Integers ke guna ki properties

1 aur −1 se guna

Tokens ka ek group khaali thaili mein ek baar daalne pe theek wahi group bachta hai, isliye har integer a ke liye, positive ho ya negative, 1 × a = a. −1 se guna group ko ek baar nikaalta hai, jisse uska ulta bachta hai: (−1) × a = −a, yaani a ka additive inverse. Jaise (−1) × 7 = −7 aur (−1) × (−7) = 7.

Kram maayne nahi rakhta: commutative

3 × (−4) = −12 ki tulna (−4) × 3 = −12 se karo, ya (−15) × (−8) = 120 ki (−8) × (−15) = 120 se. Numbers ki jagah badalne se magnitude wahi rehta hai, kyunki magnitudes bas positive numbers ka guna hain, aur sign bhi wahi rehta hai, kyunki dono sign kisi bhi kram mein wahi jodi hain. Isliye integers ka guna commutative hai: a × b = b × a.

Grouping maayne nahi rakhti: associative

5 × (−3) × 4 ko dono tarah karo: (5 × (−3)) × 4 = (−15) × 4 = −60, aur 5 × ((−3) × 4) = 5 × (−12) = −60. (5 × 4) × (−3) = 20 × (−3) = −60 bhi. Integers ka guna associative hai: (a × b) × c = a × (b × c). Commutative property ke saath iska matlab hai ki teen ya zyaada integers ka guna kisi bhi kram mein kar sakte hain: 25 × (−6) × 12 kisi bhi jodi se shuru karo, −1800 hi aata hai, aur pehle 25 × 12 = 300 karna ise aasaan bana deta hai.

Lambe guna ka sign

−1 ko baar-baar guna karo: (−1) × (−1) = 1, phir × (−1) = −1, phir 1, phir −1. Har negative factor ke saath sign palatta hai. Isliye:

• negative factors even ginti mein hon to guna positive;
• negative factors odd ginti mein hon to guna negative.

Jaise (−2) × (−1) × (−5) × (−3) mein chaar negative factors hain, isliye ye +30 hai, jabki (−7) × 4 × (−1) mein do hain, isliye ye +28 hai.

Jod pe failna: distributive

Kya 5 × (4 + (−2)) aur 5 × 4 + 5 × (−2) ek hi hain? Baayin taraf 5 × 2 = 10 hai aur daayin taraf 20 + (−10) = 10. Tokens dikhate hain kyun. Tokens ko 4 rows mein rakho, positives ke 2 columns aur negatives ke 3 columns ke saath:

Poora rectangle 4 × (2 + (−3)) hai, aur iske do hisse 4 × 2 aur 4 × (−3) hain, isliye dono barabar hone chahiye: a × (b + c) = a × b + a × c. Ye negative multiplier ke saath bhi sahi hai: (−2) × (4 + (−3)) = (−2) × 1 = −2, aur (−2) × 4 + (−2) × (−3) = −8 + 6 = −2.

Properties se hisaab aasaan

Properties lambe guna ke bina hisaab karne mein madad karti hain. (−548) × 972 = −532656 diya ho, to (−547) × 972 ki value −532656 + 972 = −531684 hai, kyunki −547, −548 se ek zyaada hai. Ye chapter ke magic grid ko bhi samjhaati hain: uske numbers ek pahaada banaate hain, aur commutative aur associative properties ki wajah se gol kiye numbers ka har set ek hi guna deta hai, −30240.

Ye kyun zaroori hai

Ye niyam negative numbers ke saath bhi hisaab ko surakshit tareeke se ulatne-palatne aur group karne dete hain, jo aage algebra ki buniyaad hai. Dobara dekho integers ka guna aur integers ka bhaag. Textbook ke sawaalon ke jawab Part 2 Chapter 2 ke NCERT solutions mein hain; poore chapter ke liye Part 2 Chapter 2 ke notes.