152:
322:, the second known calculator to be built, and the oldest surviving, use a different method: incrementing the digit from 0 to 9, cocks a mechanical device to store energy, and the next increment, which moves the digit from 9 to 0, releases this energy to increment the next digit by 1. Pascal used weights and gravity in his machine. Another notable machine using similar method is the highly successful 19th century
72:
395:, and must be accounted for when adding data words of lengths greater than that of the CPU. For subtractive operations, two (opposite) conventions are employed as most machines set the carry flag on borrow while some machines (such as the 6502 and the PIC) instead reset the carry flag on borrow (and vice versa).
311:. They face two basic difficulties: The first one stems from the fact that a carry can require several digits to change: in order to add 1 to 999, the machine has to increment 4 different digits. Another challenge is the fact that the carry can "develop" before the next digit finished the addition operation.
314:
Most mechanical calculators implement carry by executing a separate carry cycle after the addition itself. During the addition, each carry is "signaled" rather than performed, and during the carry cycle, the machine increments the digits above the "triggered" digits. This operation has to be
40:
numbers together by starting with the rightmost digits and working to the left. For example, when 6 and 7 are added to make 13, the "3" is written to the same column and the "1" is carried to the left. When used in subtraction the operation is called a
205:
Traditionally, carry is taught in the addition of multi-digit numbers in the 2nd or late first year of elementary school. However, since the late 20th century, many widely adopted curricula developed in the United States such as
329:
Some innovative machines use continuous transmission: adding 1 to any digit, advances the next one by 1/10 (which in turn advances the next one by 1/100 and so on). Some innovative early calculators, notably
342:) continued to use and improve it, and made continuous-transmission calculators with unmatched speed, into the late 1960s, to the end of the mechanical calculator era.
315:
performed sequentially, starting with the ones digit, then the tens, the hundreds, and so on, since adding the carry can generate a new carry in the next digit.
267:
247:
211:
778:
133:
from the next digit left, and then 'paid back' by adding it to the subtrahend in the column from which it was 'borrowed', giving in this example
162:
338:
implemented continuous transmission with great success, starting with the aptly named "Silent Speed" calculator. Marchant (later to become
591:
375:, the carry from the most significant bit of an arithmetic operation (or bit shifted out from a shift operation) is placed in a special
207:
110:, and the 10 is got by taking ("borrowing") 1 from the next digit to the left. There are two ways in which this is commonly taught:
760:
662:
276:
When several random numbers of many digits are added, the statistics of the carry digits bears an unexpected connection with
334:
calculator from 1870, and a design by
Selling, from 1886, used this method, but neither were successful. In the early 1930,
387:
is also generally used to indicate borrows in subtraction, though the bit's meaning is inverted due to the effects of
192:
59:
Carrying makes a few appearances in higher mathematics as well. In computing, carrying is an important function of
686:
528:
Nakano, Fumihiko; Sadahiro, Taizo (February 2014), "A generalization of carries processes and
Eulerian numbers",
492:; Fulman, Jason (October 2010), "On adding a list of numbers (and other one-dependent determinantal processes)",
214:
methods, and methods using coloring, manipulatives, and charts. Such omissions were criticized by such groups as
857:
379:
which can be used as a carry-in for multiple precision arithmetic or tested and used to control execution of a
654:
862:
281:
218:, and some states and districts have since abandoned this experiment, though it remains widely used.
724:
174:
49:
556:
Hegland, M.; Wheeler, W. W. (January 1997), "Linear
Bijections and the Fast Fourier Transform",
319:
215:
622:
392:
308:
17:
170:
270:
8:
753:
715:
677:
391:
arithmetic. Normally, a carry bit value of "1" signifies that an addition overflowed the
388:
351:
335:
226:
60:
614:
573:
501:
458:
429:
252:
232:
53:
821:
802:
738:
700:
658:
651:
Mathematics under the
Microscope: Notes on Cognitive Aspects of Mathematical Practice
646:
331:
577:
515:
291:, the carry operation for two-digit numbers can be formalized using the language of
733:
695:
606:
565:
537:
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468:
421:
380:
292:
288:
32:
of digits to another column of more significant digits. It is part of the standard
719:
681:
412:
Holte, John M. (February 1997), "Carries, Combinatorics, and an
Amazing Matrix",
361:
339:
277:
25:
489:
485:
446:
542:
472:
449:; Fulman, Jason (August 2009), "Carries, shuffling, and symmetric functions",
307:
Carry represents one of the basic challenges facing designers and builders of
851:
824:
805:
684:; Tanny, S. (May 1973), "Significance Arithmetic: The Carrying Algorithm",
569:
79:
A typical example of carry is in the following pencil-and-paper addition:
323:
296:
229:
states that the number of carries involved in adding two numbers in base
838:
295:. This viewpoint can be applied to alternative characterizations of the
618:
433:
355:
86:
122:
in the tens column. According to this method, the term "borrow" is a
829:
810:
33:
610:
425:
372:
123:
37:
506:
463:
210:
omitted instruction of the traditional carry method in favor of
56:
do not emphasize any specific method to find a correct answer.
29:
558:
Applicable
Algebra in Engineering, Communication and Computing
161:
deal primarily with the United States and do not represent a
71:
842:
592:"A Cohomological Viewpoint on Elementary School Arithmetic"
676:
713:
722:(June 1975), "The Real Numbers as a Wreath Product",
484:
255:
235:
261:
241:
118:from the next digit left, leaving in this example
249:is equal to the exponent of the highest power of
849:
819:
800:
555:
527:
445:
494:Bulletin of the American Mathematical Society
159:The examples and perspective in this section
75:Example: The addition of two decimal numbers
326:, which replaced the weights with springs.
177:, or create a new section, as appropriate.
737:
699:
541:
505:
462:
302:
193:Learn how and when to remove this message
786:. Charles Babbage Institute. p. 96.
141:
70:
754:"Chebyshev's continuous adding machine"
645:
589:
850:
751:
820:
801:
776:
411:
221:
590:Isaksen, Daniel C. (November 2002),
145:
66:
126:, since the ten is never paid back.
13:
14:
874:
794:
599:The American Mathematical Monthly
414:The American Mathematical Monthly
766:from the original on 2017-08-09.
150:
770:
745:
687:Journal of Combinatorial Theory
530:Advances in Applied Mathematics
516:10.1090/S0273-0979-2010-01306-9
451:Advances in Applied Mathematics
707:
670:
639:
583:
549:
521:
478:
439:
405:
1:
398:
28:that is transferred from one
739:10.1016/0001-8708(75)90115-2
701:10.1016/0097-3165(73)90013-7
368:is used in a similar sense.
345:
7:
282:riffle shuffle permutations
173:, discuss the issue on the
52:, while curricula based on
10:
881:
349:
85:7 + 9 = 16, and the digit
48:Carrying is emphasized in
543:10.1016/j.aam.2013.09.005
473:10.1016/j.aam.2009.02.002
780:The Calculating Machines
364:like an adder, the word
725:Advances in Mathematics
318:Some machines, notably
50:traditional mathematics
777:Ernst, Martin (1925).
752:Roegel, Denis (2015).
309:mechanical calculators
303:Mechanical calculators
280:and the statistics of
263:
243:
216:Mathematically Correct
76:
858:Elementary arithmetic
647:Borovik, Alexandre V.
570:10.1007/s002000050059
350:Further information:
264:
244:
142:Mathematics education
74:
18:elementary arithmetic
271:binomial coefficient
253:
233:
171:improve this section
863:Computer arithmetic
628:on January 16, 2014
360:When speaking of a
352:Adder (electronics)
336:Marchant calculator
320:Pascal's calculator
269:dividing a certain
212:invented arithmetic
137:in the tens column.
822:Weisstein, Eric W.
803:Weisstein, Eric W.
657:, pp. 87–88,
259:
239:
222:Higher mathematics
99:47 − 19 ---- 28
92:The opposite is a
82:27 + 59 ---- 86
77:
54:reform mathematics
664:978-0-8218-4761-9
262:{\displaystyle p}
242:{\displaystyle p}
203:
202:
195:
67:Manual arithmetic
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682:Gian-Carlo, Rota
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621:, archived from
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482:
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389:two's complement
381:computer program
293:group cohomology
289:abstract algebra
278:Eulerian numbers
268:
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260:
248:
246:
245:
240:
227:Kummer's theorem
198:
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109:
108:(10 − 9) + 7 = 8
105:
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611:10.2307/3072368
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490:Diaconis, Persi
486:Borodin, Alexei
483:
479:
447:Diaconis, Persi
444:
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426:10.2307/2974981
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362:digital circuit
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340:SCM Corporation
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795:External links
793:
790:
789:
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744:
732:(3): 278–304,
716:Metropolis, N.
706:
694:(3): 386–421,
678:Metropolis, N.
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663:
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605:(9): 796–805,
582:
564:(2): 143–163,
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500:(4): 639–670,
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457:(2): 176–196,
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420:(2): 138–149,
403:
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165:of the subject
163:worldwide view
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98:
89:is the carry.
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809:
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772:
747:
729:
723:
718:; Ross, B.;
714:Faltin, F.;
709:
691:
690:, Series A,
685:
672:
650:
641:
630:, retrieved
623:the original
602:
598:
585:
561:
557:
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529:
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450:
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328:
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297:real numbers
286:
275:
225:
204:
189:
183:January 2009
180:
160:
130:
115:
101:
93:
91:
84:
78:
58:
47:
42:
21:
15:
720:Rota, G.-C.
632:January 22,
383:. The same
324:Comptometer
135:4 − (1 + 1)
129:The ten is
114:The ten is
852:Categories
399:References
356:Carry flag
104:7 − 9 = −2
63:circuits.
830:MathWorld
811:MathWorld
536:: 28–43,
507:0904.3740
464:0902.0179
385:carry bit
377:carry bit
373:computers
346:Computing
332:Chebyshev
175:talk page
106:, so try
34:algorithm
839:Carrying
825:"Borrow"
761:Archived
649:(2010),
578:17603981
371:In most
169:You may
124:misnomer
96:, as in
806:"Carry"
619:3072368
434:2974981
661:
617:
576:
432:
131:copied
102:Here,
94:borrow
43:borrow
30:column
784:(PDF)
764:(PDF)
757:(PDF)
626:(PDF)
615:JSTOR
595:(PDF)
574:S2CID
502:arXiv
459:arXiv
430:JSTOR
366:carry
120:3 − 1
116:moved
61:adder
26:digit
24:is a
22:carry
843:nLab
659:ISBN
634:2014
354:and
208:TERC
20:, a
734:doi
696:doi
655:AMS
607:doi
603:109
566:doi
538:doi
512:doi
469:doi
422:doi
418:104
393:ALU
287:In
38:add
36:to
16:In
854::
841:-
827:.
808:.
759:.
730:16
728:,
692:14
680:;
653:,
613:,
601:,
597:,
572:,
560:,
534:53
532:,
510:,
498:47
496:,
488:;
467:,
455:43
453:,
428:,
416:,
299:.
284:.
273:.
45:.
833:.
814:.
736::
698::
609::
568::
562:8
540::
514::
504::
471::
461::
424::
257:p
237:p
196:)
190:(
185:)
181:(
167:.
87:1
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
