2025:"step drum"). The drum was turned by use of a crank on the top of the instrument. The number of cams encountered by each digit as the crank turned was determined by the value of that digit. For example, if a slide is set to its "6" position, a row of 6 cams would be encountered around the drum corresponding to that position. For subtraction, the drum was shifted slightly before it was turned, which moved a different row of cams into position. This alternate row contained the nines' complement of the digits. Thus, the row of 6 cams that had been in position for addition now had a row with 3 cams. The shifted drum also engaged one extra cam which added 1 to the result (as required for the method of complements). The always present ten's complement "overflow 1" which carried out beyond the most significant digit of the results register was, in effect, discarded.
2009:
minuend was entered. On some machine this could be done by dialing in the minuend using inner wheels of complements (i.e. without having to mentally determine the nines' complement of the minuend). In displaying that data in the complement window (red set), the operator could see the nines' complement of the nines' complement of the minuend, that is the minuend. The slat was then moved to expose the black digits (which now displayed the nines' complement of the minuend) and the subtrahend was added by dialing it in. Finally, the operator had to move the slat again to read the correct answer.
31:
1994:
1984:, the carry from the most significant digit that would normally be ignored is added, producing the correct result. And if not, the 1 is not added and the result is one less than the radix complement of the answer, or the diminished radix complement, which does not require an addition to obtain. This method is used by computers that use sign-and-magnitude to represent signed numbers.
2017:
afterwards, the operator would thus effectively add the ten's complement of the subtrahend. The operator also needed to hold down the "subtraction cutoff tab" corresponding to the leftmost digit of the answer. This tab prevented the carry from being propagated past it, the
Comptometer's method of dropping the initial 1 from the result.
1949:
are considered positive; the rest are considered negative (and their magnitude can be obtained by taking the radix complement). This works best for even radices since the sign can be determined by looking at the first digit. For example, numbers in ten's complement notation are positive if the first
2054:
The method of complements was used to correct errors when accounting books were written by hand. To remove an entry from a column of numbers, the accountant could add a new entry with the ten's complement of the number to subtract. A bar was added over the digits of this entry to denote its special
2044:
Using sign-magnitude representation requires only complementing the sign bit of the subtrahend and adding, but the addition/subtraction logic needs to compare the sign bits, complement one of the inputs if they are different, implement an end-around carry, and complement the result if there was no
1969:
than comparing and swapping the operands. But since taking the radix complement requires adding 1, it is difficult to do directly. Fortunately, a trick can be used to get around this addition: Instead of always setting a carry into the least significant digit when subtracting, the carry out of the
152:
and one is added to the sum. The leftmost digit '1' of the result is then discarded. Discarding the leftmost '1' is especially convenient on calculators or computers that use a fixed number of digits: there is nowhere for it to go so it is simply lost during the calculation. The nines' complement
2024:
used the method of complements for subtraction, and managed to hide this from the user. Numbers were entered using digit input slides along the side of the device. The number on each slide was added to a result counter by a gearing mechanism which engaged cams on a rotating "echelon drum" (a.k.a.
2008:
had two sets of result digits, a black set displaying the normal result and a red set displaying the nines' complement of this. A horizontal slat was used to cover up one of these sets, exposing the other. To subtract, the red digits were exposed and set to 0. Then the nines' complement of the
1744:
The method of complements is especially useful in binary (radix 2) since the ones' complement is very easily obtained by inverting each bit (changing '0' to '1' and vice versa). Adding 1 to get the two's complement can be done by simulating a carry into the least significant bit. For example:
2016:
had nines' complement digits printed in smaller type along with the normal digits on each key. To subtract, the operator was expected to mentally subtract 1 from the subtrahend and enter the result using the smaller digits. Since subtracting 1 before complementing is equivalent to adding 1
1907:
Ignore the issue. This is reasonable if a person is operating a calculating device that doesn't support negative numbers since comparing the two operands before the calculation so they can be entered in the proper order, and verifying that the result is reasonable, is easy for humans to
2075:, which is the nines' complement plus 1. The result of this addition is used when it is clear that the difference will be positive, otherwise the ten's complement of the addition's result is used with it marked as negative. The same technique works for subtracting on an adding machine.
1434:
The nines' complement of a decimal digit is the number that must be added to it to produce 9; the nines' complement of 3 is 6, the nines' complement of 7 is 2, and so on, see table. To form the nines' complement of a larger number, each digit is replaced by its nines' complement.
2058:
Complementing the sum is handy for cashiers making change for a purchase from currency in a single denomination of 1 raised to an integer power of the currency's base. For decimal currencies that would be 10, 100, 1,000, etc., e.g. a $ 10.00 bill.
587:
1315:
is the diminished radix complement of a number in base 5. However, the distinction is not important when the radix is apparent (nearly always), and the subtle difference in apostrophe placement is not common practice. Most writers use
1950:
digit is 0, 1, 2, 3, or 4, and negative if 5, 6, 7, 8, or 9. And it works very well in binary since the first bit can be considered a sign bit: the number is positive if the sign bit is 0 and negative if it is 1. Indeed,
86:
of any number is implemented by adding its complement. Changing the sign of any number is encoded by generating its complement, which can be done by a very simple and efficient algorithm. This method was commonly used in
1903:(1000 in this case); one cannot simply ignore a leading 1. The expected answer is −144, which isn't as far off as it seems; 856 happens to be the ten's complement of 144. This issue can be addressed in a number of ways:
1779:, logical constraints given that adding and subtracting arbitrary integers is normally done by comparing signs, adding the two or subtracting the smaller from the larger, and giving the result the correct sign.
2041:
Using ones' complement representation requires inverting the bits of the subtrahend and connecting the carry out of the most significant bit to the carry in of the least significant bit (end-around carry).
1521:
This is not yet correct. In the first step, 999 was added to the equation. Then 1000 was subtracted when the leading 1 was dropped. So, the answer obtained (654) is one less than the correct answer
2034:
Use of the method of complements is ubiquitous in digital computers, regardless of the representation used for signed numbers. However, the circuitry required depends on the representation:
1553:
Adding a 1 gives 655, the correct answer to our original subtraction problem. The last step of adding 1 could be skipped if instead the ten's complement of y was used in the first step.
326:
369:
1244:
919:
842:
791:
1827:
1092:
1053:
1947:
1118:
420:
273:
1901:
1854:
1192:
1145:
1545:
1270:
974:
945:
868:
663:
617:
395:
1674:
1654:
1625:
1605:
1579:
1507:
1487:
1165:
1014:
994:
746:
726:
706:
686:
637:
415:
240:
217:
197:
1307:, recommend using the placement of the apostrophe to distinguish between the radix complement and the diminished radix complement. In this usage, the
2038:
If two's complement representation is used, subtraction requires only inverting the bits of the subtrahend and setting a carry into the rightmost bit.
1449:
Compute the nines' complement of the minuend, 873. Add that to the subtrahend 218, then calculate the nines' complement of the result.
114:
of a number given in decimal representation is formed by replacing each digit with nine minus that digit. To subtract a decimal number
1466:
Compute the nines' complement of 218, which is 781. Because 218 is three digits long, this is the same as subtracting 218 from 999.
38:
c. 1910. The smaller numbers, for use when subtracting, are the nines' complement of the larger numbers, which are used when adding.
2001:
The method of complements was used in many mechanical calculators as an alternative to running the gears backwards. For example:
1676:, leading zeros must be added in the second method. These zeros become leading nines when the complement is taken. For example:
1766:
2146:
74:
half of the possible representations of numbers encode the positive numbers, the other half represents their respective
593:). Knowing this, the diminished radix complement of a number can be found by complementing each digit with respect to
163:); in particular, it is used on most digital computers to perform subtraction, represent negative numbers in base 2 or
282:
63:
1970:
most significant digit is used as the carry input into the least significant digit (an operation called an
2005:
331:
1288:
328:. While this seems equally difficult to calculate as the radix complement, it is actually simpler since
88:
708:
using diminished radix complements may be performed as follows. Add the diminished radix complement of
1197:
1951:
1707:
1703:
1633:
The nines' complement of 999990 is 000009. Removing the leading zeros gives 9, the desired result.
1299:
1293:
873:
796:
71:
2102:
2067:
In grade schools, students are sometimes taught the method of complements as a shortcut useful in
1914:
Represent negative numbers as radix complements of their positive counterparts. Numbers less than
30:
1957:
Complement the result if there is no carry out of the most significant digit (an indication that
751:
2113:
2055:
status. It was then possible to add the whole column of figures to obtain the corrected result.
582:{\displaystyle b^{n}-1=(b-1)\left(b^{n-1}+b^{n-2}+\cdots +b+1\right)=(b-1)b^{n-1}+\cdots +(b-1)}
1793:
1058:
1019:
2068:
104:
1917:
1097:
245:
141:. Then the nines' complement of the result obtained is formed to produce the desired result.
2021:
1972:
590:
75:
1879:
1832:
1170:
1123:
1966:
1911:
Use the same method to subtract 856 from 1000, and then add a negative sign to the result.
1524:
1249:
953:
924:
847:
642:
596:
374:
1659:
1639:
1610:
1590:
1564:
1492:
1472:
1150:
999:
979:
731:
711:
691:
671:
622:
400:
225:
202:
182:
35:
2140:
164:
100:
1993:
1790:. In that case, there will not be a "1" digit to cross out after the addition since
1771:
The method of complements normally assumes that the operands are positive and that
1304:
2013:
83:
43:
17:
275:. In practice, the radix complement is more easily obtained by adding 1 to the
2072:
119:
1561:
In the following example the result of the subtraction has fewer digits than
1303:. The naming of complements in other bases is similar. Some people, notably
59:
47:
1688:
Replacing 00391 with its nines' complement and adding 1 produces the sum:
1757:
Dropping the initial "1" gives the answer: 0100 1110 (equals decimal 78)
92:
67:
2116:, Comptometer Division, Felt and Tarrant Mfg. Co., Chicago, 1917, p. 12
1276:
127:
55:
1997:
Comptometer from the 1920s, with nines' complements marked on each key
2128:
Principles of
Arithmetic and Geometry for Elementary School Teachers
54:
is a technique to encode a symmetric range of positive and negative
2084:
1992:
1754:
0110 0100 + 1110 1001 + 1 ——————————— 10100 1110
220:
160:
29:
159:
The method of complements can be extended to other number bases (
1324:, and many style manuals leave out the apostrophe, recommending
2114:
Easy
Instructions for Operation the Controlled Key Comptometer
1311:
refers to the radix complement of a number in base four while
2071:. Subtraction is done by adding the ten's complement of the
1954:
is used in most modern computers to represent signed numbers.
78:. The pairs of mutually additive inverse numbers are called
1587:
Using the first method the sum of the nines' complement of
1694:
Dropping the leading 1 gives the correct answer: 47641.
1920:
1882:
1835:
1796:
1662:
1642:
1613:
1593:
1567:
1527:
1495:
1475:
1279:
numbering system, the radix complement is called the
1252:
1200:
1173:
1153:
1126:
1100:
1061:
1022:
1002:
982:
956:
927:
876:
850:
799:
754:
734:
714:
694:
674:
645:
625:
599:
423:
403:
377:
334:
285:
248:
228:
205:
185:
870:. Further taking the diminished radix complement of
1515:The leading "1" digit is then dropped, giving 654.
1941:
1895:
1848:
1821:
1668:
1648:
1619:
1599:
1573:
1539:
1501:
1481:
1455:Now calculate the nines' complement of the result
1264:
1238:
1186:
1159:
1139:
1112:
1086:
1047:
1008:
988:
976:may be obtained by adding the radix complement of
968:
939:
913:
862:
836:
785:
740:
720:
700:
680:
657:
631:
611:
581:
409:
389:
363:
320:
267:
234:
211:
191:
70:throughout the whole range. For a given number of
1876:way to complete the calculation by subtracting
144:In the second method, the nines' complement of
844:, which is the diminished radix complement of
133:In the first method, the nines' complement of
8:
1438:Consider the following subtraction problem:
1167:from the result is the same as subtracting
1711:
1691:48032 + 99608 + 1 ——————— 147641
1339:
950:Alternatively using the radix complement,
1931:
1925:
1919:
1887:
1881:
1840:
1834:
1813:
1795:
1661:
1641:
1612:
1592:
1566:
1547:. To fix this, 1 is added to the answer:
1526:
1494:
1474:
1251:
1230:
1217:
1199:
1178:
1172:
1152:
1131:
1125:
1120:, the result will be greater or equal to
1099:
1078:
1060:
1033:
1021:
1001:
981:
955:
926:
881:
875:
849:
804:
798:
759:
753:
733:
713:
693:
673:
644:
624:
598:
543:
486:
467:
428:
422:
402:
376:
344:
333:
295:
284:
253:
247:
227:
204:
184:
99:(as described below) is also valuable in
1297:and the diminished radix complement the
1283:and the diminished radix complement the
2095:
1630:876589 + 123401 ———————— 999990
321:{\displaystyle \left(b^{n}-1\right)-y}
1656:, has fewer digits than the minuend,
1291:, the radix complement is called the
7:
2045:carry from the most significant bit.
1965:). This is easier to implement with
364:{\displaystyle \left(b^{n}-1\right)}
58:in a way that they can use the same
2126:Carl Barnett Allendoerfer (1971).
167:and test overflow in calculation.
25:
921:results in the desired answer of
619:, i.e. subtracting each digit in
95:. The generalized concept of the
1869:185 + 670 + 1 ————— 856
1761:Negative number representations
1239:{\displaystyle x-y+b^{n}-b^{n}}
908:
896:
831:
819:
576:
564:
536:
524:
455:
443:
1:
1856:. For example, (in decimal):
1767:Signed number representations
1489:and the nines' complement of
914:{\displaystyle b^{n}-1-(x-y)}
837:{\displaystyle b^{n}-1-(x-y)}
1733:
1725:
1512:873 + 781 ————— 1654
1452:126 + 218 ————— 344
1423:
1415:
1407:
1399:
1391:
1383:
1375:
1367:
1359:
1351:
91:and is still used in modern
1872:At this point, there is no
1518:1654 -1000 ————— 654
786:{\displaystyle b^{n}-1-x+y}
277:diminished radix complement
130:) two methods may be used:
2163:
1782:Let's see what happens if
1764:
1748:0110 0100 - 0001 0110
1701:
2063:In grade school education
1822:{\displaystyle x-y+b^{n}}
1147:and dropping the leading
1087:{\displaystyle x-y+b^{n}}
1048:{\displaystyle x+b^{n}-y}
153:plus one is known as the
34:Complement numbers on an
1550:654 + 1 ————— 655
591:Geometric series Formula
417:times. This is because
1942:{\displaystyle b^{n}/2}
1113:{\displaystyle y\leq x}
268:{\displaystyle b^{n}-y}
1998:
1943:
1897:
1850:
1823:
1670:
1650:
1621:
1601:
1575:
1541:
1503:
1483:
1272:, the desired result.
1266:
1240:
1188:
1161:
1141:
1114:
1088:
1049:
1010:
990:
970:
941:
915:
864:
838:
787:
742:
722:
702:
682:
659:
633:
613:
583:
411:
391:
365:
322:
269:
236:
213:
193:
122:) from another number
89:mechanical calculators
39:
1996:
1944:
1898:
1896:{\displaystyle b^{n}}
1851:
1849:{\displaystyle b^{n}}
1824:
1671:
1651:
1622:
1602:
1576:
1542:
1504:
1484:
1267:
1241:
1189:
1187:{\displaystyle b^{n}}
1162:
1142:
1140:{\displaystyle b^{n}}
1115:
1089:
1050:
1011:
991:
971:
942:
916:
865:
839:
788:
743:
723:
703:
683:
660:
634:
614:
584:
412:
392:
366:
323:
270:
237:
214:
194:
97:radix complement
52:method of complements
33:
27:Method of subtraction
1918:
1880:
1833:
1794:
1660:
1640:
1611:
1591:
1584:123410 - 123401
1565:
1557:Magnitude of numbers
1525:
1493:
1473:
1250:
1198:
1194:, making the result
1171:
1151:
1124:
1098:
1059:
1020:
1000:
980:
954:
925:
874:
848:
797:
752:
732:
712:
692:
672:
643:
623:
597:
421:
401:
375:
371:is simply the digit
332:
283:
246:
226:
203:
183:
2147:Computer arithmetic
2006:Pascal's calculator
1636:If the subtrahend,
1540:{\displaystyle x-y}
1265:{\displaystyle x-y}
969:{\displaystyle x-y}
940:{\displaystyle x-y}
863:{\displaystyle x-y}
668:The subtraction of
658:{\displaystyle b-1}
612:{\displaystyle b-1}
390:{\displaystyle b-1}
171:Numeric complements
1999:
1939:
1893:
1866:and adding gives:
1846:
1829:will be less than
1819:
1685:48032 - 00391
1679:48032 - 391
1666:
1646:
1617:
1597:
1571:
1537:
1499:
1479:
1262:
1236:
1184:
1157:
1137:
1110:
1084:
1045:
1006:
986:
966:
937:
911:
860:
834:
783:
738:
718:
698:
678:
655:
629:
609:
579:
407:
387:
361:
318:
265:
232:
209:
189:
40:
2069:mental arithmetic
1751:becomes the sum:
1742:
1741:
1682:can be rewritten
1669:{\displaystyle x}
1649:{\displaystyle y}
1620:{\displaystyle y}
1600:{\displaystyle x}
1574:{\displaystyle x}
1502:{\displaystyle y}
1482:{\displaystyle x}
1469:Next, the sum of
1432:
1431:
1322:nine's complement
1313:fours' complement
1309:four's complement
1285:nines' complement
1160:{\displaystyle 1}
1009:{\displaystyle x}
989:{\displaystyle y}
741:{\displaystyle y}
721:{\displaystyle x}
701:{\displaystyle x}
681:{\displaystyle y}
632:{\displaystyle y}
410:{\displaystyle n}
235:{\displaystyle b}
212:{\displaystyle y}
192:{\displaystyle n}
165:binary arithmetic
155:tens' complement.
112:nines' complement
76:additive inverses
16:(Redirected from
2154:
2132:
2131:
2123:
2117:
2111:
2105:
2100:
2022:Curta calculator
1973:end-around carry
1967:digital circuits
1952:two's complement
1948:
1946:
1945:
1940:
1935:
1930:
1929:
1902:
1900:
1899:
1894:
1892:
1891:
1855:
1853:
1852:
1847:
1845:
1844:
1828:
1826:
1825:
1820:
1818:
1817:
1712:
1708:Two's complement
1704:Ones' complement
1675:
1673:
1672:
1667:
1655:
1653:
1652:
1647:
1626:
1624:
1623:
1618:
1606:
1604:
1603:
1598:
1580:
1578:
1577:
1572:
1546:
1544:
1543:
1538:
1508:
1506:
1505:
1500:
1488:
1486:
1485:
1480:
1340:
1330:nines complement
1300:ones' complement
1294:two's complement
1281:ten's complement
1271:
1269:
1268:
1263:
1245:
1243:
1242:
1237:
1235:
1234:
1222:
1221:
1193:
1191:
1190:
1185:
1183:
1182:
1166:
1164:
1163:
1158:
1146:
1144:
1143:
1138:
1136:
1135:
1119:
1117:
1116:
1111:
1093:
1091:
1090:
1085:
1083:
1082:
1054:
1052:
1051:
1046:
1038:
1037:
1015:
1013:
1012:
1007:
995:
993:
992:
987:
975:
973:
972:
967:
946:
944:
943:
938:
920:
918:
917:
912:
886:
885:
869:
867:
866:
861:
843:
841:
840:
835:
809:
808:
793:or equivalently
792:
790:
789:
784:
764:
763:
747:
745:
744:
739:
727:
725:
724:
719:
707:
705:
704:
699:
687:
685:
684:
679:
664:
662:
661:
656:
638:
636:
635:
630:
618:
616:
615:
610:
588:
586:
585:
580:
554:
553:
520:
516:
497:
496:
478:
477:
433:
432:
416:
414:
413:
408:
396:
394:
393:
388:
370:
368:
367:
362:
360:
356:
349:
348:
327:
325:
324:
319:
311:
307:
300:
299:
274:
272:
271:
266:
258:
257:
241:
239:
238:
233:
218:
216:
215:
210:
198:
196:
195:
190:
177:radix complement
21:
18:Nines complement
2162:
2161:
2157:
2156:
2155:
2153:
2152:
2151:
2137:
2136:
2135:
2125:
2124:
2120:
2112:
2108:
2101:
2097:
2093:
2081:
2065:
2052:
2032:
1991:
1921:
1916:
1915:
1883:
1878:
1877:
1870:
1860:
1836:
1831:
1830:
1809:
1792:
1791:
1769:
1763:
1755:
1749:
1721:
1716:
1710:
1702:Main articles:
1700:
1692:
1686:
1680:
1658:
1657:
1638:
1637:
1631:
1609:
1608:
1589:
1588:
1585:
1563:
1562:
1559:
1551:
1523:
1522:
1519:
1513:
1491:
1490:
1471:
1470:
1464:
1459:
1453:
1447:
1442:
1347:
1338:
1336:Decimal example
1248:
1247:
1226:
1213:
1196:
1195:
1174:
1169:
1168:
1149:
1148:
1127:
1122:
1121:
1096:
1095:
1074:
1057:
1056:
1029:
1018:
1017:
998:
997:
978:
977:
952:
951:
923:
922:
877:
872:
871:
846:
845:
800:
795:
794:
755:
750:
749:
730:
729:
710:
709:
690:
689:
670:
669:
641:
640:
621:
620:
595:
594:
539:
482:
463:
462:
458:
424:
419:
418:
399:
398:
373:
372:
340:
339:
335:
330:
329:
291:
290:
286:
281:
280:
249:
244:
243:
224:
223:
201:
200:
181:
180:
173:
28:
23:
22:
15:
12:
11:
5:
2160:
2158:
2150:
2149:
2139:
2138:
2134:
2133:
2118:
2106:
2094:
2092:
2089:
2088:
2087:
2080:
2077:
2064:
2061:
2051:
2048:
2047:
2046:
2042:
2039:
2031:
2028:
2027:
2026:
2018:
2010:
1990:
1989:Practical uses
1987:
1986:
1985:
1961:was less than
1955:
1938:
1934:
1928:
1924:
1912:
1909:
1890:
1886:
1868:
1862:Complementing
1859:185 - 329
1858:
1843:
1839:
1816:
1812:
1808:
1805:
1802:
1799:
1765:Main article:
1762:
1759:
1753:
1747:
1740:
1739:
1736:
1732:
1731:
1728:
1724:
1723:
1718:
1699:
1696:
1690:
1684:
1678:
1665:
1645:
1629:
1616:
1596:
1583:
1570:
1558:
1555:
1549:
1536:
1533:
1530:
1517:
1511:
1498:
1478:
1463:
1460:
1458:344 655
1457:
1451:
1446:
1443:
1441:873 - 218
1440:
1430:
1429:
1426:
1422:
1421:
1418:
1414:
1413:
1410:
1406:
1405:
1402:
1398:
1397:
1394:
1390:
1389:
1386:
1382:
1381:
1378:
1374:
1373:
1370:
1366:
1365:
1362:
1358:
1357:
1354:
1350:
1349:
1344:
1337:
1334:
1261:
1258:
1255:
1233:
1229:
1225:
1220:
1216:
1212:
1209:
1206:
1203:
1181:
1177:
1156:
1134:
1130:
1109:
1106:
1103:
1081:
1077:
1073:
1070:
1067:
1064:
1044:
1041:
1036:
1032:
1028:
1025:
1005:
985:
965:
962:
959:
936:
933:
930:
910:
907:
904:
901:
898:
895:
892:
889:
884:
880:
859:
856:
853:
833:
830:
827:
824:
821:
818:
815:
812:
807:
803:
782:
779:
776:
773:
770:
767:
762:
758:
737:
717:
697:
677:
654:
651:
648:
628:
608:
605:
602:
578:
575:
572:
569:
566:
563:
560:
557:
552:
549:
546:
542:
538:
535:
532:
529:
526:
523:
519:
515:
512:
509:
506:
503:
500:
495:
492:
489:
485:
481:
476:
473:
470:
466:
461:
457:
454:
451:
448:
445:
442:
439:
436:
431:
427:
406:
386:
383:
380:
359:
355:
352:
347:
343:
338:
317:
314:
310:
306:
303:
298:
294:
289:
264:
261:
256:
252:
242:is defined as
231:
208:
199:-digit number
188:
172:
169:
105:Midy's theorem
36:adding machine
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2159:
2148:
2145:
2144:
2142:
2129:
2122:
2119:
2115:
2110:
2107:
2104:
2099:
2096:
2090:
2086:
2083:
2082:
2078:
2076:
2074:
2070:
2062:
2060:
2056:
2049:
2043:
2040:
2037:
2036:
2035:
2029:
2023:
2019:
2015:
2011:
2007:
2004:
2003:
2002:
1995:
1988:
1983:
1979:
1975:
1974:
1968:
1964:
1960:
1956:
1953:
1936:
1932:
1926:
1922:
1913:
1910:
1906:
1905:
1904:
1888:
1884:
1875:
1867:
1865:
1857:
1841:
1837:
1814:
1810:
1806:
1803:
1800:
1797:
1789:
1785:
1780:
1778:
1774:
1768:
1760:
1758:
1752:
1746:
1737:
1734:
1729:
1726:
1719:
1714:
1713:
1709:
1705:
1698:Binary method
1697:
1695:
1689:
1683:
1677:
1663:
1643:
1634:
1628:
1614:
1594:
1582:
1568:
1556:
1554:
1548:
1534:
1531:
1528:
1516:
1510:
1496:
1476:
1467:
1462:Second method
1461:
1456:
1450:
1444:
1439:
1436:
1427:
1424:
1419:
1416:
1411:
1408:
1403:
1400:
1395:
1392:
1387:
1384:
1379:
1376:
1371:
1368:
1363:
1360:
1355:
1352:
1345:
1342:
1341:
1335:
1333:
1331:
1327:
1323:
1319:
1314:
1310:
1306:
1302:
1301:
1296:
1295:
1290:
1286:
1282:
1278:
1273:
1259:
1256:
1253:
1231:
1227:
1223:
1218:
1214:
1210:
1207:
1204:
1201:
1179:
1175:
1154:
1132:
1128:
1107:
1104:
1101:
1079:
1075:
1071:
1068:
1065:
1062:
1042:
1039:
1034:
1030:
1026:
1023:
1003:
983:
963:
960:
957:
948:
934:
931:
928:
905:
902:
899:
893:
890:
887:
882:
878:
857:
854:
851:
828:
825:
822:
816:
813:
810:
805:
801:
780:
777:
774:
771:
768:
765:
760:
756:
735:
715:
695:
675:
666:
652:
649:
646:
626:
606:
603:
600:
592:
573:
570:
567:
561:
558:
555:
550:
547:
544:
540:
533:
530:
527:
521:
517:
513:
510:
507:
504:
501:
498:
493:
490:
487:
483:
479:
474:
471:
468:
464:
459:
452:
449:
446:
440:
437:
434:
429:
425:
404:
384:
381:
378:
357:
353:
350:
345:
341:
336:
315:
312:
308:
304:
301:
296:
292:
287:
278:
262:
259:
254:
250:
229:
222:
206:
186:
178:
170:
168:
166:
162:
157:
156:
151:
147:
142:
140:
136:
131:
129:
125:
121:
117:
113:
108:
106:
103:, such as in
102:
101:number theory
98:
94:
90:
85:
81:
77:
73:
69:
65:
61:
57:
53:
49:
45:
37:
32:
19:
2130:. Macmillan.
2127:
2121:
2109:
2103:Florida Tech
2098:
2066:
2057:
2053:
2033:
2030:In computers
2000:
1981:
1977:
1971:
1962:
1958:
1873:
1871:
1863:
1861:
1787:
1783:
1781:
1776:
1772:
1770:
1756:
1750:
1743:
1693:
1687:
1681:
1635:
1632:
1586:
1560:
1552:
1520:
1514:
1468:
1465:
1454:
1448:
1445:First method
1437:
1433:
1329:
1325:
1321:
1317:
1312:
1308:
1305:Donald Knuth
1298:
1292:
1284:
1280:
1274:
949:
667:
276:
176:
174:
158:
154:
149:
148:is added to
145:
143:
138:
137:is added to
134:
132:
123:
115:
111:
109:
96:
79:
51:
41:
2050:Manual uses
2014:Comptometer
1722:complement
1348:complement
1094:. Assuming
279:, which is
84:subtraction
80:complements
44:mathematics
2091:References
2073:subtrahend
1509:is taken:
1016:to obtain
748:to obtain
589:(see also
120:subtrahend
1976:). So if
1801:−
1532:−
1257:−
1224:−
1205:−
1105:≤
1066:−
1040:−
961:−
932:−
903:−
894:−
888:−
855:−
826:−
817:−
811:−
772:−
766:−
650:−
604:−
571:−
559:⋯
548:−
531:−
502:⋯
491:−
472:−
450:−
435:−
397:repeated
382:−
351:−
313:−
302:−
260:−
93:computers
64:mechanism
60:algorithm
48:computing
2141:Category
2079:See also
1246:or just
82:. Thus
68:addition
56:integers
1346:Nines'
1277:decimal
1275:In the
161:radices
128:minuend
1874:simple
1720:Ones'
1717:digit
1715:Binary
1343:Digit
1289:binary
179:of an
72:places
66:) for
50:, the
2085:Curta
1786:<
1318:one's
1287:. In
688:from
639:from
221:radix
126:(the
118:(the
2020:The
2012:The
1706:and
1607:and
1328:and
1326:ones
1320:and
175:The
110:The
62:(or
46:and
1908:do.
1627:is
1055:or
996:to
728:to
219:in
42:In
2143::
1980:≤
1775:≤
1738:0
1735:1
1730:1
1727:0
1581::
1428:0
1425:9
1420:1
1417:8
1412:2
1409:7
1404:3
1401:6
1396:4
1393:5
1388:5
1385:4
1380:6
1377:3
1372:7
1369:2
1364:8
1361:1
1356:9
1353:0
1332:.
947:.
665:.
107:.
1982:x
1978:y
1963:y
1959:x
1937:2
1933:/
1927:n
1923:b
1889:n
1885:b
1864:y
1842:n
1838:b
1815:n
1811:b
1807:+
1804:y
1798:x
1788:y
1784:x
1777:x
1773:y
1664:x
1644:y
1615:y
1595:x
1569:x
1535:y
1529:x
1497:y
1477:x
1260:y
1254:x
1232:n
1228:b
1219:n
1215:b
1211:+
1208:y
1202:x
1180:n
1176:b
1155:1
1133:n
1129:b
1108:x
1102:y
1080:n
1076:b
1072:+
1069:y
1063:x
1043:y
1035:n
1031:b
1027:+
1024:x
1004:x
984:y
964:y
958:x
935:y
929:x
909:)
906:y
900:x
897:(
891:1
883:n
879:b
858:y
852:x
832:)
829:y
823:x
820:(
814:1
806:n
802:b
781:y
778:+
775:x
769:1
761:n
757:b
736:y
716:x
696:x
676:y
653:1
647:b
627:y
607:1
601:b
577:)
574:1
568:b
565:(
562:+
556:+
551:1
545:n
541:b
537:)
534:1
528:b
525:(
522:=
518:)
514:1
511:+
508:b
505:+
499:+
494:2
488:n
484:b
480:+
475:1
469:n
465:b
460:(
456:)
453:1
447:b
444:(
441:=
438:1
430:n
426:b
405:n
385:1
379:b
358:)
354:1
346:n
342:b
337:(
316:y
309:)
305:1
297:n
293:b
288:(
263:y
255:n
251:b
230:b
207:y
187:n
150:x
146:y
139:y
135:x
124:x
116:y
20:)
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