chaturthansh

ganitamadhye, chaturthansh sankhya pranali jatil sankhyancha vistar karate. ayrish ganitshastradnya william rovan hamilton yanni 1843 madhye pratham chaturthanshanche varnan kele aanee trimiteey jaaget yantrikila lagoo kele. hamiltonne trimiteey jaaget done "nirdeshit reshancha" bhagfal mhanoon kinwa, samatulya, done vectorcha bhagfal mhanoon chaturthanshachi vyakhya kelee. chaturthanshancha gunakar haah gair-vinimayatmak asato.
chaturthansh samanyatah swarupat darshavile jaatat
sootra _ 1
jethe, aanee tyaa vastavik sankhya aahet; aanee, aanee 'besis vector' kinwa 'besis element' aahet.
shuddha ganitamadhye chaturthansh vaparale jaatat, parantu upayojit ganitamadhye, visheshtah trimiteey sanganakeey graphics, sanganakeey drushti aanee sphatikshastriy rachana vishleshan yasarkhya trimiteey paribhramanancha samavesh asalelya gananesathi dekhil vyavharik upayog aahet. tyancha vapar euler coone aanee rotation matrixsarkhya rotationchya itar paddhatinsobat kinwa anuprayoganusar tyanna paryay mhanoon keela jaau shakato.
aadhunik ganiteey bhashet, chaturthansh heey vastavik sankhyanvar chaar-aayami sahyogi manak vibhajan bijganit tayar karatat aanee mhanoon eka ring, vibhajan ring aanee domain donhi asate. chaturthanshanche bijganit anekada (hamiltonsathi) darshavile jaate, kinwa blackboardmadhye thalak sootra _ 2 dware darshavile jaate, heey clifford bijganit vargikaran sootra _ 3 dware dekhil dile jaau shakate, kharey tarr, heey shodhale gelele pahile gair-vinimayatmak vibhajan bijganit hote.
frobenius prameyanusar, bijganitache sootra _ 4 heey keval done maryadit-aayami vibhajan ringspacky eka aahe jyamadhye vastavik sankhyansathi yogya suburing samaroop aahe; dusare mhanje jatil sankhya. yaa rings dekhil euclidean herwitz algebra aahet, jyapaiki chaturthansh heey sarvat mothe sahyogi bijganit (aanee mhanunach sarvat mothe ring) aahet. chaturthanshancha aanakhi vistar kelyane gair-sambandhit ashtak utpann hotat, jeey vastavik sankhyanvarche shevatche manak vibhajan bijganit aahe. (octonianscha vistar asalelya sedeniansana shoonya vibhajak asatat aanee tyamule teey eka manak vibhajan bijganit asoo shakat nahee.)
ekak chaturthanshancha vichar 3-golavareel gut sanrachanechi nivad mhanoon keela jaau shakato, joou gatala firki (3) deto, joou suu (2) sathee samaroop aahe aanee tasech (3) chya sarvatrik aavaranasathi dekhil aahe.
itihas.
1843 madhye hamiltonne chaturthanshanchi olakh karoon dilee. yaa karyachya mahattvachya purvavartimadhye yoolarchi chaar-chauras olakh (1748) aanee olinde rodriguesche chaar mapdandandware (1840) samanya paribhramananche mapdand samavisht hote, parantu yapaiki konatyahi lekhakane chaar-mapdand firanyala bijganit manale nahee. carl friedrich goss hyane 1819 madhye chaturthanshancha shodh lavala hota, parantu heey kaam 1900 paryant prakashit jhaale navhate.
hamiltonla maheet hote qee jatil sankhyancha arth samatalatil bindu mhanoon lavala jaau shakato aanee too trimiteey jaaget gunansathihi asach marg shodhat hota. antaralatil bindu tyanchya nirdeshankadware darshavile jaau shakatat, jeey sankhyanchya tippat aahet aanee anek varshanpasun tyanna sankhyanchya tippatanchi bharr aanee vaja kashee karayachi heey mahit hote. matra, matra kalapasoon teey gunakar aanee vibhajanachya samasyevar adakale hote. antaralatil done bindunchya nirdeshankanchya bhagachi ganana kashee karayachi heey tyala samajoo shakale nahee. kharey tarr, ferdinand george frobeniusne nantar 1877 madhye siddha kele qee vastavik sankhyanvaril vibhajan bijganit maryadit-aayami aanee sahyogi honyasathi, teey trimiteey asoo shakat nahee aanee asey fakt teen vibhajan algebra aaheth sootra _ 5 (jatil sankhya) aanee sootra _ 4 (chaturthansh) jyanche pariman anukrame 1,2 aanee 4 aahet.
quaterniencemadhil mothee pragati shevati somwari 16 aktoobar 1843 roji doublinmadhye jhaali, jevha hamilton parishadechya baithakichya adhyakshatesathi royal ayrish akadameela jaat hota. jevha too tyachya patnisah shahee kalvyachya topthavarun chaalat hota, tevha tyachya manat chaturthanshanmagil sankalpana aakar ghet hotya. jevha tyala uttar midale, tevha hamilton chaturthanshanche sootra koranyachya ichchecha pratikar karoo shakala nahee.
sootra _ 7
too tyavar thambla tevha too brome pulachya dagadat. jaree korivkam tevhapasoon lupt jhaale asale taree 1989 pasoon hamiltonchya shodhachya smaranarth dunsink vedhshalepasun royal canal brijparyant brijparyant shastradnya aanee ganitshastradnyansathi hamilton walk navachi varshik tirthayatra suru aahe.
prakare divashi hamiltonne tyacha mitra aanee sahkari ganitshastradnya johnn tii. yanna eka patra lihile. kabari, jyaa vicharanchya pravahamule tyacha shodh lagla tyache varnan karatat. heey patra nantar london, edinberg aanee doublin philosophical masika aanee journal auph sciencela lihilelya patrat prakashit karnyat aale; hamilton mhanatoh
hamiltonne gunakarachya yaa niyamansah chaupatala 'chaturthansh' mhatale aanee tyanni tyanche urvarit ayushya tyancha abhyas aanee shikshanasathi samarpit kele. hamiltonchi upcharpaddhati aadhunik drushtikonapeksha adhik bhaumitik aahe, jee chaturthanshanchya bijganitiy gunadharmanvar bharr dete. tyanni 'chaturthankiyavadi' yaa shalechi sthapana kelee aanee tyanni anek pustakanmadhye chaturthankiyanna lokpriy karnyacha prayatna keela. elements auph quaternians' hii tyanchi shevatchi aanee sarvat lamb pustake 800 pananchi hotee; tee tyanchya mulane sampadit kelee hotee aanee tyanchya mrutyunantar lagechach prakashit jhaali hotee.
hamiltonchya mrutyunantar, scottish ganiteey bhautikshastradnya pieter tett heey chaturthanshanche mukhya pratipadak banale. yaa velee, doublinemadhye chaturthansh haah eka anivarya parikshecha vishay hota. bhautikshastra aanee bhoomititil jyaa vishayanche aata vector vaparoon varnan kele jaeel, jase qee avkashatil kinematics aanee maxwellchi samikarane, tyanche varnan purnapane chaturthanshanchya drushtine kele gele hote. quaternians aanee itar hypercomplex sankhya pranalinchya abhyasasathi samarpit quaternians society hii eka vyavsayik sanshodhan sanstha dekhil hotee.
1880 chya dashakachya madhyapasoon, chaturthansh vector vishleshanadvare visthapit houu lagale, jeey joshiya willard gibbs, oliver havicide aanee herman wonn helmholtz yanni vikasit kele hote. vector vishleshanane chaturthanshansarkhyach ghatananche varnan kele aahe, mhanoon tyane chaturthanshanvaril sahityatun kahee kalpana aanee shabdavali udarpane ghetali aahe. tathapi, vector vishleshan sankalpanatmakdrushtya sope aanee sanketikdrushtya swachchh hote aanee akheris chaturthanshanna ganeet aanee bhautikshastratil kirkol bhoomikesathi utaravanyat aale. hya sankramanacha eka dushparinam mhanje hamiltonche kaam anek aadhunik vachakanna samajoon ghene katheen aahe. hamiltonchya mool vyakhya aparichit aahet aanee tyachi lekhanshaili shabdayukt aanee anusaran karane katheen hote.
tathapi, 20 vya shatkachya uttaradhapasun, pramukhyane avkashiy paribhramananche varnan karnyatil tyanchya upayuktatemule, chaturthanshanche punarujjivan jhaale aahe. matrixdware kelelya pratinidhitvanpeksha chaturthanshandware firanyachya pratinidhitvanchi ganana adhik sankshipt aanee jalad asate. yavyatirikt, yular konanchya ulat, teey "gimble lock" sathee sanvedanshil nasatat. yaa karanastav, sanganakeey graphics, sanganakeey drushti, yantrashastra, niyantran siddhant, sanket prakriya, vrutti niyantran, bhautikshastra, jaiv-mahitishastra, aanvik gatishilata, sanganakeey anukaran aanee kakshiy yantrikimadhye chaturthansh vaparale jaatat. udaharanarth, antaral yanachya vrutti niyantran pranalinna chaturthanshanchya sandarbhat aadesh dene samanya aahe. chaturbhujanchya swarupanshi asalelya sambandhanmule, chaturthanshanna sankhya siddhantatun aanakhi eka chalana midali aahe.
bhautikshastratil chaturthansh.
pii.aar. girard yanchya 1984 chya nibandhat 'daad quaternian group andd modern physics' madhye bhautikshastratil quaternianchya kahee bhoomikanvar charcha kelee aahe. aadhunik bijganitatil chaturthansh gatashi, vividh bhautik sahparivartan gut, mhanje laurentz gut, sapekshata gatacha samanya siddhant, clifford bijganit aanee anuroop gut kase sahajpane sambandhit asoo shakatat heey nibandh darshavito. girardne gut pratinidhitvanvar charcha karoon aanee sphatikshastrachya kahee avkash gatanche pratinidhitva karoon suruvat kelee. too kathor sharirachya gatishastrashi pudhe gela. tyanantar tyane thomas precessions vishesh sapekshatechya laurentz gatache pratinidhitva karnyasathi jatil chaturthansh (dvi-chaturthansh) vaparale. tyanni ludwick silbersteinpasoon suruvat karoon paach lekhakancha hawala deela, jyanni ekach vibhedak samikaranat maxwellchi samikarane vyakt karnyasathi eka chaturthansh chalche sambhavya karya vaparale. samanya sapekshatebabat, tyanni rung-lenz vector vyakt keela. clifford bijganitache udaharan mhanoon tyanni clifford biquaterniancha (vibhajan-sarvavyapi) ullekh keela. shevati, dwaitsambandhachya parasparsanvadache aavahan karat, girardne antaralkalavaril anuroop nakashanche varnan kele. pannas sandarbhanmadhye, girardne alekzandar mcfarlane aanee tyachya chaturthansh samajache 'bulletin' samavisht kele. 1999 madhye tyanni dakhavun dile qee einsteinchi samanya sapekshatechi samikarane clifford bijganitemadhye kashee tayar kelee jaau shakatat jee thet chaturthanshanshi jodaleli aahet.
1924 chya shodhanusar, quantum yantrikimadhye electron aanee itar dravya kananchya (spinners mhanoon olakhalya jaanarya) firkiche varnan chaturthansh vaparoon kele jaau shakate (prasiddha pauli firki matrixchya swarupat); chaturthanshanni tyanchi aavad vadhavali; 360° neey electronchi firki 720° neey kashee olakhali jaau shakate heey samajoon ghenyas chaturthanshanni madat kelee. tyanchya vaparane firatya gatanna mage takale nahee.
vyakhya.
aaa "chaturthansh" hii rupachi abhivyakti aahe
sootra _ 8
jethe,,,,, vastavik sankhya aahet aanee,,, hii chinhe aahet jyancha arth teen avkashiy akshanvar nirdeshit ekak-vahak mhanoon lavala jaau shakato. vyavharat, jarr,,, 0 asel tarr sambandhit sandnya vagalali jaate; jarr,,, serve shoonya asateel tarr, chaturthansh haah "shoonya chaturthansh" aahe, joou 0 darshavila jaato; jarr eka, 1 chya barobaricha asel tarr sambandhit sandnya fakt, kinwa lihili jaate.
hamiltonne chaturbhuj sootra _ 9 chey varnan kele aahe, jyamadhye scalar bhag aanee vector bhag asato. 10' yaa chaturthansh sutrala 'vector bhag' (kadhikadhi 'kalpanik bhag') mhanatat aanee too 'scalar bhag' (kadhikadhi 'vastavik bhag') asato. jyaa chaturthanshala tyachya vastavik bhagachya barobarine (mhanje tyacha vahak bhag shoonya aahe) mhatale jaate tyala 'scalar' kinwa 'vastavik chaturthansh' mhanatat aanee tyala sambandhit vastavik sankhyes olakhale jaate. mhanjech, vastavik sankhya chaturthanshanmadhye 'antarbhut' aahet. (adhik yogyaritya, vastavik sankhyanche kshetra chaturthanshanchya upasanchasathi samaroop aahe. jatil sankhyanche kshetra dekhil chaturthanshanchya teen upasanchansathi samaroop aahe.) tyachya vector bhagachya barobaricha chaturthansh jyala 'vector chaturthansh' mhanatat.
chaturthanshancha sanch haah vastavik sankhyanvar 4-aayami vector space aahe, sootra _ 11 aadhar mhanoon, ghatak-vaar jodanidware
sootra _ 12
aanee ghatakanihay scalar gunakar
sootra _ 13
hamilton utpadan' mhanoon olakhali jaanari gunakari gut rachana, jee jukstapojishandware darshavili jaate, tee chaturthankanvar khaleel prakare paribhashit kelee jaau shakateh
asha prakare chaturthansh eka vibhajak bijganit tayar karatat.
moolbhut ghatakancha gunakar.
aadhar ghatkansah gunakar, aanee gunakar olakh yaa vastusthitidvare paribhashit kele jaate, mhanje,
sootra _ 15
itar moolbhut ghatakanchi utpadane ashee aahet
sootra _ 16
yaa niyamancha mell,
sootra _ 17
kendra.
gair-vinimayatmak ringcha "kendra" haah pratyekasathi asha ghatakancha up-ring aahe. chaturthansh bijganitache kendra heey vastavik chaturthanshanche upkshetra aahe. kharey tarr, vastavik chaturthansh heey kendrache aahet haah vyakhyecha eka bhag aahe. yaulat, jarr teey kendrache asel tarr
sootra _ 18
aanee. tyach prakarchi ganana, tyaeivaji, asey darshavate qee ekakade dekhil aahe. tyamule haah eka 'vastavik' chaturthansh aahe.
chaturthansh eka vibhajak bijganit tayar karatat. yacha arth assa qee gunakaratil gair-sampreshakta haah ekmev gunadharm aahe joou chaturthanshanna kshetrapeksha vegale banavato. yaa gair-samyavadache kahee anpekshit parinam aahet, tyapaiki chaturthanshanvaril bahupadi samikaranamadhye bahupadichya anshapeksha adhik bhinn uttare asoo shakatat. udaharanarth, samikaranat anant pramanat anek chaturthansh uttare aahet, jee asha chaturthansh aahet. asha prakare hii-1 chee mule vector chaturthanshanchya trimiteey jaaget eka ekak goal tayar karatat.
hamilton utpadan.
done ghatkansathi aanee tyanche utpadan, jyala hamilton utpadan () () mhanatat, teey aadhar ghatakanchya utpadanandvare aanee vitaran kayadyadvare nirdharit kele jaate. vitaran kayada utpadanacha vistar karane shakya karato jenekarun too moolbhut ghatakanchya utpadananchi berries asel. yamule khaleel abhivyakti milateh
sootra _ 19
aata varr dilelya niyamancha vapar karoon aadhar ghatakancha gunakar keela jaau shakatoh
sootra _ 20
done firatya chaturthanshanche gunakar heey firanyachya nantarchya firanyachya samatulya asel. 
scalar aanee vector bhag.
jyaa rupatil eka chaturthansh, jithe eka vastavik sankhya aahe, tyala scalar mhanatat aanee jyaa rupatil eka chaturthansh, jithe vastavik sankhya aahet aanee kiman eka shoonya nasalela kinwa aahe, tyala vector chaturthansh mhanatat. jarr ekhada chaturthansh asel tarr tyala tyacha scalar bhag mhanatat aanee tyala tyacha vector bhag mhanatat. jaree pratyek chaturthanshakade chaar-aayami vector jaaget vector mhanoon pahile jaau shakate, taree tri-aayami jaaget vector bhagacha vector mhanoon sandarbh dene samanya aahe. yaa paramparenusar, vector heey vector space sootra _ 21 chya ghatakapramanech aahe.
hamiltonla vector chaturthansh aanee vastavik sankhya (shoonya vector bhagasah chaturthansh manale jaanare) scalar chaturthansh asehi mhanatat.
jarr chaturthansh heey scalar bhag aanee vector bhagat vibhagale gele asel, mhanje,
sootra _ 22
mug jodani aanee gunakarasathichi sutre ashee aahet
sootra _ 23
jethe "sootra _ 24" aanee "sootra _ 25" anukrame bindu utpadan aanee cross utpadan darshavtat.
sanyugman, niyam aanee paraspar.
chaturdhatukanche sanyugman heey jatil sankhyanchya sanyugman aanee clifford algebranchya ghatakanchya sthanantaranashi (jyala ulatya mhanoon dekhil olakhale jaate) samaan aahe. tyachi vyakhya karnyasathi, sootra _ 26 heey chaturthansh asoo dya. yache sanyugma heey chaturbhuj sootra _ 27 aahe. teey "qt", sootra _ 28, kinwa dware darshavile jaate. sanyugman haah eka antarvartan aahe, yacha arth too tyacha swatacha vyutkram aahe, mhanoon ekhadya ghatakache donada sanyugman kelyane mool ghatak parat midato. done chaturthanshanchya utpadanache sanyugma heey sanyugmanche "ulat kramane" utpadan asate. mhanje, jarr aanee chaturthansh asateel, tarr nahee.
chaturdhatukanche sanyugman, jatil mandanichya agadi ulat, gunakar aanee chaturdhatukanchya jodanisah vyakt kele jaau shakateh
sootra _ 29
sanyugman haah chaturthanshache scalar aanee vector bhag kadhanyasathi vaparla jaau shakato. yacha scalar bhag aahe aanee vector bhag aahe.
tyachya sanyugmasah chaturthanshchya utpadanachya vargamoolala tyache 'manak' mhanatat aanee teey darshavile jaate (hamiltonne yaa pramanala 'kyu' chaa 'tensor' mhatale, parantu heey 'tensor' chya aadhunik arthashi sangharsh karate). sutranmadhye heey khalilpramane vyakt kele jaateh
sootra _ 30
hii nehmich eka nakaratmak nasaleli vastavik sankhya asate aanee tee vector space sootra mhanoon manalya manalya sootra 4 vareel euclidean normsarkhich asate. vastavik sankhyene chaturthansh gunakar kelyane tyachya normla sankhyechya paripoorn moolyane mojale jaate. mhanje, jarr teey kharey asel tarr
sootra _ 33
heey eka vishesh prakaran aahe qee manak "gunak" aahe, mhanje
sootra _ 34
konatyahi done chaturthanshansathi aanee. gunakar haah utpadanachya sanyugma sutracha parinam aahe.
paryayipane teey olakhitoon anusarate
sootra _ 35
(jethe nehamiche kalpanik ekak darshavate) aanee mhanoon varg matrixchya nirdharkanchya gunakar gunadharmanpasun.
yaa nikashamule aanee tyanchya farkacha niyam mhanoon antar paribhashit karane shakya hoteh
sootra _ 36
yamule sootra 4 hii metrik jaaga banate.
sambandhit metrik topologychya sandarbhat jodani aanee gunakar satat asatat.
heey vastavik sankhyanchya sutrasathichya agadi tyach puravyasah yete, sootra 4 heey eka pramanit bijganit aahe.
ekak chaturthansh.
ekak chaturthansh haah eka chaturthansh aahe. shunyetar chaturthanshala tyachya mankane vibhajit kelyane eka ekak chaturthansh tayar hoto jyala 'versor' mhanatatah
sootra _ 40
pratyek shunyetar chaturthanshache eka adviteey dhruviy vighatan sootra _ 41 asate, tarr shoonya chaturthansh konatyahi ekak chaturthanshapasun tayar kele jaau shakate.
sanyugman aanee manak vaparoon shunyetar chaturthanshache paraspar paribhashit karane shakya hote. tyachya parasparasah chaturthankache gunakar 1 chya barobariche asave aanee vareel vicharancha arth assa aahe qee sootra _ 42 aanee sootra _ 43 chey gunakar 1 aahe (gunakarachya donhi kramansathi). mhanoon 'paraspar' ashee paribhashit kelee jaate
sootra _ 44
yamule done chaturthansh aanee done vegvegalya prakare (jevha shunyetar asel) vibhajit karane shakya hote. mhanje, tyancha bhag ektar asoo shakato kinwa; sarvasadharanpane, gunakar kramanusar tee utpadane bhinn asatat, vishesh prakaran vagalata qee aanee ekmekanche scalar gunakar aahet (jyamadhye prakaran samavisht aahe). tyamule, sanketan sandigdha aahe kaaran teey davya kinwa ujavya bajula vibhajit hote qee nahee heey nirdisht karat nahee (tyachya davya kinwa ujavya bajula gunakar hote qee nahee).
bijganitiy gunadharm.
serve chaturthanshanche sanch sootra _ 4 heey pariman 4 asalelya vastavik sankhyanvar vector space aahe. chaturthanshancha gunakar haah sahyogi asato aanee vector jodanivar vitarit hoto, parantu scalar upasanch vagalata, too parivartaneey nasato. mhanoon, chaturthansh sootra _ 4 heey vastavik sankhyanvarche gair-parasparsanvadi, sahyogi bijganit aahe. jaree sootra 4 madhye jatil sankhyanchya pratee asalya taree, teey jatil sankhyanvaril sahyogi bijganit nahee.
kaaran chaturthanshanche vibhajan karane shakya aahe, tyamule teey vibhajak bijganit tayar karatat. gunakaratil gair-samyavad vagalata hii kshetrasarkhich rachana aahe. vastavik sankhyanvar maryadit-aayami sahyogi vibhajan algebra atishay durmil aahet. frobenius pramey asey sangato qee nemke teen aaheth sootra _ 38, sootra _ 49 aanee sootra _ 4. heey manak chaturthanshanna eka manak bijganit banavate aanee vastavik sankhyanvar manak vibhajan algebras dekhil khoop durmil aaheth herwitzche pramey sangate qee keval chaar aaheth sootra _ 38, sootra _ 49, sootra _ 4 aanee sootra _ 54 (octonians). chaturthansh heey rachana bijganit aanee eksandh banach bijganitache dekhil eka udaharan aahe.
kaaran konatyahi done besis vectorche gunakar adhik kinwa vaja dusra besis vector asalyane, sanch gunakarantargat eka gut tayar karato. yaa gair-abelian gatala chaturthansh gut mhanatat aanee too darshavila jaato. vastavik samooh ring hii ring sootra 55 aahe jee sootra 56 varr aath-aayami vector jaaga dekhil aahe, sootra 57 chya pratyek ghatkasathi eka besis vector aahe, aanee ghatkandware tayar kelelya aayadiyaldware sootra 55 chya bhagakar ringsathi chaturthansh samaroop aahet, aanee. yethe pratyek farkanmadhil pahili sandnya hii eka moolbhut ghatak aahe aanee dusari sandnya hii eka moolbhut ghatak aahe, aanee, aanee chya yojak vyutkram nahee.
chaturthansh aanee antaral bhoomiti.
chaturthanshchya vector bhagacha arth sootra _ 59 madhye samanvay vector mhanoon lavala jaau shakato, mhanoon chaturthanshanchya bijganitiy karye sootra _ 21 operationschi bhoomiti pratibimbit karatat, jase qee vector dott aanee cross productschee vyakhya chaturthanshanchya drushtine kelee jaau shakate aanee yamule jithe jithe avkashiy vector udbhavtat tithe chaturthansh tantre lagoo karane shakya hote. sanganakachya chitrakaletil qee-framechya abhimukhata prakshepit karane haah chaturthanshancha eka upayukt vapar aahe.
yaa vibhagachya urvarit bhagasathi,, aanee sootra _ 4 chey teen kalpanik aadhar vahak aanee sootra _ 21 sathee aadhar darshavel, jyachya jaagi, dware aanee bhektarla tyachya yojak ulat pathavate, mhanoon vectorcha yojak ulat tyachya sanyugma samaan aahe. yaa karanastav, sanyugmanala kadhikadhi "avkashiy ulat" mhanatat.
done vector chaturthansh aanee tyanchya bindu utpadanasathi, sootra _ 63 madheel vectorshee sadrushyane aahe
sootra _ 64
heey ghatak-mukt paddhatine dekhil vyakt kele jaau shakate
sootra _ 65
heey utpadananchya scalar bhaganchya barobariche aahe. tyanche vector bhag bhinn aahet heey lakshat ghya.
kramanusar aadharadvare nirdharit kelelya abhimukhatache aanee tyachyashi sambandhit mishra utpadan, aanee
sootra _ 66
(lakshat thewa qee chinh nishchit karnyasathi abhimukhata aavashyak aahe.) heey utpadanachya vector bhagachya (chaturthansh mhanoon), tasech utpadanachya vector bhagachya samaan aahe. tyache sutrahi aahe
sootra _ 67
commutatorsathi, done vector chaturthanshanpaiki eka prapt hoto
sootra _ 68
sarvasadharanpane, chala aanee chaturthansh houya aanee liha
sootra _ 69
scalar bhag kuthe aanee kuthe aahet aanee aanee aanee chey vector bhag aahet. mug aamchyakade sootra aahe
sootra _ 70
yavarun asey disun yete qee chaturthansh gunakaratil gair-samyavad vector chaturthanshanchya gunakaratoon yeto. heey asehi darshavite qee done chaturthansh jarr aanee keval tyanche vahak bhag samarekheey asateel tarach pravas karatat. hamiltonne dakhavun dile qee heey utpadan golakar trikonachya karane) shirobinduchi ganana dilelya done shirobindu aanee tyanchya sambandhit kamani-lambivarun karate, jeey lambvartulakar bhoomititil bindunche bijganit dekhil aahe.
ekak chaturthansh heey sootra _ 71 madheel paribhramananni olakhale jaau shakatat aanee hamiltondware tyanna versorsors mhatale jaate. chaturthansh vaparoon trimiteey paribhramananchya modelingbaddal adhik mahitisathi chaturthansh aanee avkashiy paribhraman dekhil pahaa.
chaturthanshanchya drushyasathi hanson (2005) pahaa.
matricks pratinidhitva.
jyapramane jatil sankhya matricks mhanoon darshavalya jaau shakatat, tyachpramane chaturthansh dekhil darshavile jaau shakatat. matricks mhanoon chaturthanshanche pratinidhitva karnyache kiman done marg aahet, jenekarun chaturthansh jode aanee gunakar heey matricks jode aanee matricks gunakar yanchyashi sambandhit asateel. eka mhanje 2x2 jatil matricks vaparane aanee dusare mhanje 4x4 vastavik matricks vaparane. pratyek prakaranat, dilele pratinidhitva heey resheey sambandhit pratinidhitvanchya kutumbatil eka aahe. amoort bijganitachya paribhashet, heey anukrame sootra _ 4 teey matricks ringsparyantche injective homomorphism aahet.
2 × 2 jatil matricks vaparoon, chaturthansh asey darshavile jaau shakate
lakshat ghya qee jatil sankhyanche 'i' heey chaturthansh sankhyanchya 'i' peksha vegale aahe.
yaa pratinidhitvat khaleel gunadharm aaheth
4 × 4 vastavik matricks vaparoon, tyach chaturthanshala asey lihile jaau shakate
sootra _ 75
tathapi, yaat chaturthankanche pratinidhitva adviteey nahee. udaharanarth, tyach chaturthanshache pratinidhitva dekhil kele jaau shakate
sootra _ 76
yaa swarupachi 48 bhinn matricks pratinidhye astitvat aahet jyamadhye matrixpacky eka scalar bhag darshavite aanee itar teen serve tirkas-samamiteey aahet. adhik tantotantpane, yaa samamiti nirbandhansah matrixchya chaupatanche 48 sanch aahet, jase qee karya pathvane aanee chaupatatil matrixla samarupata aahe, mhanje teey matrixchya berries aanee utpadananna chaturthanshanchi berries aanee utpadane pathavate. 
yaa sadarikaranat, chaturthankache sanyugma heey matrixchya transposeshy sambandhit aahe. chaturthanshchya mankatil chauthi shakti hii sambandhit matrixchi nirdharak asate. vareel 2x2 jatil pratinidhitvapramane, gunankanna yogya prakare maryadit karoon punha jatil sankhya tayar kelya jaau shakatat; udaharanarth, done 2x2 blocks block karn matricks mhanoon settingdware.
chaturthankanche pratyek 4 × 4 matricks pratinidhitva ekak chaturthankanchya gunakar saranishi sambandhit aahe. udaharanarth, varr dilele shevatche matricks pratinidhitva gunakar taktashi sambandhit aahe.
jeey samaroop aahe-sutradvare _ 77-teey
asha konatyahi gunakar taktyala pahilya ranget aanee stambhat olakh patavanyasathi aanee panktichya shirshakanchi chinhe stambh shirshakanchya viruddha asanyasathi pratibandhit karane, nantar prakare stambhasathi 3 sambhavya paryay aahet (chinhakade durlaksh karane), karane) stambhasathi 2 sambhavya paryay (chinhakade durlaksh karane) aanee chauthya stambhasathi 1 sambhavya paryay (chinhakade durlaksh karane); jyamule 6 shakyata nirman hotat. tyanantar, dusra stambh ektar sakaratmak kinwa nakaratmak mhanoon nivadala jaau shakato, tisra stambh sakaratmak kinwa nakaratmak mhanoon nivadala jaau shakato aanee chautha stambh sakaratmak kinwa nakaratmak mhanoon nivadala jaau shakato, jyamule chinhasathi 8 shakyata aahet. aksharanchya sthitisathi aanee tyanchya chinhansathichya shakyatancha gunakar kelyas 48 milate. nantar, saha, saha aanee saha aanee pankti aanee stambh shirshake kadhoon takalyas matrixche pratinidhitva milate.
lagrangeche chaar chauras pramey.
sankhya siddhantatil lagrangechya chaar-varg prameyatil puravyanpaiki eka puravyat dekhil chaturthansh vaparale jaatat, jyamadhye asey mhatale aahe qee pratyek nakaratmak nasalela purnank haah chaar purnank varganchi berries aahe. tasech swatachya drushtine eka mohak pramey asalyane, lagrangechya chaar chauras prameyala sankhyechya siddhantachya bahereel ganitachya kshetranmadhye upayukt anuprayog aahet, jase qee sanyoji rachana siddhant. quaternion-aadharit purava herwitz quaternienscha vapar karato, serve quaterniencechya ringchi eka upparichhedika jyasathi euclidean algoridamche analogue aahe.
jatil sankhyanchya jodya mhanoon chaturthansh.
chaturdhatukanna jatil sankhyanchya jodya mhanoon darshavile jaau shakate. yaa drushtikonatun, chaturdhatuk heey jatil sankhyanvar callie-dickson rachana lagoo karnyacha parinam aahet. vastavik sankhyanchya jodya mhanoon jatil sankhyanchya rachaneche heey samanyikaran aahe.
sootra 78 heey jatil sankhyanvar dvi-aayami vector space asoo dya. done ghatakancha samavesh asalela aadhar nivada aanee. sootra _ 78 madheel vector heey aadhar ghatakanchya sandarbhat lihile jaau shakate aanee
sootra _ 80
jarr aapan paribhashit kele aanee, tarr aapan vitaran kayadyacha vapar karoon done vector gunakar karoo shakato. utpadanasathi sankshipt sanketan mhanoon vaparane heey nehamichya chaturthanshanpramanech gunakarasathi samaan niyamankade netat. mhanoon, vareel jatil sankhyancha vector chaturthanshashi sambandhit aahe. jarr aapan sootra 78 chey ghatak krambaddha jodya aanee chaturthansh chaupat mhanoon lihile, tarr patravyavahar aahe
sootra _ 82
chauras mule.
- 1 chey vargamoole.
_ 83 yaa jatil sankhyanchya sutramadhye fakt done sankhya aahet, "i" aanee-"i", jyancha varg −1 aahe. sootra _ 4 madhye vaja ekchi anant vargamoole aaheth heey pahanyasathi −1 chya vargmoolasathi chaturthansh samadhan heey sootra _ 21 madheel ekak goal aahe, eka chaturthansh asoo dya aanee tyacha varg −1 aahe asey gruheet dhara.
sootra _ 86
shevatchi teen samikarane, ektar kinwa, aanee serve 0 aahet, tee purn karane. nantarche ashakya aahe kaaran "a" hii eka vastavik sankhya aahe aanee pahile samikaran soochit karel qee mhanoon, aanee prakare shabdantah jarr aanee keval jarr teey manak 1 saha vector chaturthansh asel tarr chaturthansh varg −1 paryant asato. vyakhyenusar, asha serve vectorcha sanch ekak gola tayar karato.
keval nakaratmak vastavik chaturthanshanna anant vargamoole asatat. itar sarvankade fakt done aahet (kinwa 0 chya babatit eka).
jatil vimananche ektrikaran mhanoon.
−1 chya vargmoolanchi pratyek prati-pratikruti jodee chaturthanshanchya aat asalelya jatil sankhyanchi eka vegali prat tayar karate. jarr nantar prat karyadvare nirdharit kelee jaate
sootra _ 87
heey sootra _ 49 teey sootra _ 89 paryantche eka injective ring homomorphism aahe jeey sootra _ 90 pasoon tyachya pratimevar feeled isomorphism paribhashit karate. aanee-shhee sambandhit embeddingchya pratima sarkhyach aahet.
pratyek avastavik chaturthansh, chaturthanshanche eka up-bijganit tayar karate jeey sootra _ 83 chey samaroop aahe aanee asha prakare sootra _ 92 chey samtol upkshetra aahe, tyachya scalar bhagachi aanee tyachya vector bhagachi berries mhanoon lihah
sootra _ 93
vector bhag tyachya manak aanee tyachya viruddhache utpadan mhanoon pudhe vighatit karah
sootra _ 94
(lakshat ghya qee heey sootra 95 sarkhe nahee.) sootra 96 chya vector bhagacha versor,-1 saha tyacha varg mhanoon yogya versor aahe. eka saral padtalani darshavate qee
sootra _ 97
49 yaa sutrapasoon chaturthanshanmadhye pramanit algebranche injective homomorphism paribhashit karate. yaa samarupatekhali, complexes number formula _ 99 chee pratima aahe.
sootra 4 heey yaa serve samarupatanchya pratimanche sanyojan asalyane, yamule chaturthanshanna vastavik reshevar reshevar jatil pratalanche millen mhanoon pahanyachi parvanagi milate. yaa pratyek guntaguntichya pratalanmadhye vaja ekchya vargmoolachya golyachya prati-pratikruti bindunchi achook eka jodee asate.
parivartaneey upgolak.
sootra _ 4 chya jatil upatalanmadhye ekmekanshi chaturbhujanche sambandh dekhil olakhale jaau shakatat aanee parivartaneey upgolanchya sandarbhat vyakt kele jaau shakatat. visheshtah, done chaturthansh aanee pravas (i.eae.,) jarr teey sootra _ 4 chya ekach jatil upatalat asateel, tarr jevha konee chaturthansh ringchya serve parivartaneey upagolika shodhanyacha prayatna karato tevha jatil samatalanchya sanyog mhanoon sootra _ 4 chey swaroop udbhavate.
aniyantrit chaturthanshanchi vargamoole.
konatyahi chaturthansh sootra _ 104 (yethe scalar-vectorchya sadarikaranat darshavile aahe) madhye kiman eka vargamool sootra _ 105 aahe jeey samikaran sootra _ 106 sodavate. yaa samikaranatil scalar aanee vector bhagankade swatantrapane pahilyas done samikarane midatat, jee sodavalyavar uttare detat.
sootra _ 107
jethe sootra _ 108 heey sootra _ 109 chey manak aahe aanee sootra _ 110 heey sootra _ 111 chey manak aahe. konatyahi scalar chaturthansh sootra _ 111 sathee, sootra _ 113 chaa arth jarr aniyantrit ekak vector mhanoon lavala gela tarr heey samikaran yogya vargamoole pradan karate.
mhanoon, shunyetar, scalar nasalele chaturthansh kinwa sakaratmak scalar chaturthansh, yanchi achook done mule aahet, tarr 0 chee achook ekach mule (0) aahet aanee nakaratmak scalar chaturthanshanchi anant mule aahet, jee _ 114, i sutravar sthit vector chaturthansh aahet.eae.jethe scalar bhag shoonya aahe aanee vector bhag trijya sootra _ 115 saha 2-golavar sthit aahe.
chaturthansh chalanchi karye.
jatil chalanchya karyanpramanech, chaturthansh chalanchi karye upayukt bhautik namune suchavatat. udaharanarth, maxwellne varnan keleli mool vidyut aanee chumbakeey kshetre hii chaturthansh chalanchi karye hotee. itar karyanchya udaharananmadhye mundelbrott sanch aanee julia sanchancha 4-aayami jaaget vistar samavisht aahe.
ghatankiy, laghuganak aanee shakti karye.
chaturthansh dile,
sootra _ 116
ghatankachi ganana ashee kelee jaate
sootra _ 117
aanee laghuganak aahe
sootra _ 118
tyanantar chaturthankache dhruviy vighatan lihile jaau shakate
sootra _ 119
jethe coone sootra _ 120 aahe
sootra _ 121
aanee unit vector sootra _ 122 hyadvare paribhashit kele aaheh
sootra _ 123
konatehi ekak chaturthansh dhruviy swarupat asey vyakt kele jaau shakateh
sootra _ 124
aniyantrit (vastavik) ghatankat vadhavalelya chaturthanshachi shakti khalilpramane dilee jaateh
sootra _ 125
bhooganitiy niyam.
ekak chaturthanshanmadhil bhooganitiy antar aanee tyachi vyakhya ashee kelee jaateh
sootra _ 126
aanee golakachya mothya kamanidware aanee tyabarobar asalelya konachya ardhya sampoorn mulyachya pramanat aahe.
haah coone laghugankashivay chaturthansh bindu gunakaratoon dekhil asha prakare mojala jaau shakatoh
sootra _ 127
trimiteey aanee chaar-aayami paribhraman gut.
varr dilelya arthavyatirikt, 'sanyugman' yaa shabdacha arth ekhadya ghatkala shunyetar chaturthansh asalelya thikani nene assa dekhil asoo shakato. dilelya ghatkashi sanyugmit asalelya serve ghatakanmadhye (sanyugma shabdachya yaa arthane) vector bhagacha samaan vastavik bhag aanee samaan manak asato. (asha prakare prakare arthane sanyugma haah yaa arthane sanyugmanpaiki eka aahe.) 
asha prakare shunyetar chaturthanshancha gunakar gut sootra _ 71 chya prativar sanyugmandware karya karato jyamadhye shoonyachya barobaricha vastavik bhag asalelya chaturthanshancha samavesh asato. vastavik bhagasah ekak chaturthanshadware (nirpeksh moolya 1 chaa chaturthansh) sanyugman mhanje konatoon firane, firanyacha aksh haah sadish bhagachi disha aahe. chaturthanshanche fayade pudhilpramane aaheth
serve ekak chaturthanshancha (viruddha) sanch gunakarantargat 3-kshetra aanee eka gut (eka khote gut) tayar karato, joou nirdharak 1 chya vastavik orthogonal 3 × 3 matrixchya gatala duppat vyapato kaaran "done" ekak chaturthansh vareel patrvyavaharantargat pratyek paribhramanas anuroop asatat. plate trick pahaa.
versorschya upasamuhanchi pratima haah eka bindu gut aahe aanee yaulat, bindu gatachi purvakalpana haah versorscha upasamuh aahe. maryadit bindu gatachya purvgrahala binary upasargasah tyach navane sambodhale jaate. udaharanarth, dwaiti dwaiti samahalik gut haah dwaiti samahalik gatacha purvagrah aahe.
versorscha gut haah nirdharak 1 chya jatil ekatmak 2 × 2 matrixchya gatasathi samaroop aahe.
ektar serve purnank kinwa serve ardh-purnank asalelya swarupachya chaturthanshancha sanch asoo dya. sanch haah eka ring (kharey tarr eka domain) aanee eka jaali aahe aanee tyala herwitz chaturthanshanchi ring mhanatat. yaa ringmadhye 24 unit quaternians aahet aanee teey niyamit 24 selche shirsh aahet, jyamadhye schlafly chinh aahe, teey niyamit tetrahedronchya rotational samamiti gatachya duheri aavarnashi sambandhit aahet. tyachpramane, sclafly chinh asalelya niyamit 600 peshiche shirobindu heey niyamit icosahedronchya firatya samamiti gatachya duheri aavarnashi sambandhit, ekak icotian mhanoon ghetale jaau shakatat. niyamit octahedronchya firati samamiti gatache duheri aavaran dispanoidal 288-peshichya shirobinduche pratinidhitva pratinidhitva chaturthanshanshi sambandhit aahe.
chaturthansh algebra.
chaturthanshanche pudhil aljebramadhye samanyikaran kele jaau shakate jyala "chaturthansh algebras" mhanatat. 2 peksha bhinn vaishishtya asalele konatehi kshetra ghya aanee tyache ghatak mhanoon; chaar-aayami ekatmak sahyogi bijganit heey aadhar aanee, kuthe aanee (mhanoon) saha paribhashit kele jaau shakate.
aanee chya nivadivar avalambun, chaturthansh algebra heey 2 × 2 matrixchya bijganitashi samaroop asatat kinwa algebrache vibhajan karatat.
summ bhag mhanoon chaturthansh.
bhaumitik gananesathi chaturdhavanchi upayuktata, clifford bijganit sootra 130 chey summ bhag sootra mhanoon chaturdhavke olakhoon itar parimananmadhye samanyikrut kelee jaau shakate, heey utpadan niyam vaparoon moolbhut aadhar ghatakanpasun tayar kelele eka sahyogi bahubhajak bijganit aahe.
sootra _ 131
sootra _ 132
jarr heey moolbhut aadhar ghatak 3d jaaget vectorche pratinidhitva karnyasathi ghetale gele, tarr asey disun yete qee unit vectorla lumb asalelya pratalatil vectorche "pratibimb" lihile jaau shakateh
sootra _ 133
done paravartan done paravartan pratalanmadhil konachya duppat konatoon firavatat, mhanoon
sootra _ 134
σ" 1 aanee "σ" 2 asalelya pratalatil 180° chya paribhramanas anuroop aahe. heey sambandhit chaturthansh sutrasarkhech aahe,
sootra _ 135
kharokhar, done sanrachana sootra _ 129 aanee sootra _ 4 samaroop aahet. eka naisargik olakh aahe
sootra _ 138
aanee hamiltonche sambandh yamule tikoon rahatat yachee pushti karane sope aahe
sootra _ 139
yaa chitrat, tathakathit 'vector chaturthansh' (mhanje, shuddha kalpanik chaturthansh) heey vectorshee navhe tarr dwibhajkanshi sambandhit aahet-pariman aanee 1d 'dishanirdeshanaivaji vishisht 2d' samatalanshi 'sambandhit abhimukhata asalele praman. jatil sankhyanshi asalela sambandh dekhil spasht hotoh 2d madhye, done vector dishansah aanee, fakt eka dvibhajak aadhar ghatak aahe, mhanoon fakt eka kalpanik. parantu keval, teen vector dishansah, teen dvibhajak aadhar ghatak aahet, mhanoon teen kalpana.
haah tark aanakhi vistarato. clifford bijganit sootra _ 140 madhye sahaa dvibhajak aadhar ghatak aahet, kaaran chaar vegvegalya moolbhut vector dishanirdeshansaha, sahaa bhinn jodya aanee mhanoon sahaa bhinn resheey swatantra pratale paribhashit kelee jaau shakatat. asha jaganmadhil rotation, jyala rotter mhanatat, asha chaturthanshanchya samanyikaranancha vapar karoon, sajatiy nirdeshankancha samavesh asalelya anuprayogansathi khoop upayukt tharoo shakatat. parantu keval keval besis bivectorschi sankhya besis vectorschya sankhyeitaki asate aanee pratyek bivectorla seudovector mhanoon olakhale jaau shakate.
yaa vyapak vyavasthemadhye chaturthansh thevanyasathi anek fayade aaheth
clifford algebranchya bhaumitik vaparabaddal adhik tapashilansathi, bhaumitik bijganit pahaa.
broyer gut.
quaternians heey vastavik sankhyanvaril ekmev (kshullak nasalele) kendriya sadhe bijganit (see. ess. aii.) aahet, yaa arthane qee vastavik sankhyanvaril pratyek see. ess. aii. ektar vastavik sankhya kinwa quaterniencechya samatulya aahe. spashtapane, vastavik sankhyanchya brauer gatat done varg asatat, jeey vastavik sankhya aanee chaturthanshandware darshavile jaatat, jethe brauer gut haah serve see. ess. aii. chaa sanch asato, eka see. ess. aii. chya prakarechyyaa varr matricks ring asanyaparyantchya samatulya sambandhaparyant. artin-wederburn prameyanusar (visheshtah, wederburncha bhag), see. ess. aii. heey vibhajan bijganitevaril serve matricks algebra aahet aanee asha prakare chaturthansh heey vastavik sankhyanvarche ekmev alp vibhajan nasalele bijganit aahet.
see. ess. aii. ess.-kshetravaril maryadit aayami rings, jyaa sadhya algebra asatat (kshetranpramanech, kshullak-nasaleli dwimukh ideals nasatat) jyanche kendra agadi kshetra asate-vistar kshetranche gair-sampreshaniy analogue asatat aanee samanya ring vistaranpeksha adhik pratibandhatmak asatat. vastavik sankhyanvar (samatulyateparyant) chaturthansh heey ekmev kshullak nasalele see. ess. aii. aahet yaa vastusthitichi tulana yaa vastusthitishi kelee jaau shakate qee jatil sankhya yaa vastavik sankhyancha ekmev kshullak nasalela maryadit kshetra vistar aahe.

