
dvi-sankul sankhya

amoort bijganitemadhye, dvi-sankul sankhya hii callie-dickson prakriyedvare tayar keleli jatil sankhyanchi jodee aahe jee dvi-sankul sanyugma sootra _ 1 paribhashit karate aanee done dvi-sankul sankhyanche gunakar asey aaheh
mug bicomplexes manak dile jaate
dvi-sankul sankhya, dvi-pariman see varr eka parivartaneey bijganit tayar karatat, jeey algebranchya thet berijsathi samaroop aahe.
done dvi-samamiteey sankhyanchya gunakaramule eka dvighat swaroop moolya milate jeey sankhyanchya vaiyaktik dvighat rupanche gunakar aaheh 
utpadanachya vargakar swarupachya yaa malmattechi padtalani hii brahmagupta-fibonachi olakh darshavate. dvi-samamiteey sankhyechya dvighat swarupacha haah gunadharm soochit karato qee yaa sankhya eka rachana bijganit tayar karatat. kharey tarr, bicomplexes sankhya norm z2 saha sootra _ 4 varr aadharit callie-dixon bandhkamachya dvivruniy staravar udbhavtat.
samanya dvi-sankul sankhya matricks sutradvare darshavili jaau shakate, jyat nirdharak sootra _ 6 aahe. asha prakare, dvighat swarupacha rachana gunadharm nirdharkachya rachana gunadharmanshi sahmat hoto.
dvi-sankul sankhyanmadhye done bhinn kalpanik ekak asatat. gunakar haah sahsanyojak aanee parivartaneey asalyane, yaa kalpanik ekkanchya gunakaramadhye tyachya vargasathi sakaratmak asane aavashyak aahe. yaa utpadanasarkhya ghatkala hyperbolic unit mhatale gele aahe.
vastavik bijganit mhanoon.
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r varr tessarine 4-bijganitacha aadhar matricks dett "z" = 1 aanee "z" = − "i" nirdisht karato. 
sootra _ 7, jeey dilelya taktyanusar gunakar karate. jevha olakh matricks 1 neey olakhala jaato, tevha tessarine "tii" = "doubleu" + "zedje".
itihas.
anek kalpanik ekkanchya vishayachi 1840 chya dashakat tapasani karnyat aalee. 1844 madhye tatvik niyatkalikat suru jhalelya 'chaturthanshanvar kinwa bijganitatil kalpanikatechya naveen pranalivar' yaa dirgh maliket, william rovan hamiltonne chaturthansh gatanusar gunakar karananya pranalicha sanvad sadhala. 1848 madhye thomas kirkmanne arther kelishi kelelya patravyavaharabaddal atisankalit sankhyanchi pranali nirdharit karananya ekkanvaril samikaranambaddal ahwal deela.
tesserines.
1848 madhye james cocklene 'tatvik niyatkalik' madheel lekhanchya maliket tesserineschi olakh karoon dilee.
tessarine hii yaa swarupachi atisankalit sankhya aahe.
jethe sootra _ 9
cocklene ghatankiy maliketil hyperbolic cosine malika aanee hyperbolic sine malika vegale karnyasathi tesserinescha vapar keela. tessarinesmadhye shoonya vibhajak kase udbhavtat heey dekhil tyane dakhavale, jyamule tyala "ashakya" haah shabd vaparanyachi prerana midali. tessarines aata tyanchya vastavik tessarines sootra _ 10 chya up-bijganitasathi prasiddha aahet. 
yala vibhajit-jatil sankhya dekhil mhanatat, jeey ekak hyperbolache parametrization vyakt karatat.
dvi-sammishra sankhya.
1892 chya 'mathematische enalen' yaa shodhanibandhat, corado segrene dvi-sankul sankhya sadar kelya, jyaa tessarinsasathi eka samaroop bijganit tayar karatat.
segre reed doubleu. aar. hamiltonchi 'chaturthanshavaril vyakhyane' (1853) aanee doubleu. keey. clifford. segrene hamiltonchya kahee sanketankancha vapar karoon tyachya dvi-sankul sankhyanchi pranali vikasit keleeh 'etch' aanee 'aay' heey ghatak asoo dya jeey −1 paryant varg karatat aanee teey pravas karatat. mug, gunakaratil sahsambandh manoon, gunakar "hye" chaa varg + 1 assa asane aavashyak aahe. yaa aadharavar tayar kelela bijganit nantar james cocklechya tesarinsasarkhach asato, joou vegalya aadharadvare darshavila jaato. segrene namood kele qee ghatak
jevha dvi-sankul sankhya aadharanusar vyakt kelya jaatat, tevha tessarinshee tyanchi samanata spasht hote. yaa samaroop algebranche resheey pratinidhitva pahatana jevha nakaratmak chinh vaparale jaate tevha chauthya parimanat sahmati darshavate; resheey pratinidhitvantargat varr dilelya namuna utpadanacha vichar qaraa.
dvibhashik.
rachana algebrancha aadhunik siddhant bijganitela dusya dwibinian sanrachanevar aadharit dwibinian rachana mhanoon sthan deto, mhanoon dwibinarian. callie-dickson prakriyetil anarion patali eka kshetra asane aavashyak aahe aanee vastavik kshetrapasun suruvat karoon, nehamichya jatil sankhya vibhagani binarian mhanoon, dusare kshetra mhanoon udbhavtat. asha prakare hii prakriya punha suru houn dvibhajak tayar houu shakate. kevin macrimonne tyachya 'aaa test auph jorden algebras' (2004) yaa majkurat 'binarian' yaa shabdane pradan kelelya namkaran saralikaranachi nond kelee.
bahupadi mule.
jatil sankhyanchya krambaddha jodyanni ("yuu", "vhee") tyateel ghatak liha aanee tyanche pratinidhitva qaraa. tessarines teeche bijganit 2seeshii samaroop asalyane, bahupadi tii [x] aanee 2see [x "] chya rings dekhil samaroop aahet, tathapi nantarchya bijganitetil bahupadi vibhajit hotatah
parinami, jevha yaa bijganitetil bahupadi samikaran sootra _ 13 nishchit kele jaate, tevha teey c varr done bahupadi samikarananmadhye qamee hote. jarr degree "n" asel tarr pratyek samikaranasathi "n" mule aaheth sootra _ 14
yaa moodanchya sanchatil konatehi krambaddha jodee sootra _ 15 heey mool samikaran 2c ["x"] madhye purn karel, mhanoon tyala "n" 2 mule aahet.
t ["x"] saha samarupatemule, bahupadicha parasparsambandh aanee tyanchya mulancha parasparsambandh aahe. tyamule 'aine' chya anshanchya tessarine bahupadila dekhil 'aine' 2 mule asatat, jee mulanchi sankhya mojatat.
anuprayog.
dvi-sankul sankhya hii capsche kendra (bhautik jaageche jatil bijganit) mhanoon disate, jeey clifford bijganit sootra _ 16 aahe. kaaran capschya rekheey jagekade, {sootra _ 18} varcha chaar aayami avkash kalavadhi {sootra _ 17} mhanoon pahile jaau shakate.
digital signal prakriyet tesserinecha vapar karnyat aala aahe.
drav yantrikimadhye dvi-sankul sankhya vaparlya jaatat. dvi-jatil bijganitacha vapar jatil sankhyanchya done bhinn anuprayoganmadhye samet ghadavun aanatoh jatil pratalatil dvi-aayami sambhavya pravahache pratinidhitva aanee jatil ghatankiy karya.