अनन्त
| chitr:Disamb2.jpg anant | ek bahuvikalpi shabd hai any arthoan ke lie dekhean:- anant (bahuvikalpi) |
anant shabd ka aangreji paryay 'inaphiniti' laitin bhasha ke anh (anh) aur phinis (aant) ki sandhi hai. yah shabd un rashiyoan ke lie prayukt kiya jata hai jinaki map athava ganana unake parimit n rahane ke karan asanbhav hai. aparimit saral rekha ki lanbaee simavihin aur anant hoti hai.
- ganitiy vishleshan mean prachalit 'anant', jise ¥ dvara niroopit karate haian.
- yadi y koee char hai aur ph (y) koee y ka phalan hai, aur yadi ab char y kisi sankhya k ki or agrasar hota hai tab ph (y) is prakar badhata hi chala jata hai ki vah pratyek di huee sankhya n se b da ho jata hai aur b da hi bana rahata hai chahe kitana bhi b da ho, to kaha jata hai ki y=k ke lie ph (y) ki sima anant hai.
- bhinnoan ki paribhasha se[1] spasht hai ki bhinn v/s vah sankhya hai jo s se guna karane par gunanaphal v deti hai. yadi v, s mean 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 saant sankhya ho. ise parimey (raishanal) sankhyaoan ko shoony kaha jata hai aur gananatmak (kardinal) sankhya 0 ke saman hai. viparitat:, v/0 ek arthahin pad hai. ise anant samajhana bhool hai. yadi k/y mean k achar rahata hai, aur y ghatata jata hai, aur k,y donoan dhanatmak haian, to k/y ka man badhata jaega. yadi y shoony ki or agrasar hota hai to aantatogatva k/y kisi b di se b di sankhya se bhi b da ho jaega. ham is bat ko nimnalikhit prakar se vyakt karate haian .
- kaiantar (1845-1918) ne anant ki samasya ko doosare dhang se vyakt kiya hai. kaiantariy sankhyaean, jo anant aur saant ke viparit hone ke karan kabhi-kabhi atit (traiansaphainait) sankhyaean kahi jati haian, jyamiti aur simasiddhaant mean prachalit anant ki paribhasha se bhinn prakar ki haian. kaiantar ne laghutam atit gananatmak sankhya (traiansaphainait kardinal nanbar) (ek, do tin ityadi kardinal sankhyaean haian; pratham, dvitiy, tritiy ityadi ardinal sankhyaean haian.) a0 (akar shoony, aliph-jiro) ki vyakhya prakritik sankhyaoan 1,2,3... ke sangh (set) ki gananatmak sankhya se ki hai. yah siddh ho chuka hai ki a0+s =a0, jisamean s koee saant poorn sankhya hai. kaiantar ne keval akar shoony sankhyaoan a0,a1...ke siddhaant ko bhi vikasit kiya hai. hardi ne gananatmak sankhya a1 vale bianduoan ke sangh ki rachana karane ki vidhi bataee hai. sankhya san (=2a0)pratan (kantinuam) ki, arthat vastavik sankhyaoan ke sangh ki, gananatmak sankhya hai. ekaiki roopaantar (van tu van traiansaph aaurmeshan) dvara yah siddh kiya ja sakata hai ki aantaraj (iantaraval) (0,1) mean bhi bianduoan ke sangh ki gananatmak sankhya san hoti hai.
- vastavik sankhyaoan 1,2,3... sangh se sanbanddh atit kramik sankhya ko au (aaumega,w) likhate haian aur ise pratham atit kramik sankhya (traiansaphainait ardinal nanbar) kahate haian. kisi die hue aantaral ka kha mean ba1, ba2, ba3,... bianduoan ke ek anukram par, jo vriddhimay sankhyaoan k1, k2, k3,... ke anukram ko vyakt karata hai, vichar karean. is anukram ka ek simabiandu (limitiang p aauiant) hoga jo in samast bianduoan ke dahini or hoga; ise ham ba i dvara niroopit kar sakate haian. ab kalpana karean ki biandu b ke uparaant any biandu aise bhi haian jinhean ham ba1,... ba2,... bas,... baa... vale sangh se sanbaddh manana chaheange, tab in bianduoan ko ham baa +1,baa +2,... dvara vyakt kareange. yadi baa , ba +1,ba +2=2,... namak bianduoan ke sangh ka kaee aantim biandu n ho aur ye sab ka kha ke aantargat sthit hoan to is sangh ka ek simabiandu hoga jise ham baa +a ya ba 2,dvara vyakt kar sakate haian; ityadi. at: hamean kram sankhyaean 1,2,3,..., au, au+1, au+2k... au.2, a.2+1..., a.3,... au2,...prapt hoti haian.
- ganitiy vishleshan mean ham bahudha anant ki or agrasar honevale anukramoan (ya phalanoan) ki vriddhi ki tulana karate haian. laandaoo ne o, o,~ namak sanketalipi prachalit ki hai, jisaki vyakhya is prakar hai: yadi ph(y) aur pha (y) arrinatmak hoan aur yadi samast y>y0 ke lie ph (y)/pha(y)<ek achal rashi t ho, to y ke anant ki or agrasar hone par ph (y)=0 {pha (y)} hota he. yadi samast ke y>y0 lie pha (y)/pha (y)<t ho, jisamean t koee ichchhanusar chhoti sankhya hai, to y ke anant ki or agrasar hone par ph (y)=0 {pha (y)} hota hai, aur yadi y ke anant ki or agrasar hone par pha (y)/pha (y)®1athava koee any saant sankhya, to ham y®¥ par ph (y)~ pha (y) likhate haian. at: jab s®¥ to s2+20s+1000~ s2. samanyataya donoan anukram anant ki or agrasar hote haian aur unaki vriddhi lagabhag saman rahati he. p aaul doo bois-raimoan aur ji.ech. hardi ne phalanoan ke anukramoan ki vriddhi mean tulana karane ke lie 'anant mapaniyoan' (skels aauv inaphiniti) ki vyakhya ki hai.[2]
anant (visheshan) [nasti anto yasy n. b.]
- antarahit, aparimit, nissim, akshay,-°ratnaprabhavasy yasy-ku. 1/3,-tah 1. vishnu ki shayya sheshanag, krishna, balaram, shiv, nagoan ka pati vasuki 2. badal 3. kahani, 4. chaudah granthiyoan se yukt reshmi dora jo anant chaturdashi ke din dakshin bhuja par baandha jata hai;-ta 1. prithvi (antahin) 2. ek ki sankhya 3. parvati 4. shariva, anantamool, doorva adi paudhe-tamh 1. akash, vatavaran 2. asimata 3. moksh 4. parabrahm.[3]
sam.-tritiya vaishakh, bhadrapad aur margashirsh mas ki shukl paksh ki tij-dashtiah shiv, indr,-devah 1. sheshanag 2. narayan jo sheshanag ke oopar sota hai,-par (visheshan) asim vistarayukt, nissim,-°rankil shabdashastramh-panch. 1,-roop (visheshan) aganit roopavala, vishnu,-vijayah yudhishthir ka shankh-bhag. 1/26
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