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Disamb2.jpg anant ek bahuvikalpi shabd hai any arthon ke lie dekhen:- anant (bahuvikalpi)


annt shabd ka angreji paryay 'inaphiniti' laitin bhasha ke anh‌ (anh‌) aur phinis (ant) ki sndhi hai. yah shabd un rashiyon ke lie prayukt kiya jata hai jinaki map athava ganana unake parimit n rahane ke karan asnbhav hai. aparimit saral rekha ki lnbaee simavihin aur annt hoti hai.

  • ganitiy vishleshan men prachalit 'annt', jise ¥ dvara niroopit karate hain.
  • yadi y koee char hai aur ph (y) koee y ka phalan hai, aur yadi ab char y kisi snkhya k ki or agrasar hota hai tab ph (y) is prakar badhata hi chala jata hai ki vah pratyek di huee snkhya n se bada ho jata hai aur bada hi bana rahata hai chahe kitana bhi bada ho, to kaha jata hai ki y=k ke lie ph (y) ki sima annt hai.
  • bhinnon ki paribhasha se[1] spasht hai ki bhinn v/s vah snkhya hai jo s se guna karane par gunanaphal v deti hai. yadi v, s men se koee bhi shoony n ho to v/s ek advitiy rashi ka niroopan karata hai. phir spasht hai ki 0/s sadaiv saman rahata hai, chahe s koee bhi sant snkhya ho. ise parimey (raishanal) snkhyaon ko shoony kaha jata hai aur gananatmak (kardinal) snkhya 0 ke saman hai. viparitat:, v/0 ek arthahin pad hai. ise annt samajhana bhool hai. yadi k/y men k achar rahata hai, aur y ghatata jata hai, aur k,y donon dhanatmak hain, to k/y ka man badhata jaega. yadi y shoony ki or agrasar hota hai to antatogatva k/y kisi badi se badi snkhya se bhi bada ho jaega. ham is bat ko nimnalikhit prakar se vyakt karate hain .
  • kaintar (1845-1918) ne annt ki samasya ko doosare dhng se vyakt kiya hai. kaintariy snkhyaen, jo annt aur sant ke viparit hone ke karan kabhi-kabhi atit (trainsaphainait) snkhyaen kahi jati hain, jyamiti aur simasiddhant men prachalit annt ki paribhasha se bhinn prakar ki hain. kaintar ne laghutam atit gananatmak snkhya (trainsaphainait kardinal nnbar) (ek, do tin ityadi kardinal snkhyaen hain; pratham, dvitiy, tritiy ityadi ardinal snkhyaen hain.) a0 (akar shoony, aliph-jiro) ki vyakhya prakritik snkhyaon 1,2,3... ke sngh (set) ki gananatmak snkhya se ki hai. yah siddh ho chuka hai ki a0+s =a0, jisamen s koee sant poorn snkhya hai. kaintar ne keval akar shoony snkhyaon a0,a1...ke siddhant ko bhi vikasit kiya hai. hardi ne gananatmak snkhya a1 vale binduon ke sngh ki rachana karane ki vidhi bataee hai. snkhya sn (=2a0)pratan (kntinuam) ki, arthat vastavik snkhyaon ke sngh ki, gananatmak snkhya hai. ekaiki roopantar (van tu van trainsaphaaurmeshan) dvara yah siddh kiya ja sakata hai ki antaraj (intaraval) (0,1) men bhi binduon ke sngh ki gananatmak snkhya sn hoti hai.
  • vastavik snkhyaon 1,2,3... sngh se snbnddh atit kramik snkhya ko au (aaumega,w) likhate hain aur ise pratham atit kramik snkhya (trainsaphainait ardinal nnbar) kahate hain. kisi die hue antaral ka kha men ba1, ba2, ba3,... binduon ke ek anukram par, jo vriddhimay snkhyaon k1, k2, k3,... ke anukram ko vyakt karata hai, vichar karen. is anukram ka ek simabindu (limiting paauint) hoga jo in samast binduon ke dahini or hoga; ise ham ba i dvara niroopit kar sakate hain. ab kalpana karen ki bindu b ke uparant any bindu aise bhi hain jinhen ham ba1,... ba2,... bas,... baa... vale sngh se snbaddh manana chahenge, tab in binduon ko ham baa +1,baa +2,... dvara vyakt karenge. yadi baa , ba +1,ba +2=2,... namak binduon ke sngh ka kaee antim bindu n ho aur ye sab ka kha ke antargat sthit hon to is sngh ka ek simabindu hoga jise ham baa +a ya ba 2,dvara vyakt kar sakate hain; ityadi. at: hamen kram snkhyaen 1,2,3,..., au, au+1, au+2k... au.2, a.2+1..., a.3,... au2,...prapt hoti hain.
  • ganitiy vishleshan men ham bahudha annt ki or agrasar honevale anukramon (ya phalanon) ki vriddhi ki tulana karate hain. landaoo ne o, o,~ namak snketalipi prachalit ki hai, jisaki vyakhya is prakar hai: yadi ph(y) aur pha (y) arinatmak hon aur yadi samast y>y0 ke lie ph (y)/pha(y)<ek achal rashi t ho, to y ke annt ki or agrasar hone par ph (y)=0 {pha (y)} hota he. yadi samast ke y>y0 lie pha (y)/pha (y)<t ho, jisamen t koee ichchhanusar chhoti snkhya hai, to y ke annt ki or agrasar hone par ph (y)=0 {pha (y)} hota hai, aur yadi y ke annt ki or agrasar hone par pha (y)/pha (y)®1athava koee any sant snkhya, to ham y®¥ par ph (y)~ pha (y) likhate hain. at: jab s®¥ to s2+20s+1000~ s2. samanyataya donon anukram annt ki or agrasar hote hain aur unaki vriddhi lagabhag saman rahati he. paaul doo bois-raimon aur ji.ech. hardi ne phalanon ke anukramon ki vriddhi men tulana karane ke lie 'annt mapaniyon' (skels aauv inaphiniti) ki vyakhya ki hai.[2]


panne ki pragati avastha
adhar
prarambhik
madhyamik
poornata
shodh

tika tippani aur sndarbh

  1. (dr. snkhya)
  2. hindi vishvakosh, khand 1 |prakashak: nagari pracharini sabha, varanasi |snkalan: bharat diskavari pustakalay |prishth snkhya: 103,104 |

snbndhit lekh

shabd sndarbh kosh
kashtakar antararashtriy antarrashtriy kalpasiddhi gatha upakhyan akhyan shatabdi suee nyagrodh mahapran vynjan antahsth vynjan oshthy vynjan danty vynjan moordhany vynjan talavy vynjan kanthy vynjan avyay vats kumar kimpurush saty vah snvarttak (hal) sthavar jara mohanastr aishik shastr rat sanatan anant adbhut varah dand vimukh kukur vrishadhvaj (bahuvikalpi) yayavar satr kulapati charu shoorp yoothanath dvarapal kalabattoo dhatta kadavak klib bhring patohoo attagattar attagam achal pravritti abohan ant-ayamh barabar kshetr hirany haraniparynt halli hallikar halik haiman shasy shrani svaviryopajivin sunu soona sthan sthan many sthal sthal vritti strot chaturth pnchabhagikam gunapatr khettasamik kripyagrih kumaragadiyanak kulyavap karmantik sima ekapurushikam utpadyamanavishti avat asihar ali aghat adhavay adanak skndhavar sity siddh seva setubndh senabhakt sarvamany sarv-abhyantar-siddhi sannidhata sngrahat snkrant snkar kul snkar gram sadhy ayanangal attapatibhag asvaynkrit krishi 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