2014:"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.
1998:
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.
20:
1983:
1973:, 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.
2006:
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.
1938:
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
2043:
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
2033:
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
1958:
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
141:
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
2013:
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.
1997:
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
1733:
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:
2005:
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
1896:
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
2064:, 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.
1423:
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.
2047:
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.
576:
1304:
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
1939:
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,
75:
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
1892:(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:
1768:, 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.
2030:
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).
1510:
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
2023:
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:
1542:
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.
315:
358:
1233:
908:
831:
780:
1816:
1081:
1042:
1936:
1107:
409:
262:
1890:
1843:
1181:
1134:
1534:
1259:
963:
934:
857:
652:
606:
384:
1663:
1643:
1614:
1594:
1568:
1496:
1476:
1154:
1003:
983:
735:
715:
695:
675:
626:
404:
229:
206:
186:
1296:, recommend using the placement of the apostrophe to distinguish between the radix complement and the diminished radix complement. In this usage, the
2027:
If two's complement representation is used, subtraction requires only inverting the bits of the subtrahend and setting a carry into the rightmost bit.
1438:
Compute the nines' complement of the minuend, 873. Add that to the subtrahend 218, then calculate the nines' complement of the result.
103:
of a number given in decimal representation is formed by replacing each digit with nine minus that digit. To subtract a decimal number
1455:
Compute the nines' complement of 218, which is 781. Because 218 is three digits long, this is the same as subtracting 218 from 999.
27:
c. 1910. The smaller numbers, for use when subtracting, are the nines' complement of the larger numbers, which are used when adding.
1990:
The method of complements was used in many mechanical calculators as an alternative to running the gears backwards. For example:
1665:, leading zeros must be added in the second method. These zeros become leading nines when the complement is taken. For example:
63:
half of the possible representations of numbers encode the positive numbers, the other half represents their respective
582:). Knowing this, the diminished radix complement of a number can be found by complementing each digit with respect to
152:); in particular, it is used on most digital computers to perform subtraction, represent negative numbers in base 2 or
1755:
271:
2135:
1959:
most significant digit is used as the carry input into the least significant digit (an operation called an
320:
317:. While this seems equally difficult to calculate as the radix complement, it is actually simpler since
697:
using diminished radix complements may be performed as follows. Add the diminished radix complement of
1186:
1622:
The nines' complement of 999990 is 000009. Removing the leading zeros gives 9, the desired result.
862:
785:
2091:
2056:
In grade schools, students are sometimes taught the method of complements as a shortcut useful in
1903:
Represent negative numbers as radix complements of their positive counterparts. Numbers less than
1946:
Complement the result if there is no carry out of the most significant digit (an indication that
740:
52:
2102:
2044:
status. It was then possible to add the whole column of figures to obtain the corrected result.
571:{\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)}
1994:
1782:
1047:
1008:
1906:
1277:
1086:
234:
130:. Then the nines' complement of the result obtained is formed to produce the desired result.
77:
1868:
1821:
1159:
1112:
1900:
Use the same method to subtract 856 from 1000, and then add a negative sign to the result.
8:
1940:
1696:
1692:
1513:
1288:
1282:
1238:
942:
913:
836:
631:
585:
363:
60:
1648:
1628:
1599:
1579:
1553:
1481:
1461:
1139:
988:
968:
720:
700:
680:
660:
611:
389:
214:
191:
171:
2057:
93:
1779:. In that case, there will not be a "1" digit to cross out after the addition since
2010:
1961:
1760:
The method of complements normally assumes that the operands are positive and that
579:
64:
1955:
19:
264:. In practice, the radix complement is more easily obtained by adding 1 to the
24:
2129:
1550:
In the following example the result of the subtraction has fewer digits than
1292:. The naming of complements in other bases is similar. Some people, notably
153:
89:
1293:
1677:
Replacing 00391 with its nines' complement and adding 1 produces the sum:
2002:
1746:
Dropping the initial "1" gives the answer: 0100 1110 (equals decimal 78)
72:
32:
2105:, Comptometer Division, Felt and Tarrant Mfg. Co., Chicago, 1917, p. 12
2061:
1982:
108:
1986:
Comptometer from the 1920s, with nines' complements marked on each key
48:
36:
2117:
Principles of
Arithmetic and Geometry for Elementary School Teachers
43:
is a technique to encode a symmetric range of positive and negative
81:
56:
1743:
0110 0100 + 1110 1001 + 1 ——————————— 10100 1110
1265:
148:
The method of complements can be extended to other number bases (
116:
44:
1313:, and many style manuals leave out the apostrophe, recommending
2103:
Easy
Instructions for Operation the Controlled Key Comptometer
1300:
refers to the radix complement of a number in base four while
2073:
2060:. Subtraction is done by adding the ten's complement of the
1943:
is used in most modern computers to represent signed numbers.
209:
149:
67:. The pairs of mutually additive inverse numbers are called
1576:
Using the first method the sum of the nines' complement of
2114:
1683:
Dropping the leading 1 gives the correct answer: 47641.
1909:
1871:
1824:
1785:
1651:
1631:
1602:
1582:
1556:
1516:
1484:
1464:
1268:
numbering system, the radix complement is called the
1241:
1189:
1162:
1142:
1115:
1089:
1050:
1011:
991:
971:
945:
916:
865:
839:
788:
743:
723:
703:
683:
663:
634:
614:
588:
412:
392:
366:
323:
274:
237:
217:
194:
174:
859:. Further taking the diminished radix complement of
1504:The leading "1" digit is then dropped, giving 654.
1930:
1884:
1837:
1810:
1749:
1657:
1637:
1608:
1588:
1562:
1528:
1490:
1470:
1444:Now calculate the nines' complement of the result
1253:
1227:
1175:
1148:
1128:
1101:
1075:
1036:
997:
977:
965:may be obtained by adding the radix complement of
957:
928:
902:
851:
825:
774:
729:
709:
689:
669:
646:
620:
600:
570:
398:
378:
352:
309:
256:
223:
200:
180:
59:throughout the whole range. For a given number of
2127:
1865:way to complete the calculation by subtracting
133:In the second method, the nines' complement of
833:, which is the diminished radix complement of
122:In the first method, the nines' complement of
2051:
1427:Consider the following subtraction problem:
1156:from the result is the same as subtracting
1680:48032 + 99608 + 1 ——————— 147641
939:Alternatively using the radix complement,
1536:. To fix this, 1 is added to the answer:
1109:, the result will be greater or equal to
88:(as described below) is also valuable in
1981:
1286:and the diminished radix complement the
1272:and the diminished radix complement the
18:
1545:
2128:
1619:876589 + 123401 ———————— 999990
310:{\displaystyle \left(b^{n}-1\right)-y}
159:
1645:, has fewer digits than the minuend,
1280:, the radix complement is called the
2034:carry from the most significant bit.
1954:). This is easier to implement with
353:{\displaystyle \left(b^{n}-1\right)}
47:in a way that they can use the same
13:
2115:Carl Barnett Allendoerfer (1971).
1324:
156:and test overflow in calculation.
14:
2147:
1977:
910:results in the desired answer of
608:, i.e. subtracting each digit in
84:. The generalized concept of the
1858:185 + 670 + 1 ————— 856
1686:
1450:
2018:
1750:Negative number representations
1433:
1228:{\displaystyle x-y+b^{n}-b^{n}}
2108:
2096:
2085:
2038:
897:
885:
820:
808:
565:
553:
525:
513:
444:
432:
1:
2079:
1845:. For example, (in decimal):
1756:Signed number representations
1478:and the nines' complement of
903:{\displaystyle b^{n}-1-(x-y)}
826:{\displaystyle b^{n}-1-(x-y)}
1722:
1714:
1501:873 + 781 ————— 1654
1441:126 + 218 ————— 344
1412:
1404:
1396:
1388:
1380:
1372:
1364:
1356:
1348:
1340:
80:and is still used in modern
7:
2067:
1861:At this point, there is no
1507:1654 -1000 ————— 654
775:{\displaystyle b^{n}-1-x+y}
266:diminished radix complement
119:) two methods may be used:
10:
2152:
1771:Let's see what happens if
1753:
1737:0110 0100 - 0001 0110
1690:
2052:In grade school education
1811:{\displaystyle x-y+b^{n}}
1136:and dropping the leading
1076:{\displaystyle x-y+b^{n}}
1037:{\displaystyle x+b^{n}-y}
142:plus one is known as the
23:Complement numbers on an
1539:654 + 1 ————— 655
580:Geometric series Formula
406:times. This is because
1931:{\displaystyle b^{n}/2}
1102:{\displaystyle y\leq x}
257:{\displaystyle b^{n}-y}
1987:
1932:
1886:
1839:
1812:
1659:
1639:
1610:
1590:
1564:
1530:
1492:
1472:
1261:, the desired result.
1255:
1229:
1177:
1150:
1130:
1103:
1077:
1038:
999:
979:
959:
930:
904:
853:
827:
776:
731:
711:
691:
671:
648:
622:
602:
572:
400:
380:
354:
311:
258:
225:
202:
182:
111:) from another number
78:mechanical calculators
28:
1985:
1933:
1887:
1885:{\displaystyle b^{n}}
1840:
1838:{\displaystyle b^{n}}
1813:
1660:
1640:
1611:
1591:
1565:
1531:
1493:
1473:
1256:
1230:
1178:
1176:{\displaystyle b^{n}}
1151:
1131:
1129:{\displaystyle b^{n}}
1104:
1078:
1039:
1000:
980:
960:
931:
905:
854:
828:
777:
732:
712:
692:
672:
649:
623:
603:
573:
401:
381:
355:
312:
259:
226:
203:
183:
86:radix complement
41:method of complements
22:
16:Method of subtraction
1907:
1869:
1822:
1783:
1649:
1629:
1600:
1580:
1573:123410 - 123401
1554:
1546:Magnitude of numbers
1514:
1482:
1462:
1239:
1187:
1183:, making the result
1160:
1140:
1113:
1087:
1048:
1009:
989:
969:
943:
914:
863:
837:
786:
741:
721:
701:
681:
661:
632:
612:
586:
410:
390:
364:
360:is simply the digit
321:
272:
235:
215:
192:
172:
2136:Computer arithmetic
1995:Pascal's calculator
1625:If the subtrahend,
1529:{\displaystyle x-y}
1254:{\displaystyle x-y}
958:{\displaystyle x-y}
929:{\displaystyle x-y}
852:{\displaystyle x-y}
657:The subtraction of
647:{\displaystyle b-1}
601:{\displaystyle b-1}
379:{\displaystyle b-1}
160:Numeric complements
1988:
1928:
1882:
1855:and adding gives:
1835:
1818:will be less than
1808:
1674:48032 - 00391
1668:48032 - 391
1655:
1635:
1606:
1586:
1560:
1526:
1488:
1468:
1251:
1225:
1173:
1146:
1126:
1099:
1073:
1034:
995:
975:
955:
926:
900:
849:
823:
772:
727:
707:
687:
667:
644:
618:
598:
568:
396:
376:
350:
307:
254:
221:
198:
178:
29:
2058:mental arithmetic
1740:becomes the sum:
1731:
1730:
1671:can be rewritten
1658:{\displaystyle x}
1638:{\displaystyle y}
1609:{\displaystyle y}
1589:{\displaystyle x}
1563:{\displaystyle x}
1491:{\displaystyle y}
1471:{\displaystyle x}
1458:Next, the sum of
1421:
1420:
1311:nine's complement
1302:fours' complement
1298:four's complement
1274:nines' complement
1149:{\displaystyle 1}
998:{\displaystyle x}
978:{\displaystyle y}
730:{\displaystyle y}
710:{\displaystyle x}
690:{\displaystyle x}
670:{\displaystyle y}
621:{\displaystyle y}
399:{\displaystyle n}
224:{\displaystyle b}
201:{\displaystyle y}
181:{\displaystyle n}
154:binary arithmetic
144:tens' complement.
101:nines' complement
65:additive inverses
2143:
2121:
2120:
2112:
2106:
2100:
2094:
2089:
2011:Curta calculator
1962:end-around carry
1956:digital circuits
1941:two's complement
1937:
1935:
1934:
1929:
1924:
1919:
1918:
1891:
1889:
1888:
1883:
1881:
1880:
1844:
1842:
1841:
1836:
1834:
1833:
1817:
1815:
1814:
1809:
1807:
1806:
1701:
1700:
1697:Two's complement
1693:Ones' complement
1664:
1662:
1661:
1656:
1644:
1642:
1641:
1636:
1615:
1613:
1612:
1607:
1595:
1593:
1592:
1587:
1569:
1567:
1566:
1561:
1535:
1533:
1532:
1527:
1497:
1495:
1494:
1489:
1477:
1475:
1474:
1469:
1329:
1328:
1319:nines complement
1289:ones' complement
1283:two's complement
1270:ten's complement
1260:
1258:
1257:
1252:
1234:
1232:
1231:
1226:
1224:
1223:
1211:
1210:
1182:
1180:
1179:
1174:
1172:
1171:
1155:
1153:
1152:
1147:
1135:
1133:
1132:
1127:
1125:
1124:
1108:
1106:
1105:
1100:
1082:
1080:
1079:
1074:
1072:
1071:
1043:
1041:
1040:
1035:
1027:
1026:
1004:
1002:
1001:
996:
984:
982:
981:
976:
964:
962:
961:
956:
935:
933:
932:
927:
909:
907:
906:
901:
875:
874:
858:
856:
855:
850:
832:
830:
829:
824:
798:
797:
782:or equivalently
781:
779:
778:
773:
753:
752:
736:
734:
733:
728:
716:
714:
713:
708:
696:
694:
693:
688:
676:
674:
673:
668:
653:
651:
650:
645:
627:
625:
624:
619:
607:
605:
604:
599:
577:
575:
574:
569:
543:
542:
509:
505:
486:
485:
467:
466:
422:
421:
405:
403:
402:
397:
385:
383:
382:
377:
359:
357:
356:
351:
349:
345:
338:
337:
316:
314:
313:
308:
300:
296:
289:
288:
263:
261:
260:
255:
247:
246:
230:
228:
227:
222:
207:
205:
204:
199:
187:
185:
184:
179:
166:radix complement
2151:
2150:
2146:
2145:
2144:
2142:
2141:
2140:
2126:
2125:
2124:
2113:
2109:
2101:
2097:
2090:
2086:
2082:
2070:
2054:
2041:
2021:
1980:
1920:
1914:
1910:
1908:
1905:
1904:
1876:
1872:
1870:
1867:
1866:
1859:
1849:
1829:
1825:
1823:
1820:
1819:
1802:
1798:
1784:
1781:
1780:
1758:
1752:
1744:
1738:
1710:
1705:
1699:
1691:Main articles:
1689:
1681:
1675:
1669:
1650:
1647:
1646:
1630:
1627:
1626:
1620:
1601:
1598:
1597:
1581:
1578:
1577:
1574:
1555:
1552:
1551:
1548:
1540:
1515:
1512:
1511:
1508:
1502:
1483:
1480:
1479:
1463:
1460:
1459:
1453:
1448:
1442:
1436:
1431:
1336:
1327:
1325:Decimal example
1240:
1237:
1236:
1219:
1215:
1206:
1202:
1188:
1185:
1184:
1167:
1163:
1161:
1158:
1157:
1141:
1138:
1137:
1120:
1116:
1114:
1111:
1110:
1088:
1085:
1084:
1067:
1063:
1049:
1046:
1045:
1022:
1018:
1010:
1007:
1006:
990:
987:
986:
970:
967:
966:
944:
941:
940:
915:
912:
911:
870:
866:
864:
861:
860:
838:
835:
834:
793:
789:
787:
784:
783:
748:
744:
742:
739:
738:
722:
719:
718:
702:
699:
698:
682:
679:
678:
662:
659:
658:
633:
630:
629:
613:
610:
609:
587:
584:
583:
532:
528:
475:
471:
456:
452:
451:
447:
417:
413:
411:
408:
407:
391:
388:
387:
365:
362:
361:
333:
329:
328:
324:
322:
319:
318:
284:
280:
279:
275:
273:
270:
269:
242:
238:
236:
233:
232:
216:
213:
212:
193:
190:
189:
173:
170:
169:
162:
17:
12:
11:
5:
2149:
2139:
2138:
2123:
2122:
2107:
2095:
2083:
2081:
2078:
2077:
2076:
2069:
2066:
2053:
2050:
2040:
2037:
2036:
2035:
2031:
2028:
2020:
2017:
2016:
2015:
2007:
1999:
1979:
1978:Practical uses
1976:
1975:
1974:
1950:was less than
1944:
1927:
1923:
1917:
1913:
1901:
1898:
1879:
1875:
1857:
1851:Complementing
1848:185 - 329
1847:
1832:
1828:
1805:
1801:
1797:
1794:
1791:
1788:
1754:Main article:
1751:
1748:
1742:
1736:
1729:
1728:
1725:
1721:
1720:
1717:
1713:
1712:
1707:
1688:
1685:
1679:
1673:
1667:
1654:
1634:
1618:
1605:
1585:
1572:
1559:
1547:
1544:
1538:
1525:
1522:
1519:
1506:
1500:
1487:
1467:
1452:
1449:
1447:344 655
1446:
1440:
1435:
1432:
1430:873 - 218
1429:
1419:
1418:
1415:
1411:
1410:
1407:
1403:
1402:
1399:
1395:
1394:
1391:
1387:
1386:
1383:
1379:
1378:
1375:
1371:
1370:
1367:
1363:
1362:
1359:
1355:
1354:
1351:
1347:
1346:
1343:
1339:
1338:
1333:
1326:
1323:
1250:
1247:
1244:
1222:
1218:
1214:
1209:
1205:
1201:
1198:
1195:
1192:
1170:
1166:
1145:
1123:
1119:
1098:
1095:
1092:
1070:
1066:
1062:
1059:
1056:
1053:
1033:
1030:
1025:
1021:
1017:
1014:
994:
974:
954:
951:
948:
925:
922:
919:
899:
896:
893:
890:
887:
884:
881:
878:
873:
869:
848:
845:
842:
822:
819:
816:
813:
810:
807:
804:
801:
796:
792:
771:
768:
765:
762:
759:
756:
751:
747:
726:
706:
686:
666:
643:
640:
637:
617:
597:
594:
591:
567:
564:
561:
558:
555:
552:
549:
546:
541:
538:
535:
531:
527:
524:
521:
518:
515:
512:
508:
504:
501:
498:
495:
492:
489:
484:
481:
478:
474:
470:
465:
462:
459:
455:
450:
446:
443:
440:
437:
434:
431:
428:
425:
420:
416:
395:
375:
372:
369:
348:
344:
341:
336:
332:
327:
306:
303:
299:
295:
292:
287:
283:
278:
253:
250:
245:
241:
231:is defined as
220:
197:
188:-digit number
177:
161:
158:
94:Midy's theorem
25:adding machine
15:
9:
6:
4:
3:
2:
2148:
2137:
2134:
2133:
2131:
2118:
2111:
2104:
2099:
2093:
2088:
2084:
2075:
2072:
2071:
2065:
2063:
2059:
2049:
2045:
2032:
2029:
2026:
2025:
2024:
2012:
2008:
2004:
2000:
1996:
1993:
1992:
1991:
1984:
1972:
1968:
1964:
1963:
1957:
1953:
1949:
1945:
1942:
1925:
1921:
1915:
1911:
1902:
1899:
1895:
1894:
1893:
1877:
1873:
1864:
1856:
1854:
1846:
1830:
1826:
1803:
1799:
1795:
1792:
1789:
1786:
1778:
1774:
1769:
1767:
1763:
1757:
1747:
1741:
1735:
1726:
1723:
1718:
1715:
1708:
1703:
1702:
1698:
1694:
1687:Binary method
1684:
1678:
1672:
1666:
1652:
1632:
1623:
1617:
1603:
1583:
1571:
1557:
1543:
1537:
1523:
1520:
1517:
1505:
1499:
1485:
1465:
1456:
1451:Second method
1445:
1439:
1428:
1425:
1416:
1413:
1408:
1405:
1400:
1397:
1392:
1389:
1384:
1381:
1376:
1373:
1368:
1365:
1360:
1357:
1352:
1349:
1344:
1341:
1334:
1331:
1330:
1322:
1320:
1316:
1312:
1308:
1303:
1299:
1295:
1291:
1290:
1285:
1284:
1279:
1275:
1271:
1267:
1262:
1248:
1245:
1242:
1220:
1216:
1212:
1207:
1203:
1199:
1196:
1193:
1190:
1168:
1164:
1143:
1121:
1117:
1096:
1093:
1090:
1068:
1064:
1060:
1057:
1054:
1051:
1031:
1028:
1023:
1019:
1015:
1012:
992:
972:
952:
949:
946:
937:
923:
920:
917:
894:
891:
888:
882:
879:
876:
871:
867:
846:
843:
840:
817:
814:
811:
805:
802:
799:
794:
790:
769:
766:
763:
760:
757:
754:
749:
745:
724:
704:
684:
664:
655:
641:
638:
635:
615:
595:
592:
589:
581:
562:
559:
556:
550:
547:
544:
539:
536:
533:
529:
522:
519:
516:
510:
506:
502:
499:
496:
493:
490:
487:
482:
479:
476:
472:
468:
463:
460:
457:
453:
448:
441:
438:
435:
429:
426:
423:
418:
414:
393:
373:
370:
367:
346:
342:
339:
334:
330:
325:
304:
301:
297:
293:
290:
285:
281:
276:
267:
251:
248:
243:
239:
218:
211:
195:
175:
167:
157:
155:
151:
146:
145:
140:
136:
131:
129:
125:
120:
118:
114:
110:
106:
102:
97:
95:
92:, such as in
91:
90:number theory
87:
83:
79:
74:
70:
66:
62:
58:
54:
50:
46:
42:
38:
34:
26:
21:
2119:. Macmillan.
2116:
2110:
2098:
2092:Florida Tech
2087:
2055:
2046:
2042:
2022:
2019:In computers
1989:
1970:
1966:
1960:
1951:
1947:
1862:
1860:
1852:
1850:
1776:
1772:
1770:
1765:
1761:
1759:
1745:
1739:
1732:
1682:
1676:
1670:
1624:
1621:
1575:
1549:
1541:
1509:
1503:
1457:
1454:
1443:
1437:
1434:First method
1426:
1422:
1318:
1314:
1310:
1306:
1301:
1297:
1294:Donald Knuth
1287:
1281:
1273:
1269:
1263:
938:
656:
265:
165:
163:
147:
143:
138:
137:is added to
134:
132:
127:
126:is added to
123:
121:
112:
104:
100:
98:
85:
68:
40:
30:
2039:Manual uses
2003:Comptometer
1711:complement
1337:complement
1083:. Assuming
268:, which is
73:subtraction
69:complements
33:mathematics
2080:References
2062:subtrahend
1498:is taken:
1005:to obtain
737:to obtain
578:(see also
109:subtrahend
1965:). So if
1790:−
1521:−
1246:−
1213:−
1194:−
1094:≤
1055:−
1029:−
950:−
921:−
892:−
883:−
877:−
844:−
815:−
806:−
800:−
761:−
755:−
639:−
593:−
560:−
548:⋯
537:−
520:−
491:⋯
480:−
461:−
439:−
424:−
386:repeated
371:−
340:−
302:−
291:−
249:−
82:computers
53:mechanism
49:algorithm
37:computing
2130:Category
2068:See also
1235:or just
71:. Thus
57:addition
45:integers
1335:Nines'
1266:decimal
1264:In the
150:radices
117:minuend
1863:simple
1709:Ones'
1706:digit
1704:Binary
1332:Digit
1278:binary
168:of an
61:places
55:) for
39:, the
2074:Curta
1775:<
1307:one's
1276:. In
677:from
628:from
210:radix
115:(the
107:(the
2009:The
2001:The
1695:and
1596:and
1317:and
1315:ones
1309:and
164:The
99:The
51:(or
35:and
1897:do.
1616:is
1044:or
985:to
717:to
208:in
31:In
2132::
1969:≤
1764:≤
1727:0
1724:1
1719:1
1716:0
1570::
1417:0
1414:9
1409:1
1406:8
1401:2
1398:7
1393:3
1390:6
1385:4
1382:5
1377:5
1374:4
1369:6
1366:3
1361:7
1358:2
1353:8
1350:1
1345:9
1342:0
1321:.
936:.
654:.
96:.
1971:x
1967:y
1952:y
1948:x
1926:2
1922:/
1916:n
1912:b
1878:n
1874:b
1853:y
1831:n
1827:b
1804:n
1800:b
1796:+
1793:y
1787:x
1777:y
1773:x
1766:x
1762:y
1653:x
1633:y
1604:y
1584:x
1558:x
1524:y
1518:x
1486:y
1466:x
1249:y
1243:x
1221:n
1217:b
1208:n
1204:b
1200:+
1197:y
1191:x
1169:n
1165:b
1144:1
1122:n
1118:b
1097:x
1091:y
1069:n
1065:b
1061:+
1058:y
1052:x
1032:y
1024:n
1020:b
1016:+
1013:x
993:x
973:y
953:y
947:x
924:y
918:x
898:)
895:y
889:x
886:(
880:1
872:n
868:b
847:y
841:x
821:)
818:y
812:x
809:(
803:1
795:n
791:b
770:y
767:+
764:x
758:1
750:n
746:b
725:y
705:x
685:x
665:y
642:1
636:b
616:y
596:1
590:b
566:)
563:1
557:b
554:(
551:+
545:+
540:1
534:n
530:b
526:)
523:1
517:b
514:(
511:=
507:)
503:1
500:+
497:b
494:+
488:+
483:2
477:n
473:b
469:+
464:1
458:n
454:b
449:(
445:)
442:1
436:b
433:(
430:=
427:1
419:n
415:b
394:n
374:1
368:b
347:)
343:1
335:n
331:b
326:(
305:y
298:)
294:1
286:n
282:b
277:(
252:y
244:n
240:b
219:b
196:y
176:n
139:x
135:y
128:y
124:x
113:x
105:y
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
