swaabhaavika samkhya

ganithashaasthrathil, 1,2,3 muthalaaya samkhyakalaanu swaabhaavika samkhyakal.orupakshe 0 ulppede. staanderd aiso 80000-2 ulppedeyulla chila nirvachanangal, noonn-negatteevu poornnasamkhyakalumaayi poruthappedunna swaabhaavika samkhyakal aarambikkunnu, mattullava aarambhikkunnath, swaabhaavika samkhyakalil ninnu poojyathe ozhivaakkunna positteev poornnasamkhyakalude vaachakangalumayi poruthappedunnathinaal chilappol poojyathe muzhuvan samkhyakalaayi soochippikkunnu, mattu rachanakalil, aaa padam poornnasamkhyakalkku (negatteevu poornnasamkhyakal ulppede) pakaram upayogikkunnu. pothuvaaya bhashayil, prathyekichu primery school vidyaabhyaasathil, negatteevu poornnasamkhyakalum poojyavum avabodhapoorvvam ozhivaakkunnathinum ennunnathinte vyathyaasathe alaveduppinte thudarchayumaayi thaarathamyam cheiyunnathinum swaabhaavika samkhyakale ennal samkhyakal ennu vilikkam-ithu yadhaartha samkhyakalude oru mukhamudra savisheshathayaanu.
swaabhaavika samkhyakal ennunnathinu upayogikkaam ("meshappurathu aarr" nanayangal ullathupole), eee saahacharyathil avaa "cardinal nambarukalaayi" varthikkunnu. oordar cheiyunnathinum avaa upayogikkaam ("ithu raajyathe moonnaamathe" valia nagaramaan "ennathupole), eee saahacharyathil avaa" oordinal nambarukalaayi "varthikkunnu. ganithashaasthraparamaaya ardhathil samkhyakalude gunangalonnumillaatha (udha.jii. sports jezhsi nambarukal).
swaabhaavika samkhyakal oru koottam roopappeduthunnu, palappozhum soothravaakyam _ 1 aayi pratheekappeduthunnu. swaabhaavika samkhyakalude koottam thudarchayaayi vipuleekarichukondaan mattu niravadhi samkhyaasamghangal nirmmikkunnath poornnasamkhyakal, oru additteev aidantitti 0 (ithuvare illengil), oro poojyamallaatha swaabhaavika samkhyaykkum oru additteev invezhs enniva ulppeduthunnathiloode; yukthisahamaya samkhyakal, oro poojyamallaatha poornnasamkhyaykkum oru gunana vipareeta soothravaakyam _ 2 ulppeduthunnathiloode (koodaathe poornnasamkhyakalaal eee vipareethangalude gunanafalavum); yukthikalude kauchi seekwansukalude paridhikal ulppeduthunnathiloode yadhaartha samkhyakal; sankeernna samkhyakal, yadhaartha samkhyakalkku sameepam oru varggamoolavum (athinte thukayum uthppannangalum); thudangiyava. eee vipuleekarana shringhala mattu samkhyaasampradaayangalil swaabhaavika samkhyakale niyamaparamaayi ulcherkkunnu.
swaabhaavika samkhyakalude gunangal, athaayath vibhajyatha, avibhaajyasamkhyakalude vitharanam enniva samkhyaa sidhaanthathil padikkunnu. partitioning, enumeration thudangiya ennavum kramavum sambandhicha prasnangal kompinettoriksil padikkunnu.
charithram.
purathana verukal.
oru swaabhaavika samkhyaye prathinidheekarikkunnathinu ettavum aadima reethi oro vastuvinum oru adayaalam iduka ennathaanu. pinneedu, oru koottam vasthukkal samathvam, amitata allengil kuravu ennivaykkaayi pareekshikkaan kazhiyum-oru adayaalam adichu settil ninnu oru vasthu neekkam cheythukondu.
amoortheekaranathile aadyathe pradhana munnettam samkhyakale prathinidheekarikkunnathin akkangalude upayogamaayirunnu. ithu valia ennam record cheiyunnathinulla samvidaanangal vikasippikkaan anuvadichu. purathana eejipthukar 1,10 ennathinte prathyeka hyroglyphukalum 10 muthal 10 dasalaksham vareyulla ellah shakthikalum ulla sakthamaaya oru samkhyaasampradaayam vikasippichu. bisi 1500 muthal ippol paareesile lauvaril sthithicheyyunna karnakkil ninnulla oru kallu kothupaniyil 276 ennam noorukanakkinu, 7 patthukal, 6 ennam enningane chithreekarichirikkunnu; athupole 4,622 enna samkhyaykkum. babyloniakkarkku adisthaana arupath upayogichu 1,2,10 ennee akkangale adisthaanamaakkiyulla oru sthala-moolya samvidaanam undaayirunnu, angane arupathinte chihnam onninte chinnathinu thulyamaayirunnu-athinte moolyam sandarbhathil ninnu nirnnayikkappedunnu.
valare pilkkaalatthe munnettam, svanthamaayi oru samkhyayulla oru samkhyayaayi kanakkakkavunna aashayathinte vikasamaayirunnu. sthaana-moolya notteshanil (mattu samkhyakalkkullil) 0 akkatthinte upayogam bisi 700 muthal babyloniakkaar upayogichirunnu, avar akkatthile avasana chihnamaayirikkumbol atharam akkam ozhivaakki. oolmeku, maaya samskaarangal 0 neey oru prathyeka samkhyayaayi upayogichirunnuvenkilum eee upayogam mesoamericaic appurathekku vyaapichilla. aadhunika kaalath 0 enna samkhyayude upayogam 628 sii. iii. yill indian ganithashaasthranjanaaya brahmaguptanil ninnaan uthbhavichath. ennirunnaalum, madhyakala computasil (eesttarinte theeyathi kanakkaakkunnath) 0 oru samkhyayaayi upayogichirunnu, 525 sii. iii. yill dionysius exeguesil aarambhich, oru samkhyayaal soochippikkappedaathe. saadhaarana roman akkangalk 0 ennathinu oru chihnamilla; pakaram, "onnumilla" ennathinte laattin padamaaya "nulla" (allengil janithaka roopam "nalle") oru 0 moolyathe soochippikkaan upayogichu.
amoorthathakalaayi samkhyakalekkurichulla aadyathe chittayaya patanam saadhaaranayaayi greak thathwachinthakaraaya paithagorasinum archemediesinum nalkappedunnu. chila greak ganithashaasthranjar 1 enna samkhyaye valia samkhyakalekkaal vyathyasthamayi kanakkakki, chilappol oru samkhyayenna nilayil polum. udaaharanathinu, euclid aadyam oru unitineyum pinneedu oru samkhyayeyum yoonittukalude oru koottamaayi nirvachichu, athinaal addehathinte nirvachanamanusarichu, oru yoonit oru samkhyayalla, koodaathe advitheeya samkhyakalilla (udha.jii.anischithamaayi niravadhi yoonittukalil ninnulla eathenkilum rund unitukal a2 aanu).
india, chaina, mesoamerica ennividangalil ekadesham ore samayam akkangalekkurichulla swatantra padanangal nadannu.
aadhunika nirvachanangal.
pathonpathaam noottaandil europil swaabhaavika samkhyakalude krithyamaaya swabhaavathekkurichu ganithashaasthraparavum daarshanikavumaaya charchakal nadannirunnu. svayamsidhavaakyangal avayude parimithamaaya prayogatthil maathrame prakadippikkaan kazhiyoo ennu henry poincare prasthaavichu, athe pravrithiyude anischithakaala aavarthanam sangalppikkaan anuvadikkunnathu "manassinte sakthiyaanu" ennu nigamanam cheythu. leopold chronecker thante vishwaasathe samgrahichathu "daivam poornnasamkhyakale srishtichu, backiellam manushyante pravrithiyaanu" ennaanu.
ganithashaasthrathinte aditharayile yukthisahamaya kaadinyam mechappeduthendathinte aavasyakatha kanstrakttivisttukal kandu. 1860-kalil, herman grassman swaabhaavika samkhyakalkku oru aavarthana nirvachanam nirddeshichu, angane avaa yadhaarthathil swaabhaavikamallennu prasthaavichu-marichu nirvachanangalude anantharafalamaanu. pinneedu, atharam aupachaarika nirvachanangalude rund klaasukal nirmikkappettu; pinneedu avaa mikka praayogika prayogangalilum thulyamaanennu kaanichu.
swaabhaavika samkhyakalude settu-saidhaanthika nirvachanangalkk thudakkamittath freej aanu. oru prathyeka settumaayi parasparam bandhapetta ellah settukaludeyum klaas ennaanu addeham thudakkathil oru swaabhaavika samkhyaye nirvachichathu. ennirunnaalum, eee nirvachanam rassalinte vairudhyam ulppedeyulla vairudhyangalilekku nayichu. atharam vairudhyangal ozhivakkaan, oru swaabhaavika samkhyaye oru prathyeka ganamaayi nirvachikkunnathinayi aupachaarikatha parishkkarichu, aaa ganavumaayi parasparam bandhappedutthaan kazhiyunna eathoru ganathinum aaa samkhyakal undennu parayappedunnu.
randaam klaas nirvachanangal avatharippichathu chaals saandezhs peezhs aanu, richard dedekind parishkarikkukayum giuseppe peeno kooduthal paryavekshanam cheyyukayum cheythu; eee sameepanathe ippol peeno ganithashaasthram ennu vilikkunnu. ithu oordinal nambarukalude gunangalude svayamsidheekaranathe adisthaanamaakkiyullathaanu oro swaabhaavika samkhyaykkum oru pingamiyundu, poojyamallaatha oro swaabhaavika samkhyaykkum oru advitheeya mungamiyund. peeno ganithashaasthram settu sidhaanthathinte niravadhi durbalamaaya samvidaanangalumaayi thulyamaanu. atharam oru samvidaanam ananthathayude svayamsidhathaykku pakaram athinte nishedham upayogichu maattisthaapikkunna zfc aanu. zfc yill theliyikkaan kazhiyunnathum ennaal peeno sidhaanthangal upayogichu theliyikkaan kazhiyathathumaaya sidhaanthangalil goodstein sidhaantham ulppedunnu.
eee nirvachanangalellam upayogichu, 0 (shoonyamaaya settumaayi poruthappedunna) oru swaabhaavika samkhyayaayi ulppeduthunnathu soukaryapradamaanu. 0 ulppeduthunnathu ippol settu sidhaanthakarkkum yukthivaadikalkkum idayil pothuvaaya kanvenshanaanu. mattu ganithashaasthranjaril 0 umm ulppedunnu, lupe kaundarukal, string-allengil shreni-gadakangal polulla enangal kanakkaakkumbol computer bhashakal palappozhum poojyathil ninnu aarambikkunnu. maruvashathu, pala ganithashaasthranjarum 1 neey aadyathe swaabhaavika samkhyayaayi edukkunna pazhaya paarambaryam nilanirthiyittundu.
notteshan.
ellah swaabhaavika samkhyakaludeyum koottam staanderdaayi soochippikkunnu allengil formula 3 pazhaya grandhangal idaykkide eee koottatthinte chihnamaayi upayogichittundu.
swaabhaavika samkhyakalil adangiyirikkaam allengil illa ennathinaal, eathu pathippaanu paraamarshikkunnathenn ariyendathu pradhaanamaan. ithu palappozhum sandarbhathaal vyakthamaakkappedunnu, ennaal oru subscripto allengil notteshanil oru superscripto upayogichum ithu cheyyaam, udaaharanathina
pakaramaayi, swaabhaavika samkhyakal swaabhaavikamaayi poornnasamkhyakalude oru upavibhagam roopappedunnathinaal (palappozhum avaye yathaakramam positteev allengil noonn-negatteevu poornnasamkhyakal ennu vilikkam. 0 ulppeduthiyittundo illayo ennathinekkurichu avyakthathayundakaan, chilappol muun kesil oru subscript (allengil superscript) "0" cherkkukayum, pinneedulla kesil oru superscript "cherkkukayum cheiyunnu
swathukkal.
kootticherkkal.
swaabhaavika samkhyakalude settu formula _ 8 umm oro swaabhaavika samkhyayeyum aduthathilekku ayaykkunna pingamiyaaya function formula _ 9 umm nalkiyaal, oraalkk swaabhaavika samkhyakalude kootticherkkal krameekarichukondu aavarthichu nirvachikkaan kazhiyum. appol formula _ 10 ennathu aidantitti ghadakam 0 ulla oru commutative monoidaanu. ithu oru janarettarile oru swatantra monoidaanu. eee commutative monoid radhaakkal property thripthippeduthunnu, athinaal ithu oru groupil ulcherkkaan kazhiyum. swaabhaavika samkhyakal adangiyirikkunna ettavum cheria group poornnasamkhyakalaanu.
1 ennu nirvachichittundenkil, appol. athaayath, avan athinte pingamiyaanu.
gunanam.
samaanamaayi, kootticherkkal nirvachichirikkunnathil, oru gunana oopparettar formula _ 11 aandu vazhi nirvachikkaam. ithu formula _ 12 neey aidantitti ghadakam 1 ulla oru free commutative monoidaayi maattunnu; eee monoidinulla oru janarettar settu prime nambarukalude koottamaanu.
koottalum gunanavum thammilulla bandham.
kootticherkkalum gunanavum anuyojyamaanu, ithu vitharana niyamathil prakadippikkunnu kootticherkkalinteyum gunanathinteyum eee gunangal swaabhaavika samkhyakale oru commutative semiringinte udaaharanamaakkunnu. gunanam commutative aayirikkanamennilla ennathinaal swaabhaavika samkhyakalude beejaganita saamaanyavalkkaranamaanu semiringukal. additteev invezhsukalude abhavam, athaayath formula _ 8 kuraykkunnathinu keezhil adachittilla enna vasthuthaykk thulyamaanu (athaayath, onnu mattonnil ninnu swaabhaavikamaayi kuraykkunnathu ellaayppozhum mattoru swaabhaavikathaykku kaaranamakilla), athaayath formula _ 8 oru valayamalla; pakaram ithu oru semiring aanu (rigg ennum ariyappedunnu).
swaabhaavika samkhyakale "0 ozhivakkal", "1 ll aarambhikkunnath" enningane edukkukayaanengil, +, × ennivayude nirvachanangal mukalil paranjathupole aayirikkum, avaa "and" ll aarambikkunnuvennu maathram. koodaathe, formula _ 15 nuu aidantitti ghadakam illa.
oordar.
eee vibhaagathil, uthppannathe soochippikkunnathupollu samyuktha variabilukalum pravartthanangalude staanderd oordarum anumaanikkappedunnu.
mattoru swaabhaavika samkhya undengil maathram anuvadikkunnathiloodeyaanu swaabhaavika samkhyakalude motham kramam nirvachikkappedunnathu. eee kramam inipparayunna ardhathil ganitha pravarthanangalumaayi poruthappedunnu iff, aandu aandu aandu aandu, aandu aandu.
swaabhaavika samkhyakalude oru pradhana savisheshatha avaa nannayi krameekarichirikkunnu ennathaanu shoonyamallaatha oro swaabhaavika samkhyakalkkum ettavum kuranja moolakamundu. nannayi krameekaricha settukalkkidayile rank oru oordinal nambar upayogichu prakadippikkunnu; swaabhaavika samkhyakalkku, ithu (omega) aayi soochippikkunnu.
vibhajanam.
eee vibhaagathil, uthppannathe soochippikkunnathupollu samyuktha variabilukalum pravartthanangalude staanderd oordarum anumaanikkappedunnu.
oru swaabhaavika samkhyaye mattonn kondu harikanum athinte falamaayi oru swaabhaavika samkhya nedaanum pothuve saadhyamallengilum, "sheshikkunnathodo vibhajichatho" allengil euclidean vibhajanam enna prakriya oru pakaramaayi labhyamaanu eathenkilum rund swaabhaavika samkhyakalkkum swaabhaavika samkhyakal ullavarkkum.
eee samkhyaye "koshyant" ennu vilikkukayum bye ennathinte vibhajanathinte "bakkiyullava" ennu vilikkukayum cheiyunnu. samkhyakal aandu ennivayaal advitheeyamaayi nirnnayikkappedunnu. eee euclidean vibhajanam mattu niravadhi gunangalkkum (vibhajyatha), algorithangalkkum (euclidean algoritam polullava), samkhya sidhaanthathile ashayangalkkum pradhaanamaan.
swaabhaavika samkhyakalaal thripthippedunna beejaganita gunangal.
mukalil nirvachichathupole swaabhaavika samkhyakalil koottal (+), gunanam (×) pravarthanangalkku niravadhi beejaganita gunangalunda
saamaanyavalkkaranangal.
ennalinteyum kramappeduthalinteyum rund upayogangalil ninnaan swaabhaavika samkhyakalude rund pradhana saamaanyavalkkaranangal undaakunnatha cardinal nambarukalum oordinal nambarukalum.
cardinalityude ettavum kuranja oordinal (athaayath, prarambha oordinal) ennaal cardinal nambarulla nannayi oordar cheytha pala settukalkkum athinekkaal valia oordinal nambar undu.
parimithamaaya nannayi krameekaricha settukalkku, oordinal, cardinal samkhyakal thammil oru parasparabandhamund; athinaal avaa randum ore swaabhaavika samkhyayaal prakadippikkaan kazhiyum, athaayath settinte gadakangalude ennam. oru valia parimithamaaya allengil ananthamaaya shreniyil oru moolakatthinte sthaanam vivarikkaanum eee samkhya upayogikkaam.
peeno ganithathe (athaayath phastt-oordar peeno axiums) thripthippeduthunna ganithatthinte kanakkakkavunna staanderd allatha oru model 1933 ll scholim vikasippicheduthu. ultrapower nirmmaanatthiloode saadhaarana swaabhaavika samkhyakalil ninnu nirmmikkaan kazhiyunna ennaanaavaatha oru modalaan hypernatural nambarukal.
nishkalanka poornnasamkhyakal niraykkunnilla" enna soothravaakyam prakopanaparamaayi avakashappedaan georges reeb upayogikkunnu. mattu saamaanyavalkkaranangal akkangalekkurichulla lekhanathil charcha cheythittundu.
aupachaarika nirvachanangal.
swaabhaavika samkhyakale aupachaarikamaayi nirvachikkunnathin rund staanderd reethikalundu. aadyathethu, giuseppe peenoyude peril, peeno aaksiyomukal ennu vilikkunna chila sidhaanthangale adisthaanamaakkiyulla peeno ganithashaasthram enna svayambharana svayamsidha sidhaantham ulkkollunnu.
randaamathe nirvachanam settu sidhaanthathe adisthaanamaakkiyullathaanu. ithu swaabhaavika samkhyakale nirddishta settukalaayi nirvachikkunnu. kooduthal krithyamaayi paranjaal, oro swaabhaavika samkhyayeyum vyakthamaayi nirvachicha oru koottamaayi nirvachikkunnu, athinte gadakangal mattu settukalude gadakangal kanakkakkaan anuvadikkunnu, "oru settinu gadakangalundu" enna ardhathil rund settukalkkidayil onnu muthal onnu vare correspondence undennaanu ardhamaakkunnathu.
swaabhaavika samkhyakale nirvachikkaan upayogikkunna settukal peeno aaksiyamkale thripthippeduthunnu. peeno ganithathil prasthaavikkaanum theliyikkaanum kazhiyunna oro sidhaanthavum settu sidhaanthathilum theliyikkaan kazhiyumennu ithu pinthudarunnu. ennirunnaalum, rund nirvachanangalum thulyamalla, kaaranam peeno ganithatthinte adisthaanathil prasthaavikkaanum settu sidhaanthathil theliyikkaanum kazhiyunna sidhaanthangalundu, avaa peeno ganithathil "theliyikkanaavilla". fermaattinte avasana sidhaantham oru saadhyamaaya udaaharanamaanu.
poornnasamkhyakale peeno aaksiyomukale thripthippeduthunna settukalaayi nirvachikkunnath settu sidhaanthathinullile peeno ganithatthinte oru maathruka nalkunnu. oru pradhana anantharafalam, settu sidhaantham sthiramaanenkil (saadhaaranayaayi oohikkunnathupole), peeno ganithashaasthram sthiramaanu ennathaanu. mattoru vidhathil paranjaal, oru vairudhyam peeno ganithathil theliyikkaan kazhiyumenkil, settu sidhaantham parasparavirudhamaayirikkum, koodaathe settu sidhaanthathinte oro sidhaanthavum shariyum thettum aayirikkum. 
peeno svayamsidhathakal.
anchu peeno sidhaanthangal inipparayunnavayaanu
iva payano prasidheekaricha yadhaartha soothravaakyangalalla, marichu addehathinte bahumaanaartham namakaranam cheyyappettavayaanu. peeno svayamsidhavaakyangalude chila roopangalkku 0 nuu pakaram 1 undu. saadhaarana ganithathil, soothravaakyathinte pingamiyaaya formula _ 20 formula _ 23 aanu.
settu-theoritic nirvachanam.
avabodhajanyamaayi, moolakangalulla ellah settukaludeyum pothu swathaanu swaabhaavika samkhya. athinaal, "onnu muthal onnu vareyulla kathidapaadukalil nirmmikkaan kazhiyum" enna bandhathinu keezhil oru thulyatha klaas aayi nirvachikkunnath swaabhaavikamaanennu thonnunnu. nirbhagyavashaal, ithu settu sidhaanthathil pravarthikkunnilla, kaaranam atharamoru thulyatha klaas oru settu aayirikkilla (rassalinte vairudhyam kaaranam). swaabhaavika samkhya ennu vilikkappedunna moolakangalulla oru prathyeka koottatthe nirvachikkuka ennathaanu saadhaarana pariharam.
1916-ll sermeloyude prasidheekarikkappedaathu kruthiyaanu eee ashayathe levi attribute cheiyunnathenkilum, thaazhepparayunna nirvachanam aadyamaayi prasidheekarichathu joonn voonn neuman aanu. eee nirvachanam oordinal nambarinte nirvachanamaayi ananthamaaya settilekku vyaapikkunnathinal, chuvade pariganikkunna settukale chilappol voonn neuman oordinalukal ennu vilikkunnu.
nirvachanam inipparayunna reethiyil thudarunnu
swaabhaavika samkhyakale inipparayunna reethiyil aavarthichu nirvachikkunnu
swaabhaavika samkhyakal peeno soothrangale thripthippeduthunnuvennu parisodhikkaam.
eee nirvachanam upayogichu, oru swaabhaavika samkhya nalkiyaal, "oru koottatthinu gadakangalundu" enna vaakyathe aupachaarikamaayi nirvachikkaan kazhiyum "muthal muthal vare oru bijection nilavilundu. ithu gadakangale "ennunna" pravartthanatthe aupachaarikamaakkunnu. koodaathe, oru upavibhaagamaanengil maathram. mattoru vidhathil paranjaal, settu inclusion swaabhaavika samkhyakalil saadhaarana motham kramatthe nirvachikkunnu. eee oordar oru nalla oordaraanu.
oro swaabhaavika samkhyayum athinekkaal kurravulla ellah swaabhaavika samkhyakaludeyum koottatthinu thulyamaanenna nirvachanathil ninnu ithu pinthudarunnu. eee nirvachanam, ananthamaaya samkhyakal ulppede ellah oordinal nambarukalum nirvachikkunnathinulla oordinalukalude voonn neuman nirvachanathilekku vyaapippikkam "oro oordinalum ellah cheria oordinalukaludeyum nannayi oordar cheytha koottamaanu."
oraal ananthathayude soothravaakyam amgeekarikkunnillenkil, swaabhaavika samkhyakal oru koottam roopappeduthanamennilla. ennirunnaalum, swaabhaavika samkhyakale ippozhum mukalil paranjathupole vyakthigathamayi nirvachikkaan kazhiyum, avaa ippozhum peeno soothrangale thripthippeduthunnu.
mattu settu saidhaanthika nirmmithikalund. prathyekichum, ernst sermelo innu charitraparamaaya thaalpparyamullathum chilappol 'ennu vilikkappedunnathumaaya oru nirmmaanam nalki. shoonyamaaya settu ennu nirvachikkunnathil ithu adangiyirikkunnu, oppam.
eee nirvachanathode oro swaabhaavika samkhyayum oru singleton settaanu. athinaal, kaardinaalittikale prathinidheekarikkunna swaabhaavika samkhyakalude property nerittu axus cheyyaan kazhiyilla; oordinal property (oru shreniyude onnaamathe ghadakam enna nilayil) maathramaan udanadi. voonn nyoomaante nirmmaanatthil ninnu vyathyasthamayi, sermelo oordinalukal ananthamaaya oordinalukal vare vyapikkunnilla.

