1406:(BCD) are mixed base systems where bits (binary digits) are used to express decimal digits. E.g., in 1001 0011, each group of four bits may represent a decimal digit (in this example 9 and 3, so the eight bits combined represent decimal 93). The weights associated with these 8 positions are 80, 40, 20, 10, 8, 4, 2 and 1. Uniqueness is ensured by requiring that, in each group of four bits, if the first bit is 1, the next two must be 00.
22:
1424:
where each digit can have different bases, usually non-integer. In these, not only are the bases of a given digit different, they can be also nonuniform and altered in an asymmetric way to encode information more efficiently. They are optimized for chosen non-uniform probability distributions of
1386:
numeral system was a mixed-radix system, since one of its positions represents a multiplication by 18 rather than 20, in order to fit a 360-day calendar. Also, giving an angle in degrees, minutes and seconds (with decimals), or a time in days, hours, minutes and seconds, can be interpreted as
1043:, which can be classified as standard systems of base 60 and 10, respectively, counting the space representing zero as a numeral, can also be classified as non-standard systems, more specifically, mixed-base systems with unary components, considering the primitive repeated
1003:
909:
1050:
However, most of the non-standard systems listed below have never been intended for general use, but were devised by mathematicians or engineers for special academic or technical use.
1379:, the weights form a sequence where each weight is an integer multiple of the previous one, and the number of permitted digit values varies accordingly from position to position.
1337:
40:
1029:
This article summarizes facts on some non-standard positional numeral systems. In most cases, the polynomial form in the description of standard systems still applies.
1496:
762:
1394:
an integer multiple of the previous weight may also be used, but then every integer may not have a unique representation. For example,
1250:. Due to the properties of negative numbers raised to powers, all integers, positive and negative, can be represented without a sign.
315:
58:
938:
1355:
It is sometimes convenient to consider positional numeral systems where the weights associated with the positions do not form a
106:
1070:
different numerals to represent all non-negative integers. However, the numerals have values 1, 2, 3, etc. up to and including
844:
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330:
1443:
1402:(1, 2, 3, 5, 8, ...); a unique representation of all non-negative integers may be ensured by forbidding consecutive 1s.
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1096: = 1. In unary, one numeral is used to represent all positive integers. The value of the digit string
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1178: = 3, and the numerals have the values ā1, 0 and +1 (rather than 0, 1 and 2 as in the standard
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540:
1316:
1032:
Some historical numeral systems may be described as non-standard positional numeral systems. E.g., the
1403:
1130:
The value of a digit does not depend on its position. Thus, one can easily argue that unary is not a
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Introducing a radix point in this system will not enable representation of non-integer values.
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The value 0 cannot be represented (or is implicitly represented by an empty digit string).
933: = 16), using the numerals A for 10, B for 11 etc., the digit string 7A3F means
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In some systems, while the base is a positive integer, negative digits are allowed.
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The use of asymmetric numeral systems as an accurate replacement for
Huffman coding
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In non-integer bases, the number of different numerals used clearly cannot be
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The reflected binary code, also known as the Gray code, is closely related to
1018:
452:
210:
1242:, with bases −2, −3, and −10 respectively; in base −
1191:
462:
820: ā 1, but the value is weighted according to the position of the
1455:
1140:
The single numeral represents the value 1, not the value 0 =
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447:
432:
437:
1291:. It can be generalized to other complex bases, giving rise to the
1044:
791:
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404:
365:
1204:
are inverted, depending on the parity of the higher order bits.
1212:
A few positional systems have been suggested in which the base
1201:
15:
998:{\displaystyle 7\times 16^{3}+10\times 16^{2}+3\times 16+15}
1456:
Expansions in non-integer bases: the top order and the tail
904:{\displaystyle p\times b^{3}+q\times b^{2}+r\times b+s}
36:
1010:
which written in our normal decimal notation is 31295.
1319:
1182:, or 1, 2 and 3 as in the bijective ternary system).
941:
847:
1100:
given by the polynomial form can be simplified into
1398:uses the digits 0 and 1, weighted according to the
31:
may be too technical for most readers to understand
1331:
997:
903:
918:The numbers written in superscript represent the
1126:. Non-standard features of this system include:
1092:Unary is the bijective numeral system with base
1420:Asymmetric numeral systems are systems used in
1475:J. Duda, K. Tahboub, N. J. Gadil, E. J. Delp,
824:in a number. The value of a digit string like
1280:, the standard set of digits consists of the
812:. The standard set of numerals contains the
790:In a standard positional numeral system, the
756:
8:
1326:
1320:
1025:can be represented up to arbitrary accuracy.
1347:), uses the 2 different numerals 0 and 1.
763:
749:
99:
70:
1318:
1246:the number of different numerals used is
1166:is a particular system where the base is
971:
952:
940:
877:
858:
846:
59:Learn how and when to remove this message
43:, without removing the technical details.
1425:symbols, using on average approximately
1497:Non-standard positional numeral systems
1467:
776:Non-standard positional numeral systems
82:
41:make it understandable to non-experts
7:
1208:Bases that are not positive integers
14:
1480:, Picture Coding Symposium, 2015.
1332:{\displaystyle \lfloor b\rfloor }
782:that may loosely be described as
1272:is an integer larger than 1 and
20:
1390:Sequences where each weight is
1082:Base one (unary numeral system)
1230:Negative-base systems include
1:
1313:. Instead, the numerals 0 to
1054:Bijective numeration systems
816:values 0, 1, 2, etc., up to
1264:In a purely imaginary base
1216:is not a positive integer.
1158:Signed-digit representation
1152:Signed-digit representation
797:is a positive integer, and
1513:
1416:Asymmetric numeral systems
1413:
1410:Asymmetric numeral systems
1305:Non-integer representation
1302:
1257:
1223:
1189:
1155:
1085:
805:are used to represent all
483:Non-standard radices/bases
1382:For calendrical use, the
1444:KomornikāLoreti constant
1060:bijective numeral system
1047:making up the numerals.
1439:List of numeral systems
1377:factorial number system
1339:are used. For example,
1170: = 2. In the
1122: = 1 for all
739:List of numeral systems
1333:
1076:decimal without a zero
999:
905:
1387:mixed-radix systems.
1334:
1260:Quater-imaginary base
1000:
906:
107:HinduāArabic numerals
1404:Binary-coded decimal
1317:
1293:complex-base systems
1174:system, the base is
1088:Unary numeral system
939:
845:
636:Prehistoric counting
412:Common radices/bases
94:Place-value notation
1375:system such as the
1037:Babylonian notation
1013:Upon introducing a
583:Sign-value notation
1400:Fibonacci sequence
1357:geometric sequence
1329:
1284:numbers from 0 to
995:
901:
784:positional systems
239:East Asian systems
1429:bits per symbol.
1341:golden ratio base
1164:Non-adjacent form
925:For instance, in
922:of the base used.
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1422:computer science
1396:Fibonacci coding
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1299:Non-integer base
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1172:balanced ternary
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1039:and the Chinese
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1427:Shannon entropy
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834:polynomial form
780:numeral systems
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1303:Main article:
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1268:system, where
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1224:Main article:
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1198:binary numbers
1190:Main article:
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1180:ternary system
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49:November 2023
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29:This article
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1254:Complex base
1247:
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1195:
1175:
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1041:rod numerals
1031:
1028:
1023:real numbers
930:
833:
829:
825:
817:
813:
807:non-negative
798:
794:
775:
774:
542:
503:Signed-digit
482:
378:Contemporary
245:Contemporary
55:
46:
30:
1373:mixed-radix
1351:Mixed bases
1240:negadecimal
1236:negaternary
1200:, but some
1034:sexagesimal
1015:radix point
927:hexadecimal
681:Akį¹£arapallÄ«
651:Tally marks
550:Non-integer
1462:References
1232:negabinary
1132:positional
1062:with base
1019:minus sign
1017:"." and a
801:different
718:Glagolitic
691:Kaį¹apayÄdi
659:Alphabetic
563:Asymmetric
405:radix/base
346:Cistercian
331:Babylonian
278:Vietnamese
133:Devanagari
1327:⌋
1321:⌊
1192:Gray code
1186:Gray code
984:×
965:×
946:×
890:×
871:×
852:×
686:Äryabhaį¹a
631:Kharosthi
523:factorial
490:Bijective
391:(IƱupiaq)
221:Sundanese
216:Mongolian
163:Malayalam
1491:Category
1433:See also
828:in base
810:integers
803:numerals
713:Georgian
703:Cyrillic
671:Armenian
626:Etruscan
621:Egyptian
529:Negative
389:Kaktovik
384:Cherokee
361:Pentadic
285:Historic
268:Japanese
201:Javanese
191:Balinese
178:Dzongkha
143:Gurmukhi
138:Gujarati
76:a series
74:Part of
1345:phinary
616:Chuvash
534:Complex
324:Ancient
316:History
263:Hokkien
251:Chinese
196:Burmese
186:Tibetan
173:Kannada
153:Sinhala
128:Bengali
35:Please
1118:since
1045:glyphs
920:powers
728:Hebrew
698:Coptic
611:Brahmi
596:Aegean
553:
537:
519:
506:
493:
356:Muisca
296:Tangut
273:Korean
256:Suzhou
168:Telugu
1384:Mayan
1066:uses
1021:"ā",
822:digit
723:Greek
708:GeŹ½ez
666:Abjad
646:Roman
606:Aztec
601:Attic
516:Mixed
474:table
366:Quipu
351:Mayan
206:Khmer
158:Tamil
1276:the
1238:and
1202:bits
1098:pqrs
826:pqrs
792:base
371:Rumi
226:Thai
148:Odia
1392:not
1359:1,
1289:ā 1
403:By
211:Lao
39:to
1493::
1367:,
1363:,
1295:.
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