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562:), number format. In the decimal system, there are 10 digits, 0 through 9, which combine to form numbers. In an octal system, there are only 8 digits, 0 through 7. That is, the value of an octal "10" is the same as a decimal "8", an octal "20" is a decimal "16", and so on. In a hexadecimal system, there are 16 digits, 0 through 9 followed, by convention, with A through F. That is, a hexadecimal "10" is the same as a decimal "16" and a hexadecimal "20" is the same as a decimal "32". An example and comparison of numbers in different bases is described in the chart below.
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1269:{\displaystyle {\begin{aligned}&{\text{octal }}756\\={}&(7\times 8^{2})+(5\times 8^{1})+(6\times 8^{0})\\={}&(7\times 64)+(5\times 8)+(6\times 1)\\={}&448+40+6\\={}&{\text{decimal }}494\end{aligned}}\qquad {\begin{aligned}&{\text{hex }}\mathrm {3b2} \\={}&(3\times 16^{2})+(11\times 16^{1})+(2\times 16^{0})\\={}&(3\times 256)+(11\times 16)+(2\times 1)\\={}&768+176+2\\={}&{\text{decimal }}946\end{aligned}}}
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the computer effectively discards it. Analyzing the effect of limited precision is a well-studied problem. Estimates of the magnitude of round-off errors and methods to limit their effect on large calculations are part of any large computation project. The precision limit is different from the range limit, as it affects the significand, not the exponent.
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different internal representations of numerical values and different conventions are used for integer and real numbers. Most calculations are carried out with number formats that fit into a processor register, but some software systems allow representation of arbitrarily large numbers using multiple words of memory.
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The significand is a binary fraction that doesn't necessarily perfectly match a decimal fraction. In many cases a sum of reciprocal powers of 2 does not match a specific decimal fraction, and the results of computations will be slightly off. For example, the decimal fraction "0.1" is equivalent to an
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The relation between numbers and bit patterns is chosen for convenience in computer manipulation; eight bytes stored in computer memory may represent a 64-bit real, two 32-bit reals, or four signed or unsigned integers, or some other kind of data that fits into eight bytes. The only difference is how
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The representation has a limited precision. For example, only 15 decimal digits can be represented with a 64-bit real. If a very small floating-point number is added to a large one, the result is just the large one. The small number was too small to even show up in 15 or 16 digits of resolution, and
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requires the programmer to keep track of the representation of numbers. Where the processor does not support a required mathematical operation, the programmer must work out a suitable algorithm and instruction sequence to carry out the operation; on some microprocessors, even integer multiplication
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and hexadecimal encoding are convenient ways to represent binary numbers, as used by computers. Computer engineers often need to write out binary quantities, but in practice writing out a binary number such as 1001001101010001 is tedious and prone to errors. Therefore, binary quantities are written
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The advantage of this scheme is that by using the exponent we can get a much wider range of numbers, even if the number of digits in the significand, or the "numeric precision", is much smaller than the range. Similar binary floating-point formats can be defined for computers. There is a number of
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Each of these number systems is a positional system, but while decimal weights are powers of 10, the octal weights are powers of 8 and the hexadecimal weights are powers of 16. To convert from hexadecimal or octal to decimal, for each digit one multiplies the value of the digit by the value of its
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and words. The encoding between numerical values and bit patterns is chosen for convenience of the operation of the computer; the encoding used by the computer's instruction set generally requires conversion for external use, such as for printing and display. Different types of processors may have
487:. A single bit allows only two value-combinations, two bits combined can make four separate values, three bits for eight, and so on, increasing with the formula 2. The amount of possible combinations doubles with each binary digit added as illustrated in Table 2.
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While both unsigned and signed integers are used in digital systems, even a 32-bit integer is not enough to handle all the range of numbers a calculator can handle, and that's not even including fractions. To approximate the greater range and precision of
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The eight's bit is followed by the four's bit, then the two's bit, then the one's bit. The fractional bits continue the pattern set by the integer bits. The next bit is the half's bit, then the quarter's bit, then the ⅛'s bit, and so on. For example:
2223:, will invoke different object code appropriate to the representation of the numerical type; mathematical operations on any number—whether signed, unsigned, rational, floating-point, fixed-point, integral, or complex—are written exactly the same way.
511:, the atom of addressability, say. For example, even though 64-bit processors may address memory sixty-four bits at a time, they may still split that memory into eight-bit pieces. This is called byte-addressable memory. Historically, many
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The number of bits needed for the precision and range desired must be chosen to store the fractional and integer parts of a number. For instance, using a 32-bit format, 16 bits may be used for the integer and 16 for the fraction.
503:. On most modern computers, this is an eight bit string. Because the definition of a byte is related to the number of bits composing a character, some older computers have used a different bit length for their byte. In many
536:, the nibble was named as a play on words. A person may need several nibbles for one bite from something; similarly, a nybble is a part of a byte. Because four bits allow for sixteen values, a nibble is sometimes known as a
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the computer interprets them. If the computer stored four unsigned integers and then read them back from memory as a 64-bit real, it almost always would be a perfectly valid real number, though it would be junk data.
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Some programs also use 32-bit floating-point numbers. The most common scheme uses a 23-bit significand with a sign bit, plus an 8-bit exponent in "excess-127" format, giving seven valid decimal digits.
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Only a finite range of real numbers can be represented with a given number of bits. Arithmetic operations can overflow or underflow, producing a value too large or too small to be represented.
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an 11-bit binary exponent, using "excess-1023" format. Excess-1023 means the exponent appears as an unsigned binary integer from 0 to 2047; subtracting 1023 gives the actual signed value
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where "S" denotes the sign bit, "x" denotes an exponent bit, and "m" denotes a significand bit. Once the bits here have been extracted, they are converted with the computation:
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When typing numbers, formatting characters are used to describe the number system, for example 000_0000B or 0b000_00000 for binary and 0F8H or 0xf8 for hexadecimal numbers.
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2219:. Mathematical operations are carried out by library routines provided by the implementation of the language. A given mathematical symbol in the source code, by
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Computers represent data in sets of binary digits. The representation is composed of bits, which in turn are grouped into larger sets such as bytes.
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may be used to represent larger values. For example, a string of three bits can represent up to eight distinct values as illustrated in Table 1.
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read data in some multiple of eight bits. Because the byte size of eight bits is so common, but the definition is not standardized, the term
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A 32-bit float value is sometimes called a "real32" or a "single", meaning "single-precision floating-point value".
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A 64-bit float is sometimes called a "real64" or a "double", meaning "double-precision floating-point value".
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which means "1.1030402 times 1 followed by 5 zeroes". We have a certain numeric value (1.1030402) known as a "
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a 52-bit significand, also an unsigned binary number, defining a fractional value with a leading implied "1"
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Such floating-point numbers are known as "reals" or "floats" in general, but with a number of variations:
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This scheme provides numbers valid out to about 15 decimal digits, with the following range of numbers:
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Inside the machine: an illustrated introduction to microprocessors and computer architecture
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This form of encoding cannot represent some values in binary. For example, the fraction
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In the decimal system, we are familiar with floating-point numbers of the form (
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2.3434E−6 = 2.3434 × 10 = 2.3434 × 0.000001 = 0.0000023434
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1555:", multiplied by a power of 10 (E5, meaning 10 or 100,000), known as an "
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While a single bit, on its own, is able to represent only two values, a
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2115: in this section. Unsourced material may be challenged and removed.
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standard specification defines a 64 bit floating-point format with:
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is a bit string containing the number of bits needed to represent a
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1438:, 0.2 in decimal, the closest approximations would be as follows:
2013:<sign> × (1 + <fractional significand>) × 2
1833:<sign> × (1 + <fractional significand>) × 2
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The bits are converted to a numeric value with the computation:
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is sometimes used to explicitly describe an eight bit sequence.
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425:. The concept of a bit can be understood as a value of either
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Internal representation of numeric values in a digital computer
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offer an abstract number that may be an expanded type such as
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formatting can be useful to represent fractions in binary.
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1.1030402 × 10 = 1.1030402 × 100000 = 110304.02
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infinitely repeating binary fraction: 0.000110011 ...
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2280:The initial version of this article was based on a
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1568:such schemes, the most popular has been defined by
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60:. Unsourced material may be challenged and removed.
2350:"Microprocessor and CPU (Central Processing Unit)"
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1570:Institute of Electrical and Electronics Engineers
825:position and then adds the results. For example:
574:Table 3: Comparison of values in different bases
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532:), is a number composed of four bits. Being a
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144:. Numerical values are stored as groupings of
250:Table 2: Number of values for a bit string.
2017:leading to the following range of numbers:
1590:With the bits stored in 8 bytes of memory:
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1586:a sign bit, giving the sign of the number.
2175:Learn how and when to remove this message
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120:Learn how and when to remove this message
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14:
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820:Positional notation (base conversion)
2352:. Network Dictionary. Archived from
2113:adding citations to reliable sources
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1416:00000000 00000111.01100000 00000000
1382:00000000 00000001.01000000 00000000
1348:00000000 00000000.10000000 00000000
544:Octal and hexadecimal number display
58:adding citations to reliable sources
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1467:00000000 00000000.00110011 00110100
1450:00000000 00000000.00110011 00110011
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2081:Numbers in programming languages
1280:Representing fractions in binary
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2307:. No Starch Press. p. 66.
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171:Table 1: Binary to octal
2286:Greg Goebel's Vectorsite
2226:Some languages, such as
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1867:-4.940656458412465E-324
1473:0.2000122... in decimal
1456:0.1999969... in decimal
569:Converting between bases
255:Length of bit string (b)
69:"Computer number format"
2254:Binary-to-text encoding
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1856:1.797693134862231E+308
483:combinations increases
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1515:Floating-point numbers
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2221:operator overloading
2109:improve this article
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2447:Computer arithmetic
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2301:Jon Stokes (2007).
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1285:Fixed-point numbers
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673:
670:
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664:
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645:
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639:
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632:
631:
628:
625:
622:
618:
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614:
611:
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589:
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583:
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567:
545:
542:
470:string of bits
465:of some kind.
404:
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392:
389:
384:
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369:
358:
347:
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344:
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336:
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158:
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128:
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42:
40:
33:
26:
9:
6:
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2464:
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2330:
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2316:
2310:
2306:
2305:
2297:
2293:
2290:
2289:
2287:
2284:article from
2283:
2282:public domain
2270:
2267:
2265:
2262:
2260:
2259:Binary number
2257:
2255:
2252:
2250:
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2222:
2218:
2214:
2210:
2206:
2202:
2198:
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2190:
2179:
2176:
2168:
2165:December 2018
2157:
2154:
2150:
2147:
2143:
2140:
2136:
2133:
2129:
2126: –
2125:
2121:
2120:Find sources:
2114:
2110:
2104:
2103:
2098:This section
2096:
2092:
2087:
2086:
2078:
2074:
2070:
2067:
2063:
2060:
2057:
2049:
2046:
2043:
2042:
2039:2.802597E-45
2038:
2036:3.402823E+38
2035:
2032:
2031:
2027:
2024:
2022:
2021:
2018:
2012:
2011:
2010:
2002:
1999:
1996:
1993:
1990:
1987:
1984:
1981:
1978:
1977:
1973:
1970:
1967:
1964:
1961:
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1601:
1598:
1595:
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1591:
1585:
1582:
1579:
1578:
1577:
1575:
1574:IEEE 754-2008
1571:
1562:
1561:
1560:
1558:
1554:
1546:
1545:
1544:
1538:
1537:
1536:
1534:
1529:
1527:
1523:
1512:
1472:
1469:
1466:
1463:
1461:13108 / 65536
1460:
1459:
1455:
1452:
1449:
1446:
1444:13107 / 65536
1443:
1442:
1439:
1412:
1392:
1389:
1386:
1385:
1378:
1358:
1355:
1352:
1351:
1344:
1327:
1324:
1321:
1320:
1316:
1313:
1311:
1309:
1307:
1305:
1304:
1301:
1297:
1293:
1291:
1259:
1256:decimal
1247:
1240:
1237:
1234:
1231:
1228:
1221:
1211:
1208:
1205:
1199:
1193:
1190:
1187:
1181:
1175:
1172:
1169:
1164:
1157:
1145:
1141:
1137:
1134:
1128:
1120:
1116:
1112:
1109:
1103:
1095:
1091:
1087:
1084:
1079:
1072:
1064:
1058:
1039:
1036:decimal
1027:
1020:
1017:
1014:
1011:
1008:
1001:
991:
988:
985:
979:
973:
970:
967:
961:
955:
952:
949:
944:
937:
925:
921:
917:
914:
908:
900:
896:
892:
889:
883:
875:
871:
867:
864:
859:
852:
845:
828:
827:
826:
821:
811:
808:
805:
802:
801:
797:
794:
791:
788:
787:
783:
780:
777:
774:
773:
769:
766:
763:
760:
759:
755:
752:
749:
746:
745:
741:
738:
735:
732:
731:
727:
724:
721:
718:
717:
713:
710:
707:
704:
703:
699:
696:
693:
690:
689:
685:
682:
679:
676:
675:
671:
668:
665:
662:
661:
657:
654:
651:
648:
647:
643:
640:
637:
634:
633:
629:
626:
623:
620:
619:
615:
612:
609:
606:
605:
601:
598:
595:
592:
591:
587:
584:
581:
578:
577:
566:
563:
561:
556:
551:
541:
539:
535:
531:
527:
526:
520:
518:
514:
510:
506:
502:
498:
497:
491:
488:
486:
485:exponentially
482:
478:
473:
471:
466:
464:
460:
456:
452:
448:
444:
440:
436:
432:
428:
424:
420:
417:
413:
412:
390:
387:
382:
378:
370:
356:
349:
348:
345:
342:
341:
337:
334:
333:
329:
326:
325:
321:
318:
317:
313:
310:
309:
305:
302:
301:
297:
294:
293:
289:
286:
285:
281:
278:
277:
273:
270:
269:
265:
262:
261:
257:
254:
253:
242:
239:
238:
234:
231:
230:
226:
223:
222:
218:
215:
214:
210:
207:
206:
202:
199:
198:
194:
191:
190:
186:
183:
182:
178:
176:Binary string
175:
174:
168:
164:
154:
151:
147:
143:
139:
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:
2425:10 September
2423:. Retrieved
2413:
2401:. Retrieved
2391:
2379:. Retrieved
2370:
2358:. Retrieved
2354:the original
2344:
2332:. Retrieved
2323:
2303:
2296:
2279:
2278:
2225:
2216:
2212:
2208:
2194:
2186:
2171:
2162:
2152:
2145:
2138:
2131:
2119:
2107:Please help
2102:verification
2099:
2075:
2071:
2068:
2064:
2061:
2058:
2055:
2016:
2008:
1884:
1877:
1875:
1836:
1828:
1589:
1572:(IEEE). The
1566:
1550:
1542:
1530:
1522:real numbers
1518:
1478:
1421:
1314:integer bits
1298:
1294:
1288:
823:
588:Hexadecimal
564:
559:
553:
529:
523:
521:
494:
492:
489:
480:
476:
474:
467:
454:
450:
446:
442:
438:
434:
430:
426:
409:
407:
179:Octal value
166:
133:
131:
116:
110:October 2022
107:
97:
90:
83:
76:
64:
52:Please help
47:verification
44:
2195:High-level
1553:significand
1547:1.1030402E5
1511:precisely.
1290:Fixed-point
842:octal
528:(sometimes
142:calculators
2441:Categories
2135:newspapers
1528:" format.
548:See also:
161:See also:
148:, such as
80:newspapers
2264:Gray code
2044:negative
2033:positive
1864:negative
1853:positive
1209:×
1191:×
1173:×
1138:×
1113:×
1088:×
1054:hex
989:×
971:×
953:×
918:×
893:×
868:×
534:half-byte
501:character
138:computers
2397:"Nybble"
2334:24 April
2238:See also
2209:rational
2199:such as
2028:minimum
2025:maximum
1848:minimum
1845:maximum
1557:exponent
1394:7
1360:1
2217:complex
2149:scholar
1979:byte 3
1950:byte 2
1921:byte 1
1892:byte 0
1799:byte 7
1770:byte 6
1741:byte 5
1712:byte 4
1683:byte 3
1654:byte 2
1625:byte 1
1596:byte 0
1509:
1497:
1493:
1481:
1436:
1424:
1409:
1375:
1341:
1329:
579:Decimal
459:encoded
94:scholar
2311:
2213:bignum
2205:Python
2151:
2144:
2137:
2130:
2122:
806:001111
792:001110
778:001101
764:001100
750:001011
736:001010
722:001001
708:001000
694:000111
680:000110
666:000101
652:000100
638:000011
624:000010
610:000001
596:000000
582:Binary
550:Base64
530:nybble
525:nibble
463:toggle
423:states
416:binary
96:
89:
82:
75:
67:
2403:3 May
2381:3 May
2360:1 May
2215:, or
2156:JSTOR
2142:books
1387:7.375
1353:1.250
1322:0.500
585:Octal
555:Octal
517:octet
457:, or
455:false
419:digit
414:is a
338:1024
150:bytes
101:JSTOR
87:books
2427:2012
2405:2012
2383:2012
2362:2012
2336:2012
2309:ISBN
2232:Java
2230:and
2228:REXX
2203:and
2201:Ruby
2128:news
1945:m16
1879:NaNs
1765:m16
1736:m24
1707:m32
1678:m40
1649:m48
513:CPUs
496:byte
479:and
451:true
330:512
322:256
314:128
146:bits
140:and
73:news
2111:by
2003:m0
1974:m8
1968:m10
1965:m11
1962:m12
1959:m13
1956:m14
1953:m15
1942:m17
1939:m18
1936:m19
1933:m20
1930:m21
1927:m22
1916:x1
1823:m0
1794:m8
1788:m10
1785:m11
1782:m12
1779:m13
1776:m14
1773:m15
1762:m17
1759:m18
1756:m19
1753:m20
1750:m21
1747:m22
1744:m23
1733:m25
1730:m26
1727:m27
1724:m28
1721:m29
1718:m30
1715:m31
1704:m33
1701:m34
1698:m35
1695:m36
1692:m37
1689:m38
1686:m39
1675:m41
1672:m42
1669:m43
1666:m44
1663:m45
1660:m46
1657:m47
1646:m49
1643:m50
1640:m51
1620:x4
1602:x10
1535:):
1260:946
1235:176
1229:768
1176:256
1040:494
1009:448
846:756
812:0F
798:0E
784:0D
770:0C
756:0B
742:0A
728:09
714:08
700:07
686:06
672:05
658:04
644:03
630:02
616:01
602:00
560:hex
453:or
445:or
443:yes
439:off
437:or
429:or
411:bit
343:...
306:64
298:32
290:16
240:111
232:110
224:101
216:100
208:011
200:010
192:001
184:000
56:by
2443::
2288:.
2211:,
2000:m1
1997:m2
1994:m3
1991:m4
1988:m5
1985:m6
1982:m7
1971:m9
1924:x0
1913:x2
1910:x3
1907:x4
1904:x5
1901:x6
1898:x7
1820:m1
1817:m2
1814:m3
1811:m4
1808:m5
1805:m6
1802:m7
1791:m9
1637:x0
1634:x1
1631:x2
1628:x3
1617:x5
1614:x6
1611:x7
1608:x8
1605:x9
1194:16
1188:11
1142:16
1117:16
1110:11
1092:16
1015:40
956:64
809:17
803:15
795:16
789:14
781:15
775:13
767:14
761:12
753:13
747:11
739:12
733:10
725:11
711:10
697:07
683:06
669:05
655:04
641:03
627:02
613:01
599:00
540:.
522:A
493:A
449:,
447:no
441:,
435:on
433:,
408:A
335:10
282:8
274:4
266:2
243:7
235:6
227:5
219:4
211:3
203:2
195:1
187:0
132:A
2429:.
2407:.
2385:.
2364:.
2338:.
2317:.
2178:)
2172:(
2167:)
2163:(
2153:·
2146:·
2139:·
2132:·
2105:.
1895:S
1599:S
1506:3
1503:/
1500:1
1490:3
1487:/
1484:1
1470:=
1464:=
1453:=
1447:=
1433:5
1430:/
1427:1
1413:=
1406:8
1403:/
1400:3
1396:+
1390:=
1379:=
1372:4
1369:/
1366:1
1362:+
1356:=
1345:=
1338:2
1335:/
1332:1
1325:=
1248:=
1241:2
1238:+
1232:+
1222:=
1215:)
1212:1
1206:2
1203:(
1200:+
1197:)
1185:(
1182:+
1179:)
1170:3
1167:(
1158:=
1151:)
1146:0
1135:2
1132:(
1129:+
1126:)
1121:1
1107:(
1104:+
1101:)
1096:2
1085:3
1082:(
1073:=
1065:2
1062:b
1059:3
1028:=
1021:6
1018:+
1012:+
1002:=
995:)
992:1
986:6
983:(
980:+
977:)
974:8
968:5
965:(
962:+
959:)
950:7
947:(
938:=
931:)
926:0
922:8
915:6
912:(
909:+
906:)
901:1
897:8
890:5
887:(
884:+
881:)
876:2
872:8
865:7
862:(
853:=
719:9
705:8
691:7
677:6
663:5
649:4
635:3
621:2
607:1
593:0
481:1
477:0
431:0
427:1
391:N
388:=
383:b
379:2
357:b
327:9
319:8
311:7
303:6
295:5
287:4
279:3
271:2
263:1
123:)
117:(
112:)
108:(
98:·
91:·
84:·
77:·
50:.
20:)
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