Integers ka bhaag

Bhaag guna ko ulatta hai

Tum jaante ho ki 20 ÷ 4 = 5 kyunki 4 × 5 = 20. Bhaag ek gum factor dhoondhta hai. Yahi vichaar negative numbers pe bhi chalta hai: (−100) ÷ 25 ke liye poochho kis number ko 25 se guna karne pe −100 milega? Kyunki 25 × (−4) = −100, isliye (−100) ÷ 25 = −4.

Isi tarah:

• (−100) ÷ (−4): kya guna −4 se −100 deta hai? (−4) × 25 = −100, isliye jawab 25 hai.
• 50 ÷ (−25): kya guna −25 se 50 deta hai? (−25) × (−2) = 50, isliye jawab −2 hai.

Bhaag ke sign ke niyam

Kyunki bhaag guna ko ulatta hai, iske sign bhi wahi niyam maante hain. Positive integers a aur b ke liye (b, 0 nahi):

Bhaag	Nateeja
a ÷ (−b)	−(a ÷ b), negative
(−a) ÷ b	−(a ÷ b), negative
(−a) ÷ (−b)	a ÷ b, positive

Chhote mein: magnitudes ka bhaag karo; ek jaise sign se quotient positive, alag sign se negative. 628 CE ke Brahmagupta ke niyam bhaag pe bhi lagte hain: do karz ka quotient dhan hai, aur dhan aur karz ka quotient karz hai.

Hal kiye gaye udaharan

• 36 ÷ (−18) = −2 (alag sign).
• (−46) ÷ (−23) = 2 (ek jaise sign).
• (−27) ÷ 9 = −3, aur (−56) ÷ (−2) = 28.
• Khaali jagah: ___ ÷ (−8) = 7 ka matlab hai (−8) × 7 = ___, isliye khaali jagah −56 hai; ___ ÷ 12 = −11 ka matlab 12 × (−11), isliye −132.

Rozaana ke sawaalon mein bhaag

• Taapmaan: ek hill station pe 6 ghante mein taapmaan ek jaisi raftaar se 18 °C gira. Har ghante ka badlaav (−18) ÷ 6 = −3 °C hai.
• Baanta hua nuksaan: ek stall chalaane waale chaar dost ₹1,200 ka nuksaan barabar baantte hain. Har ek ka hissa (−1200) ÷ 4 = −300 hai, yaani ₹300 ka nuksaan.
• Neeche jaana: ek lift ek jaisi raftaar se 60 minute mein zameen se −180 m tak jaati hai. Har minute wo (−180) ÷ 60 = −3 m chalti hai, yaani 3 m neeche.

Dhyaan rakhne ki baatein

Bhaag commutative nahi hai: 12 ÷ (−3) = −4, par (−3) ÷ 12 = −1/4. Do integers ka quotient integer ho, ye zaroori nahi; jaise 32 ÷ ((−36) × (−18)) = 32 ÷ 648 = 4/81, ek positive fraction, kyunki do negative factors divisor ko positive bana dete hain. Aur kisi bhi number ka zero se bhaag nahi ho sakta: koi number 0 se guna hokar non-zero jawab nahi deta, isliye is sawaal ka koi jawab nahi.

Ye kyun zaroori hai

Integers ka bhaag “har baar kitna?” ya “har ek ko kitna?” jaise sawaalon ka jawab deta hai jab rakam nuksaan, girawat ya karz ho. Bhaag check karne ke liye waapas guna karo: quotient guna divisor dividend dena chahiye. Dobara dekho integers ka guna aur integers ke guna ki properties. Textbook ke sawaalon ke jawab Part 2 Chapter 2 ke NCERT solutions mein hain; poore chapter ke liye Part 2 Chapter 2 ke notes.