30:, while optionally ignoring any 9s or digits which sum to 9 or a multiple of 9. The result of this procedure is a number which is smaller than the original whenever the original has more than one digit, leaves the same remainder as the original after division by nine, and may be obtained from the original by subtracting a multiple of 9 from it. The name of the procedure derives from this latter property.
568:
67:
More generally, when casting out nines by summing digits, any set of digits which add up to 9, or a multiple of 9, can be ignored. In the number 3264, for example, the digits 3 and 6 sum to 9. Ignoring these two digits, therefore, and summing the other two, we get 2 + 4 = 6. Since 6 = 3264 − 362 ×
2842:
Any non-negative integer can be written as 9×n + a, where 'a' is a single digit from 0 to 8, and 'n' is some non-negative integer. Thus, using the distributive rule, (9×n + a)×(9×m + b)= 9×9×n×m + 9(am + bn) + ab. Since the first two factors are multiplied by 9, their sums will end up being 9 or 0,
2838:
A trick to learn to add with nines is to add ten to the digit and to count back one. Since we are adding 1 to the tens digit and subtracting one from the units digit, the sum of the digits should remain the same. For example, 9 + 2 = 11 with 1 + 1 = 2. When adding 9 to itself, we would thus expect
2851:
While extremely useful, casting out nines does not catch all errors made while doing calculations. For example, the casting-out-nines method would not recognize the error in a calculation of 5 × 7 which produced any of the erroneous results 8, 17, 26, etc. (that is, any result congruent to 8 modulo
857:
The number 12565, for instance, has digit sum 1+2+5+6+5 = 19, which, in turn, has digit sum 1+9=10, which, in its turn has digit sum 1+0=1, a single-digit number. The digital root of 12565 is therefore 1, and its computation has the effect of casting out (12565 - 1)/9 = 1396 lots of 9 from 12565.
866:
To check the result of an arithmetical calculation by casting out nines, each number in the calculation is replaced by its digital root and the same calculations applied to these digital roots. The digital root of the result of this calculation is then compared with that of the result of the
44:
in which the above-mentioned procedures are used to check for errors in arithmetical calculations. The test is carried out by applying the same sequence of arithmetical operations to the digital roots of the operands as are applied to the operands themselves. If no mistakes are made in the
64:. The digit sum of 2946, for example is 2 + 9 + 4 + 6 = 21. Since 21 = 2946 − 325 × 9, the effect of taking the digit sum of 2946 is to "cast out" 325 lots of 9 from it. If the digit 9 is ignored when summing the digits, the effect is to "cast out" one more 9 to give the result 12.
289:
2839:
the sum of the digits to be 9 as follows: 9 + 9 = 18, (1 + 8 = 9) and 9 + 9 + 9 = 27, (2 + 7 = 9). Let us look at a simple multiplication: 5 × 7 = 35, (3 + 5 = 8). Now consider (7 + 9) × 5 = 16 × 5 = 80, (8 + 0 = 8) or 7 × (9 + 5) = 7 × 14 = 98, (9 + 8 = 17), (1 + 7 = 8).
827:
2826:
If a calculation was correct before casting out, casting out on both sides will preserve correctness. However, it is possible that two previously unequal integers will be identical modulo 9 (on average, a ninth of the time).
1689:
2206:
2091:
703:
563:{\displaystyle {\begin{aligned}&10^{n}d_{n}+10^{n-1}d_{n-1}+\cdots +d_{0}-\left(d_{n}+d_{n-1}+\cdots +d_{0}\right)\\={}&\left(10^{n}-1\right)d_{n}+\left(10^{n-1}-1\right)d_{n-1}+\cdots +9d_{1}.\end{aligned}}}
1565:
1064:
1276:
1787:
3205:
1385:
3078:
160:
967:
2995:
2823:(for example, modulo 7) will always have the same sum, difference or product as their originals. This property is also preserved for the 'digit sum' where the base and the modulus differ by 1.
294:
1967:
2387:
1163:
281:
2856:, such as 1324 instead of 1234. In other words, the method only catches erroneous results whose digital root is one of the 8 digits that is different from that of the correct result.
216:
2892:
were used to compute a unique "root" between 1 and 9. Neither of them displayed any awareness of how the procedure could be used to check the results of arithmetical computations.
846:
If the procedure described in the preceding paragraph is repeatedly applied to the result of each previous application, the eventual result will be a single-digit number from which
854:
of the original. The exception occurs when the original number has a digital root of 9, whose digit sum is itself, and therefore will not be cast out by taking further digit sums.
2233:
1814:
1412:
2670:
2314:
1891:
1484:
1209:
1013:
867:
original calculation. If no mistake has been made in the calculations, these two digital roots must be the same. Examples in which casting-out-nines has been used to check
33:
Repeated application of this procedure to the results obtained from previous applications until a single-digit number is obtained. This single-digit number is called the "
2112:
1988:
1710:
1586:
1297:
1184:
1085:
988:
2623:
2600:
2577:
2554:
2259:
1840:
1438:
45:
calculations, the digital roots of the two resultants will be the same. If they are different, therefore, one or more mistakes must have been made in the calculations.
2843:
leaving us with 'ab'. In our example, 'a' was 7 and 'b' was 5. We would expect that in any base system, the number before that base would behave just like the nine.
604:
2715:
2408:
1616:
2471:
2429:
2293:
1870:
1463:
1321:
1109:
2739:
2694:
2516:
2495:
1734:
3125:
3105:
2781:
2760:
2649:
2450:
2335:
2133:
2009:
1912:
1505:
711:
37:" of the original. If a number is divisible by 9, its digital root is 9. Otherwise, its digital root is the remainder it leaves after being divided by 9.
2799:
The method works because the original numbers are 'decimal' (base 10), the modulus is chosen to differ by 1, and casting out is equivalent to taking a
1635:
2144:
2024:
609:
1525:
1024:
2895:
The earliest known surviving work which describes how casting out nines can be used to check the results of arithmetical computations is the
1220:
2907:) (c.980–1037), also gave full details of what he called the "Hindu method" of checking arithmetical calculations by casting out nines.
1745:
3565:
3423:
3139:
1332:
3012:
74:
920:
3431:
2935:
2888:. Both Hippolytus's and Iamblichus's descriptions, though, were limited to an explanation of how repeated digital sums of
3210:
So we can use the remainder from casting out nine hundred ninety nines to get the remainder of division by thirty seven.
2787:
In other words, perform the same procedure as in a multiplication, only backwards. 8x4=32 which is 5, 5+3 = 8. And 8=8.
1932:
2358:
3371:
3083:
So we can use the remainder from casting out ninety nines to get the remainder of division by eleven. This is called
60:
To "cast out nines" from a single number, its decimal digits can be simply added together to obtain its so-called
2830:
The operation does not work on fractions, since a given fractional number does not have a unique representation.
1120:
900:, cross out all 9s and pairs of digits that total 9, then add together what remains. These new values are called
221:
3087:. The same result can also be calculated directly by alternately adding and subtracting the digits that make up
850:
9s, with the possible exception of one, have been "cast out". The resulting single-digit number is called the
165:
2214:
1795:
1393:
2880:
2870:
2655:
2299:
1876:
1469:
3254:
3476:
3402:
2864:
A form of casting out nines known to ancient Greek mathematicians was described by the Roman bishop
1190:
994:
1916:
The difference between the minuend and the subtrahend excesses should equal the difference excess.
2097:
1973:
1695:
1571:
1282:
1169:
1070:
973:
3453:
3303:
3287:
3246:
2926:
This method can be generalized to determine the remainders of division by certain prime numbers.
2608:
2585:
2562:
2539:
2244:
1825:
1423:
3481:, vol. V, American reprint of the Edinburgh edition, New York, NY: Charles Scribner's Sons
880:
3000:
So we can use the remainder from casting out nines to get the remainder of division by three.
576:
3359:
3230:
2700:
2266:
27:
3511:
3345:
2393:
1592:
8:
3560:
3472:
3380:
2853:
2456:
2414:
2276:
1853:
1446:
1303:
1091:
3003:
Casting out ninety nines is done by adding groups of two digits instead just one digit.
2721:
2676:
2501:
2477:
1716:
3398:
3110:
3090:
2865:
2766:
2745:
2634:
2627:
Add up all uncrossed digits from each value until one digit is reached for each value.
2435:
2320:
2118:
1994:
1897:
1490:
822:{\displaystyle {\frac {10^{n}-1}{9}}d_{n}+{\frac {10^{n-1}-1}{9}}d_{n-1}+\cdots +d_{1}}
3478:
The Ante-Nicene
Fathers. Translations of The Writings of the Fathers down to A.D. 325.
1846:
Because subtracting 2 from zero gives a negative number, borrow a 9 from the minuend.
3497:
3494:
3468:
3457:
3437:
3427:
3367:
904:. Add up leftover digits for each addend until one digit is reached. Now process the
16:
Arithmetic procedure of verifying operations using modulo characteristics of digit 9
3417:
3130:
Casting out nine hundred ninety nines is done by adding groups of three digits.
705:), replacing the original number by its digit sum has the effect of casting out
3419:
An
Episodic History of Mathematics—Mathematical Culture through Problem Solving
3384:
2889:
2220:
1801:
1684:{\displaystyle {\underline {-\ 2\ {\bcancel {8}}{\bcancel {9}}{\bcancel {1}}}}}
1399:
876:
3525:
3349:
3554:
1818:
Now follow the same procedure with the difference, coming to a single digit.
3517:
2339:
The excess from the product should equal the final excess from the factors.
2900:
2201:{\displaystyle {{\bcancel {3}}\ 4\ \ 4\ {\bcancel {6}}{\bcancel {9}}\ 2\ }}
841:
34:
2086:{\displaystyle {\underline {\ \ \ \ \ \ \times \ 6\ \ 2\ {\bcancel {9}}}}}
3536:
2785:
The dividend excess should equal the final excess from the other values.
2137:
Add up leftover digits for each multiplicand until one digit is reached.
2014:
872:
698:{\displaystyle 10^{i}-1=9\times \left(10^{i-1}+10^{i-2}+\cdots +1\right)}
68:
9, this computation has resulted in casting out 362 lots of 9 from 3264.
41:
1560:{\displaystyle {\bcancel {5}}{\bcancel {6}}{\bcancel {4}}{\bcancel {3}}}
1509:
The excess from the sum should equal the final excess from the addends.
1059:{\displaystyle {\bcancel {8}}{\bcancel {4}}{\bcancel {1}}{\bcancel {5}}}
2885:
2875:
1625:
1271:{\displaystyle {\underline {+{\bcancel {3}}\ 2\ \ 0\ {\bcancel {6}}}}}
3502:
2911:
2800:
2529:
905:
61:
55:
2903:(c.920–c.1000). Writing about 1020, the Persian polymath, Ibn Sina (
2237:
Multiply the two excesses, and then add until one digit is reached.
2904:
2525:
868:
283:. The difference between the original number and its digit sum is
3441:
1738:
Add up leftover digits for each value until one digit is reached.
2521:
1621:
3461:
2899:, written around 950 by the Indian mathematician and astronomer
1782:{\displaystyle {\bcancel {2}}\ 7\ {\bcancel {5}}{\bcancel {2}}}
897:
3185:
3169:
3147:
3058:
3042:
3020:
2975:
2962:
2943:
861:
3366:(New ed.), New York, NY: Macmillan Publishing Company,
3200:{\displaystyle n{\bmod {3}}7=(n{\bmod {9}}99){\bmod {3}}7.}
1380:{\displaystyle {\bcancel {1}}\ 7\ {\bcancel {8}}\ 3\ \ 1\ }
3492:
3073:{\displaystyle n{\bmod {1}}1=(n{\bmod {9}}9){\bmod {1}}1.}
2874:, and more briefly by the Syrian Neoplatonist philosopher
162:, normally represented by the sequence of decimal digits,
3317:
The
Penguin Dictionary of Curious and Interesting Numbers
155:{\displaystyle 10^{n}d_{n}+10^{n-1}d_{n-1}+\cdots +d_{0}}
2013:
First, cross out all 9s and digits that total 9 in each
1620:
First, cross out all 9s and digits that total 9 in both
962:{\displaystyle {\bcancel {3}}\ \ 2\ {\bcancel {6}}\ 4\ }
2990:{\displaystyle n{\bmod {3}}=(n{\bmod {9}}){\bmod {3}}.}
3364:
Synergetics: Explorations in the
Geometry of Thinking
3142:
3113:
3093:
3015:
2938:
2769:
2748:
2724:
2703:
2679:
2658:
2637:
2611:
2588:
2565:
2542:
2504:
2480:
2459:
2438:
2417:
2396:
2361:
2323:
2302:
2279:
2247:
2217:
2147:
2121:
2100:
2027:
1997:
1976:
1935:
1900:
1879:
1856:
1828:
1798:
1748:
1719:
1698:
1638:
1595:
1574:
1528:
1493:
1472:
1449:
1426:
1396:
1335:
1306:
1285:
1223:
1193:
1172:
1123:
1094:
1073:
1027:
997:
976:
923:
714:
612:
579:
292:
224:
168:
77:
2852:
9). In particular, casting out nines does not catch
3397:
3250:
3199:
3119:
3099:
3072:
2989:
2846:
2775:
2754:
2733:
2709:
2688:
2664:
2643:
2617:
2594:
2571:
2548:
2510:
2489:
2465:
2444:
2423:
2402:
2381:
2329:
2308:
2287:
2253:
2227:
2200:
2127:
2106:
2085:
2003:
1982:
1961:
1906:
1885:
1864:
1834:
1808:
1781:
1728:
1704:
1683:
1610:
1580:
1559:
1499:
1478:
1457:
1432:
1406:
1379:
1315:
1291:
1270:
1203:
1178:
1157:
1103:
1079:
1058:
1007:
982:
961:
821:
697:
598:
562:
275:
210:
154:
3319:. Middlesex, England: Penguin Books, p. 74, 1986.
2833:
2370:
2363:
2182:
2175:
2150:
2071:
1962:{\displaystyle {\bcancel {5}}{\bcancel {4}}\ 8\ }
1944:
1937:
1773:
1766:
1750:
1669:
1662:
1655:
1551:
1544:
1537:
1530:
1353:
1337:
1256:
1231:
1195:
1131:
1050:
1043:
1036:
1029:
999:
944:
925:
3552:
3467:
3408:
2520:Cross out all 9s and digits that total 9 in the
2382:{\displaystyle {\bcancel {27}}{\bcancel {54}}62}
2345:8 times 8 is 64; 6 and 4 are 10; 1 and 0 are 1.
1416:6, 0, 3 and 2 make 11; 1 and 1 add up to 2.
3351:History of Hindu Mathematics: A Source Book
1158:{\displaystyle 2\ {\bcancel {9}}\ 4\ \ 6\ }
1113:8+1=9 and 4+5=9; there are no digits left.
276:{\displaystyle d_{n}+d_{n-1}+\cdots +d_{0}}
3344:
3299:
3283:
862:Checking calculations by casting out nines
2269:, crossing out 9s and getting one digit.
22:is any of three arithmetical procedures:
3450:Elementary Introduction to Number Theory
3127:if and only if eleven divides that sum.
3424:The Mathematical Association of America
3266:The Greek term used by Hippolytus was "
2878:(c.245–c.325) in his commentary on the
2803:. In general any two 'large' integers,
211:{\displaystyle d_{n}d_{n-1}\dots d_{0}}
3553:
3415:
3358:
3226:
3493:
3379:
3242:
1213:2, 4, and 6 make 12; 1 and 2 make 3.
3447:
3328:
3407:, translated by MacMahon, J.H., In
3348:; Singh, Avadhesh Narayan (1962) ,
2228:{\displaystyle {\bigg \Downarrow }}
1809:{\displaystyle {\bigg \Downarrow }}
1407:{\displaystyle {\bigg \Downarrow }}
13:
14:
3577:
3523:
3486:
3393:, Oxford: Oxford University Press
2921:
1922:
606:are always divisible by 9 (since
2665:{\displaystyle \Leftrightarrow }
2309:{\displaystyle \Leftrightarrow }
1886:{\displaystyle \Leftrightarrow }
1479:{\displaystyle \Leftrightarrow }
835:
3354:, Bombay: Asia Publishing House
2910:The procedure was described by
2847:Limitation to casting out nines
2794:
908:and also the excesses to get a
26:Adding the decimal digits of a
3566:Error detection and correction
3404:The Refutation of all Heresies
3386:A History of Greek Mathematics
3322:
3309:
3293:
3277:
3260:
3236:
3220:
3181:
3162:
3054:
3035:
2971:
2955:
2871:The Refutation of all Heresies
2834:A variation on the explanation
2728:
2680:
2659:
2612:
2589:
2566:
2543:
2303:
2248:
2101:
1977:
1880:
1829:
1699:
1605:
1599:
1575:
1515:
1473:
1427:
1286:
1204:{\displaystyle {\bcancel {3}}}
1173:
1074:
1008:{\displaystyle {\bcancel {6}}}
977:
1:
3409:Roberts & Donaldson (1919
3338:
49:
2273:
2241:
2107:{\displaystyle \Rightarrow }
1983:{\displaystyle \Rightarrow }
1850:
1822:
1705:{\displaystyle \Rightarrow }
1581:{\displaystyle \Rightarrow }
1443:
1420:
1292:{\displaystyle \Rightarrow }
1179:{\displaystyle \Rightarrow }
1080:{\displaystyle \Rightarrow }
983:{\displaystyle \Rightarrow }
573:Because numbers of the form
7:
3452:(2nd ed.), Lexington:
2811:, expressed in any smaller
2618:{\displaystyle \Downarrow }
2595:{\displaystyle \Downarrow }
2572:{\displaystyle \Downarrow }
2549:{\displaystyle \Downarrow }
2348:
2254:{\displaystyle \Downarrow }
2141:
2021:
1929:
1835:{\displaystyle \Downarrow }
1742:
1632:
1522:
1433:{\displaystyle \Downarrow }
1329:
1217:
1117:
1021:
917:
891:
886:
10:
3582:
3416:Krantz, Steven G. (2010),
2881:Introduction to Arithmetic
2859:
2784:
2626:
2519:
839:
53:
2211:
1792:
1390:
71:For an arbitrary number,
3514:by R. Buckminster Fuller
3448:Long, Calvin T. (1972),
3251:Hippolytus of Rome (1919
3213:
599:{\displaystyle 10^{i}-1}
3454:D. C. Heath and Company
3300:Datta & Singh (1962
3284:Datta & Singh (1962
2710:{\displaystyle \times }
3360:Fuller, R. Buckminster
3201:
3121:
3101:
3074:
2991:
2777:
2756:
2735:
2711:
2690:
2666:
2645:
2619:
2596:
2573:
2550:
2512:
2491:
2467:
2446:
2425:
2404:
2383:
2331:
2310:
2289:
2255:
2229:
2202:
2129:
2108:
2087:
2005:
1984:
1963:
1908:
1887:
1866:
1836:
1810:
1783:
1730:
1706:
1685:
1612:
1582:
1561:
1501:
1480:
1459:
1434:
1408:
1381:
1317:
1293:
1272:
1205:
1180:
1159:
1105:
1081:
1060:
1009:
984:
963:
823:
699:
600:
564:
277:
212:
156:
3391:From Thales to Euclid
3202:
3122:
3102:
3075:
2992:
2778:
2757:
2736:
2712:
2691:
2667:
2646:
2620:
2597:
2574:
2551:
2513:
2492:
2468:
2447:
2426:
2405:
2403:{\displaystyle \div }
2384:
2332:
2311:
2290:
2265:Do the same with the
2256:
2230:
2203:
2130:
2109:
2088:
2006:
1985:
1964:
1909:
1888:
1867:
1837:
1811:
1784:
1731:
1707:
1686:
1613:
1583:
1562:
1502:
1481:
1460:
1435:
1409:
1382:
1318:
1294:
1273:
1206:
1181:
1160:
1106:
1082:
1061:
1017:2 and 4 add up to 6.
1010:
985:
964:
824:
700:
601:
565:
278:
213:
157:
28:positive whole number
3518:"Paranormal Numbers"
3346:Datta, Bibhatibhusan
3140:
3111:
3091:
3013:
2936:
2886:Nicomachus of Gerasa
2854:transposition errors
2767:
2746:
2722:
2701:
2677:
2656:
2635:
2609:
2586:
2563:
2540:
2502:
2478:
2457:
2436:
2415:
2394:
2374:
2367:
2359:
2321:
2300:
2277:
2245:
2215:
2186:
2179:
2154:
2145:
2119:
2098:
2075:
2025:
1995:
1974:
1948:
1941:
1933:
1898:
1877:
1854:
1826:
1796:
1777:
1770:
1754:
1746:
1717:
1696:
1673:
1666:
1659:
1636:
1611:{\displaystyle 0(9)}
1593:
1572:
1555:
1548:
1541:
1534:
1526:
1491:
1470:
1447:
1424:
1394:
1357:
1341:
1333:
1304:
1283:
1260:
1235:
1221:
1199:
1191:
1170:
1135:
1121:
1092:
1071:
1054:
1047:
1040:
1033:
1025:
1003:
995:
974:
948:
929:
921:
712:
610:
577:
290:
222:
166:
75:
3526:"Casting Out Nines"
3498:"Casting Out Nines"
3133:Since 37·27 = 999,
3085:casting out elevens
2466:{\displaystyle 314}
2424:{\displaystyle 877}
2288:{\displaystyle {1}}
1865:{\displaystyle {7}}
1458:{\displaystyle {2}}
1316:{\displaystyle \ 2}
1104:{\displaystyle \ 0}
218:, the digit sum is
3495:Weisstein, Eric W.
3469:Roberts, Alexander
3399:Hippolytus of Rome
3197:
3117:
3097:
3070:
2987:
2773:
2752:
2734:{\displaystyle 8)}
2731:
2707:
2689:{\displaystyle (4}
2686:
2662:
2641:
2615:
2592:
2569:
2546:
2511:{\displaystyle 84}
2508:
2490:{\displaystyle r.}
2487:
2463:
2442:
2421:
2400:
2379:
2327:
2306:
2285:
2251:
2225:
2198:
2125:
2104:
2083:
2081:
2001:
1980:
1959:
1904:
1883:
1862:
1832:
1806:
1779:
1729:{\displaystyle -2}
1726:
1702:
1681:
1679:
1608:
1578:
1557:
1497:
1476:
1455:
1430:
1404:
1377:
1313:
1289:
1268:
1266:
1201:
1176:
1155:
1101:
1077:
1056:
1005:
980:
959:
819:
695:
596:
560:
558:
273:
208:
152:
3433:978-0-88385-766-3
3411:, pp. 9–153)
3120:{\displaystyle n}
3107:. Eleven divides
3100:{\displaystyle n}
3006:Since 11·9 = 99,
2792:
2791:
2776:{\displaystyle 3}
2755:{\displaystyle +}
2644:{\displaystyle 8}
2445:{\displaystyle =}
2343:
2342:
2330:{\displaystyle 1}
2196:
2190:
2173:
2167:
2164:
2158:
2128:{\displaystyle 8}
2069:
2063:
2060:
2054:
2048:
2045:
2042:
2039:
2036:
2033:
2029:
2004:{\displaystyle 8}
1958:
1952:
1920:
1919:
1907:{\displaystyle 7}
1764:
1758:
1653:
1647:
1640:
1513:
1512:
1500:{\displaystyle 2}
1376:
1370:
1367:
1361:
1351:
1345:
1309:
1254:
1248:
1245:
1239:
1225:
1154:
1148:
1145:
1139:
1129:
1097:
958:
952:
942:
936:
933:
883:are given below.
782:
738:
20:Casting out nines
3573:
3547:
3545:
3543:
3530:
3520:by Paul Niquette
3508:
3507:
3482:
3473:Donaldson, James
3464:
3444:
3412:
3394:
3376:
3355:
3332:
3326:
3320:
3313:
3307:
3297:
3291:
3281:
3275:
3264:
3258:
3240:
3234:
3224:
3206:
3204:
3203:
3198:
3193:
3192:
3177:
3176:
3155:
3154:
3126:
3124:
3123:
3118:
3106:
3104:
3103:
3098:
3079:
3077:
3076:
3071:
3066:
3065:
3050:
3049:
3028:
3027:
2996:
2994:
2993:
2988:
2983:
2982:
2970:
2969:
2951:
2950:
2782:
2780:
2779:
2774:
2761:
2759:
2758:
2753:
2740:
2738:
2737:
2732:
2716:
2714:
2713:
2708:
2695:
2693:
2692:
2687:
2671:
2669:
2668:
2663:
2650:
2648:
2647:
2642:
2624:
2622:
2621:
2616:
2601:
2599:
2598:
2593:
2578:
2576:
2575:
2570:
2555:
2553:
2552:
2547:
2517:
2515:
2514:
2509:
2496:
2494:
2493:
2488:
2472:
2470:
2469:
2464:
2451:
2449:
2448:
2443:
2430:
2428:
2427:
2422:
2409:
2407:
2406:
2401:
2388:
2386:
2385:
2380:
2375:
2368:
2353:
2352:
2336:
2334:
2333:
2328:
2315:
2313:
2312:
2307:
2294:
2292:
2291:
2286:
2284:
2260:
2258:
2257:
2252:
2234:
2232:
2231:
2226:
2224:
2223:
2207:
2205:
2204:
2199:
2197:
2194:
2188:
2187:
2180:
2171:
2165:
2162:
2156:
2155:
2134:
2132:
2131:
2126:
2113:
2111:
2110:
2105:
2092:
2090:
2089:
2084:
2082:
2077:
2076:
2067:
2061:
2058:
2052:
2046:
2043:
2040:
2037:
2034:
2031:
2010:
2008:
2007:
2002:
1989:
1987:
1986:
1981:
1968:
1966:
1965:
1960:
1956:
1950:
1949:
1942:
1927:
1926:
1913:
1911:
1910:
1905:
1892:
1890:
1889:
1884:
1871:
1869:
1868:
1863:
1861:
1841:
1839:
1838:
1833:
1815:
1813:
1812:
1807:
1805:
1804:
1788:
1786:
1785:
1780:
1778:
1771:
1762:
1756:
1755:
1735:
1733:
1732:
1727:
1711:
1709:
1708:
1703:
1690:
1688:
1687:
1682:
1680:
1675:
1674:
1667:
1660:
1651:
1645:
1617:
1615:
1614:
1609:
1587:
1585:
1584:
1579:
1566:
1564:
1563:
1558:
1556:
1549:
1542:
1535:
1520:
1519:
1506:
1504:
1503:
1498:
1485:
1483:
1482:
1477:
1464:
1462:
1461:
1456:
1454:
1439:
1437:
1436:
1431:
1413:
1411:
1410:
1405:
1403:
1402:
1386:
1384:
1383:
1378:
1374:
1368:
1365:
1359:
1358:
1349:
1343:
1342:
1322:
1320:
1319:
1314:
1307:
1298:
1296:
1295:
1290:
1277:
1275:
1274:
1269:
1267:
1262:
1261:
1252:
1246:
1243:
1237:
1236:
1210:
1208:
1207:
1202:
1200:
1185:
1183:
1182:
1177:
1164:
1162:
1161:
1156:
1152:
1146:
1143:
1137:
1136:
1127:
1110:
1108:
1107:
1102:
1095:
1086:
1084:
1083:
1078:
1065:
1063:
1062:
1057:
1055:
1048:
1041:
1034:
1014:
1012:
1011:
1006:
1004:
989:
987:
986:
981:
968:
966:
965:
960:
956:
950:
949:
940:
934:
931:
930:
915:
914:
828:
826:
825:
820:
818:
817:
799:
798:
783:
778:
771:
770:
754:
749:
748:
739:
734:
727:
726:
716:
704:
702:
701:
696:
694:
690:
677:
676:
658:
657:
622:
621:
605:
603:
602:
597:
589:
588:
569:
567:
566:
561:
559:
552:
551:
530:
529:
514:
510:
503:
502:
479:
478:
469:
465:
458:
457:
441:
432:
428:
427:
426:
408:
407:
389:
388:
371:
370:
352:
351:
336:
335:
317:
316:
307:
306:
296:
282:
280:
279:
274:
272:
271:
253:
252:
234:
233:
217:
215:
214:
209:
207:
206:
194:
193:
178:
177:
161:
159:
158:
153:
151:
150:
132:
131:
116:
115:
97:
96:
87:
86:
3581:
3580:
3576:
3575:
3574:
3572:
3571:
3570:
3551:
3550:
3541:
3539:
3528:
3489:
3475:, eds. (1919),
3434:
3389:, vol. I:
3374:
3341:
3336:
3335:
3327:
3323:
3314:
3310:
3298:
3294:
3282:
3278:
3265:
3261:
3241:
3237:
3225:
3221:
3216:
3188:
3184:
3172:
3168:
3150:
3146:
3141:
3138:
3137:
3112:
3109:
3108:
3092:
3089:
3088:
3061:
3057:
3045:
3041:
3023:
3019:
3014:
3011:
3010:
2978:
2974:
2965:
2961:
2946:
2942:
2937:
2934:
2933:
2929:Since 3·3 = 9,
2924:
2862:
2849:
2836:
2797:
2768:
2765:
2764:
2747:
2744:
2743:
2723:
2720:
2719:
2702:
2699:
2698:
2678:
2675:
2674:
2657:
2654:
2653:
2636:
2633:
2632:
2610:
2607:
2606:
2587:
2584:
2583:
2564:
2561:
2560:
2541:
2538:
2537:
2503:
2500:
2499:
2479:
2476:
2475:
2458:
2455:
2454:
2437:
2434:
2433:
2416:
2413:
2412:
2395:
2392:
2391:
2369:
2362:
2360:
2357:
2356:
2351:
2322:
2319:
2318:
2301:
2298:
2297:
2280:
2278:
2275:
2274:
2246:
2243:
2242:
2219:
2218:
2216:
2213:
2212:
2181:
2174:
2149:
2148:
2146:
2143:
2142:
2120:
2117:
2116:
2099:
2096:
2095:
2070:
2030:
2028:
2026:
2023:
2022:
1996:
1993:
1992:
1975:
1972:
1971:
1943:
1936:
1934:
1931:
1930:
1925:
1899:
1896:
1895:
1878:
1875:
1874:
1857:
1855:
1852:
1851:
1827:
1824:
1823:
1800:
1799:
1797:
1794:
1793:
1772:
1765:
1749:
1747:
1744:
1743:
1718:
1715:
1714:
1697:
1694:
1693:
1668:
1661:
1654:
1641:
1639:
1637:
1634:
1633:
1594:
1591:
1590:
1573:
1570:
1569:
1550:
1543:
1536:
1529:
1527:
1524:
1523:
1518:
1492:
1489:
1488:
1471:
1468:
1467:
1450:
1448:
1445:
1444:
1425:
1422:
1421:
1398:
1397:
1395:
1392:
1391:
1352:
1336:
1334:
1331:
1330:
1325:2 and 0 are 2.
1305:
1302:
1301:
1284:
1281:
1280:
1255:
1230:
1226:
1224:
1222:
1219:
1218:
1194:
1192:
1189:
1188:
1171:
1168:
1167:
1130:
1122:
1119:
1118:
1093:
1090:
1089:
1072:
1069:
1068:
1049:
1042:
1035:
1028:
1026:
1023:
1022:
998:
996:
993:
992:
975:
972:
971:
943:
924:
922:
919:
918:
894:
889:
864:
844:
838:
813:
809:
788:
784:
760:
756:
755:
753:
744:
740:
722:
718:
717:
715:
713:
710:
709:
666:
662:
647:
643:
642:
638:
617:
613:
611:
608:
607:
584:
580:
578:
575:
574:
557:
556:
547:
543:
519:
515:
492:
488:
487:
483:
474:
470:
453:
449:
448:
444:
442:
440:
434:
433:
422:
418:
397:
393:
384:
380:
379:
375:
366:
362:
341:
337:
325:
321:
312:
308:
302:
298:
293:
291:
288:
287:
267:
263:
242:
238:
229:
225:
223:
220:
219:
202:
198:
183:
179:
173:
169:
167:
164:
163:
146:
142:
121:
117:
105:
101:
92:
88:
82:
78:
76:
73:
72:
58:
52:
17:
12:
11:
5:
3579:
3569:
3568:
3563:
3549:
3548:
3524:Grime, James.
3521:
3515:
3509:
3488:
3487:External links
3485:
3484:
3483:
3465:
3445:
3432:
3413:
3395:
3377:
3372:
3362:(April 1982),
3356:
3340:
3337:
3334:
3333:
3321:
3308:
3292:
3276:
3259:
3235:
3218:
3217:
3215:
3212:
3208:
3207:
3196:
3191:
3187:
3183:
3180:
3175:
3171:
3167:
3164:
3161:
3158:
3153:
3149:
3145:
3116:
3096:
3081:
3080:
3069:
3064:
3060:
3056:
3053:
3048:
3044:
3040:
3037:
3034:
3031:
3026:
3022:
3018:
2998:
2997:
2986:
2981:
2977:
2973:
2968:
2964:
2960:
2957:
2954:
2949:
2945:
2941:
2923:
2922:Generalization
2920:
2890:Greek numerals
2861:
2858:
2848:
2845:
2835:
2832:
2796:
2793:
2790:
2789:
2783:
2772:
2762:
2751:
2741:
2730:
2727:
2717:
2706:
2696:
2685:
2682:
2672:
2661:
2651:
2640:
2629:
2628:
2625:
2614:
2604:
2602:
2591:
2581:
2579:
2568:
2558:
2556:
2545:
2534:
2533:
2518:
2507:
2497:
2486:
2483:
2473:
2462:
2452:
2441:
2431:
2420:
2410:
2399:
2389:
2378:
2373:
2366:
2350:
2347:
2341:
2340:
2337:
2326:
2316:
2305:
2295:
2283:
2271:
2270:
2263:
2261:
2250:
2239:
2238:
2235:
2222:
2210:
2208:
2193:
2185:
2178:
2170:
2161:
2153:
2139:
2138:
2135:
2124:
2114:
2103:
2093:
2080:
2074:
2066:
2057:
2051:
2019:
2018:
2017:(italicized).
2011:
2000:
1990:
1979:
1969:
1955:
1947:
1940:
1924:
1923:Multiplication
1921:
1918:
1917:
1914:
1903:
1893:
1882:
1872:
1860:
1848:
1847:
1844:
1842:
1831:
1820:
1819:
1816:
1803:
1791:
1789:
1776:
1769:
1761:
1753:
1740:
1739:
1736:
1725:
1722:
1712:
1701:
1691:
1678:
1672:
1665:
1658:
1650:
1644:
1630:
1629:
1628:(italicized).
1618:
1607:
1604:
1601:
1598:
1588:
1577:
1567:
1554:
1547:
1540:
1533:
1517:
1514:
1511:
1510:
1507:
1496:
1486:
1475:
1465:
1453:
1441:
1440:
1429:
1418:
1417:
1414:
1401:
1389:
1387:
1373:
1364:
1356:
1348:
1340:
1327:
1326:
1323:
1312:
1299:
1288:
1278:
1265:
1259:
1251:
1242:
1234:
1229:
1215:
1214:
1211:
1198:
1186:
1175:
1165:
1151:
1142:
1134:
1126:
1115:
1114:
1111:
1100:
1087:
1076:
1066:
1053:
1046:
1039:
1032:
1019:
1018:
1015:
1002:
990:
979:
969:
955:
947:
939:
928:
893:
890:
888:
885:
877:multiplication
863:
860:
840:Main article:
837:
834:
830:
829:
816:
812:
808:
805:
802:
797:
794:
791:
787:
781:
777:
774:
769:
766:
763:
759:
752:
747:
743:
737:
733:
730:
725:
721:
693:
689:
686:
683:
680:
675:
672:
669:
665:
661:
656:
653:
650:
646:
641:
637:
634:
631:
628:
625:
620:
616:
595:
592:
587:
583:
571:
570:
555:
550:
546:
542:
539:
536:
533:
528:
525:
522:
518:
513:
509:
506:
501:
498:
495:
491:
486:
482:
477:
473:
468:
464:
461:
456:
452:
447:
443:
439:
436:
435:
431:
425:
421:
417:
414:
411:
406:
403:
400:
396:
392:
387:
383:
378:
374:
369:
365:
361:
358:
355:
350:
347:
344:
340:
334:
331:
328:
324:
320:
315:
311:
305:
301:
297:
295:
270:
266:
262:
259:
256:
251:
248:
245:
241:
237:
232:
228:
205:
201:
197:
192:
189:
186:
182:
176:
172:
149:
145:
141:
138:
135:
130:
127:
124:
120:
114:
111:
108:
104:
100:
95:
91:
85:
81:
54:Main article:
51:
48:
47:
46:
38:
31:
15:
9:
6:
4:
3:
2:
3578:
3567:
3564:
3562:
3559:
3558:
3556:
3538:
3534:
3527:
3522:
3519:
3516:
3513:
3510:
3505:
3504:
3499:
3496:
3491:
3490:
3480:
3479:
3474:
3470:
3466:
3463:
3459:
3455:
3451:
3446:
3443:
3439:
3435:
3429:
3425:
3421:
3420:
3414:
3410:
3406:
3405:
3400:
3396:
3392:
3388:
3387:
3382:
3381:Heath, Thomas
3378:
3375:
3373:0-02-065320-4
3369:
3365:
3361:
3357:
3353:
3352:
3347:
3343:
3342:
3331:, p. 83)
3330:
3325:
3318:
3312:
3305:
3301:
3296:
3289:
3285:
3280:
3273:
3269:
3263:
3256:
3252:
3248:
3244:
3239:
3232:
3228:
3223:
3219:
3211:
3194:
3189:
3178:
3173:
3165:
3159:
3156:
3151:
3143:
3136:
3135:
3134:
3131:
3128:
3114:
3094:
3086:
3067:
3062:
3051:
3046:
3038:
3032:
3029:
3024:
3016:
3009:
3008:
3007:
3004:
3001:
2984:
2979:
2966:
2958:
2952:
2947:
2939:
2932:
2931:
2930:
2927:
2919:
2917:
2913:
2908:
2906:
2902:
2898:
2897:Mahâsiddhânta
2893:
2891:
2887:
2883:
2882:
2877:
2873:
2872:
2868:(170–235) in
2867:
2857:
2855:
2844:
2840:
2831:
2828:
2824:
2822:
2818:
2814:
2810:
2806:
2802:
2788:
2770:
2763:
2749:
2742:
2725:
2718:
2704:
2697:
2683:
2673:
2652:
2638:
2631:
2630:
2605:
2603:
2582:
2580:
2559:
2557:
2536:
2535:
2531:
2527:
2523:
2505:
2498:
2484:
2481:
2474:
2460:
2453:
2439:
2432:
2418:
2411:
2397:
2390:
2376:
2371:
2364:
2355:
2354:
2346:
2338:
2324:
2317:
2296:
2281:
2272:
2268:
2264:
2262:
2240:
2236:
2209:
2191:
2183:
2176:
2168:
2159:
2151:
2140:
2136:
2122:
2115:
2094:
2078:
2072:
2064:
2055:
2049:
2020:
2016:
2012:
1998:
1991:
1970:
1953:
1945:
1938:
1928:
1915:
1901:
1894:
1873:
1858:
1849:
1845:
1843:
1821:
1817:
1790:
1774:
1767:
1759:
1751:
1741:
1737:
1723:
1720:
1713:
1692:
1676:
1670:
1663:
1656:
1648:
1642:
1631:
1627:
1623:
1619:
1602:
1596:
1589:
1568:
1552:
1545:
1538:
1531:
1521:
1508:
1494:
1487:
1466:
1451:
1442:
1419:
1415:
1388:
1371:
1362:
1354:
1346:
1338:
1328:
1324:
1310:
1300:
1279:
1263:
1257:
1249:
1240:
1232:
1227:
1216:
1212:
1196:
1187:
1166:
1149:
1140:
1132:
1124:
1116:
1112:
1098:
1088:
1067:
1051:
1044:
1037:
1030:
1020:
1016:
1000:
991:
970:
953:
945:
937:
926:
916:
913:
911:
907:
903:
899:
884:
882:
878:
874:
870:
859:
855:
853:
849:
843:
836:Digital roots
833:
814:
810:
806:
803:
800:
795:
792:
789:
785:
779:
775:
772:
767:
764:
761:
757:
750:
745:
741:
735:
731:
728:
723:
719:
708:
707:
706:
691:
687:
684:
681:
678:
673:
670:
667:
663:
659:
654:
651:
648:
644:
639:
635:
632:
629:
626:
623:
618:
614:
593:
590:
585:
581:
553:
548:
544:
540:
537:
534:
531:
526:
523:
520:
516:
511:
507:
504:
499:
496:
493:
489:
484:
480:
475:
471:
466:
462:
459:
454:
450:
445:
437:
429:
423:
419:
415:
412:
409:
404:
401:
398:
394:
390:
385:
381:
376:
372:
367:
363:
359:
356:
353:
348:
345:
342:
338:
332:
329:
326:
322:
318:
313:
309:
303:
299:
286:
285:
284:
268:
264:
260:
257:
254:
249:
246:
243:
239:
235:
230:
226:
203:
199:
195:
190:
187:
184:
180:
174:
170:
147:
143:
139:
136:
133:
128:
125:
122:
118:
112:
109:
106:
102:
98:
93:
89:
83:
79:
69:
65:
63:
57:
43:
39:
36:
32:
29:
25:
24:
23:
21:
3542:13 September
3540:. Retrieved
3532:
3512:"Numerology"
3501:
3477:
3449:
3418:
3403:
3390:
3385:
3363:
3350:
3324:
3316:
3311:
3295:
3279:
3271:
3267:
3262:
3238:
3227:Krantz (2010
3222:
3209:
3132:
3129:
3084:
3082:
3005:
3002:
2999:
2928:
2925:
2915:
2909:
2901:Aryabhata II
2896:
2894:
2879:
2869:
2863:
2850:
2841:
2837:
2829:
2825:
2820:
2816:
2812:
2808:
2804:
2798:
2795:How it works
2786:
2344:
909:
901:
895:
865:
856:
852:digital root
851:
847:
845:
842:Digital root
831:
572:
70:
66:
59:
35:digital root
19:
18:
3537:Brady Haran
3286:, pp.
3253:, pp.
3245:, pp.
3243:Heath (1921
3229:, pp.
2916:Liber Abaci
1516:Subtraction
873:subtraction
832:lots of 9.
42:sanity test
3561:Arithmetic
3555:Categories
3442:2010921168
3339:References
3329:Long (1972
3315:Wells, D.
3302:, p.
2876:Iamblichus
2866:Hippolytus
1626:subtrahend
50:Digit sums
3503:MathWorld
3401:(1919) ,
2912:Fibonacci
2801:digit sum
2705:×
2660:⇔
2613:⇓
2590:⇓
2567:⇓
2544:⇓
2530:remainder
2398:÷
2304:⇔
2249:⇓
2102:⇒
2079:_
2050:×
1978:⇒
1881:⇔
1830:⇓
1721:−
1700:⇒
1677:_
1643:−
1576:⇒
1474:⇔
1428:⇓
1287:⇒
1264:_
1174:⇒
1075:⇒
978:⇒
804:⋯
793:−
773:−
765:−
729:−
682:⋯
671:−
652:−
636:×
624:−
591:−
535:⋯
524:−
505:−
497:−
460:−
413:⋯
402:−
373:−
357:⋯
346:−
330:−
258:⋯
247:−
196:…
188:−
137:⋯
126:−
110:−
62:digit sum
56:Digit sum
3462:77171950
3383:(1921),
2905:Avicenna
2526:quotient
2349:Division
2221:⇓
1802:⇓
1400:⇓
912:excess.
902:excesses
896:In each
892:Addition
887:Examples
881:division
869:addition
3533:YouTube
3529:(video)
3288:180–184
3272:pythmen
3247:113–117
2914:in his
2860:History
2813:modulus
2522:divisor
2267:product
1622:minuend
3460:
3440:
3430:
3370:
3268:πυθμήν
2528:, and
2195:
2189:
2172:
2166:
2163:
2157:
2068:
2062:
2059:
2053:
2047:
2044:
2041:
2038:
2035:
2032:
2015:factor
1957:
1951:
1763:
1757:
1652:
1646:
1375:
1369:
1366:
1360:
1350:
1344:
1308:
1253:
1247:
1244:
1238:
1153:
1147:
1144:
1138:
1128:
1096:
957:
951:
941:
935:
932:
898:addend
879:, and
3255:30–32
3231:67–70
3214:Notes
910:final
3544:2017
3458:LCCN
3438:LCCN
3428:ISBN
3368:ISBN
3270:" ("
2884:of
2819:and
2807:and
1624:and
3304:184
3274:").
3249:),
3186:mod
3170:mod
3148:mod
3059:mod
3043:mod
3021:mod
2976:mod
2963:mod
2944:mod
2821:y'
2815:as
2461:314
2419:877
906:sum
848:all
40:A
3557::
3535:.
3531:.
3500:.
3471:;
3456:,
3436:,
3426:,
3422:,
3257:).
3195:7.
3179:99
3068:1.
2918:.
2817:x'
2532:.
2524:,
2506:84
2377:62
2372:54
2365:27
875:,
871:,
758:10
720:10
664:10
645:10
615:10
582:10
490:10
451:10
323:10
300:10
103:10
80:10
3546:.
3506:.
3306:)
3290:)
3233:)
3190:3
3182:)
3174:9
3166:n
3163:(
3160:=
3157:7
3152:3
3144:n
3115:n
3095:n
3063:1
3055:)
3052:9
3047:9
3039:n
3036:(
3033:=
3030:1
3025:1
3017:n
2985:.
2980:3
2972:)
2967:9
2959:n
2956:(
2953:=
2948:3
2940:n
2809:y
2805:x
2771:3
2750:+
2729:)
2726:8
2684:4
2681:(
2639:8
2485:.
2482:r
2440:=
2325:1
2282:1
2192:2
2184:9
2177:6
2169:4
2160:4
2152:3
2123:8
2073:9
2065:2
2056:6
1999:8
1954:8
1946:4
1939:5
1902:7
1859:7
1775:2
1768:5
1760:7
1752:2
1724:2
1671:1
1664:9
1657:8
1649:2
1606:)
1603:9
1600:(
1597:0
1553:3
1546:4
1539:6
1532:5
1495:2
1452:2
1372:1
1363:3
1355:8
1347:7
1339:1
1311:2
1258:6
1250:0
1241:2
1233:3
1228:+
1197:3
1150:6
1141:4
1133:9
1125:2
1099:0
1052:5
1045:1
1038:4
1031:8
1001:6
954:4
946:6
938:2
927:3
815:1
811:d
807:+
801:+
796:1
790:n
786:d
780:9
776:1
768:1
762:n
751:+
746:n
742:d
736:9
732:1
724:n
692:)
688:1
685:+
679:+
674:2
668:i
660:+
655:1
649:i
640:(
633:9
630:=
627:1
619:i
594:1
586:i
554:.
549:1
545:d
541:9
538:+
532:+
527:1
521:n
517:d
512:)
508:1
500:1
494:n
485:(
481:+
476:n
472:d
467:)
463:1
455:n
446:(
438:=
430:)
424:0
420:d
416:+
410:+
405:1
399:n
395:d
391:+
386:n
382:d
377:(
368:0
364:d
360:+
354:+
349:1
343:n
339:d
333:1
327:n
319:+
314:n
310:d
304:n
269:0
265:d
261:+
255:+
250:1
244:n
240:d
236:+
231:n
227:d
204:0
200:d
191:1
185:n
181:d
175:n
171:d
148:0
144:d
140:+
134:+
129:1
123:n
119:d
113:1
107:n
99:+
94:n
90:d
84:n
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