36:
226:
values when they were transmitted from a register to the math unit and then converted back to sign–magnitude when the result was transmitted back to the register. The electronics required more gates than the other systems – a key concern when the cost and packaging of discrete transistors were critical. IBM was one of the early supporters of sign–magnitude, with their
246:(−0). Negative zero behaves exactly like positive zero: when used as an operand in any calculation, the result will be the same whether an operand is positive or negative zero. The disadvantage is that the existence of two forms of the same value necessitates two comparisons when checking for equality with zero. Ones' complement subtraction can also result in an
222:, the system that is dominant today. Another camp supported ones' complement, where a negative value is formed by inverting all of the bits in its positive equivalent. A third group supported sign–magnitude, where a value is changed from positive to negative simply by toggling the word's highest-order bit.
277:
Two's complement is the easiest to implement in hardware, which may be the ultimate reason for its widespread popularity. Processors on the early mainframes often consisted of thousands of transistors, so eliminating a significant number of transistors was a significant cost savings. Mainframes such
2686:
is almost a sign bit; zero has the same least significant bit (0) as all the negative numbers. This choice results in the largest magnitude representable positive number being one higher than the largest magnitude negative number, unlike in two's complement or the
Protocol Buffers zig-zag encoding.
225:
There were arguments for and against each of the systems. Sign–magnitude allowed for easier tracing of memory dumps (a common process in the 1960s) as small numeric values use fewer 1 bits. These systems did ones' complement math internally, so numbers would have to be converted to ones' complement
217:
The early days of digital computing were marked by competing ideas about both hardware technology and mathematics technology (numbering systems). One of the great debates was the format of negative numbers, with some of the era's top experts expressing very strong and differing opinions. One camp
1548:
The range of numbers that can be represented is asymmetric. If the word has an even number of bits, the magnitude of the largest negative number that can be represented is twice as large as the largest positive number that can be represented, and vice versa if the word has an odd number of bits.
1048:
of the ones' complement representation. This can also be thought of as the most significant bit representing the inverse of its value in an unsigned integer; in an 8-bit unsigned byte, the most significant bit represents the 128ths place, where in two's complement that bit would represent −128.
1118:
lowest significant bits of a product (value of multiplication). For instance, a two's-complement addition of 127 and −128 gives the same binary bit pattern as an unsigned addition of 127 and 128, as can be seen from the 8-bit two's complement table.
250:(described below). It can be argued that this makes the addition and subtraction logic more complicated or that it makes it simpler, as a subtraction requires simply inverting the bits of the second operand as it is passed to the adder. The
241:
Ones' complement allowed for somewhat simpler hardware designs, as there was no need to convert values when passed to and from the math unit. But it also shared an undesirable characteristic with sign–magnitude: the ability to represent
582:
Addition and subtraction require different behavior depending on the sign bit, whereas ones' complement can ignore the sign bit and just do an end-around carry, and two's complement can ignore the sign bit and depend on the overflow
2917:
1052:
In two's-complement, there is only one zero, represented as 00000000. Negating a number (whether negative or positive) is done by inverting all the bits and then adding one to that result. This actually reflects the
2936:
837:
binary decimal 11111110 −1 + 00000010 +2 ─────────── ── 1 00000000 0 ← Not the correct answer 1 +1 ← Add carry ─────────── ── 00000001 1 ← Correct answer
205:
There is no definitive criterion by which any of the representations is universally superior. For integers, the representation used in most current computing devices is two's complement, although the
883:
the magnitude (inverting all the bits after the first). For example, the decimal number −125 with its sign–magnitude representation 11111101 can be represented in ones' complement form as 10000010.
1533:, is 2; thus the rightmost bit represents 2, the next bit represents 2, the next bit 2, and so on. However, a binary number system with base −2 is also possible. The rightmost bit represents
1104:
1044:(i.e. the "complement") of the positive number plus one, i.e. to the ones' complement plus one. It circumvents the problems of multiple representations of 0 and the need for the
586:
Comparison also requires inspecting the sign bit, whereas in two's complement, one can simply subtract the two numbers, and check if the outcome is positive or negative.
1359:
is represented by an all-zero bit pattern. This can be seen as a slight modification and generalization of the aforementioned two's-complement, which is virtually the
841:
In the previous example, the first binary addition gives 00000000, which is incorrect. The correct result (00000001) only appears when the carry is added back in.
593:
This approach is directly comparable to the common way of showing a sign (placing a "+" or "−" next to the number's magnitude). Some early binary computers (e.g.,
53:
788:(i.e. the "complement") of the positive number. Like sign–magnitude representation, ones' complement has two representations of 0: 00000000 (+0) and 11111111 (
834:
back into the resulting sum. To see why this is necessary, consider the following example showing the case of the addition of −1 (11111110) to +2 (00000010):
864:
from the ones' complement representation of zero that is a long sequence of ones (−0). Two's complement arithmetic, on the other hand, forms the negation of
314:, etc.) also chose to use two's complement math. As IC technology advanced, two's complement technology was adopted in virtually all processors, including
2886:
597:) use this representation, perhaps because of its natural relation to common usage. Sign–magnitude is the most common way of representing the
100:
72:
1529:
representation, a signed number is represented using a number system with base −2. In conventional binary number systems, the base, or
79:
3051:
879:
Note that the ones' complement representation of a negative number can be obtained from the sign–magnitude representation merely by
119:
2986:
86:
1545:
and so on, with alternating sign. The numbers that can be represented with four bits are shown in the comparison table below.
57:
547:, only seven bits represent the magnitude, which can range from 0000000 (0) to 1111111 (127). Thus numbers ranging from −127
826:
To add two numbers represented in this system, one does a conventional binary addition, but it is then necessary to do an
68:
3066:
2675:
876:
to +0. Therefore, ones' complement and two's complement representations of the same negative value will differ by one.
2695:
2248:
Same table, as viewed from "given these binary bits, what is the number as interpreted by the representation system":
133:
571:
0101011. Using sign–magnitude representation has multiple consequences which makes them more intricate to implement:
1067:
46:
2950:
Shedletsky, John J. (1977). "Comment on the
Sequential and Indeterminate Behavior of an End-Around-Carry Adder".
2679:
602:
2664:
1186:
194:. Some of the alternative methods use implicit instead of explicit signs, such as negative binary, using the
2853:
2794:
206:
93:
2725:
2730:
2683:
2671:
2660:
1178:
Invert all the bits through the number. This computes the same result as subtracting from negative one.
171:
2890:
1557:
The following table shows the positive and negative integers that can be represented using four bits.
539:, set to 0 for a positive number and to 1 for a negative number), and the magnitude of the number (or
158:, negative numbers in any base are represented by prefixing them with a minus sign ("−"). However, in
3025:
2768:
2720:
1367:
536:
531:, a signed number is represented by the bit pattern corresponding to the sign of the number for the
892:
873:
614:
219:
199:
2667:
encoding intended for nonnegative (unsigned) integers to be used efficiently for signed integers.
3005:
2967:
2822:
1058:
1054:
880:
163:
3047:
2757:"Two's complement computation sharing multiplier and its applications to high performance DFE"
1335:, a signed number is represented by the bit pattern corresponding to the unsigned number plus
327:
247:
3001:
2706:
also expressed preference for such modified decimal numbers to reduce errors in computation.
2959:
2800:
2776:
2715:
2656:
1390:
1382:
1111:
1045:
335:
2703:
2691:
263:
148:
1040:
representation, a negative number is represented by the bit pattern corresponding to the
784:
representation, a negative number is represented by the bit pattern corresponding to the
2772:
2702:
advocated reducing expressions to "small numbers", numerals 1, 2, 3, 4, and 5. In 1840,
589:
The minimum negative number is −127, instead of −128 as in the case of two's complement.
1374:
1106:. Addition of a pair of two's-complement integers is the same as addition of a pair of
540:
279:
255:
3060:
1414:
1398:
1209:
844:
A remark on terminology: The system is referred to as "ones' complement" because the
283:
243:
175:
2971:
2915:, "Array multiplier operating in one's complement format", issued 1981-03-10
1122:
An easier method to get the negation of a number in two's complement is as follows:
2931:
1061:
2912:
2699:
2633:
2467:
1980:
1394:
1189:
1041:
853:
785:
598:
555:
can be represented once the sign bit (the eighth bit) is added. For example, −43
267:
202:, whether positive, negative, fractional, or other elaborations on such themes.
155:
35:
17:
2735:
1386:
1107:
831:
804:
311:
2663:
to represent the sign and has a single representation of zero. This allows a
2963:
2862:
2780:
347:
323:
140:
1402:
1378:
845:
594:
532:
259:
235:
2659:"zig-zag encoding" is a system similar to sign–magnitude, but uses the
1364:
343:
339:
231:
227:
2934:, "One's complement cryptographic combiner", issued 1999-12-11
789:
576:
170:, without extra symbols. The four best-known methods of extending the
1185:
Example: for +2, which is 00000010 in binary (the ~ character is the
310:
machines. The architects of the early integrated-circuit-based CPUs (
303:
291:
2756:
2831:
1530:
1373:
Biased representations are now primarily used for the exponent of
1114:, if that is done); the same is true for subtraction and even for
331:
299:
295:
287:
251:
575:
There are two ways to represent zero, 00000000 (0) and 10000000 (
238:
series computers being perhaps the best-known systems to use it.
544:
319:
271:
315:
307:
167:
159:
29:
2824:
Intel 64 and IA-32 Architectures
Software Developer's Manual
795:
As an example, the ones' complement form of 00101011 (43
823:
with zero being either 00000000 (+0) or 11111111 (−0).
543:) for the remaining bits. For example, in an eight-bit
294:
use two's complement, as did minicomputers such as the
1401:. It also had use for binary-coded decimal numbers as
2755:
Choo, Hunsoo; Muhammad, K.; Roy, K. (February 2003).
1070:
27:
Encoding of negative numbers in binary number systems
1192:operator, so ~X means "invert all the bits in X"):
60:. Unsourced material may be challenged and removed.
1156:2. Invert all of the bits to the left of that "1"
1098:
2990:, Volume 2: Seminumerical Algorithms, chapter 4.1
807:numbers using ones' complement is represented by
1199:11111101 + 1 → 11111110 (−2 in two's complement)
1138:1. Starting from the right, find the first "1"
166:, numbers are represented only as sequences of
1099:{\displaystyle \mathbb {Z} /2^{N}\mathbb {Z} }
274:computer use ones' complement representation.
815:and ±0. A conventional eight-bit byte is −127
8:
2682:to negative numbers. In that extension, the
198:. Corresponding methods can be devised for
2796:GE-625 / 635 Programming Reference Manual
1974:
1971:
1968:
1959:
1092:
1091:
1085:
1076:
1072:
1071:
1069:
872:from a single large power of two that is
187:
183:
120:Learn how and when to remove this message
2250:
1559:
1418:
1213:
1124:
896:
618:
357:
2747:
2761:IEEE Transactions on Signal Processing
1393:(64-bit) exponent field is an 11-bit
7:
2887:"Computer Science 315 Lecture Notes"
860:) can also be formed by subtracting
58:adding citations to reliable sources
179:
2676:High Efficiency Video Coding/H.265
25:
3026:Protocol Buffers: Signed Integers
2690:Another approach is to give each
1561:Four-bit integer representations
209:mainframes use ones' complement.
3037:The Logic of Computer Arithmetic
2680:extend exponential-Golomb coding
2670:A similar method is used in the
1381:defines the exponent field of a
1379:IEEE 754 floating-point standard
906:Two's complement interpretation
628:Ones' complement interpretation
559:encoded in an eight-bit byte is
34:
2987:The Art of Computer Programming
2678:video compression standards to
69:"Signed number representations"
45:needs additional citations for
3044:Computer Arithmetic Algorithms
2952:IEEE Transactions on Computers
367:Sign–magnitude interpretation
207:Unisys ClearPath Dorado series
1:
2885:Bacon, Jason W. (2010–2011).
830:: that is, add any resulting
191:
145:signed number representations
3000:Thomas Finley (April 2000).
2224:−11
2201:−10
1385:(32-bit) number as an 8-bit
1324:representation, also called
1164:
1158:
1146:
1140:
523:representation, also called
195:
2696:signed-digit representation
2672:Advanced Video Coding/H.264
2223:
2200:
2178:−9
2177:
2155:−8
2154:
2132:−7
2131:
2109:−6
2108:
2086:−5
2085:
2063:−4
2062:
2040:−3
2039:
2017:−2
2016:
1994:−1
1993:
1979:
1956:
1933:
1910:
1887:
1864:
1841:
1818:
1795:
1772:
1749:
1727:10
1726:
1704:11
1703:
1681:12
1680:
1658:13
1657:
1635:14
1634:
1612:15
1611:
1589:16
1588:
1351:. Thus 0 is represented by
898:Eight-bit two's complement
620:Eight-bit ones' complement
134:Signed-digit representation
3083:
1957:0
1934:1
1911:2
1888:3
1865:4
1842:5
1819:6
1796:7
1773:8
1750:9
1537:, the next bit represents
1412:
1223:Excess-128 interpretation
1207:
1057:structure on all integers
890:
886:
612:
151:in binary number systems.
131:
2698:. For instance, in 1726,
1983:
1110:(except for detection of
608:
359:Eight-bit sign–magnitude
2665:variable-length quantity
1431:Unsigned interpretation
1226:Unsigned interpretation
909:Unsigned interpretation
631:Unsigned interpretation
370:Unsigned interpretation
132:Not to be confused with
2964:10.1109/TC.1977.1674817
2781:10.1109/TSP.2002.806984
1428:Base −2 interpretation
799:) becomes 11010100 (−43
147:are required to encode
3046:, A.K. Peters (2002),
3039:, Prentice-Hall (1963)
2855:Power ISA Version 2.07
2726:Computer number format
1100:
2731:Method of complements
2694:a sign, yielding the
2684:least significant bit
2661:least significant bit
1215:Eight-bit excess-128
1101:
881:bitwise complementing
172:binary numeral system
2721:Binary-coded decimal
1368:most significant bit
1363:representation with
1196:~00000010 → 11111101
1068:
852:(represented as the
848:of a positive value
537:most significant bit
54:improve this article
3067:Computer arithmetic
2893:on 14 February 2020
2773:2003ITSP...51..458C
1562:
1421:
1216:
899:
621:
360:
3006:Cornell University
3002:"Two's Complement"
1581:Excess-8 (biased)
1560:
1420:Eight-bit base −2
1419:
1214:
1096:
897:
619:
525:sign-and-magnitude
358:
2648:
2647:
2246:
2245:
1578:Two's complement
1575:Ones' complement
1523:
1522:
1318:
1317:
1172:
1171:
1034:
1033:
778:
777:
517:
516:
248:end-around borrow
130:
129:
122:
104:
16:(Redirected from
3074:
3028:
3023:
3017:
3016:
3014:
3012:
2997:
2991:
2982:
2976:
2975:
2947:
2941:
2940:
2939:
2935:
2928:
2922:
2921:
2920:
2916:
2909:
2903:
2902:
2900:
2898:
2889:. Archived from
2882:
2876:
2874:
2872:
2870:
2860:
2850:
2844:
2843:
2841:
2839:
2829:
2819:
2813:
2812:
2810:
2808:
2801:General Electric
2791:
2785:
2784:
2752:
2716:Balanced ternary
2657:Protocol Buffers
2266:Two's complement
2263:Ones' complement
2251:
1563:
1553:Comparison table
1544:
1540:
1536:
1422:
1391:double-precision
1383:single-precision
1362:
1217:
1125:
1108:unsigned numbers
1105:
1103:
1102:
1097:
1095:
1090:
1089:
1080:
1075:
1046:end-around carry
1038:two's complement
900:
893:Two's complement
887:Two's complement
828:end-around carry
814:
810:
803:). The range of
782:ones' complement
622:
615:Ones' complement
609:Ones' complement
563:0101011 while 43
529:signed magnitude
361:
220:two's complement
188:two's complement
184:ones' complement
149:negative numbers
125:
118:
114:
111:
105:
103:
62:
38:
30:
21:
18:End-around carry
3082:
3081:
3077:
3076:
3075:
3073:
3072:
3071:
3057:
3056:
3032:
3031:
3024:
3020:
3010:
3008:
2999:
2998:
2994:
2983:
2979:
2949:
2948:
2944:
2937:
2930:
2929:
2925:
2918:
2911:
2910:
2906:
2896:
2894:
2884:
2883:
2879:
2868:
2866:
2858:
2852:
2851:
2847:
2837:
2835:
2834:. Section 4.2.1
2827:
2821:
2820:
2816:
2806:
2804:
2793:
2792:
2788:
2754:
2753:
2749:
2744:
2712:
2704:Augustin Cauchy
2653:
1572:Sign–magnitude
1555:
1542:
1541:, the next bit
1538:
1534:
1417:
1411:
1360:
1358:
1354:
1342:
1338:
1329:
1212:
1206:
1081:
1066:
1065:
895:
889:
871:
868:by subtracting
867:
863:
859:
851:
839:
822:
818:
812:
808:
802:
798:
617:
611:
566:
558:
554:
550:
356:
264:CDC 6000 series
215:
137:
126:
115:
109:
106:
63:
61:
51:
39:
28:
23:
22:
15:
12:
11:
5:
3080:
3078:
3070:
3069:
3059:
3058:
3055:
3054:
3042:Israel Koren,
3040:
3030:
3029:
3018:
2992:
2984:Donald Knuth:
2977:
2958:(3): 271–272.
2942:
2923:
2904:
2877:
2845:
2814:
2803:. January 1966
2786:
2767:(2): 458–469.
2746:
2745:
2743:
2740:
2739:
2738:
2733:
2728:
2723:
2718:
2711:
2708:
2652:
2649:
2646:
2645:
2642:
2639:
2636:
2631:
2628:
2625:
2621:
2620:
2617:
2614:
2611:
2608:
2605:
2602:
2598:
2597:
2594:
2591:
2588:
2585:
2582:
2579:
2575:
2574:
2571:
2568:
2565:
2562:
2559:
2556:
2552:
2551:
2548:
2545:
2542:
2539:
2536:
2533:
2529:
2528:
2525:
2522:
2519:
2516:
2513:
2510:
2506:
2505:
2502:
2499:
2496:
2493:
2490:
2487:
2483:
2482:
2479:
2476:
2473:
2470:
2465:
2462:
2458:
2457:
2454:
2451:
2448:
2445:
2442:
2439:
2435:
2434:
2431:
2428:
2425:
2422:
2419:
2416:
2412:
2411:
2408:
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2402:
2399:
2396:
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2389:
2388:
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2382:
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2370:
2366:
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2359:
2356:
2353:
2350:
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2343:
2342:
2339:
2336:
2333:
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2327:
2324:
2320:
2319:
2316:
2313:
2310:
2307:
2304:
2301:
2297:
2296:
2293:
2290:
2287:
2284:
2281:
2278:
2274:
2273:
2270:
2267:
2264:
2261:
2260:Sign–magnitude
2258:
2255:
2244:
2243:
2240:
2237:
2234:
2231:
2228:
2225:
2221:
2220:
2217:
2214:
2211:
2208:
2205:
2202:
2198:
2197:
2194:
2191:
2188:
2185:
2182:
2179:
2175:
2174:
2171:
2168:
2165:
2162:
2159:
2156:
2152:
2151:
2148:
2145:
2142:
2139:
2136:
2133:
2129:
2128:
2125:
2122:
2119:
2116:
2113:
2110:
2106:
2105:
2102:
2099:
2096:
2093:
2090:
2087:
2083:
2082:
2079:
2076:
2073:
2070:
2067:
2064:
2060:
2059:
2056:
2053:
2050:
2047:
2044:
2041:
2037:
2036:
2033:
2030:
2027:
2024:
2021:
2018:
2014:
2013:
2010:
2007:
2004:
2001:
1998:
1995:
1991:
1990:
1987:
1984:
1977:
1976:
1973:
1970:
1967:
1964:
1961:
1958:
1954:
1953:
1950:
1947:
1944:
1941:
1938:
1935:
1931:
1930:
1927:
1924:
1921:
1918:
1915:
1912:
1908:
1907:
1904:
1901:
1898:
1895:
1892:
1889:
1885:
1884:
1881:
1878:
1875:
1872:
1869:
1866:
1862:
1861:
1858:
1855:
1852:
1849:
1846:
1843:
1839:
1838:
1835:
1832:
1829:
1826:
1823:
1820:
1816:
1815:
1812:
1809:
1806:
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1800:
1797:
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1792:
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1786:
1783:
1780:
1777:
1774:
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1769:
1766:
1763:
1760:
1757:
1754:
1751:
1747:
1746:
1743:
1740:
1737:
1734:
1731:
1728:
1724:
1723:
1720:
1717:
1714:
1711:
1708:
1705:
1701:
1700:
1697:
1694:
1691:
1688:
1685:
1682:
1678:
1677:
1674:
1671:
1668:
1665:
1662:
1659:
1655:
1654:
1651:
1648:
1645:
1642:
1639:
1636:
1632:
1631:
1628:
1625:
1622:
1619:
1616:
1613:
1609:
1608:
1605:
1602:
1599:
1596:
1593:
1590:
1586:
1585:
1582:
1579:
1576:
1573:
1570:
1567:
1554:
1551:
1521:
1520:
1517:
1514:
1510:
1509:
1506:
1503:
1499:
1498:
1495:
1492:
1488:
1487:
1484:
1481:
1477:
1476:
1473:
1470:
1466:
1465:
1462:
1459:
1455:
1454:
1451:
1448:
1444:
1443:
1440:
1437:
1433:
1432:
1429:
1426:
1410:
1407:
1375:floating-point
1356:
1352:
1340:
1336:
1327:
1316:
1315:
1312:
1309:
1305:
1304:
1301:
1298:
1294:
1293:
1290:
1287:
1283:
1282:
1279:
1276:
1272:
1271:
1268:
1265:
1261:
1260:
1257:
1254:
1250:
1249:
1246:
1243:
1239:
1238:
1235:
1232:
1228:
1227:
1224:
1221:
1208:Main article:
1205:
1202:
1201:
1200:
1197:
1183:
1182:
1179:
1170:
1169:
1163:
1157:
1153:
1152:
1145:
1139:
1135:
1134:
1131:
1128:
1094:
1088:
1084:
1079:
1074:
1032:
1031:
1028:
1025:
1021:
1020:
1017:
1014:
1010:
1009:
1006:
1003:
999:
998:
995:
992:
988:
987:
984:
981:
977:
976:
973:
970:
966:
965:
962:
959:
955:
954:
951:
948:
944:
943:
940:
937:
933:
932:
929:
926:
922:
921:
918:
915:
911:
910:
907:
904:
891:Main article:
888:
885:
869:
865:
861:
857:
849:
836:
820:
816:
800:
796:
776:
775:
772:
769:
765:
764:
761:
758:
754:
753:
750:
747:
743:
742:
739:
736:
732:
731:
728:
725:
721:
720:
717:
714:
710:
709:
706:
703:
699:
698:
695:
692:
688:
687:
684:
681:
677:
676:
673:
670:
666:
665:
662:
659:
655:
654:
651:
648:
644:
643:
640:
637:
633:
632:
629:
626:
613:Main article:
610:
607:
603:floating-point
591:
590:
587:
584:
580:
564:
556:
552:
548:
541:absolute value
521:sign–magnitude
515:
514:
511:
508:
504:
503:
500:
497:
493:
492:
489:
486:
482:
481:
478:
475:
471:
470:
467:
464:
460:
459:
456:
453:
449:
448:
445:
442:
438:
437:
434:
431:
427:
426:
423:
420:
416:
415:
412:
409:
405:
404:
401:
398:
394:
393:
390:
387:
383:
382:
379:
376:
372:
371:
368:
365:
355:
354:Sign–magnitude
352:
280:IBM System/360
256:CDC 160 series
214:
211:
180:sign–magnitude
176:signed numbers
128:
127:
42:
40:
33:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3079:
3068:
3065:
3064:
3062:
3053:
3052:1-56881-160-8
3049:
3045:
3041:
3038:
3035:Ivan Flores,
3034:
3033:
3027:
3022:
3019:
3007:
3003:
2996:
2993:
2989:
2988:
2981:
2978:
2973:
2969:
2965:
2961:
2957:
2953:
2946:
2943:
2933:
2927:
2924:
2914:
2908:
2905:
2892:
2888:
2881:
2878:
2865:. Section 1.4
2864:
2857:
2856:
2849:
2846:
2833:
2826:
2825:
2818:
2815:
2802:
2798:
2797:
2790:
2787:
2782:
2778:
2774:
2770:
2766:
2762:
2758:
2751:
2748:
2741:
2737:
2734:
2732:
2729:
2727:
2724:
2722:
2719:
2717:
2714:
2713:
2709:
2707:
2705:
2701:
2697:
2693:
2688:
2685:
2681:
2677:
2673:
2668:
2666:
2662:
2658:
2651:Other systems
2650:
2643:
2640:
2637:
2635:
2632:
2629:
2626:
2623:
2622:
2618:
2615:
2612:
2609:
2606:
2603:
2600:
2599:
2595:
2592:
2589:
2586:
2583:
2580:
2577:
2576:
2572:
2569:
2566:
2563:
2560:
2557:
2554:
2553:
2549:
2546:
2543:
2540:
2537:
2534:
2531:
2530:
2526:
2523:
2520:
2517:
2514:
2511:
2508:
2507:
2503:
2500:
2497:
2494:
2491:
2488:
2485:
2484:
2480:
2477:
2474:
2471:
2469:
2466:
2463:
2460:
2459:
2455:
2452:
2449:
2446:
2443:
2440:
2437:
2436:
2432:
2429:
2426:
2423:
2420:
2417:
2414:
2413:
2409:
2406:
2403:
2400:
2397:
2394:
2391:
2390:
2386:
2383:
2380:
2377:
2374:
2371:
2368:
2367:
2363:
2360:
2357:
2354:
2351:
2348:
2345:
2344:
2340:
2337:
2334:
2331:
2328:
2325:
2322:
2321:
2317:
2314:
2311:
2308:
2305:
2302:
2299:
2298:
2294:
2291:
2288:
2285:
2282:
2279:
2276:
2275:
2271:
2268:
2265:
2262:
2259:
2256:
2253:
2252:
2249:
2241:
2238:
2235:
2232:
2229:
2226:
2222:
2218:
2215:
2212:
2209:
2206:
2203:
2199:
2195:
2192:
2189:
2186:
2183:
2180:
2176:
2172:
2169:
2166:
2163:
2160:
2157:
2153:
2149:
2146:
2143:
2140:
2137:
2134:
2130:
2126:
2123:
2120:
2117:
2114:
2111:
2107:
2103:
2100:
2097:
2094:
2091:
2088:
2084:
2080:
2077:
2074:
2071:
2068:
2065:
2061:
2057:
2054:
2051:
2048:
2045:
2042:
2038:
2034:
2031:
2028:
2025:
2022:
2019:
2015:
2011:
2008:
2005:
2002:
1999:
1996:
1992:
1988:
1985:
1982:
1978:
1965:
1962:
1955:
1951:
1948:
1945:
1942:
1939:
1936:
1932:
1928:
1925:
1922:
1919:
1916:
1913:
1909:
1905:
1902:
1899:
1896:
1893:
1890:
1886:
1882:
1879:
1876:
1873:
1870:
1867:
1863:
1859:
1856:
1853:
1850:
1847:
1844:
1840:
1836:
1833:
1830:
1827:
1824:
1821:
1817:
1813:
1810:
1807:
1804:
1801:
1798:
1794:
1790:
1787:
1784:
1781:
1778:
1775:
1771:
1767:
1764:
1761:
1758:
1755:
1752:
1748:
1744:
1741:
1738:
1735:
1732:
1729:
1725:
1721:
1718:
1715:
1712:
1709:
1706:
1702:
1698:
1695:
1692:
1689:
1686:
1683:
1679:
1675:
1672:
1669:
1666:
1663:
1660:
1656:
1652:
1649:
1646:
1643:
1640:
1637:
1633:
1629:
1626:
1623:
1620:
1617:
1614:
1610:
1606:
1603:
1600:
1597:
1594:
1591:
1587:
1583:
1580:
1577:
1574:
1571:
1568:
1565:
1564:
1558:
1552:
1550:
1546:
1532:
1528:
1518:
1515:
1512:
1511:
1507:
1504:
1501:
1500:
1496:
1493:
1490:
1489:
1485:
1482:
1479:
1478:
1474:
1471:
1468:
1467:
1463:
1460:
1457:
1456:
1452:
1449:
1446:
1445:
1441:
1438:
1435:
1434:
1430:
1427:
1425:Binary value
1424:
1423:
1416:
1415:Negative base
1408:
1406:
1404:
1400:
1399:exponent bias
1396:
1392:
1388:
1384:
1380:
1377:numbers. The
1376:
1371:
1369:
1366:
1350:
1346:
1345:biasing value
1334:
1330:
1323:
1322:offset binary
1313:
1310:
1307:
1306:
1302:
1299:
1296:
1295:
1291:
1288:
1285:
1284:
1280:
1277:
1274:
1273:
1269:
1266:
1263:
1262:
1258:
1255:
1252:
1251:
1247:
1244:
1241:
1240:
1236:
1233:
1230:
1229:
1225:
1222:
1220:Binary value
1219:
1218:
1211:
1210:Offset binary
1204:Offset binary
1203:
1198:
1195:
1194:
1193:
1191:
1188:
1180:
1177:
1176:
1175:
1167:
1161:
1155:
1154:
1150:
1144:
1137:
1136:
1132:
1129:
1127:
1126:
1123:
1120:
1117:
1113:
1109:
1086:
1082:
1077:
1063:
1060:
1056:
1050:
1047:
1043:
1039:
1029:
1026:
1023:
1022:
1018:
1015:
1012:
1011:
1007:
1004:
1001:
1000:
996:
993:
990:
989:
985:
982:
979:
978:
974:
971:
968:
967:
963:
960:
957:
956:
952:
949:
946:
945:
941:
938:
935:
934:
930:
927:
924:
923:
919:
916:
913:
912:
908:
905:
903:Binary value
902:
901:
894:
884:
882:
877:
875:
855:
847:
842:
835:
833:
829:
824:
806:
793:
791:
787:
783:
773:
770:
767:
766:
762:
759:
756:
755:
751:
748:
745:
744:
740:
737:
734:
733:
729:
726:
723:
722:
718:
715:
712:
711:
707:
704:
701:
700:
696:
693:
690:
689:
685:
682:
679:
678:
674:
671:
668:
667:
663:
660:
657:
656:
652:
649:
646:
645:
641:
638:
635:
634:
630:
627:
625:Binary value
624:
623:
616:
606:
604:
600:
596:
588:
585:
581:
578:
574:
573:
572:
570:
562:
546:
542:
538:
534:
530:
526:
522:
512:
509:
506:
505:
501:
498:
495:
494:
490:
487:
484:
483:
479:
476:
473:
472:
468:
465:
462:
461:
457:
454:
451:
450:
446:
443:
440:
439:
435:
432:
429:
428:
424:
421:
418:
417:
413:
410:
407:
406:
402:
399:
396:
395:
391:
388:
385:
384:
380:
377:
374:
373:
369:
366:
364:Binary value
363:
362:
353:
351:
349:
345:
341:
337:
333:
329:
325:
321:
317:
313:
309:
305:
301:
297:
293:
289:
285:
284:GE-600 series
281:
275:
273:
269:
265:
261:
257:
253:
249:
245:
244:negative zero
239:
237:
233:
229:
223:
221:
212:
210:
208:
203:
201:
197:
193:
192:offset binary
189:
185:
181:
177:
174:to represent
173:
169:
165:
161:
157:
152:
150:
146:
142:
135:
124:
121:
113:
102:
99:
95:
92:
88:
85:
81:
78:
74:
71: –
70:
66:
65:Find sources:
59:
55:
49:
48:
43:This article
41:
37:
32:
31:
19:
3043:
3036:
3021:
3011:15 September
3009:. Retrieved
2995:
2985:
2980:
2955:
2951:
2945:
2926:
2907:
2895:. Retrieved
2891:the original
2880:
2867:. Retrieved
2854:
2848:
2836:. Retrieved
2823:
2817:
2805:. Retrieved
2795:
2789:
2764:
2760:
2750:
2689:
2669:
2654:
2247:
1556:
1547:
1526:
1524:
1372:
1348:
1344:
1332:
1325:
1321:
1319:
1184:
1174:Method two:
1173:
1165:
1159:
1148:
1142:
1121:
1115:
1051:
1037:
1035:
878:
843:
840:
827:
825:
794:
781:
779:
592:
568:
560:
528:
524:
520:
518:
276:
270:series, and
240:
224:
216:
204:
153:
144:
138:
116:
107:
97:
90:
83:
76:
64:
52:Please help
47:verification
44:
2897:21 February
2869:November 2,
2700:John Colson
1397:field; see
1395:excess-1023
1389:field. The
1190:bitwise NOT
1042:bitwise NOT
854:bitwise NOT
786:bitwise NOT
599:significand
535:(often the
268:UNIVAC 1100
200:other bases
156:mathematics
2932:US 6760440
2913:US 4484301
2807:August 15,
2742:References
2736:Signedness
1413:See also:
1387:excess-127
1361:excess-(2)
1343:being the
1133:Example 2
1130:Example 1
312:Intel 8080
286:, and the
218:supported
110:April 2013
80:newspapers
2863:Power.org
2838:August 6,
2655:Google's
1569:Unsigned
1543:(−2) = +4
1539:(−2) = −2
1535:(−2) = +1
874:congruent
583:behavior.
348:DEC Alpha
324:Power ISA
164:registers
141:computing
3061:Category
2972:14661474
2710:See also
2272:Base −2
2269:Excess-8
2257:Unsigned
1584:Base −2
1566:Decimal
1513:11111111
1491:10000001
1480:10000000
1469:01111111
1447:00000001
1436:00000000
1403:excess-3
1308:11111111
1286:10000001
1275:10000000
1264:01111111
1242:00000001
1231:00000000
1112:overflow
1024:11111111
1013:11111110
991:10000010
980:10000001
969:10000000
958:01111111
947:01111110
925:00000001
914:00000000
846:negation
809:−(2 − 1)
768:11111111
757:11111110
746:11111101
724:10000010
713:10000001
702:10000000
691:01111111
680:01111110
669:01111101
647:00000001
636:00000000
605:values.
595:IBM 7090
533:sign bit
507:11111111
496:11111110
485:11111101
463:10000010
452:10000001
441:10000000
430:01111111
419:01111110
408:01111101
386:00000001
375:00000000
302:and the
262:series,
260:CDC 3000
2769:Bibcode
1527:base −2
1525:In the
1409:Base −2
1365:negated
1355:, and −
1339:, with
1326:excess-
1320:In the
1181:Add one
1160:1101011
1141:0010100
1036:In the
819:to +127
813:(2 − 1)
780:In the
551:to +127
519:In the
344:PA-RISC
340:Itanium
278:as the
213:History
196:base −2
162:or CPU
94:scholar
3050:
2970:
2938:
2919:
2254:Binary
1349:offset
1333:biased
1059:modulo
805:signed
346:, and
304:PDP-11
292:PDP-10
282:, the
190:, and
96:
89:
82:
75:
67:
2968:S2CID
2859:(PDF)
2832:Intel
2828:(PDF)
2692:digit
2219:1010
2196:1011
2173:1000
2170:0000
2167:1000
2150:1001
2147:0001
2144:1001
2141:1000
2138:1111
2127:1110
2124:0010
2121:1010
2118:1001
2115:1110
2104:1111
2101:0011
2098:1011
2095:1010
2092:1101
2081:1100
2078:0100
2075:1100
2072:1011
2069:1100
2058:1101
2055:0101
2052:1101
2049:1100
2046:1011
2035:0010
2032:0110
2029:1110
2026:1101
2023:1010
2012:0011
2009:0111
2006:1111
2003:1110
2000:1001
1989:1111
1986:1000
1975:0000
1972:1000
1969:0000
1966:0000
1963:0000
1960:0000
1952:0001
1949:1001
1946:0001
1943:0001
1940:0001
1937:0001
1929:0110
1926:1010
1923:0010
1920:0010
1917:0010
1914:0010
1906:0111
1903:1011
1900:0011
1897:0011
1894:0011
1891:0011
1883:0100
1880:1100
1877:0100
1874:0100
1871:0100
1868:0100
1860:0101
1857:1101
1854:0101
1851:0101
1848:0101
1845:0101
1834:1110
1831:0110
1828:0110
1825:0110
1822:0110
1811:1111
1808:0111
1805:0111
1802:0111
1799:0111
1776:1000
1753:1001
1730:1010
1707:1011
1684:1100
1661:1101
1638:1110
1615:1111
1531:radix
1166:11010
1147:00101
832:carry
332:SPARC
300:PDP-8
296:PDP-5
288:PDP-6
252:PDP-1
178:are:
101:JSTOR
87:books
3048:ISBN
3013:2015
2899:2020
2871:2023
2840:2013
2809:2013
2674:and
2624:1111
2601:1110
2578:1101
2555:1100
2532:1011
2527:−10
2509:1010
2486:1001
2461:1000
2438:0111
2415:0110
2392:0101
2369:0100
2346:0011
2323:0010
2300:0001
2277:0000
1519:255
1497:129
1494:−127
1486:128
1483:−128
1475:127
1314:255
1292:129
1281:128
1270:127
1245:−127
1234:−128
1168:100
1055:ring
1030:255
1019:254
997:130
994:−126
986:129
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