
sankhya rekha

praathamik ganit mein, ack sankhya rekha ack snaatak seedhi rekha kii ack tasveer haye joe vaastavik sankhyaon key drishya pratinidhitva key roop mein kaary karati haye. ack sankhya rekha kaa pratyek bindu ack vaastavik sankhya key anuroop maanaa jaataa haye, aur pratyek vaastavik sankhya ack bindu key anuroop maani jaati haye.
purnankon koo aksar rekha parr samaan roop seey chihnit vishesh binduon key roop mein dikhaaya jaataa haye. haalaanki chhavi kewal-3 seey 3 taka key purnank dikhaati haye, rekha mein sabhi vaastavik sankhyaaen shamil hain, joe pratyek disha mein hamesha key liye jaari rahati hain, aur way sankhyaaen bhee joe purnankon key beach hain. iska upayog aksar saral jodd aur ghataav key shikshan mein sahaayata key roop mein kiya jaataa haye, vishesh roop seey nakaratmak sankhya shamil haye.
unnat ganit mein, sankhya rekha koo vaastavik rekha yaa vaastavik sankhya rekha kahaa jaa sakata haye, jise aupchaarik roop seey sabhi vaastavik sankhyaon key samooh key roop mein paribhaashit kiya jaataa haye. ise ack jyaamitiya sthaan key roop mein dekhaa jaataa haye, arthaat ayam ack kaa vaastavik samanvaya sthaan, yaa ayam ack kaa euclidean sthaan-euclidean rekha. ise ack sadish sthaan (yaa sanlagn sthaan), ack metrick sthaan, ack sthalvaigyaanik sthaan, ack maap sthaan, yaa ack raikhik nirantarta key roop mein bhee socha jaa sakata haye.
vaastavik sankhyaon key samooh kii tarah, vaastavik rekha koo aamtaur parr prateek (yaa vaikalpik roop seey, sootra _ 1, blackboard bolled mein akshar "r") dwara darshaaya jaataa haye. haalaanki, ise kabhi-kabhi pahale vaastavik sthaan yaa pahale euclidean sthaan key roop mein apani bhoomika parr jore dene key liye darshaaya jaataa haye.
itihas.
operation uddeshyon key liye upayog kii jaane vaali sankhya rekha kaa pahala ullekh johnn wallis key "beejganit key granth" mein paaya jaataa haye. apane granth mein, valis ack vyakti key chalane key roopak key tahat, aage aur peechhe kii or badhane key sandarbh mein ack sankhya rekha parr jodd aur ghataav kaa varnan kartaa haye.
operation kaa ullekh kiye binaa ack poorv chitran, halanki, johnn napier key "laghuganak kii prashansaniya talika key vivaran" mein paaya jaataa haye, joe 1 seey 12 key moolyon koo baaen seey daaen pankti mein dikhaata haye.
lokpriya dhaarana key vipareet, rene descartes key mool laa gametry mein ack sankhya rekha naheen haye, jise aaj humm iska upayog karate hain, halanki yeh ack samanvaya pranali kaa upayog kartaa haye. vishesh roop seey, descart key kaam mein vishisht sankhyaaen naheen hoti hain joe rekhaaon parr maanchitrit kii jaati hain, kewal amoort maatraaen hoti hain.
sankhya rekha banaaen.
ack sankhya rekha koo aamtaur parr kshaitij honey key roop mein darshaaya jaataa haye, lekin ack cartasian nirdeshank tala mein urdhvaadhar aksh (y-aksh) bhee ack sankhya rekha haye. ack parampara key anusaar, sakaratmak sankhyaaen hamesha shoonya key daaeen or hoti hain, nakaratmak sankhyaaen hamesha shoonya key baain or hoti hain, aur rekha key donon siron parr teeron kaa matlab yeh haye kii rekha sakaratmak aur nakaratmak dishaaon mein anishchit kaal taka jaari rahati haye. ack anya parampara kewal ack teer kaa upayog karati haye joe usa disha koo ingit karati haye jisamen sankhya badhati haye. jyaamiti key niyamon key anusaar rekha anishchit kaal taka sakaratmak aur nakaratmak dishaaon mein jaari rahati haye joe ack "anant rekha" key roop mein anta binduon key binaa ack rekha koo paribhaashit karati haye, ack antim bindu key saath ack rekha "kiran" key roop mein, aur dou antim binduon key saath ack rekha "rekha khand" key roop mein.
sankhyaon kii tulanaa karein.
yadi koee vishesh sankhya doosari sankhya kii tulanaa mein sankhya rekha parr daaeen or adhik door haye, tou pahali sankhya doosari seey adhik haye (samatulya roop seey, doosari sankhya pahale seey cumm haye). unake beach kii doori unake antar kaa parimaan haye-yaani, yeh pahali sankhya koo doosari sankhya seey ghataakar, yaa doosari sankhya key poorna maan koo pahali sankhya seey ghataakar maapata haye. iss antar koo lena ghataav kii prakriya haye.
iss prakaar, udaaharan key liye, 0 aur kuchh anya sankhya key beach ack rekha khand kii lambai baad kii sankhya key parimaan koo darshaati haye.
dou sankhyaon koo 0 seey ack sankhya taka kii lambai koo "oopar" karke jodaa jaa sakata haye, aur ise dusare sankhya key sheersh parr 0 key anta key saath phir seey neeche rakhaa jaa sakata haye.
iss udaaharan mein dou sankhyaon koo gunaa kiya jaa sakata haih 5 × 3 koo gunaa karane key liye, dhyaan dein kii yeh 5 + 5 + 5 key samaan haye, isaliye lambai koo 0 seey 5 taka uthaayen aur ise 5 key daaeen or rakhein, aur phir usa lambai koo phir seey uthaayen aur ise pichhale parinam key daaeen or rakhein. yeh ack parinam detaa haye joe pratyek kii 5 kii 3 sanyukt lambai haye; choonki prakriya 15 parr samaapt hoti haye, humm paate hain kii 5 × 3 = 15.
vibhajan koo nimnalikhit udaaharan key roop mein kiya jaa sakata haih 6 koo 2 seey vibhajit karane key liye-yaani, yeh pataa lagaane key liye kii 2 kitni baar 6 mein jaataa haye-dhyaan dein kii 0 seey 2 kii lambai 0 seey 6 taka kii lambai kii shuruat mein hoti haye; poorv kii lambai koo uthaayen aur ise phir seey apani mool sthiti key daaeen or rakhein, jisamen poorv mein 0 parr anta abb 2 parr rakhaa gayaa haye, aur phir lambai koo apani navintam sthiti key daaeen or leey jaaen. yeh lambai 2 key daahine chhor koo lambai key daahine chhor parr 0 seey 6 taka rakhta haye. kyonki 2 kii teen lambai 6 seey bharee jaati haye, 2 6 mein teen baar jaataa haye (yaani, 6 ρ2 = 3).
sankhya rekha key bhaag.
dou sankhyaon key beach kii sankhya rekha key khand koo antaral kahaa jaataa haye. yadi khand mein donon sankhyaaen shamil hain tou ise ack bund antaral kahaa jaataa haye, jabaki yadi yeh donon sankhyaon koo baahar kartaa haye tou ise ack khula antaral kahaa jaataa haye. yadi isamen ack sankhya shamil haye lekin doosari naheen haye, tou ise aadhaa khula antaral kahaa jaataa haye.
ack vishesh bindu seey ack disha mein hamesha key liye faile sabhi binduon koo ack saath ack kiran key roop mein jaanaa jaataa haye. yadi kiran mein vishesh bindu shamil haye, tou yeh ack bund kiran haye; anyatha yeh ack khuli kiran haye.
avadharana kaa vistaar.
laghuganak paimana.
sankhya rekha parr, dou binduon key beach kii doori ikai kii lambai haye yadi aur kewal tabhi jaba pratinidhitva kii gaee sankhyaon kaa antar 1 key barabar hoe. anya vikalp sambhav hain. 
sabse aam vikalpon mein seey ack "laghuganak paimana" haye, joe ack rekha parr "sakaratmak" sankhyaon kaa pratinidhitva kartaa haye, jaise kii dou binduon kii doori ikai kii lambai haye, yadi darshaae gaye sankhyaon key anupaat kaa ack nishchit moolya haye, aamtaur parr 10. aise laghuganak paimaane mein, mool 1 kaa pratinidhitva kartaa haye; daaeen or ack inch, ack key paas 10 haye, ack key daaeen or ack inch haye, tou, phir, aadi. isi tarah, 1 key baain or ack inch, ack key paas haye, phir, aadi.
yeh drishtikon upayogi haye, jaba koee ack hii aakriti parr, parimaan key bahut alag crum key saath moolyon kaa pratinidhitva karna chaahata haye. udaaharan key liye, brahmand mein maujood vibhinn nikaayon key aakar kaa pratinidhitva karane key liye ack logridmic paimaane kii aavashyakta hoti haye, aamtaur parr, ack photon, ack electron, ack paramanu, ack anu, ack maanav, prithvi, saur mandal, ack akashganga, aur drishya brahmand.
laghuganak taraju kaa upayog slide niyamon mein laghuganak taraju parr lambai jodkar yaa ghataakar sankhyaon koo gunaa yaa vibhajit karane key liye kiya jaataa haye.
sankhya rekhaaon kaa sanyojan.
vaastavik sankhya rekha key samkon parr mool key madhyam seey kheenchi gaee rekha kaa upayog kaalpanik sankhyaon kaa pratinidhitva karane key liye kiya jaa sakata haye. yeh rekha, jise kaalpanik rekha kahaa jaataa haye, sankhya rekha koo ack jatil sankhya tala taka vistaarit karati haye, jisamen jatil sankhyaon kaa pratinidhitva karane vaale bindu hotey hain.
vaikalpik roop seey, ack vaastavik sankhya rekha koo ack vaastavik sankhya key sambhaavit moolyon koo niroopit karane key liye kshaitij roop seey kheencha jaa sakata haye, jise aamtaur parr "x" kahaa jaataa haye, aur doosari vaastavik sankhya rekha koo doosari vaastavik sankhya key sambhaavit moolyon koo niroopit karane key liye urdhvaadhar roop seey kheencha jaa sakata haye, jise aamtaur parr "y" kahaa jaataa haye. inn rekhaaon koo ack saath jodkar cartasian samanvaya pranali key roop mein jaanaa jaataa haye, aur samtal mein koee bhee bindu vaastavik sankhyaon kii ack jodee key moolya kaa pratinidhitva kartaa haye. aage, cartasian samanvaya pranali koo "screen (yaa prishth) seey baahar aaney vaali teesari sankhya rekha" koo "jaid" naamak teesare charr koo maapate hue dekhaa jaa sakata haye. sakaratmak sankhyaaen screen kii tulanaa mein darshak kii aankho key karib hoti hain, jabaki nakaratmak sankhyaaen "screen key peechhe" hoti hain; badee sankhyaaen screen seey door hoti hain. phir tri-ayami sthaan mein koee bhee bindu jisamen humm rahate hain, vaastavik sankhyaon kii tikadi key moolyon kaa pratinidhitva kartaa haye.
unnat avadhaaranaaen.
ack raikhik nirantarta key roop mein.
vaastavik rekha maanak crum key tahat ack raikhik nirantarta haye. vishesh roop seey, vaastavik rekha raikhik roop seey krambaddh haye, aur yeh crum ghanaa haye aur isamen sabse cumm-oopari-baadhya guna haye.
uparokt gunon key alawa, vaastavik rekha mein koee adhikatam yaa nyuntam tatva naheen haye. isamen ack ganana yogya ghanaa upasamuchchay bhee haye, arthaat parimeya sankhyaon kaa samooh. yeh ack pramey haye kii ganana yogya ghaney upasamuchchay key saath koee bhee raikhik nirantarta aur koee adhikatam yaa nyuntam tatva vaastavik rekha key liye crum-samroop naheen haye.
vaastavik rekha ganana yogya shrinkhala sthiti koo bhee santusht karati haih parasparik roop seey vichhedit, gair-khaali khule antaral kaa pratyek sangrah ganana yogya haye. crum siddhant mein, prasiddh saslin samasya poochhati haye kii kyaa pratyek raikhik nirantarta ganana yogya shrinkhala sthiti koo santusht karati haye jisamen koee adhikatam yaa nyuntam tatva naheen haye, yeh aavashyak roop seey crum-samroop haye. yeh kathan sett siddhant kii maanak swayamsiddh pranali seey swatantra dikhaaya gayaa haye jise jaid. aff. see. key roop mein jaanaa jaataa haye.
ack metrick sthaan key roop mein.
vaastavik rekha ack metrick sthaan banaati haye, jisamen doori falan nirpeksh antar dwara diyaa jaataa haih
metrick tensor spasht roop seey 1-ayami euclidean metrick haye. choonki-ayami euclidean metrick koo matricks key roop mein-dwara-pehchaan matricks key roop mein darshaaya jaa sakata haye, isaliye vaastavik rekha parr metrick kewal 1-dwara-1 pehchaan matricks haye, i.ee. 1.
yadi aur, tou kendra mein-gend kewal khula antaral haye.
iss vaastavik rekha mein metrick sthaan key roop mein kaee mahatvapurna guna hainh
ack sthalvaigyaanik sthaan key roop mein.
vaastavik rekha mein ack maanak topology hoti haye, jise dou alag-alag, samkaksh tarikon seey pesh kiya jaa sakata haye.
pahala, choonki vaastavik sankhyaaen poori tarah seey krambaddh hain, way ack crum topology rakhate hain. doosara, vaastavik sankhyaaen oopar paribhaashit metrick seey ack metrick topology viraasat mein praapt karati hain. crum topology aur metrick topology samaan hain. ack sthalvaigyaanik sthaan key roop mein, vaastavik rekha khule antaral key liye samroop haye.
vaastavik rekha mamuli roop seey ayam kaa ack sthalvaigyaanik manifold haye. homeomorphism taka, yeh kewal dou alag-alag judey hue 1-manifolds mein seey ack haye, joe binaa seema key haye, doosara vritt haye. iss parr ack maanak vibhedak sanrachana bhee haye, joe ise ack vibhedak kaee gunaa banaati haye. (difiomorphism taka, kewal ack hii differentiable sanrachana haye jiska topological space samarthan kartaa haye.)
vaastavik rekha sthaaniya roop seey ack saghan sthaan aur ack paracompact sthaan haye, saath hii saath doosari-ganana yogya aur samanya haye. yeh path seey bhee judaa huaa haye, aur isaliye yeh bhee judaa huaa haye, halanki ise kissy bhee ack bindu koo hataakar disconnect kiya jaa sakata haye. vaastavik rekha bhee samvidaatmak haye, aur iss tarah isake sabhi homotopy samooh aur cumm homology samooh shoonya hain.
sthaaniya roop seey ack saghan sthaan key roop mein, vaastavik rekha koo kaee alag-alag tarikon seey saghan kiya jaa sakata haye. ack-bindu sampeedan ack vritt (arthaat, vaastavik prakshepya rekha) haye, aur atirikt bindu koo ack hastaaksharit anant key roop mein socha jaa sakata haye. vaikalpik roop seey, vaastavik rekha key dou chhor hotey hain, aur parinami anta sampeedan vistaarit vaastavik rekha haye. vaastavik rekha kaa patthar-shack sampeedan bhee haye, jisamen anant sankhya mein atirikt ank jodana shamil haye.
kuchh sandarbhon mein, vaastavik sankhyaon key samooh parr anya topology koo rakhna sahayak hotaa haye, jaise kii nimn seema topology yaa zariski topology. vaastavik sankhyaon key liye, baad vaalaa seemit poorak topology key samaan haye.
ack vector sthaan key roop mein.
vaastavik rekha vaastavik sankhyaon key kshetra (yaani, swayam parr) key ayam parr ack sadish sthaan haye. isamen ack aantarik utpaad key roop mein samanya gunan hotaa haye, jisase yeh ack euclidean vector sthaan bunn jaataa haye. iss aantarik utpaad dwara paribhaashit maanak kewal nirpeksh moolya haye.
ack maap sthaan key roop mein.
vaastavik rekha mein ack vihit maap hotaa haye, arthaat labsgue maap. iss upaay koo ack boral maap key puuraa honey key roop mein paribhaashit kiya jaa sakata haye, jahaan kissy bhee antaral kaa maap antaral kii lambai haye.
vaastavik rekha parr labsgue maap sthaaniya roop seey compact samooh parr ack haar maap key sabse saral udaharanon mein seey ack haye.
vaastavik algebras mein.
jaba "a" ack ekmusht vaastavik beejganit haye, tou 1 key saath vaastavik sankhyaon key utpaad beejganit key bheetar ack vaastavik rekha haye. udaaharan key liye, jatil tala "z" = "x" + i "y" mein, upasthaan {"z": "y" = 0} ack vaastavik rekha haye. isi tarah, chaturthak kaa beejganit
upasthaan mein ack vaastavik rekha haih "x" = "y" = "z" = 0}.
jaba vaastavik beejganit ack pratyaksh yog sootra _ 3 hotaa haye tou "a" parr ack sanyugman upasthaan "v" key maanchitran sootra _ 4 dwara pesh kiya jaataa haye. iss tarah vaastavik rekha mein sanyugman key nishchit bindu hotey hain.
ack ayam "aine" key liye, varg matricks ack ring banaate hain jisamen ring mein pehchaan matricks key saath vaastavik utpaadon key roop mein ack vaastavik rekha hoti haye.

