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Bhartiya ganit ke fraction sawaal

Short mein jawab

Bhartiya ganitagyon ne lagbhag do hazaar saal tak fractions ko aage badhaaya. Shulbasutra (c. 800 BCE) ne eenton ki ginti ke liye fractions ka bhaag kiya, Brahmagupta (628 CE) ne chaaron kriyaaon ke aam niyam diye, aur Bhaskara II ki Lilavati (1150 CE) mein ek kanjoos ka fractions waala daan ek kauri nikla.

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Sabse purani geometry mein fractions

Non-unit fractions ke kuch sabse purane udaaharan Shulbasutra mein hain, jo yagya vediyan banaane ke geometry granth hain. Baudhayana ka Shulbasutra (c. 800 BCE) poochhta hai: 1/5 unit side waali kitni square eentein 7 1/2 square units dhakengi? Har eent ka area 1/5 × 1/5 = 1/25 square unit hai, aur eenton ki ginti × ek eent ka area = kul area. Isliye eenton ki ginti 15/2 ÷ 1/25 = 15/2 × 25 = 375/2 hai, yaani 187 1/2 eentein.

Baudhayana ki eentein: ek unit square mein 1/5 side ki 25 eentein; 7 1/2 square units ke liye 15/2 ÷ 1/25 = 375/2 eentein
Baudhayana ki eentein (c. 800 BCE)

Chaar fawwaare

Brahmagupta ki rachna par teeka likhne waale Chaturveda Prithudakasvami (c. 860 CE) ne ye paheli di. Chaar fawwaare ek kund bharte hain: pehla ek din mein, doosra aadhe din mein, teesra chauthaai din mein aur chautha din ke paanchve hisse mein. Saath mein kitna samay lagega? Gino ki har ek din mein kund kitni baar bharta hai: 1 ÷ 1 = 1, 1 ÷ 1/2 = 2, 1 ÷ 1/4 = 4 aur 1 ÷ 1/5 = 5. Saath mein ve din mein 12 baar bharte hain, isliye unhe sirf din ka 1/12 hissa chahiye.

Chaar fawwaare din mein kund ko 1, 2, 4 aur 5 baar bharte hain; saath mein 12 baar, isliye din ke 1/12 hisse mein
Chaar fawwaare

Ek kanjoos ka daan

1150 CE mein likhi Lilavati mein Bhaskara II padhne waale ko chhedte hain: ek kanjoos ne ek bhikhaari ko chaandi ke sikke dramma ke 3/4 ke 2/3 ke 1/2 ke 1/4 ke 1/16 ka 1/5 diya. ‘Ka’ ka matlab guna hai, isliye daan 1/2 × 2/3 × 3/4 × 1/5 × 1/16 × 1/4 = 6/7680 = 1/1280 dramma hai. Kahaani mein ek dramma 1280 kauriyon ka tha, isliye bhikhaari ko theek ek kauri mili, sabse kam keemat ka sikka.

Dramma ka daan: fractions ki kadi ka guna 1/1280 dramma, yaani ek kauri
Dramma waala daan

Sikke aur fractions

12th century ke aas paas kai sikke chalte the: bade len den aur bachat ke liye sone ke dinar ya gadyana aur huna, roz ke vyapaar ke liye chaandi ke dramma ya tanka, chhoti khareed ke liye taambe ke kasu ya pana aur mashaka, aur chhutte ke liye kauri. Badalne ki dar ilaake aur sikkon ke vazan aur shuddhta ke hisaab se badalti thi. Kitaab ki maani gayi daron se 1 gold dinar = 12 dramma, 1 dramma = 4 pana aur 1 pana = 30 kauri, isliye 1 pana = 1/12 × 1/4 = 1/48 dinar aur 1 kauri = 1/30 pana = 1/1440 dinar.

12th century ke sikke: 1 dinar = 12 dramma = 48 pana = 1440 kauri, isliye pana 1/48 dinar aur kauri 1/1440 dinar
12th century ke sikke

Niyam kisne likhe

Fractions jaldi hi Bharat ki roz ki sanskriti mein aa gaye: Jaina vidwaan Umasvati (c. 150 CE) ne darshan ki ek rachna mein fraction ko lowest terms mein laane ki upma di. Brahmagupta ne apne Brahmasphutasiddhanta (628 CE) mein pehli baar fractions ke jod, ghataav, guna aur bhaag ke aam niyam likhe, lagbhag waise hi jaise hum aaj karte hain. Aryabhata par teeka likhte hue Bhaskara I (629 CE) ne guna ko square ke andar rectangles ki tarah samjhaaya; Shridharacharya (c. 750 CE), jinki Patiganita 1 ÷ 1/6 + 1 ÷ 1/10 + 1 ÷ 1/13 + 1 ÷ 1/9 + 1 ÷ 1/2 ka jod poochhti hai (jawaab 40), Mahaviracharya (c. 850 CE), Prithudakasvami aur Bhaskara II ne vishay ko aage badhaaya. Arab duniya aur North Africa ke ganitagyon, jaise Morocco ke al-Hassar (c. 1192 CE), ne ise aage le jaakar badhaaya, aur Europe mein ye lagbhag 17th century mein aam hua.

Fraction ke niyam kaise bane aur phaile: Shulbasutra, Umasvati, Brahmagupta, Bhaskara I, baad ke Bhartiya ganitagya, Bhaskara II, al-Hassar aur Europe
Fraction ke niyam kaise phaile

Ye kyun zaroori hai

Ye sawaal dikhaate hain ki fractions ka ganit asli kaamon se bana, jaise vediyan banaana, kund bharna aur sikke ginna, aur iska bada hissa pehle Bharat mein hal hua. Niyam khud fractions ka guna aur fractions ka bhaag mein hain. Har sawaal ke jawaab Chapter 8 ke NCERT solutions mein hain; poore Chapter 8 ke notes bhi dekho.

Practice karo

  1. 1/5 unit side waali kitni eentein 7 1/2 square units dhakengi?

    1. (a) 75/2
    2. (b) 375/2
    3. (c) 15/2
    4. (d) 25
    Jawab dekho

    Jawab: (b) 375/2

  2. Chaar fawwaare din mein kund ko 1, 2, 4 aur 5 baar bharte hain. Saath mein kitna samay lagega?

    1. (a) 12 din
    2. (b) 1/12 din
    3. (c) 1/4 din
    4. (d) 3 din
    Jawab dekho

    Jawab: (b) 1/12 din

  3. Lilavati ki paheli mein kanjoos ne dramma ka kitna hissa diya?

    1. (a) 1/1280
    2. (b) 1/640
    3. (c) 6/16
    4. (d) 1/20
    Jawab dekho

    Jawab: (a) 1/1280

  4. 1 pana = 30 kauri aur 1 pana = 1/48 gold dinar ho, to 1 kauri dinar ka kitna hissa hai?

    Jawab dekho

    Jawab: 1 kauri = 1/30 pana = 1/30 × 1/48 = 1/1440 gold dinar.

  5. Sridharacharya ki Patiganita 1 ÷ 1/6 + 1 ÷ 1/10 + 1 ÷ 1/13 + 1 ÷ 1/9 + 1 ÷ 1/2 ka jod poochhti hai. Jawaab kya hai?

    Jawab dekho

    Jawab: Har 1 ÷ 1/n, n ke baraabar hai, isliye jod 6 + 10 + 13 + 9 + 2 = 40 hai.

Students ke sawal

Shulbasutra kya hain aur fractions ke liye kyun zaroori hain?

Shulbasutra lagbhag 800 BCE tak purane Bhartiya geometry granth hain, jo yagya vediyan banaane ke liye likhe gaye. Inme non-unit fractions ka khulkar istemaal hai, fractions ke bhaag samet, jaise Baudhayana ka sawaal ki 1/5 unit side waali eenton se 7 1/2 square units kaise dhakein.

Lilavati ki kanjoos waali paheli kya hai?

Lilavati (1150 CE) mein Bhaskara II ek kanjoos ki baat karte hain jisne ek bhikhaari ko dramma ke 3/4 ke 2/3 ke 1/2 ke 1/4 ke 1/16 ka 1/5 diya. Guna karne par 6/7680 = 1/1280 dramma aata hai, aur ek dramma 1280 kauriyon ka tha, isliye daan sirf ek kauri tha, sabse chhota sikka.

Bhartiya fraction tareeke duniya tak kaise pahunche?

Brahmagupta ke baad Bhaskara I, Shridharacharya, Mahaviracharya, Prithudakasvami aur Bhaskara II ne vishay ko aage badhaaya. Arab duniya aur North Africa ke ganitagyon, jaise Morocco ke al-Hassar (c. 1192 CE), ne inhe apnaaya, phir ye Arabon ke zariye Europe pahunche aur wahan lagbhag 17th century mein aam hue.

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