Difference between "≈", "≃", and "≅" - Mathematics Stack ...

≈ : ALMOST EQUAL TO (U+2248) · ≃ : ASYMPTOTICALLY EQUAL TO (U+2243) · ≅ : APPROXIMATELY EQUAL TO (U+2245). MathematicsStackExchangeisaquestionandanswersiteforpeoplestudyingmathatanylevelandprofessionalsinrelatedfields.Itonlytakesaminutetosignup. Signuptojointhiscommunity Anybodycanaskaquestion Anybodycananswer Thebestanswersarevotedupandrisetothetop Home Public Questions Tags Users Unanswered Teams StackOverflowforTeams –Collaborateandshareknowledgewithaprivategroup. CreateafreeTeam WhatisTeams? Teams CreatefreeTeam Teams Q&Aforwork Connectandshareknowledgewithinasinglelocationthatisstructuredandeasytosearch. Learnmore Differencebetween"≈","≃",and"≅" AskQuestion Asked 7years,4monthsago Active 1year,1monthago Viewed 226ktimes 60 33 $\begingroup$ Inmathematicalnotation,whataretheusagedifferencesbetweenthevariousapproximately-equalsigns"≈","≃",and"≅"? TheUnicodestandardlistsalloftheminsidetheMathematicalOperatorsBlock. ≈:ALMOSTEQUALTO(U+2248) ≃:ASYMPTOTICALLYEQUALTO(U+2243) ≅:APPROXIMATELYEQUALTO(U+2245) notationapproximation Share Cite Follow askedJul11'14at20:52 GOTO0GOTO0 1,28611goldbadge1515silverbadges2323bronzebadges $\endgroup$ 9 51 $\begingroup$ Didanyoneelsenoticethisquestionisbasicallyhowdifferentnotionsof"approximateequality"areonlyapproximatelyequal? $\endgroup$ – DavidH Jul11'14at21:01 4 $\begingroup$ @DavidH:Godisinthedetails.;-) $\endgroup$ – Lucian Jul11'14at21:36 7 $\begingroup$ @LucianIalwaystookforgrantedthatthe"approximately"qualifieractedidempotently,sowhilewecandistinguishbetween'exactequality'and'approximateequality','exactapproximateequality'isthesamethingas'approximateapproximateequality'.Aworldinwhichthisisnottruemakesmewanttostressvomit.Isittildesallthewaythedown!? $\endgroup$ – DavidH Jul11'14at21:51 8 $\begingroup$ @DavidHOTOH,somepeoplewillstartusing$\stackrel{\approx}{\approx}$for"approximatelyapproximately"andallhellwillbreakloose. $\endgroup$ – AdamHughes Jul12'14at6:38 4 $\begingroup$ AndhereIwasthinkingthat==and.equals()injavaweretoomuch... $\endgroup$ – Klesun Feb23at10:23  |  Show4morecomments 5Answers 5 Active Oldest Votes 27 $\begingroup$ Thenotations$\cong$and$\simeq$arenottotallystandardized.Bothareusuallyusedfor"isomorphic"whichmeans"thesameinwhatevercontextweare."Forexample"geometricallyisomorphic"usuallymeans"congruent,""topologicallyisomorphic"means"homeomorphic,"etcetera:itmeansthey'resomehowthe"same"forthestructureyou'reconsidering,insomesensestheyare"equivalent,"thoughnotalways"equal:"youcouldhavetwocongruenttrianglesatdifferentplacesinaplane,sotheywouldn'tliterallybe"thesame"buttheirintrinsicpropertiesarethesame.I'veseencolleaguesusebothforisomorphic,andsome(mostlythestablehomotopytheoristsIhangoutwith)willuse$\cong$for"homeomorphic"and$\simeq$for"uptohomotopyequivalence,"butthenotherswillusethesametwosymbols,forthesamepurposes,butreversingwhichgetswhichsymbol. The$\approx$isusedmostlyintermsofnumericalapproximations,meaningthatthevaluesinquestionsare"close"toeachotherinwhatevercontextoneisworking,andoftenitislesspreciseexactlyhow"close."Topologistsalsohaveatendencytouse$\approx$forhomeomorphic. Themaintake-awayfromthisanswer:notationisnotalwaysstandardized,andit'simportanttomakesureyouunderstandinwhatevercontextyou'reworking. Share Cite Follow editedJul11'14at21:29 answeredJul11'14at20:59 AdamHughesAdamHughes 34.7k1010goldbadges4545silverbadges7777bronzebadges $\endgroup$ 4 1 $\begingroup$ professionalsoftenuse=alsoforisomorphic.$\approx$isVERYOFTENusedforhomeomorphic. $\endgroup$ – ZlatanderZechpreller Jul11'14at21:27 1 $\begingroup$ I'lladdthe$\approx$oneinaswell,hedidn'taskabout=,soIdidn'tmentionit. $\endgroup$ – AdamHughes Jul11'14at21:29 $\begingroup$ Whencomparing$\cong$and$\simeq$andhavingtwokindsofisomorphismsinourcontext,Ithinkitismorenaturalto$\cong$mean“moreisomorphic”that$\simeq$. $\endgroup$ – user87690 Jul13'14at10:15 $\begingroup$ @user87690Itendtoagreewithyou,andthat'smypersonalconvention,butIhaveseenalgebraistsusethelatterastheirstandard,soIdidn'twanttopriviledgemyownaestheticsinceit'snotstandard. $\endgroup$ – AdamHughes Jul13'14at10:17 Addacomment  |  13 $\begingroup$ $\approx$isusedmostlyfortheapproximate(read:calculated)valueofamathematicalexpressionlike$\pi\approx3.14$InLaTeXitiscodedas\approx. $\cong$isusedtoshowacongruencybetweentwomathematicalexpressions,whichcouldbegeometrical,topological,andwhenusingmoduloarithmeticyoucangetdifferentnumbersthatarecongruent,e.g.,$5mod3\cong11mod3$(althoughthisisalsowrittenas$\equiv$).InLaTeXitiscodedas\cong. $\sim$isasimilarityingeometryandcanbeusedtoshowthattwothingsareasymptoticallyequal(theybecomemoreequalasyouincreaseavariablelike$n$).Thisisaweakerstatementthantheothertwo.InLaTeXitiscodedas\sim. $\simeq$ismoreofagrab-bagofmeaning.InLaTeXitiscodedas\simeqwhichmeans"similarequal"soitcanbeeither,whichmightbeappropriateinacertainsituations. Share Cite Follow answeredJul11'19at20:48 guitarphishguitarphish 18511silverbadge77bronzebadges $\endgroup$ 2 $\begingroup$ Greatanswer.But,humorme...Canyouclarifywhy5mod3=11mod3islessaccuratethan5mod3≅11mod3. $\endgroup$ – Eddie Nov17'20at16:51 1 $\begingroup$ @Eddie:5mod3=11mod3isnothowmathematicianswouldwrite.Theywouldwrite5≡11(mod3),read"5isequivalentto11,modulo3."Wenormallydon'tconsider"5mod3"tobeanythingbyitself(althoughyoucouldsay"Theresidueclassof5modulo3",whichwouldmeanthesetofallintegersequivalentto5,mod3.) $\endgroup$ – NickMatteo May1at16:48 Addacomment  |  4 $\begingroup$ ≈isfornumericaldata,homeomorphism ≃isforhomotopyequivalence ≅isforisomorphism,congruence,etc Thesearejustmyownconventions. Share Cite Follow answeredJul11'14at21:00 Kaa1elKaa1el 1,8601111silverbadges2020bronzebadges $\endgroup$ Addacomment  |  1 $\begingroup$ Aslongasthecategorytheoryisconcerned: Thesymbol≅isusedforisomorphismofobjectsofacategory,andinparticularforisomorphismofcategories(whichareobjectsofCAT).Thesymbol≃isusedforequivalenceofcategories.Atleast,thisistheconventionusedinthisbookandbymostcategorytheorists,althoughitisfarfromuniversalinmathematicsatlarge. (Warning1.3.16,BasicCategoryTheorybyTomLeinster). Share Cite Follow answeredAug11'20at6:30 teikakazurateikakazura 21511silverbadge66bronzebadges $\endgroup$ Addacomment  |  1 $\begingroup$ Inmywork"="istheidentityofanumbersoitstatesanequivalence. 1=1,2x=10iex=5. Theapproximationsign"≈"Iusefordecimalapproximationswithtilde"~"beingarougherapproximation. Tilde"~"Iusetostateageometricshapeissimilartoanotheroneieatriangleofsides3/4/5issimilartoatrianglewithsides30/40/50.Iwrite▲ABC~▲A'B'C'where▲A'B'C'isadilatedversionofthepre-image. Foraclosersimilarity"≃"mightmeanatrianglealmostcongruentbutonlyROUGHLYsimilar,suchastwotriangles3/4/5and3.1/4.1/5.1while"≅"meanscongruent.Reallifetrianglesuseapproximationsandhaveroundingerrors.3/4doesnotequal3.1/4.1butcouldberoughapproximationsforsomethingalreadyconstructed.Assuchthisisnotapopularuseandpuristsandrigorousmathprofsdisdainitbecausetheydonothaveawayofusingitordefiningitsoundly.Thatiswhylaymenorserviceprofessionalsarefreetoexploreitandacademicsprefersomethingmoreclearlydefinedunlessdeformationsareallowedasinmycase. Triangle3/4/5is≅totriangle4/5/3thedifferencebeingarotationchangingthecoordinatesoftheanglesbutpreservingangleandsidelength. AlsoIusethedefinitionsymbol"≡"todefinefunctions.Simplestcaseisf(x)≡y.Iamusingittostatearelationshiptofindsumsinsteadofstatingitasthesumthatwearederivinghoweveritcanbeusedtostateequivalence.ForaquadraticIwouldsetf(x)tozerobutwouldnotdefinef(x)aszero.Iuse≡forallcases,notonlyimmediateones.Settingf(x)tozerocreatestheequivalencyf(x)=0forthecoordinateyouaretryingtosolvebutisnottrueforallcoordinatesthataresolvable. Share Cite Follow answeredSep28'20at0:30 JosephMaloneJosephMalone 1122bronzebadges $\endgroup$ Addacomment  |  YourAnswer ThanksforcontributingananswertoMathematicsStackExchange!Pleasebesuretoanswerthequestion.Providedetailsandshareyourresearch!Butavoid…Askingforhelp,clarification,orrespondingtootheranswers.Makingstatementsbasedonopinion;backthemupwithreferencesorpersonalexperience.UseMathJaxtoformatequations.MathJaxreference.Tolearnmore,seeourtipsonwritinggreatanswers. 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