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780:{\displaystyle {\begin{array}{ccccccccccccc}&1&&4&&2&&8&&5&&7\\\searrow &&\searrow &&\searrow &&\searrow &&\searrow &&\searrow &&\searrow \\&7&&1&&4&&2&&8&&5\end{array}}}
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is a somewhat informal term that means declaring things equivalent that otherwise would be considered distinct. For example, suppose the sequence 1 4 2 8 5 7 is to be regarded as the same as the sequence 7 1 4 2 8 5, because each is a
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which itself means "a small measure") is often used to assert that two distinct mathematical objects can be regarded as equivalent—if their difference is accounted for by an additional factor. It was initially introduced into
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in 1801. Since then, the term has gained many meanings—some exact and some imprecise (such as equating "modulo" with "except for"). For the most part, the term often occurs in statements of the form:
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as applied to functional programming, "operating modulo" is special jargon which refers to mapping a functor to a category by highlighting or defining remainders.
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Bullynck, Maarten (2009-02-01). "Modular arithmetic before C.F. Gauss: Systematizations and discussions on remainder problems in 18th-century
Germany".
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is finite, that is, you can remove a finite piece from the first subset, then add a finite piece to it, and get the second subset as a result.
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The term has gained many meanings over the years—some exact and some imprecise. The most general precise definition is simply in terms of an
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The term "modulo" can be used differently—when referring to different mathematical structures. For example:
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This article is about the general term in mathematics. For the operation, see
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out a normal subgroup (or an ideal) from a group (or ring) is often called "
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13 − 63 is a multiple of 10 (equiv., 13 and 63 differ by a multiple of 10).
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are the same—except for differences accounted for or explained by
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Gauss originally intended to use "modulo" as follows: given the
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as being one space modulo another; thus, for example, that a
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operation: given two numbers (either integer or real),
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71:. Unsourced material may be challenged and removed.
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579:, if the difference between them is in the ideal.
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440:both leave the same remainder when divided by
311:both share the same remainder when divided by
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328:, which itself means "a small measure."
131:Learn how and when to remove this message
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560:is a member of the normal subgroup (see
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642:cyclicly-shifted version of the other:
14:
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575:or an algebra are congruent modulo an
945:. London: Prentice Hall. p. 22.
943:Category Theory for Computing Science
614:of maps leads to the definition of a
149:("with respect to a modulus of", the
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27:Word with multiple distinct meanings
599:Two subsets of an infinite set are
34:. For the mathematical system, see
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811:List of mathematical jargon
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582:Used as a verb, the act of
245:Disquisitiones Arithmeticae
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428:is an integer multiple of
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299:is an integer multiple of
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238:that was introduced into
143:In mathematics, the term
80:"Modulo" mathematics
989:Mathematical terminology
897:10.1016/j.hm.2008.08.009
601:equal modulo finite sets
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840:Encyclopedia Britannica
549:are congruent modulo a
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481:, it is typically the
792:In that case, one is
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941:Barr; Wells (1996).
885:Historia Mathematica
836:"Modular arithmetic"
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612:short exact sequence
605:symmetric difference
333:equivalence relation
252:in 1801. Given the
250:Carl Friedrich Gauss
173:Carl Friedrich Gauss
65:improve this article
923:The Free Dictionary
626:modulo exact forms.
603:precisely if their
590:the..." or "we now
566:isomorphism theorem
432:, or equivalently,
303:, or equivalently,
236:mathematical jargon
806:Essentially unique
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167:in the context of
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867:. Retrieved
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635:In general,
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63:Please help
58:verification
55:
974:Jargon File
638:modding out
631:Modding out
588:modding out
452:means that
240:mathematics
165:mathematics
928:2019-11-21
869:2019-11-21
845:2019-11-21
822:References
620:cohomology
568:for more).
529:Structures
348:congruent)
344:equivalent
91:newspapers
905:0315-0860
733:↘
727:↘
721:↘
715:↘
709:↘
703:↘
697:↘
584:factoring
503:remainder
479:computing
467:computing
461:Computing
983:Category
919:"modulo"
864:catb.org
860:"modulo"
800:See also
594:the...".
507:division
383:integers
338:, where
320:ablative
254:integers
154:ablative
972:in the
592:mod out
501:is the
497:modulo
416:modulo
354:modulo
325:modulus
287:modulo
227:History
186:modulo
159:modulus
105:scholar
970:Modulo
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483:modulo
232:Modulo
146:modulo
107:
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32:Modulo
816:Up to
577:ideal
547:group
545:of a
404:(mod
366:Usage
317:Latin
275:(mod
234:is a
202:up to
151:Latin
112:JSTOR
98:books
947:ISBN
901:ISSN
573:ring
564:and
541:and
489:and
469:and
436:and
392:and
346:(or
307:and
263:and
214:and
84:news
893:doi
796:".
520:In
513:by
509:of
477:In
465:In
360:aRb
358:if
350:to
342:is
322:of
248:by
171:by
156:of
67:by
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899:.
889:36
887:.
862:.
838:.
610:A
558:ab
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493:,
400:≡
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271:≡
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765:8
759:2
753:4
747:1
741:7
690:7
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678:8
672:2
666:4
660:1
543:b
539:a
515:n
511:a
499:n
495:a
491:n
487:a
442:n
438:b
434:a
430:n
426:b
422:a
418:n
414:b
410:a
406:n
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398:a
394:n
390:b
386:a
356:R
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340:a
336:R
313:n
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289:n
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265:n
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222:.
220:C
216:B
212:A
205:C
199:B
195:A
188:C
184:B
180:A
134:)
128:(
123:)
119:(
109:·
102:·
95:·
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61:.
38:.
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