:
901:
898:
894:
893:
891:
889:
886:
883:
880:
877:
874:
871:
868:
864:
863:
861:
859:
856:
853:
850:
847:
844:
841:
837:
836:
834:
832:
829:
826:
823:
820:
817:
813:
812:
810:
808:
805:
802:
799:
796:
792:
791:
789:
787:
784:
781:
778:
774:
773:
771:
769:
766:
763:
759:
758:
756:
754:
751:
745:
730:
729:
726:
723:
720:
717:
714:
711:
708:
705:
702:
699:
696:
693:
689:
688:
685:
682:
679:
676:
673:
670:
667:
664:
661:
658:
655:
652:
648:
647:
644:
641:
638:
635:
632:
629:
626:
623:
620:
617:
614:
611:
607:
606:
603:
600:
597:
594:
591:
588:
585:
582:
579:
576:
573:
570:
566:
565:
562:
559:
556:
553:
550:
547:
544:
541:
538:
535:
532:
529:
525:
524:
521:
518:
515:
512:
509:
506:
503:
500:
497:
494:
491:
488:
484:
483:
480:
477:
474:
471:
468:
465:
462:
459:
456:
453:
450:
447:
443:
442:
439:
436:
433:
430:
427:
424:
421:
418:
415:
412:
409:
406:
402:
401:
398:
395:
392:
389:
386:
383:
380:
377:
374:
371:
368:
365:
361:
360:
357:
354:
351:
348:
345:
342:
339:
336:
333:
330:
327:
324:
320:
319:
316:
313:
310:
307:
304:
301:
298:
295:
292:
289:
286:
283:
279:
278:
275:
272:
269:
266:
263:
260:
257:
254:
251:
248:
245:
242:
238:
237:
234:
231:
228:
225:
222:
219:
216:
213:
210:
207:
204:
201:
195:
178:
175:
171:Roman numerals
160:British Museum
140:mathematician
127:Napier's bones
116:Warring States
97:Warring States
84:
81:
79:
76:
61:multiplication
15:
9:
6:
4:
3:
2:
3250:
3239:
3236:
3234:
3231:
3229:
3226:
3225:
3223:
3211:
3207:
3202:
3194:
3190:
3184:
3176:
3169:
3162:
3156:
3149:
3148:0-486-20429-4
3145:
3141:
3135:
3129:
3125:
3124:
3117:
3111:
3106:
3098:
3094:
3089:
3084:
3080:
3076:
3072:
3066:
3064:
3062:
3053:
3049:
3045:
3041:
3034:
3030:
3019:
3016:
3014:
3011:
3010:
3004:
3001:
2997:
2993:
2989:
2986:In 1989, the
2974:
2968:
2961:
2959:
2955:
2941:
2938:
2936:
2933:
2931:
2928:
2926:
2923:
2921:
2920:ku go shijūgo
2918:
2916:
2913:
2911:
2908:
2906:
2905:ku ni jūhachi
2903:
2901:
2900:ku ichi ga ku
2898:
2896:
2892:
2891:
2888:
2885:
2883:
2880:
2878:
2875:
2873:
2870:
2868:
2865:
2863:
2860:
2858:
2855:
2853:
2850:
2848:
2845:
2843:
2839:
2838:
2835:
2832:
2830:
2827:
2825:
2822:
2820:
2817:
2815:
2812:
2810:
2807:
2805:
2802:
2800:
2797:
2795:
2792:
2790:
2786:
2785:
2782:
2781:rokku gojūshi
2779:
2777:
2774:
2772:
2769:
2767:
2764:
2762:
2761:roku go sanjū
2759:
2757:
2754:
2752:
2749:
2747:
2744:
2742:
2739:
2737:
2733:
2732:
2729:
2728:gokku shijūgo
2726:
2724:
2721:
2719:
2716:
2714:
2713:go roku sanjū
2711:
2709:
2706:
2704:
2701:
2699:
2696:
2694:
2691:
2689:
2688:go ichi ga go
2686:
2684:
2680:
2679:
2676:
2673:
2671:
2668:
2666:
2663:
2661:
2658:
2656:
2653:
2651:
2648:
2646:
2643:
2641:
2638:
2636:
2633:
2631:
2627:
2626:
2623:
2620:
2618:
2617:sanpa nijūshi
2615:
2613:
2610:
2608:
2605:
2603:
2600:
2598:
2595:
2593:
2590:
2588:
2585:
2583:
2580:
2578:
2574:
2573:
2570:
2569:ni ku jūhachi
2567:
2565:
2562:
2560:
2557:
2555:
2552:
2550:
2547:
2545:
2542:
2540:
2537:
2535:
2534:ni nin ga shi
2532:
2530:
2529:ni ichi ga ni
2527:
2525:
2521:
2520:
2517:
2514:
2512:
2509:
2507:
2504:
2502:
2499:
2497:
2494:
2492:
2489:
2487:
2484:
2482:
2479:
2477:
2474:
2472:
2468:
2467:
2464:
2460:
2458:
2454:
2452:
2448:
2446:
2442:
2440:
2436:
2434:
2430:
2428:
2424:
2422:
2418:
2416:
2412:
2409:
2408:
2402:
2400:
2396:
2392:
2388:
2384:
2379:
2367:
2349:
2345:
2341:
2336:
2326:
2324:
2320:
2310:
2302:
2299:
2296:
2293:
2290:
2287:
2286:
2282:
2279:
2276:
2273:
2270:
2267:
2266:
2262:
2259:
2256:
2253:
2250:
2247:
2246:
2242:
2239:
2236:
2233:
2230:
2227:
2226:
2222:
2219:
2216:
2213:
2210:
2207:
2206:
2202:
2199:
2196:
2193:
2190:
2187:
2186:
2183:
2176:
2173:
2170:
2167:
2164:
2161:
2160:
2156:
2153:
2150:
2147:
2144:
2141:
2140:
2136:
2133:
2130:
2127:
2124:
2121:
2120:
2116:
2113:
2110:
2107:
2104:
2101:
2100:
2096:
2093:
2090:
2087:
2084:
2081:
2080:
2076:
2073:
2070:
2067:
2064:
2061:
2060:
2057:
2056:
2052:
2050:
2043:
2039:
2035:
2031:
2027:
2025:
2018:
2013:
2011:
2010:Cayley tables
2007:
2003:
1999:
1995:
1982:
1979:
1976:
1973:
1970:
1967:
1964:
1961:
1960:
1959:
1956:
1950:
1945:
1931:
1924:
1919:
1916:
1914:
1911:
1909:
1908:
1900:
1896:
1894:
1891:
1888:
1885:
1882:
1881:
1878:
1876:
1874:
1871:
1868:
1865:
1864:
1857:
1855:
1852:
1843:
1840:
1837:
1833:
1822:
1818:
1810:
1802:
1800:
1798:
1740:
1738:
1680:
1678:
1620:
1619:
1616:
1558:
1556:
1498:
1496:
1438:
1436:
1378:
1376:
1318:
1317:
1314:
1256:
1254:
1196:
1194:
1136:
1134:
1076:
1074:
1016:
1015:
1012:
1010:
1008:
1006:
1004:
1003:
999:
997:
996:rote learning
986:
983:
980:
977:
974:
971:
968:
965:
962:
959:
958:
954:
951:
948:
945:
942:
939:
936:
933:
930:
927:
926:
923:
920:
917:
914:
911:
908:
905:
902:
899:
896:
895:
892:
890:
887:
884:
881:
878:
875:
872:
869:
866:
865:
860:
857:
854:
851:
848:
845:
842:
839:
838:
833:
830:
827:
824:
821:
818:
815:
814:
809:
806:
803:
800:
797:
794:
793:
788:
785:
782:
779:
776:
775:
770:
767:
764:
761:
760:
755:
752:
749:
748:
744:
742:
738:
727:
724:
721:
718:
715:
712:
709:
706:
703:
700:
697:
694:
691:
690:
686:
683:
680:
677:
674:
671:
668:
665:
662:
659:
656:
653:
650:
649:
645:
642:
639:
636:
633:
630:
627:
624:
621:
618:
615:
612:
609:
608:
604:
601:
598:
595:
592:
589:
586:
583:
580:
577:
574:
571:
568:
567:
563:
560:
557:
554:
551:
548:
545:
542:
539:
536:
533:
530:
527:
526:
522:
519:
516:
513:
510:
507:
504:
501:
498:
495:
492:
489:
486:
485:
481:
478:
475:
472:
469:
466:
463:
460:
457:
454:
451:
448:
445:
444:
440:
437:
434:
431:
428:
425:
422:
419:
416:
413:
410:
407:
404:
403:
399:
396:
393:
390:
387:
384:
381:
378:
375:
372:
369:
366:
363:
362:
358:
355:
352:
349:
346:
343:
340:
337:
334:
331:
328:
325:
322:
321:
317:
314:
311:
308:
305:
302:
299:
296:
293:
290:
287:
284:
281:
280:
276:
273:
270:
267:
264:
261:
258:
255:
252:
249:
246:
243:
240:
239:
235:
232:
229:
226:
223:
220:
217:
214:
211:
208:
205:
202:
199:
198:
194:
191:
188:
184:
174:
172:
168:
163:
161:
157:
153:
152:
147:
143:
139:
135:
128:
123:
119:
117:
113:
110:
106:
98:
94:
89:
75:
72:
67:
65:
62:
58:
54:
50:
46:
37:
31:
24:
23:
3205:
3201:
3193:the original
3183:
3174:
3168:
3160:
3155:
3139:
3134:
3122:
3116:
3105:
3078:
3046:(1): 21–25,
3043:
3039:
3033:
3013:Vedic square
3000:Focal Points
2985:
2951:
2939:
2934:
2929:
2924:
2919:
2914:
2909:
2904:
2899:
2894:
2886:
2881:
2876:
2871:
2866:
2861:
2856:
2851:
2846:
2841:
2833:
2828:
2823:
2818:
2813:
2808:
2803:
2798:
2793:
2788:
2780:
2775:
2770:
2765:
2760:
2755:
2750:
2746:roku ni jūni
2745:
2740:
2735:
2727:
2722:
2717:
2712:
2708:go go nijūgo
2707:
2702:
2697:
2692:
2687:
2682:
2674:
2669:
2664:
2659:
2654:
2649:
2645:shi san jūni
2644:
2639:
2634:
2629:
2621:
2616:
2611:
2606:
2601:
2597:san shi jūni
2596:
2591:
2586:
2581:
2576:
2568:
2563:
2558:
2554:ni roku jūni
2553:
2548:
2543:
2538:
2533:
2528:
2523:
2515:
2510:
2505:
2500:
2495:
2491:inshi ga shi
2490:
2486:insan ga san
2485:
2480:
2475:
2470:
2462:
2456:
2450:
2444:
2438:
2432:
2426:
2420:
2414:
2398:
2394:
2344:Heijō Palace
2338:
2316:
2308:
2045:
2040:table. (See
2037:
2033:
2029:
2023:
2020:
2016:
2014:
2004:, and other
1991:
1957:
1953:
1816:
1808:
1741:
1681:
1621:
1559:
1499:
1439:
1379:
1319:
1257:
1197:
1137:
1077:
1017:
993:
734:
192:
182:
180:
177:Modern times
164:
149:
131:
102:
68:
52:
48:
42:
21:
3128:John Farrar
3079:Nature News
2723:go ha shijū
2703:go shi nijū
2698:go san jūgo
2655:shi go nijū
2602:san go jūgo
2592:sazan ga ku
737:commutative
187:John Leslie
142:Nichomachus
138:Greco-Roman
105:Babylonians
53:times table
45:mathematics
3222:Categories
3025:References
2516:inku ga ku
2496:ingo ga go
2481:inni ga ni
2391:tens digit
134:Pythagoras
95:, Chinese
3097:130132289
3071:Qiu, Jane
64:operation
3208:article
3052:40247697
3018:IBM 1620
3007:See also
2693:go ni jū
2549:ni go jū
2399:saburoku
2395:san roku
2323:octonion
118:period.
2387:soroban
2383:suanpan
1928:
1922:
1827:
1788:
1783:
1778:
1773:
1768:
1763:
1758:
1753:
1748:
1743:
1728:
1723:
1718:
1713:
1708:
1703:
1698:
1693:
1688:
1683:
1668:
1663:
1658:
1653:
1648:
1643:
1638:
1633:
1628:
1623:
1606:
1601:
1596:
1591:
1586:
1581:
1576:
1571:
1566:
1561:
1546:
1541:
1536:
1531:
1526:
1521:
1516:
1511:
1506:
1501:
1486:
1481:
1476:
1471:
1466:
1461:
1456:
1451:
1446:
1441:
1426:
1421:
1416:
1411:
1406:
1401:
1396:
1391:
1386:
1381:
1366:
1361:
1356:
1351:
1346:
1341:
1336:
1331:
1326:
1321:
1304:
1299:
1294:
1289:
1284:
1279:
1274:
1269:
1264:
1259:
1244:
1239:
1234:
1229:
1224:
1219:
1214:
1209:
1204:
1199:
1184:
1179:
1174:
1169:
1164:
1159:
1154:
1149:
1144:
1139:
1124:
1119:
1114:
1109:
1104:
1099:
1094:
1089:
1084:
1079:
1064:
1059:
1054:
1049:
1044:
1039:
1034:
1029:
1024:
1019:
109:Chinese
78:History
71:decimal
55:) is a
3206:Nature
3146:
3095:
3050:
2789:shichi
2451:shichi
2340:Mokkan
2321:, and
2309:
1998:fields
1994:groups
3093:S2CID
3048:JSTOR
2842:hachi
2397:with
2385:or a
2319:group
2051:are:
2002:rings
156:Greek
3144:ISBN
2996:TERC
2736:roku
2445:roku
2415:ichi
728:144
687:132
646:120
605:108
91:The
69:The
47:, a
3126:by
3083:doi
2630:shi
2577:san
2433:shi
2427:san
2401:).
2036:by
955:81
725:132
722:120
719:108
692:12
684:121
681:110
651:11
643:110
640:100
610:10
564:96
523:84
482:72
441:60
400:48
359:36
318:24
277:12
236:12
233:11
230:10
43:In
3224::
3091:.
3081:.
3077:.
3060:^
3042:,
2895:ku
2893:9
2840:8
2787:7
2734:6
2683:go
2681:5
2628:4
2575:3
2524:ni
2522:2
2471:in
2469:1
2463:ku
2461:9
2457:ha
2455:8
2449:7
2443:6
2439:go
2437:5
2431:4
2425:3
2421:ni
2419:2
2413:1
2410:×
2378:ga
2366:dé
2325:.
2303:1
2288:4
2283:2
2268:3
2263:3
2248:2
2243:4
2228:1
2223:0
2208:0
2203:4
2200:3
2197:2
2194:1
2191:0
2188:×
2177:3
2162:4
2157:2
2142:3
2137:1
2122:2
2117:0
2102:1
2097:4
2082:0
2077:4
2074:3
2071:2
2068:1
2065:0
2062:+
2012:.
2000:,
1996:,
1925:0
1917:5
1905:←
1902:←
1897:8
1892:6
1889:9
1886:8
1883:7
1872:6
1861:↓
1858:4
1853:2
1850:↑
1847:↓
1844:3
1841:2
1838:1
1835:↑
1830:→
1824:→
987:9
984:8
981:7
978:6
975:5
972:4
969:3
966:2
963:1
960:×
952:72
949:63
946:54
943:45
940:36
937:27
934:18
928:9
921:64
918:56
915:48
912:40
909:32
906:24
903:16
897:8
888:49
885:42
882:35
879:28
876:21
873:14
867:7
858:36
855:30
852:24
849:18
846:12
840:6
831:25
828:20
825:15
822:10
816:5
807:16
804:12
795:4
777:3
762:2
750:1
743:.
716:96
713:84
710:72
707:60
704:48
701:36
698:24
695:12
678:99
675:88
672:77
669:66
666:55
663:44
660:33
657:22
654:11
637:90
634:80
631:70
628:60
625:50
622:40
619:30
616:20
613:10
602:99
599:90
596:81
593:72
590:63
587:54
584:45
581:36
578:27
575:18
569:9
561:88
558:80
555:72
552:64
549:56
546:48
543:40
540:32
537:24
534:16
528:8
520:77
517:70
514:63
511:56
508:49
505:42
502:35
499:28
496:21
493:14
487:7
479:66
476:60
473:54
470:48
467:42
464:36
461:30
458:24
455:18
452:12
446:6
438:55
435:50
432:45
429:40
426:35
423:30
420:25
417:20
414:15
411:10
405:5
397:44
394:40
391:36
388:32
385:28
382:24
379:20
376:16
373:12
364:4
356:33
353:30
350:27
347:24
344:21
341:18
338:15
335:12
323:3
315:22
312:20
309:18
306:16
303:14
300:12
297:10
282:2
274:11
271:10
241:1
227:9
224:8
221:7
218:6
215:5
212:4
209:3
206:2
203:1
200:×
162:.
3099:.
3085::
3055:.
3044:1
2375:(
2372:が
2363:(
2360:得
2353:如
2300:2
2297:3
2294:4
2291:0
2280:4
2277:1
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2260:1
2257:4
2254:2
2251:0
2240:3
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2234:1
2231:0
2220:0
2217:0
2214:0
2211:0
2174:2
2171:1
2168:0
2165:4
2154:1
2151:0
2148:4
2145:3
2134:0
2131:4
2128:3
2125:2
2114:4
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2108:2
2105:1
2094:3
2091:2
2088:1
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2048:5
2046:Z
2038:n
2034:n
2030:n
2024:n
2021:Z
2017:n
1912:0
1869:5
1866:4
931:9
900:8
870:7
843:6
819:5
801:8
798:4
786:9
783:6
780:3
768:4
765:2
753:1
572:9
531:8
490:7
449:6
408:5
370:8
367:4
332:9
329:6
326:3
294:8
291:6
288:4
285:2
268:9
265:8
262:7
259:6
256:5
253:4
250:3
247:2
244:1
32:.
25:.
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