658:
Other names are occasionally used for this purpose, notably 'characteristic' and 'mantissa'; but it is an abuse of terminology to call the fraction part a mantissa, since that term has quite a different meaning in connection with logarithms. Furthermore the
English word mantissa means 'a worthless
540:. The mantissa is sometimes termed the characteristic and a version of the exponent also has this title from some authors. It is hoped that the terms here will be unambiguous. e use a value which is shifted by half the binary range of the number. This special form is sometimes referred to as a
308:
To understand both terms, notice that in binary, 1 + mantissa ≈ significand, and the correspondence is exact when storing a power of two. This fact allows for a fast approximation of the base-2 logarithm, leading to algorithms e.g. for computing the
216:= 53 for the double-precision format), thus in a way independent from the encoding, and the term to express what is encoded (that is, the significand without its leading bit) is
208:
is commonly described as having either a 53-bit significand, including the hidden bit, or a 52-bit significand, excluding the hidden bit. IEEE 754 defines the precision
301:
in 1967 and is the word used in the IEEE standard as the coefficient in front of a scientific notation number discussed above. The fractional part is called the
842:
958:
694:(Technical report, Institute for Advanced Study, Princeton, New Jersey, USA). Collected Works of John von Neumann. Vol. 5. New York, USA:
205:
430:
1025:
544:, since it is the conventional value plus a constant. Some authors have called it a characteristic, but this term should not be used, since
393:
with a different meaning: it is the fractional part of the significand, i.e. the significand without its explicit or implicit leading bit.
787:
736:
962:
852:
712:
317:. The implicit leading 1 is nothing but the hidden bit in IEEE 754 floating point, and the bitfield storing the remainder is thus the
95:
917:
888:
801:
760:
651:
481:
310:
716:
954:
641:
578:
341:
137:
719:—e.g. 123.45 would be carried in the machine as (0.12345,03), where the 3 is the exponent of 10 associated with the number.
489:
1052:
324:
However, whether or not the implicit 1 is included is a major point of confusion with both terms—and especially so with
188:, this constraint uniquely determines this digit to always be 1. As such, it is not explicitly stored, being called the
614:
748:
464:
Gosling, John B. (1980). "6.1 Floating-Point
Notation / 6.8.5 Exponent Representation". In Sumner, Frank H. (ed.).
141:
233:
201:, and depending on the context, the hidden bit may or may not be counted toward the width. For example, the same
59:
707:
Several of the digital computers being built or planned in this country and
England are to contain a so-called "
545:
469:
314:
145:
1047:
473:
229:
933:
708:
978:
864:
695:
537:
294:
796:(1 (reprint) ed.). Malabar, Florida, USA: Robert E. Krieger Publishing Company. pp. 204–205.
686:
110:
with the significand 1.2345 as a fractional coefficient, and +2 as the exponent (and 10 as the base):
419:
366:
828:
674:
107:
55:
993:
908:. Prentice-Hall Series in Automatic Computation (1 ed.). Englewood Cliffs, New Jersey, USA:
638:
283:
98:, where −2 is the exponent (and 10 is the base). Its value is given by the following arithmetic:
752:
17:
423:
1021:
982:
945:
913:
884:
868:
848:
797:
756:
647:
477:
298:
181:
75:
63:
567:
English
Electric KDF9: Very high speed data processing system for Commerce, Industry, Science
328:. In keeping with the original usage in the context of log tables, it should not be present.
1013:
880:
682:
678:
574:
271:
248:, showing the need for a fixed-sized significand as currently used for floating point data.
122:
Schmid, however, called this representation with a significand ranging between 1.0 and 10 a
937:
541:
336:
212:
to be the number of digits in the significand, including any implicit leading bit (e.g.,
741:
1041:
909:
876:
390:
332:
185:
168:
Schmid called this representation with a significand ranging between 0.1 and 1.0 the
565:
264:
252:
688:
Preliminary discussion of the logical design of an electronic computing instrument
552:' representation, where, for example, - is 64 for a 7-bit exponent (2 = 64).
468:. Macmillan Computer Science Series (1 ed.). Department of Computer Science,
1017:
791:
190:
357:
contexts. In particular, the current IEEE 754 standard does not mention it.
938:"What Every Computer Scientist Should Know About Floating-Point Arithmetic"
606:
815:
549:
548:
and others use this term for the mantissa. It is also referred to as an '
529:
202:
153:
67:
149:
87:
71:
278:
is the integer part of the logarithm (i.e. the exponent), and the
184:, the most significant digit is always non-zero. When working in
136:
Finally, the value can be represented in the format given by the
1009:
140:
standard and several programming language standards, including
198:
831:, pp. 575–583, Revista de Obras Públicas, 19 November 1914.
829:
Automática: Complemento de la Teoría de las Máquinas, (pdf)
555:(NB. Gosling does not mention the term significand at all.)
875:. Automatic Computation (1st ed.). New Jersey, USA:
814:(NB. At least some batches of this reprint edition were
448:
is the significand or coefficient or (wrongly) mantissa
282:
is the fractional part. The usage remains common among
1006:
754-2019 - IEEE Standard for
Floating-Point Arithmetic
263:
to describe the two parts of a floating-point number (
711:". This is a mechanism for expressing each word as a
577:. c. 1961. Publication No. DP/103. 096320WP/RP0961.
90:
floating-point number with the integer 12345 as the
331:For those contexts where 1 is considered included,
740:
197:The significand is characterized by its width in
466:Design of Arithmetic Units for Digital Computers
992:(NB. A newer edited version can be found here:
424:"Names for Standardized Floating-Point Formats"
459:
457:
339:, prominent computer programmer and author of
129:For base 2, this 1.xxxx form is also called a
873:Computer Solution of Linear Algebraic Systems
782:
780:
778:
731:
729:
727:
600:
598:
8:
985:, and has generally replaced the older term
669:
667:
414:
412:
410:
349:. This has led to declining use of the term
633:
631:
106:The same value can also be represented in
86:The number 123.45 can be represented as a
66:. Depending on the interpretation of the
54:) is the first (left) part of a number in
959:Association for Computing Machinery, Inc.
747:(1 ed.). Binghamton, New York, USA:
685:(1963) . "5.3.". In Taub, A. H. (ed.).
406:
378:
27:Part of a number in scientific notation
496:is represented by two signed numbers
270:) by analogy with the then-prevalent
7:
957:(PARC), Palo Alto, California, USA:
70:, the significand may represent an
62:representation, consisting of its
25:
94:and a 10 power term, also called
613:. A Note on Field Designations.
335:, lead creator of IEEE 754, and
176:The hidden bit in floating point
18:Fraction (floating point number)
968:from the original on 2016-07-13
955:Xerox Palo Alto Research Center
904:Sterbenz, Pat H. (1974-05-01).
643:The Art of Computer Programming
617:from the original on 2018-07-03
584:from the original on 2020-07-27
436:from the original on 2023-12-27
342:The Art of Computer Programming
240:, where he proposed the format
138:Language Independent Arithmetic
818:with defective pages 115–146.)
1:
490:floating-point representation
1018:10.1109/IEEESTD.2019.8766229
977:This term was introduced by
646:. Vol. 2. p. 214.
605:Savard, John J. G. (2018) .
476:. pp. 74, 91, 137–138.
847:Springer, pp. 84–85, 2017.
749:John Wiley & Sons, Inc.
1069:
906:Floating-Point Computation
827:Torres Quevedo, Leonardo.
218:trailing significand field
234:floating-point arithmetic
607:"Floating-Point Formats"
470:University of Manchester
315:fast inverse-square-root
124:modified normalized form
474:The Macmillan Press Ltd
230:Leonardo Torres Quevedo
224:Floating-point mantissa
206:double-precision format
58:or related concepts in
865:Forsythe, George Elmer
709:floating decimal point
131:normalized significand
42:, or more ambiguously
843:Numbers and Computers
696:The Macmillan Company
420:Kahan, William Morton
345:, condemn the use of
679:Goldstine, Herman H.
675:Burks, Arthur Walter
367:Mantissa (logarithm)
238:Essays on Automatics
170:true normalized form
102:123.45 = 12345 × 10.
1053:Computer arithmetic
840:Ronald T. Kneusel.
793:Decimal Computation
743:Decimal Computation
284:computer scientists
108:scientific notation
56:scientific notation
871:(September 1967).
869:Moler, Cleve Barry
472:, Manchester, UK:
293:was introduced by
64:significant digits
1027:978-1-5044-5924-2
946:Computing Surveys
683:von Neumann, John
573:(Product flyer).
182:normalized number
76:fractional number
16:(Redirected from
1060:
1032:
1031:
1002:
996:
991:
974:
973:
967:
942:
930:
924:
923:
901:
895:
894:
881:Englewood Cliffs
861:
855:
838:
832:
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811:
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784:
773:
772:
770:
769:
746:
733:
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721:
704:
703:
693:
671:
662:
661:
639:Knuth, Donald E.
635:
626:
625:
623:
622:
602:
593:
592:
590:
589:
583:
575:English Electric
572:
562:
556:
554:
461:
452:
450:
442:
441:
435:
428:
416:
394:
383:
311:fast square-root
272:common logarithm
21:
1068:
1067:
1063:
1062:
1061:
1059:
1058:
1057:
1038:
1037:
1036:
1035:
1028:
1004:
1003:
999:
971:
969:
965:
940:
934:Goldberg, David
932:
931:
927:
920:
903:
902:
898:
891:
863:
862:
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839:
835:
826:
822:
808:
806:
804:
788:Schmid, Hermann
786:
785:
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767:
765:
763:
737:Schmid, Hermann
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734:
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699:
691:
673:
672:
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654:
637:
636:
629:
620:
618:
604:
603:
596:
587:
585:
581:
570:
564:
563:
559:
542:biased exponent
484:
463:
462:
455:
439:
437:
433:
426:
418:
417:
408:
403:
398:
397:
384:
380:
375:
363:
337:Donald E. Knuth
295:George Forsythe
255:used the terms
226:
199:(binary) digits
178:
96:characteristics
84:
28:
23:
22:
15:
12:
11:
5:
1066:
1064:
1056:
1055:
1050:
1048:Floating point
1040:
1039:
1034:
1033:
1026:
997:
936:(March 1991).
925:
918:
896:
889:
856:
853:978-3319505084
833:
820:
802:
774:
761:
723:
713:characteristic
663:
652:
627:
594:
557:
482:
453:
422:(2002-04-19).
405:
404:
402:
399:
396:
395:
377:
376:
374:
371:
370:
369:
362:
359:
276:characteristic
261:characteristic
225:
222:
177:
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120:
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104:
103:
83:
80:
60:floating-point
52:characteristic
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
1065:
1054:
1051:
1049:
1046:
1045:
1043:
1029:
1023:
1019:
1015:
1011:
1007:
1001:
998:
994:
990:
988:
984:
980:
964:
960:
956:
952:
948:
947:
939:
935:
929:
926:
921:
919:0-13-322495-3
915:
911:
910:Prentice Hall
907:
900:
897:
892:
890:0-13-165779-8
886:
882:
878:
877:Prentice-Hall
874:
870:
866:
860:
857:
854:
850:
846:
844:
837:
834:
830:
824:
821:
817:
805:
803:0-89874-318-4
799:
795:
794:
789:
783:
781:
779:
775:
764:
762:0-471-76180-X
758:
754:
750:
745:
744:
738:
732:
730:
728:
724:
720:
718:
714:
710:
697:
690:
689:
684:
680:
676:
670:
668:
664:
660:
655:
653:0-201-89684-2
649:
645:
644:
640:
634:
632:
628:
616:
612:
608:
601:
599:
595:
580:
576:
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543:
539:
535:
531:
527:
523:
519:
515:
511:
507:
503:
499:
495:
491:
485:
483:0-333-26397-9
479:
475:
471:
467:
460:
458:
454:
449:
447:
432:
425:
421:
415:
413:
411:
407:
400:
392:
391:IEEE 754-1985
388:
382:
379:
372:
368:
365:
364:
360:
358:
356:
352:
348:
344:
343:
338:
334:
333:William Kahan
329:
327:
322:
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316:
312:
306:
304:
300:
296:
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287:
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147:
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139:
134:
132:
127:
125:
117:
113:
112:
111:
109:
101:
100:
99:
97:
93:
89:
81:
79:
77:
73:
69:
65:
61:
57:
53:
49:
45:
41:
37:
33:
19:
1005:
1000:
986:
976:
970:. Retrieved
950:
944:
928:
905:
899:
872:
859:
841:
836:
823:
807:. Retrieved
792:
766:. Retrieved
742:
706:
700:. Retrieved
698:. p. 42
687:
657:
642:
619:. Retrieved
610:
586:. Retrieved
566:
560:
533:
525:
521:
517:
513:
509:
505:
501:
497:
493:
487:
465:
445:
444:
438:. Retrieved
386:
381:
354:
350:
346:
340:
330:
325:
323:
318:
307:
302:
290:
288:
279:
275:
274:tables: the
267:
260:
256:
253:Arthur Burks
250:
245:
241:
237:
227:
217:
213:
209:
196:
189:
179:
169:
167:
161:
135:
130:
128:
123:
121:
115:
105:
91:
85:
51:
47:
43:
39:
38:, sometimes
35:
31:
29:
659:addition.'
492:, a number
389:is used in
299:Cleve Moler
291:significand
232:introduced
164:12345 × 10.
92:significand
36:coefficient
32:significand
1042:Categories
972:2016-07-13
809:2016-01-03
768:2016-01-03
702:2016-02-07
621:2018-07-16
588:2020-07-27
504:such that
440:2023-12-27
401:References
191:hidden bit
160:123.45 = 0
118:2345 × 10.
114:123.45 = 1
816:misprints
790:(1983) .
611:quadibloc
451:(8 pages)
385:The term
289:The term
251:In 1946,
228:In 1914,
1012:. 2019.
987:mantissa
979:Forsythe
963:Archived
751:p.
739:(1974).
717:mantissa
615:Archived
579:Archived
550:excess -
530:exponent
522:mantissa
508: =
431:Archived
387:fraction
361:See also
351:mantissa
347:mantissa
326:mantissa
319:mantissa
303:fraction
280:mantissa
257:mantissa
203:IEEE 754
154:Modula-2
68:exponent
48:fraction
44:mantissa
40:argument
520:is the
512:·
286:today.
236:in his
150:Fortran
88:decimal
82:Example
72:integer
1024:
916:
887:
851:
800:
759:
755:-205.
715:and a
650:
516:where
480:
268:et al.
186:binary
180:For a
34:(also
983:Moler
966:(PDF)
961:: 7.
953:(1).
941:(PDF)
692:(PDF)
582:(PDF)
571:(PDF)
434:(PDF)
427:(PDF)
373:Notes
265:Burks
156:, as
74:or a
50:, or
1022:ISBN
1010:IEEE
981:and
914:ISBN
885:ISBN
849:ISBN
798:ISBN
757:ISBN
648:ISBN
538:base
536:the
532:and
528:the
500:and
478:ISBN
313:and
297:and
259:and
152:and
30:The
1014:doi
753:204
546:CDC
488:In
355:all
353:in
142:Ada
1044::
1020:.
1008:.
989:.
975:.
951:23
949:.
943:.
912:.
883:.
879:,
867:;
777:^
726:^
705:.
681:;
677:;
666:^
656:.
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597:^
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244:;
220:.
194:.
172:.
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845:,
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534:b
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518:m
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510:m
506:x
502:e
498:m
494:x
446:m
246:m
242:n
214:p
210:p
162:.
146:C
116:.
20:)
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