Class 7 Maths Part 2 Chapter 2: Operations with Integers (Hinglish Notes)

Ye chapter kis baare mein hai?

Class 6 mein tumne integers jode aur ghataaye. Operations with Integers aage badhta hai: tokens ki thaili aur number patterns se negative numbers ke guna aur bhaag ko matlab deta hai, sign ke niyam dhoondhta hai, aur dekhta hai ki guna ki kaunsi properties abhi bhi sahi hain. Saath mein puzzles, carrom ki goti, negative marking waala exam, khaan ki lift, magic grid aur pattern machines hain.

Ek chhota recap

Rakesh do numbers sochta hai jinka jod 25 aur antar 11 hai: jodiyan aazmaane pe 18 aur 7 milte hain. −11 antar ke liye inhe adla-badli karo: 7 aur 18. Integers gati bhi batate hain. Agar carrom ki goti positive numbers pe daayin aur negative pe baayin chale, to a aur b ki do chaalon ke baad wo P = a + b pe hoti hai: 5 aur phir −7 ki chaal se wo −2 pe pahunchti hai.

Tokens mein hara token +1, laal token −1 hai, aur zero pair (har ek ka ek) 0 hai. 7 − 18 ke liye 11 zero pairs daalo taaki 18 positives nikaal sako; 11 negatives bachte hain, isliye 7 − 18 = −11. Kisi number ko ghatana uske additive inverse ko jodne jaisa hai: 7 − 18 = 7 + (−18) aur 4 − (−12) = 4 + 12.

Tokens se integers ka guna

Khaali thaili se shuru karo. a × b mein multiplier a batata hai kitni baar, aur multiplicand b batata hai tokens ka kaunsa group. Positive multiplier groups daalta hai; negative multiplier groups nikaalta hai.

Isliye 4 × (−2) mein 2 negatives 4 baar daalte hain: −8. (−4) × 2 mein thaili khaali hai, isliye har baar 2 zero pairs daalo aur 2 positives nikaalo, 4 baar; 8 negatives bachte hain: −8. (−4) × (−2) mein zero pairs se 2 negatives 4 baar nikaalo; 8 positives bachte hain: +8. Aur padho integers ka guna mein.

Patterns aur sign ke niyam

Jab multiplicand 3 hai, multiplier ke har kadam neeche aane pe guna 3 ghatta hai, aur pattern zero ke paar chalta rehta hai: (−1) × 3 = −3, (−2) × 3 = −6. Jab multiplicand −3 hai, har kadam neeche aane pe guna 3 badhta hai: (−1) × (−3) = 3, (−2) × (−3) = 6. Isliye guna ka size hamesha sizes ka guna hota hai, aur:

• Ek jaise sign (dono positive ya dono negative): guna positive.
• Alag sign: guna negative.

Saath hi 1 × a = a, aur (−1) × a = −a, yaani a ka additive inverse. 628 CE mein Brahmagupta ki Brahmasphutasiddhanta ne ye niyam pehli baar diye, positive numbers ko dhan (dhana) aur negative ko karz (rina) kehkar: do dhan ya do karz ka guna ya bhaag dhan hai, aur dhan aur karz ka karz.

Kaam mein integers

Ek exam mein sahi jawab pe +5 aur galat pe −2 milta hai; Mala ke 30 sahi aur 20 galat jawab se 30 × 5 + 20 × (−2) = 150 + (−40) = 110 marks bante hain. Khaan ki lift jo har minute 3 m neeche jaati hai, har minute apni jagah −3 badalti hai: zameen se ek ghante baad wo 60 × (−3) = −180 pe hai, aur zameen se 15 m upar se 45 minute baad 15 + 45 × (−3) = −120 pe.

Magic grid

4 × 4 grid mein ek number gol karo, uski row aur column kaat do, aur tab tak dohraao jab tak koi number na bache; phir gol kiye numbers ka guna karo. Kuch bhi chuno, guna −30240 hi aata hai. Raaz ye hai ki grid ek pahaada hai: har chune hue set mein har row number (2, −7, 3, 5) aur har column number (4, −2, 6, −3) theek ek baar aata hai.

Integers ka bhaag

Bhaag guna ko ulatta hai. (−100) ÷ 25 poochhta hai ki kis number ko 25 se guna karne pe −100 milega; kyunki 25 × (−4) = −100, jawab −4 hai. Isi tarah (−100) ÷ (−4) = 25 aur 50 ÷ (−25) = −2. Magnitudes ka bhaag karo aur wahi sign ke niyam lagaao: ek jaise sign se quotient positive, alag sign se negative. Dekho integers ka bhaag.

Guna ki properties

Integers ka guna commutative (3 × (−4) = (−4) × 3) aur associative ((5 × (−3)) × 4 = 5 × ((−3) × 4) = −60) hai, isliye kai integers ka guna kisi bhi kram mein kar sakte hain. −1 ko baar-baar guna karne pe 1, −1, 1, −1 milta hai: jis guna mein negative factors odd ginti mein hon, wo negative hai. Guna jod pe distributive bhi hai: 4 × (2 + (−3)) = 4 × 2 + 4 × (−3). Padho integers ke guna ki properties.

Chapter pattern machines ke saath khatam hota hai jo teen numbers lekar ek jawab deti hain, jaise a + b − c, jise tumhe pehchaanna hai. Har sawaal ka jawab Part 2 Chapter 2 ke NCERT solutions mein hai.