1.1.2
Intuition Ek line mein badi baatWahi digit ek alag amount represent karti hai depending on kahan hai woh. Digit ki asli value = (uska face value) × (jis place par hai) . Yahi tarika hai jisse hum sirf 10 symbols se infinitely many numbers bana lete hain.
Yeh KAUNSI problem solve karta hai? Socho agar tally marks se bhedein ginein: 4728 marks padhna mushkil hai. Har quantity ke liye naya symbol banaane ke bajay, hum 0 9 position ko extra information carry karne dete hain.
Position KYUN? Kyunki hum das ke groups mein count karte hain (humari das ungliyan hoti hain). Jab bhi koi "column" 9 0 ek ko baayein column mein carry karta hai. Isliye baayein wala har column apne daayein wale column se 10 guna zyada worth ka hota hai.
Definition Face value vs Place value
Face value 547 5 5 Place value 547 5 5 × 100 = 500
Sabse daayein column se shuru karo aur baayein jaate jaate 10 se multiply karte jao:
Units = 1 0 0 = 1 Tens = 1 0 1 = 10 Hundreds = 1 0 2 = 100 Thousands = 1 0 3 = 1000
Toh 3 485 place values ka sum hai:
3485 = 3 × 1 0 3 + 4 × 1 0 2 + 8 × 1 0 1 + 5 × 1 0 0
Indian place value system mein digits ko international system se alag group kiya jaata hai. Hundreds ke baad yeh 3-2-2-2... grouping mein chalta hai.
Place
Power of 10
Value
Units
1 0 0 1
Tens
1 0 1 10
Hundreds
1 0 2 100
Thousands
1 0 3 1,000
Ten Thousands
1 0 4 10,000
Lakhs 1 0 5 1,00,000
Ten Lakhs
1 0 6 10,00,000
Crores 1 0 7 1,00,00,000
Ten Crores
1 0 8 10,00,00,000
Intuition Commas kahan lagte hain (Indian system)Pehla comma daayein se 3 digits ke baad (thousands), phir har 2 digits ke baad: 7 , 54 , 32 , 109 padh sako.
International se Bridge: 1 = 100 1 = 10 naming/grouping badal jaati hai.
Worked example Example 1 — Chosen digit ka place value6 , 73 , 428 7 ka place value nikalo.
Step: 7 ka column locate karo. Yeh step KYUN? Place value sirf position par depend karta hai.Daayein se padhte hain: 8(units), 2(tens), 4(hundreds), 3(thousands), 7(ten thousands ), 6(lakhs).
7 ten thousands (1 0 4
Place value = 7 × 10000 = 70000 KYUN? Kyunki place value = face value × place.
Worked example Example 2 — Dono taraf expanded form50 , 204
5 × 1 0 4 + 0 × 1 0 3 + 2 × 1 0 2 + 0 × 1 0 1 + 4 × 1 0 0 = 50000 + 0 + 200 + 0 + 4 = 50204 Zeros KYUN matter karte hain: thousands mein 0 placeholder hai — woh column khula rakhta hai taaki 5
Worked example Example 3 — Place aur face value ka difference9 , 08 , 675 8 ka (place value −
8 thousands (1 0 3 KYUN? Daayein se baayein: 5,7,6,8 → 4th place.
Place value = 8 × 1000 = 8000 = 8
Difference = 8000 − 8 = 7992
Common mistake "0 ka place value 0 hai, toh iska koi fark nahi padta."KYUN sahi lagta hai: 0 × = 0 place value sach mein 0 Fix yeh hai: 0 place value 0 placeholder ki tarah iska kaam bahut bada hai — isko hata do aur 5 0 4 54 sum mein 0 positions ke liye sab kuch karta hai.
Common mistake Face value aur place value mein confusion.KYUN sahi lagta hai: tum digit 7 dekhte ho, toh "value = 7" natural lagta hai.
Fix yeh hai: pehle "kaunsa column?" poochho. Face value column ignore karta hai; place value column ki power of ten se multiply karta hai.
Common mistake Indian system mein har 3 digits par commas lagana.KYUN sahi lagta hai: international system mein aisa hi hota hai (thousands, millions).
Fix yeh hai: Indian system = 3 phir 2-2-2 . 1234567 → 12 , 34 , 567 na ki 1 , 234 , 567
Recall Feynman: ek 12 saal ke bacche ko explain karoNumber columns ko shelves par boxes ki tarah socho, har box apne daayein wale box se 10 guna bada. Chhota box ones rakhta hai (jaise coins). Agli box tens rakhti hai (jaise ₹10 notes), agli hundreds, phir thousands... Wahi coin coins box mein chhota lagta hai lekin "thousands" box mein baitha ho toh bahut zyada represent karta hai — kyunki woh box thousands worth ka hai. 0 matlab "yeh box khali hai" — lekin tum phir bhi khali box rakhte ho taaki baaki boxes galat jagah slide na kar jaayein!
Mnemonic Indian ladder padhna"U T H — Th Th — L L — Cr Cr"
Units, Tens, Hundreds → Thousand, Ten-Thousand → Lakh, Ten-Lakh → Crore, Ten-Crore. 3 ke baad comma, phir 2-2 par hop karo .
Digit ka face value kya hota hai? Digit khud, uski position ignore karke.
Digit ka place value kya hota hai? Face value × uski place ki value (power of ten).
3,52,000 mein 5 ka place value? 5 ten-thousands mein hai = 5 × 1 0 4 = 50000
"Lakhs" place ki value power of 10 mein? 1 0 5 = 1 , 00 , 000
"Crores" place ki value power of 10 mein? 1 0 7 = 1 , 00 , 00 , 000
1 crore mein kitne lakh hote hain? 100 lakh.
Hum baayein column mein carry KYUN karte hain? Har baayein column apne daayein wale se 10× worth ka hai, isliye ek column mein 10 = agli column mein 1.
50,204 mein thousands ka 0 kya kaam karta hai? Placeholder jisse 5 ten-thousands place mein rahe; sum mein 0 contribute karta hai.
4085 ka expanded form? 4 × 1 0 3 + 0 × 1 0 2 + 8 × 1 0 1 + 5 × 1 0 0
Indian comma grouping rule kya hai? Pehla comma 3 digits ke baad, phir har 2 digits par.
9,08,675 mein 8 ka place value minus face value? 8000 − 8 = 7992
Number Systems — base and digits Expanded form and standard form Rounding and Estimation Comparing and Ordering Numbers International vs Indian Number System Powers of Ten and Exponents
Worth 10x column on right
Place value = face x place
Expanded form N = sum d_i x 10^i
