bakhshaali paandulipi

bakhshaali paandulipi mein prayog kiye gaye ank

bakshaali paandulipi ya bakhshaali paandulipi (Bakhshali Manuscript) praacheen Bhaarat ki ganit se sambandhit paandulipi hai. yeh bhojapatr par likhi hai. yeh san 1881 mein bakhshaali gaaainv (tatkaaleen pashchimottar seema praant; ab Pakistan mein, takshashila se lagbhag 70 kimi door ) mein mili thi. yeh shaarada lipi mein hai evam gaatha boli (sanskrut evam praakrut ka milaajula roop) mein hai. yeh paandulipi apoorn hai. ismein keval 70 'panne' (ya pattiyaaain) hi hain jinmein se bahut hi bekaar ho chuki hain. bahut se panne apraapya hain.

anukram

parichay

shaayad yeh sanskrut mein ankaganit par likhi gayi sabse puraani rachana hai jise sambhavat: saataveen shataabdi mein sankalit kiya gaya hoga. ise kisne sankalit kiya, iska koi ullekh naheen hai. Kolkata ki eshiyaatik society ne 'ganitaavali' naamak granth ka prakaashan karavaaya hai. is granth ki pushpika se itna jaroor pata chalta hai ki sukhadaas naamak ek kaayasth ne raamapaaladev ke shaasanakaal mein shak samvat 1615 ya 1715 mein yeh poori saamagri kaheen se haasil ki thi. is saamagri ki ek adviteeya prati eshiyaatik society ke sangrah mein surakshit hai. granth ke shuruaati pannon mein kai khaamiyaan hain, haalaanki baad ke pannon mein adhikaansh saamagri suvaachya hai.

sampoornaanand sanskrut vishvavidyaalaya Varanasi ke sarasvati bhavan ke vibhootibhooshan Bhattacharya ne is saamagri ka sampaadan kiya, lekin durbhaagya yeh raha ki unka dehaant ho gaya. antat: maanabeindu Bannerjee aur pradeep Kumar majoomadaar ne ankaganit, praarambhik rekhaaganit aur kshetramiti (maapan) ki saamagri vaale is granth ka sampaadan kiya. yeh jyotishiyon ki haindabuk hai jismein ganit va khagol shaastr ke kuchh vishay shaamil hain. is granth ke agyaat lekhakon ne apne shuruaati vaakyon mein saaf kar diya hai ki yeh pustak un kaayasthon ya hisaab-kitaab rakhane vaalon ke liye hai jo ganit ka behad praarambhik gyaan haasil karna chaahate hain.

is paandulipi mein kya-kya hai? Joseph likhte hain-

The Bakhshali manuscript is a handbook of rules and illustrative examples together with their solutions. It is devoted mainly to arithmetic and algebra, with just a few problems on geometry and mensuration. Only parts of it have been restored, so we cannot be certain about the balance between different topics.
(anuvaad) bakhshaali paandulipi niyamon aur vyaakhyaatmak udaaharanon ke saath-saath unke samaadhaan ki ek hastapustika hai. yeh mukhyat: ankaganit aur beejaganit ko samarpit hai, jismein kuchh prashn jyaamiti aur kshetramiti par hain. iske keval kuchh bhaag hi bahaal ho sake hain, isliye ham vibhinn vishyon ke santulan ke baare mein sunishchit naheen ho sakte.

sanket paddhati

  • ismein bhinnon ko lagbhag vaise hi darshaaya gaya hai jaise aajkal kiya jaata hai, arthaat ek sankhya ke neeche doosari sankhya (kintu koi rekha naheen)
  • ghataane ki kriya ko ghataayi jaane waali sankhya ke daanye + likhkar nirupit kiya gaya hai,
  • bhaajan (bhaag) ko bha de darshaaya gaya hai,
  • bhinnon ke yog ke liye yu (=yukt) ka prayog kiya gaya hai,
  • '=' ke liye f (fal) ka prayog kiya gaya hai,
  • sameekaranon ko nirupit karne ke liye ek bada bindu (dot) ka prayog kiya gaya hai jo 'agyaat char' ko niroopit karta hai. (shoonya bhi kai jagah isi tarah ke dot se niroopit kiya gaya hai)[1]

sangathan

yeh granth kaarika ke roop mein likha gaya hai aur chhah adhyaayon mein vibhaajit hai. sabse pehle adhyaaya mein prastaavana di gayi hai jo is granth ko likhe jaane ke uddeshya par prakaash daalati hai. doosra adhyaaya 'sankhyaavidhaan' hai jismein vibhinn sankhyaaon mein maapan ka ullekh kiya gaya hai. iski shuruaat 'yav' (jau ke beej) se hoti hai. fir 'anguli', vitsti ya baalisht, hast danda (lattha) jaise maapon ka varnan kiya gaya hai. fir kos aur yojan ka ullekh hai. iske baad vajan ke maapon ka varnan hai, khaaskar chaaval ke vajan karne vaale maap. sunaaron dvaara istemaal kiye jaane vaale maapon jaise gunja, maasha, karsha aur pal ka bhi ullekh kiya gaya hai.

samay ki maap ke baare mein kaafi rookhe dhang se varnan kiya gaya hai. iske anusaar ek din mein 30 muhoort hote hain, jabki 30 dinon se ek maah aur 12 maah se ek saal banta hai. ismein kuchh aisi maapein bhi di gayi hain jo ab prachalan mein naheen hain jaise varaataka, gandak ya gandik, kaakini, pan, dallak ityaadi. rochak baat hai ki ghateemaan, drammakedaar aur deenaar ka bhi ullekh hai. teesare adhyaaya ka naam hai 'paribhaashaavidhi' jismein khaaskar rekhaaganit aur ankaganit se judi takaneeki shabdaavali di gayi hai. jaise kaaya (sampoorn pind), anshak (bhaag), ched, vishkanbh, paridhi, bhaajak, harak ityaadi. chautha adhyaaya 'parikramaavidhi' vibhinn ankaganit sameekaranon va sootron ko samarpit hai. paaainchavein adhyaaya ka naam 'vyavahaaravidhi' hai jismein kai prakaar ke niyam jaise teen ka niyam, paanch ka niyam, gunan, chakravruddhi byaaj ki ganana aadi ko samajhaaya gaya hai. saath hi chaturbhuj, tribhuj, vrutt ityaadi ke baare mein bhi sankshep mein bataaya gaya hai. chhathavein adhyaaya mein udaaharanon ke jariye vibhinn niyamon ko samajhaane ka prayaas kiya gaya hai.

is granth mein kai kamiyaaain nikaali ja sakti hain. iske baavajood kehna padega ki yeh granth hajaaron saal pehle uttar Bhaarat mein prachalit ganit ke adhyayan ki ek jhalak to dikhlaata hi hai.

visheshataaeain

vargamool ki ganana

is paandulipi mein vargamool ki ganana ke liye nimnalikhit sootr kai baar aaya hai-

akrute shlishtakrutyunaat sheshachhedo dvisangun:.
tadvargadal sanshlishtahruti shuddhikruti kshay:. [2]

maana kisi sankhya ka vargamool nikaalna hai jahaaain Q poornvarg naheen hai. yadi, a1, a2, a3 kramash: adhik shuddh vargamool hon to, ise teen charanon mein nimn prakaar se kiya ja sakta hai-[3]

  • pratham charan: Q ko nimnalikhit prakaar se likhein .
  • doosra charan:
  • teesara charan:

udaaharan ke liye, maana 889 ka vargamool nikaalna hai. ham pehle a1 = 29, b = 48 lete hain to,

889 ka vargamool lagbhag 29,8160303 hai. at: uparokt parinaam dashamlav ke 4 ankon tak shuddh hai. ab ham a1 = 30, b = -11 lekar dekh sakte hain ki adhik shuddh parinaam milta hai-

yeh parinaam dashamlav ke 6 ankon tak shuddh hai.

sandarbh

baahari kadiyaaain