Integers ka guna

Token model

Hare tokens +1 aur laal tokens −1 hain; ek hara aur ek laal milkar zero pair banaate hain, jo 0 hai. Guna ke liye khaali thaili se shuru karo. a × b mein multiplier a batata hai kitni baar, aur multiplicand b batata hai tokens ka kaunsa group:

• Agar multiplier positive hai, group ko utni baar thaili mein daalo.
• Agar multiplier negative hai, group ko utni baar thaili se nikaalo.

Chaar case

• 4 × 2: 2 positives 4 baar daalo: 8 positives, yaani +8.
• 4 × (−2): 2 negatives 4 baar daalo: 8 negatives, yaani −8.
• (−4) × 2: 2 positives 4 baar nikaalo. Thaili khaali hai, isliye har baar pehle 2 zero pairs daalo, phir 2 positives nikaalo. 8 negatives bachte hain: −8.
• (−4) × (−2): 2 negatives 4 baar nikaalo, phir se zero pairs se. 8 positives bachte hain: +8.

Positives nikaalna negatives daalne jaisa hi hai, isliye (−4) × 2 ko 2 negatives 4 baar jodna bhi maan sakte hain. Aur number ke tokens kaise bhi dikhein, jawab wahi aata hai: 2 laal tokens, ya 4 laal aur 2 hare, dono −2 hain, aur dono ka 4 guna −8 hai.

Zero ke neeche bhi chalne waale patterns

Likho 3 × 3 = 9, 2 × 3 = 6, 1 × 3 = 3, 0 × 3 = 0. Har baar multiplier 1 ghatta hai to guna 3 ghatta hai, isliye agle hain (−1) × 3 = −3 aur (−2) × 3 = −6. −3 multiplicand ke saath, multiplier ke har ghatne pe guna 3 badhta hai: 0 × (−3) = 0, phir (−1) × (−3) = 3 aur (−2) × (−3) = 6. Patterns tokens se mel khaate hain.

Sign ke niyam

Guna ka magnitude kabhi signs pe depend nahi karta: wo magnitudes ka guna hai. Sign in niyamon se chalta hai:

Dono numbers ke sign	Guna ka sign
dono positive, ya dono negative	positive
ek positive, ek negative	negative

Do khaas case yaad rakho: 1 × a = a, aur (−1) × a = −a, yaani a ka additive inverse. 628 CE mein Brahmagupta ne yahi niyam shabdon mein diye: do dhan ya do karz ka guna dhan hai; dhan aur karz ka guna karz hai. Ye pehli baar tha ki kisi ne inhe likha.

Hal kiye gaye udaharan

• Negative marking: ek exam mein sahi jawab pe +5 aur galat pe −2 milta hai. Mala ke 30 sahi aur 20 galat hain: 30 × 5 + 20 × (−2) = 150 + (−40) = 110 marks. Sabse zyaada sambhav marks 50 × 5 = 250 hain aur sabse kam, saare 50 galat hone pe, 50 × (−2) = −100.
• Khaan ki lift: har minute 3 m neeche jaana har minute −3 hai, isliye 60 minute mein badlaav 60 × (−3) = −180 hai, yaani zameen se 180 m neeche.
• Bina hisaab: agar 123 × 456 = 56088, to (−123) × 456 = −56088 aur (−123) × (−456) = 56088, kyunki sirf sign badalta hai.

Ye kyun zaroori hai

Integers ka guna baar-baar hone waale nuksaan, taapmaan ke girne, karz aur shuruaati jagah se neeche ki chaal ko ek hi kadam mein sambhaal leta hai. Aage seekho integers ka bhaag 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.