Bijaganita: pracheen Bharat mein algebra

Bijaganita ka matlab kya hai?

Pracheen Bharat mein nishaanon se sankhyayein likhna aur equations solve karna ganit ka bada hissa tha. Is shaakha ko bijaganita kehte the, jise aaj hum algebra kehte hain. Bija matlab beej. Ek beej mein poora ped chhupa hota hai, aur ek anjaan sankhya mein sawaal ka jawab; sawaal hal karna us ped ko ek ek kadam badhne dene jaisa hai.

Brahmagupta aur unknowns

Apni kitaab Brahmasphutasiddhanta (628 CE) ke Chapter 18 mein Brahmagupta ne nishaanon se likhi anjaan sankhyaon ko jodna, ghataana aur guna karna samjhaaya, bilkul waise jaise aaj hum x aur y lete hain. Ye duniya mein algebra ke sabse puraane jaane maane kaamon mein se ek hai. Fields Medal jeetne waale ganitagya David Mumford ne Brahmagupta ko aaj ke algebra ko banaane mein sabse mukhya vyakti kaha.

Vichaar kaise faile

8vi sadi mein Bharatiya ganit ke vichaaron ka Arabic mein anuvaad hua. Inka asar Al-Khwarizmi pe pada, jo aaj ke Iraq mein rehte the. Lagbhag 825 CE mein unhone Hisab al-jabr wal-muqabala likhi, jiska matlab hai “jodne aur santulan karne ki ganana”. 12vi sadi mein unki kitaab ka Latin mein anuvaad hua aur wo Europe mein mashhoor hui, aur al-jabr se wo algebra shabd bana jo hum aaj bhi bolte hain.

Puraana Bharatiya notation

Brahmagupta ke samay se Bharatiya ganitagya unknowns ke liye chhote akshar lete the. ya, yavat-tavat ka chhota roop tha, yaani “jitna”, aur pehle unknown ke liye lagta tha. ka aur ni rangon ke naam kalaka (kaala) aur nilaka (neela) se aaye, aur pi aur lo bhi lagte the. (Extra: in dono ko aam taur pe pitaka, peela, aur lohita, laal, se jodte hain.) Jaani hui sankhya ko ru likhte the, rupa yaani roop se.

Har term mein nishaan pehle aata tha, to 2x + 1 ko ya 2 ru 1 likhte the. Sankhya ke upar bindu batata tha ki wo negative hai, to 2x − 8 tha ya 2 ru 8, 8 ke upar bindu ke saath. Equation mein barabar ke nishaan ki jagah dono taraf ek ke neeche ek likhe jaate the.

Brahmagupta ka niyam

Ek unknown waale sawaal hal karne ka vyavasthit tareeka sabse pehle Aryabhata (499 CE) ne diya, aur Brahmagupta ne use 628 CE mein samjhaaya. Ax + B = Cx + D jaisi kisi bhi equation ka jawab hai

x = (D − B) ÷ (A − C).

Kyun? Dono taraf se Cx aur B ghataao: Ax − Cx = D − B, yaani (A − C)x = D − B, phir A − C se bhaag do. 5x + 4 = 3x + 8 ke liye niyam deta hai (8 − 4) ÷ (5 − 3) = 2. Bachat waali equation 650m + 4000 = 500m + 5050 ke liye ye seedha (5050 − 4000) ÷ (650 − 500) = 7 deta hai.

Bhaskaracharya ka ghode waala sawaal

Bhaskaracharya ne 1150 CE mein Bijaganita naam ki kitaab likhi. Uska ek sawaal: ek aadmi ke paas ₹300 aur 6 ghode hain; doosre ke paas 10 ghode hain par us pe ₹100 ka karz hai. Agar dono barabar ameer hain aur har ghode ki keemat ek hai, to ek ghoda kitne ka hai? Ghode ki keemat ₹x maano. Tab 300 + 6x = 10x − 100. Dono taraf 100 jodo to 400 + 6x = 10x, aur dono taraf se 6x ghataao to 400 = 4x, to x = 100. Ek ghoda ₹100 ka hai.

Bakhshali Manuscript ka baantne waala sawaal

Bakhshali Manuscript, jise chapter 300 CE ka batata hai, mein baantne ki ek paheli hai. Pehle ko anjaan raqam milti hai, doosre ko uska dugna, teesre ko doosre ka teen guna aur chauthe ko teesre ka chaar guna, aur kul 132 baante jaate hain. Pehle hisse ko a maano, to hisse a, 2a, 6a aur 24a hain, to 33a = 132 aur a = 4.

Ye kyun zaroori hai

Bharatiya ganitagyon ne mushkil vichaaron ko aise saral tareekon mein badla jinhe koi bhi follow kar sake, aur algebra aaj bhi patterns batane, “odd + odd = even” jaise daave saabit karne aur kai tarah ke sawaal hal karne mein kaam aata hai. Tareekon ka abhyaas equations kaise solve karein aur word problems se equation kaise banayein mein karo. Textbook ke sawaalon ke jawab Part 2 Chapter 7 ke NCERT solutions mein hain; poore Part 2 Chapter 7 ke notes dekho.