Number System Mein Base Kya Hai? | Class 8 Maths Book 1 Ch 3

📘 Class 8 · Ganita Prakash Book 1Ch 3 · A Story of NumbersPost 4 / 8⏱ 25–30 min

Number System Mein Base Kya Hai? 🏺

IS POST MEIN

Pehle Yeh Aata Hai?Kahani Concept Deep Concept Examples Real Life Galti 10 MCQs FAQs Practice HOTS Olympiad Recap

✅ Pehle Yeh Aata Hai?

Yeh post shuru karne se pehle, in teen sawaalon ko 1 minute mein khud try kijiye:

Roman landmark numbers kaun se?
10 × 10 × 10 = ?
5³ = ?

Answers dekho

1, 5, 10, 50, 100, 500, 1,000
1,000 = 10³
125

Teeno sahi? Badhiya, aage chaliye! Koi galat hua? Koi baat nahi, Post 3 — Roman Numerals Kaise Likhein? aur Book 1 Ch 2 — Exponent Aur Base Kya Hain? dobara dekh lijiye.

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🎬 Kahani: Misr Ke Mandir Ki Deewar

Museum mein Ishaan ne Misr (Egypt) ki ek patthar ki patti dekhi. Usme chinh: 2 kamal ke phool, 3 rassi ke gole, 1 ghoda-naal, 5 khadi lakeerein. Guide: “Yeh ek number hai, 5,000 saal purana.” 🏺

Ishaan: “Phool aur rassi number kaise?” Guide muskuraaya: “Inka raaz ek hi shabd mein hai: base.”

Kaunsa number? Aur Roman se yeh behtar kyun tha? Chaliye raaz kholte hain. 🔍

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📖 Concept: Group Ke Group: Base Ka Idea

💡 Is post ki ek badi baat: Base-n system mein landmark numbers hain 1, n, n², n³… (har agla pichhle ka n guna). Kyunki sab landmark ek hi number ki powers hain, do landmark ka guna bhi landmark hota hai: yahi hisaab aasaan banaata hai.

Step 1: Misr ke landmark: 10-10 ka group

Kankad lo. 10 kankad = ek chhota dher. 10 chhote dher = ek bada dher (100). 10 bade = 1,000…

Landmark	Chinh (naam)
1	\| (khadi lakeer)
10	∩ (ghoda-naal / haddi)
100	🌀 (rassi ka gola)
1,000	🪷 (kamal)
10,000	☝️ (ungli)
1,00,000	🐸 (mendhak)
10,00,000	🙌 (haath uthaaye devta)

WHY ise base-10 kehte hain? Har agla landmark pichhle ka 10 guna: 1, 10, 10², 10³… Sab 10 ki powers.

Step 2: Likhna: har landmark kitni baar?

2,135 = 1000 + 1000 + 100 + 10 + 10 + 10 + 5 → 🪷🪷 🌀 ∩∩∩ |||||

WHY koi chinh 10 baar nahi aata? 10 baar aaya to woh agle landmark ka ek chinh ban jaata (10 | = ∩). Isliye har chinh 0 se 9 baar.

Step 3: Base 10 hi kyun? Base 5 bhi!

Ab 5-5 ka group: 1, 5, 25, 125, 625… Apne chinh: ● = 1, ▲ = 5, ■ = 25, ⬟ = 125.

89 = 25 + 25 + 25 + 5 + 5 + 4 → ■■■ ▲▲ ●●●●.

Paribhaasha: pehla landmark 1, har agla pichhle ka n guna → base-n. Landmark = n⁰, n¹, n², n³… Base-10 ko decimal kehte hain.

Step 4: Base-n mein koi chinh n se kam baar

Base-5 mein ● 5 baar = ▲. Isliye har chinh 0 se 4 baar. Base-7 mein 0 se 6 baar. Base-n mein 0 se (n − 1) baar.

WHY yeh niyam kaam ka? Har number ka ek hi roop banta hai: koi confusion nahi. Roman mein IIII aur IV dono chalte the!

Step 5: Guna aasaan kyun?

Base-10 mein: ∩ × 🌀 = 10 × 100 = 1,000 = 🪷. Ek landmark × ek landmark = ek landmark.

$10^a \times 10^b = 10^{a+b}$

WHY? Exponent ke niyam (Book 1 Ch 2 — Exponent Ke Niyam: Guna Aur Power): powers ka guna = exponents ka jod. Roman mein V × L = CCL (landmark nahi), yahan hamesha landmark!

Shortcut: Misr mein kisi number ko ∩ (10) se guna karna = har chinh ko agla chinh bana do (| → ∩, ∩ → 🌀…).

🪜 Atak Gaye? Aise Socho

Base pata karo: har landmark pichhle ka kitna guna? Woh n hai.
Landmark likho: 1, n, n², n³… jab tak number se bada na ho.
Sabse bade landmark se shuru karo: kitni baar fit hota hai?
Bacha hua agle chhote landmark mein bhaago.
Check: har chinh n se kam baar aaya? Kul jod wahi number?

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🔐 Deep Concept: Ek Hi Number Ki Powers Kyun? (Base Ka Jaadu)

Roman landmark 1, 5, 10, 50… do alag guna (× 5, × 2) se bane. Misr ke landmark ek hi guna (× 10) se. Bas itna sa farak, par asar bahut bada:

Har roop ek: har chinh 0–9 baar, koi do tareeke nahi.
Jod mein carry ek jaisa: hamesha “10 chhote = 1 bada”, chahe kisi bhi landmark pe ho. Roman mein kabhi 5, kabhi 2!
Guna mein landmark × landmark = landmark: n^a × n^b = n^a+b.

Socho kyun “n” koi bhi ho sakta hai: 10 hamare 10 ungliyon ki wajah se. Agar 8 ungliyaan hoti to shayad base-8. Computer ki “ungliyaan” sirf on/off hain, isliye base-2! Maths ko farak nahi padta, niyam wahi.

Unifying idea: base = “kitne chhote milke ek bada” ka ek fixed jawab. Yahi ek jawab poore system ko saaf aur hisaab ko aasaan banaata hai. 🎯

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✏️ Solved Examples

Example 1 (Easy): Ishaan ki patti: 2 kamal, 3 rassi, 1 ghoda-naal, 5 lakeer. Number?

2 × 1,000 + 3 × 100 + 1 × 10 + 5 × 1 = 2,315.

Check: waapas todo: 2,315 = 2000 + 300 + 10 + 5 → 🪷🪷 🌀🌀🌀 ∩ ||||| ✅

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Example 2 (Medium): Base-7 ke pehle 5 landmark, aur 200 ko base-7 landmarks mein todo.

Landmark: 1, 7, 49, 343, 2,401. 200 = 4 × 49 + 0 × 7 + 4 → 4 baar 49 + 4 baar 1.

Check: 196 + 4 = 200 ✅, aur har landmark 7 se kam baar (4, 0, 4) ✅

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Example 3 (NCERT style): Base-5 chinh mein jodo: (■ ▲▲▲ ●●●) + (■■ ▲▲ ●●●●)

■ 3, ▲ 5, ● 7. ● 7 = ▲ + ●● → ▲ 6, ● 2. ▲ 6 = ■ + ▲ → ■ 4, ▲ 1.

Jawab: ■■■■ ▲ ●● = 100 + 5 + 2 = 107.

Check: 43 + 64 = 107 ✅

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Museum ka raaz: patti pe 2,315 likha tha! Ishaan ne guide ko bataaya: “Kamal = 1,000, rassi = 100, naal = 10, lakeer = 1. Har ek 10 guna!” Guide: “Tum to Misr ke munshi ban gaye!” 📜

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🏠 Base Ka Idea Roz Ki Zindagi Mein

💵 Note-sikke: ₹1, ₹10, ₹100 (aur ₹2,000 jaise beech ke): har 10 ke ek bada note, base-10 jaisa.
⏰ Samay: 60 second = 1 minute, 60 minute = 1 ghanta: base-60 (Post 7 — Place Value Ki Kahani)!
🥚 Darjan aur gross: 12 = darjan, 12 darjan = 1 gross (144): base-12 ke landmark.
💻 Computer: sirf 0 aur 1: base-2; 1 KB ≈ 2¹⁰ byte.
📏 Metric: mm, cm, m, km: har kadam 10 ya 1,000 guna. Isliye conversion aasaan.

Jahan bhi “itne chhote = ek bada” fixed ho, wahan base chhupa hai.

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🧭 Kab Use Karein, Kab Nahi

Situation	Kya karein	Kyun
Har landmark same guna	✅ Base-n system	1, n, n², n³…
Landmark alag guna (× 5, × 2)	❌ Base nahi (Roman)	Guna mushkil
Chinh n ya zyada baar	❌ Agla landmark banao	n chhote = 1 bada
Landmark × landmark	✅ Naya landmark	nᵃ × nᵇ = nᵃ⁺ᵇ
Bahut bada number (10¹²)	❌ Misr mein naya chinh chahiye	Agla kadam: place value

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❌→✅ Worked Mistake

Sawaal: Base-5 mein 30 ko landmark (1, 5, 25) mein likho.

❌ Galat: “30 = ▲▲▲▲▲▲ (6 baar 5).” ▲ 5 baar se zyada nahi ho sakta!

✅ Sahi: 30 = 25 + 5 → ■ ▲. Check: 25 + 5 = 30 ✅, har chinh 5 se kam baar ✅

Doosri common galti: base-5 ke landmark 1, 5, 10, 15 maan lena (jod se). Landmark guna se bante hain: 1, 5, 25, 125.

🧠 Rule of thumb: Landmark = base ki powers (guna se, jod se nahi). Koi chinh base jitni baar nahi aata.

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📝 10 MCQs

Q1. Base-10 ke landmark?

(a) 1, 10, 20, 30

(b) 1, 10, 100, 1,000

(c) 1, 5, 10, 50

(d) 10, 20, 40

Solution dekho

Sahi jawab: (b)

(a) galat: Jod se bane: landmark guna se bante.
(b) sahi: 10 ki powers: 1, 10, 100… ✅
(c) galat: Yeh Roman landmark hain, base nahi.
(d) galat: Yeh doubling hai, 10 ki powers nahi.

Q2. Base-5 ka chautha landmark (1 se shuru)?

(a) 20

(b) 25

(c) 125

(d) 625

Solution dekho

Sahi jawab: (c)

(a) galat: 20 = 4 × 5: jod ka pattern hai.
(b) galat: 25 teesra landmark hai (5²).
(c) sahi: 1, 5, 25, 125: chautha 5³ = 125 ✅
(d) galat: 625 paanchva landmark hai (5⁴).

Q3. Misr mein kamal = ?

(a) 10

(b) 100

(c) 1,000

(d) 10,000

Solution dekho

Sahi jawab: (c)

(a) galat: 10 ka chinh ghoda-naal hai.
(b) galat: 100 ka chinh rassi ka gola hai.
(c) sahi: Kamal ka phool = 1,000 ✅
(d) galat: 10,000 ka chinh ungli hai.

Q4. Base-7 mein koi chinh max kitni baar?

(a) 5

(b) 6

(c) 7

(d) 10

Solution dekho

Sahi jawab: (b)

(a) galat: 5 ke baad bhi ek aur aa sakta hai.
(b) sahi: 0 se 6 baar, 7 baar = agla landmark ✅
(c) galat: 7 baar aaye to agla chinh banta.
(d) galat: 10 sirf base-10 ki baat hai.

Q5. Base-5: ■■ ▲ ●●● = ?

(a) 53

(b) 58

(c) 23

(d) 28

Solution dekho

Sahi jawab: (b)

(a) galat: ■ = 25 hai, 20 nahi; dobara jodo.
(b) sahi: ■■ = 50, phir 50 + 5 + 3 = 58 ✅
(c) galat: ■ ko 10 maan liya, 25 hai.
(d) galat: ▲ ko 0 maana, woh 5 hai.

Q6. Base system mein guna aasaan kyun?

(a) Chinh sundar

(b) Landmark × landmark = landmark

(c) Zero hai

(d) Kam chinh

Solution dekho

Sahi jawab: (b)

(a) galat: Sundarta se hisaab nahi hota.
(b) sahi: nᵃ × nᵇ = nᵃ⁺ᵇ, phir landmark ✅
(c) galat: Misr mein zero tha hi nahi.
(d) galat: Chinh ki ginti asli wajah nahi.

Q7. Misr ke kisi number ko 10 se guna: kya karte?

(a) Har chinh agla chinh

(b) Ek | jodo

(c) Kuch nahi

(d) Chinh ulte

Solution dekho

Sahi jawab: (a)

(a) sahi: | → ∩, ∩ → rassi: sab ek kadam upar ✅
(b) galat: Ek | jodna to +1 hai, × 10 nahi.
(c) galat: Guna se number badalta hai.
(d) galat: Chinh ulta karne ka koi niyam nahi.

Q8. Base-n ke landmark?

(a) n, 2n, 3n

(b) 1, n, n², n³

(c) n, n+1, n+2

(d) 1, 2, 3

Solution dekho

Sahi jawab: (b)

(a) galat: Yeh jod ka pattern hai, guna nahi.
(b) sahi: n ki powers, 1 = n⁰ se shuru ✅
(c) galat: Har baar +1, koi grouping nahi.
(d) galat: Yeh sirf natural numbers ki ginti.

Q9. Base-5 mein ● 5 baar aa gaya. Kya karein?

(a) Chhod do

(b) Ek ▲ bana do

(c) Ek ■ bana do

(d) Mita do

Solution dekho

Sahi jawab: (b)

(a) galat: Aise chhodne se roop galat rahega.
(b) sahi: Paanch ● milke ek ▲ (5) ✅
(c) galat: ■ = 25, 5 ● sirf 5 hain.
(d) galat: Mitaane se number badal jaayega.

Q10. Base-10 ko kya kehte?

(a) Binary

(b) Decimal

(c) Roman

(d) Sexagesimal

Solution dekho

Sahi jawab: (b)

(a) galat: Binary = base-2 hota hai.
(b) sahi: Base-10 ko decimal kehte hain ✅
(c) galat: Roman base system nahi hai.
(d) galat: Sexagesimal = base-60 hota hai.

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❓ FAQs (Bachchon Ke Asli Sawaal)

1. Kya Misr ke paas zero tha?

Nahi. Jis landmark ki zarurat nahi, uska chinh bas nahi likhte. Isliye place value nahi tha.

2. Misr mein 10 lakh se bada?

Naya chinh chahiye tha. Yahi is system ki kami thi (Post 6 — Abacus Aur Decimal System).

3. Base 1 ho sakta hai?

Landmark 1, 1, 1… sab same! Grouping hi nahi hoti: yeh tally jaisa hai. Isliye base kam se kam 2.

4. Base-12 ke fayde?

12 ko 2, 3, 4, 6 se bhaag sakte: aadha, tihaai, chauthaai sab poore. Isliye darjan itna lokpriya.

5. Chinh ka kram matter karta hai?

Misr mein nahi: 🪷 ∩ | aur | ∩ 🪷 dono 1,011. Har chinh ki value fixed. Place value mein kram hi sab kuch (Post 7 — Place Value Ki Kahani).

6. Base aur exponent (Ch 2) ka rishta?

Base-n ke landmark = n ke exponent 0, 1, 2… Wahi “base” shabd: n^k mein n base hai!

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🏋️ Practice Zone (12 Sawaal)

Q1. Easy Base-10 ke pehle 5 landmark?

Solution dekho

1, 10, 100, 1,000, 10,000. Check: har ek pichhle ka 10 guna ✅

Q2. Easy Base-4 ke pehle 4 landmark?

Solution dekho

1, 4, 16, 64. Check: 16 × 4 = 64 ✅

Q3. Easy Misr: 🪷 🌀🌀 ∩∩∩∩ ||| = ?

Solution dekho

1000 + 200 + 40 + 3 = 1,243. Check: waapas likh ke dekho ✅

Q4. Easy Base-5 chinh: ■ ●● = ?

Solution dekho

25 + 2 = 27. Check: ▲ nahi hai, to 5 ka hissa 0 ✅

Q5. Medium Misr mein 4,082 likho.

Solution dekho

🪷🪷🪷🪷 ∩∩∩∩∩∩∩∩ ||. Check: 4000 + 80 + 2 ✅, rassi 0 baar

Q6. Medium Base-5 mein 74 (●, ▲, ■ se).

Solution dekho

74 = 2 × 25 + 4 × 5 + 4 → ■■ ▲▲▲▲ ●●●●. Check: 50 + 20 + 4 = 74 ✅

Q7. Medium Base-7 landmarks se 100 todo.

Solution dekho

100 = 2 × 49 + 0 × 7 + 2 → 2 baar 49 + 2 baar 1. Check: 98 + 2 = 100 ✅

Q8. Medium Misr mein jodo: (🌀 ∩∩∩∩∩∩ ||||||||) + (∩∩∩∩∩ ||||).

Solution dekho

| 12 → ∩ + || ; ∩ 12 → 🌀 + ∩∩ → 🌀🌀 ∩∩ || = 222. Check: 168 + 54 = 222 ✅

Q9. NCERT style Misr ke number ko 100 se guna karna ho to shortcut?

Solution dekho

Har chinh do kadam upar: | → 🌀, ∩ → 🪷… WHY: 100 = 10², exponent +2. Check: 3 × 100 = 300 (||| → 🌀🌀🌀) ✅

Q10. NCERT style Base-5 mein ■ × ▲ = ? (landmark rehta hai?)

Solution dekho

25 × 5 = 125 = ⬟: haan, landmark! WHY: 5² × 5¹ = 5³. Check: 125 = 5³ ✅

Q11. NCERT style Base-3 mein sabse chhota number jiske liye teesra landmark (9) chahiye?

Solution dekho

Landmark 1, 3, 9 → 9 (pehli baar 9 wala landmark). Check: 8 = 2 × 3 + 2 (sirf do landmark) ✅

Q12. NCERT style Base-6 (landmark 1, 6, 36) mein 1–50 ke beech kis number ko sabse zyada chinh chahiye?

Solution dekho

Har landmark max 5 baar. 35 = 5 × 6 + 5 → 10 chinh sabse lamba. Check: 5 + 5 = 10, aur 47 = 36 + 6 + 5 → 7 ✅

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🧠 HOTS (Sochne Wale Sawaal)

Q1. Ek system ke landmark 1, 3, 9, 27. Base kya? 50 kaise?

Solution dekho

Base 3. 50 = 1 × 27 + 2 × 9 + 1 × 3 + 2 × 1. Check: 27 + 18 + 3 + 2 = 50 ✅

Q2. Misr mein 9,999 likhne mein kitne chinh? Hamare system mein?

Solution dekho

9 + 9 + 9 + 9 = 36 chinh; hamare mein 4 ank. WHY: place value se har landmark ek ank mein (Post 7 — Place Value Ki Kahani).

Q3. Kya do alag base ke systems mein ek hi number ka roop same ho sakta hai?

Solution dekho

Haan, chhote number: 3 base-5 mein bhi ●●● aur base-10 mein bhi |||. Par base se chhota hona chahiye. Check: 3 < 5 aur 3 < 10 ✅

Q4. Base-5 mein kisi number ko 5 se guna karne ka shortcut?

Solution dekho

Har chinh agla: ● → ▲, ▲ → ■, ■ → ⬟. WHY: 5^k × 5 = 5^k+1.

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🏆 Olympiad Corner

Q1. Base-4 mein sabse bada number jo sirf pehle 3 landmark (1, 4, 16) se bane?

Solution dekho

3 × 16 + 3 × 4 + 3 = 63 (= 4³ − 1).

Q2. Misr mein 1 se 100 tak sab number likhne mein ∩ kitni baar?

Solution dekho

Dahaai ank ka jod: 10 × (1 + … + 9) = 450; 100 mein ∩ nahi → 450.

Q3. Base-n ka 4th landmark 343. n?

Solution dekho

n³ = 343 → n = 7.

Q4. Base-2 landmarks se 13 todo.

Solution dekho

8 + 4 + 1 → 1 × 8 + 1 × 4 + 0 × 2 + 1 × 1.

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💗 CharuMam Motivation

Ishaan ko kamal aur rassi pehle ajeeb lage. Ab woh samjha ki chinh kuch bhi ho sakte hain: asli taakat “har baar 10 guna” wale niyam mein hai. 💡

Aapne aaj woh idea samjha jo poore aadhunik hisaab ki neev hai. Seena taan ke chalo! 💪

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🟢 Master Recap

💡 Chaabi: Base-n system mein landmark numbers hain 1, n, n², n³… (har agla pichhle ka n guna). Kyunki sab landmark ek hi number ki powers hain, do landmark ka guna bhi landmark hota hai: yahi hisaab aasaan banaata hai.
Base-n: landmark 1, n, n², n³… (har agla n guna).
Misr: base-10, chinh lakeer, naal, rassi, kamal, ungli, mendhak, devta.
Har chinh 0 se (n − 1) baar; n baar = agla landmark.
Landmark × landmark = landmark (nᵃ × nᵇ = nᵃ⁺ᵇ).
Isliye jod-guna Roman se aasaan.
Base-10 = decimal.
Kami: bade landmark ke liye naye chinh.

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👉 Aage Kya?

Ab ek hi number ko alag base mein likhna seekhte hain: 45 base-2 mein kaisa? Base-5 mein? Aur waapas kaise? Agle post mein: base conversion! 🔄

← Pichla: Post 3 — Roman Numerals Kaise Likhein?📚 Chapter 3 ke saare posts Agla: Post 5 — Base 2, Base 5: Number Badlo →

Koi doubt? CharuMam AI Tutor se poochho, ya comment mein likho! 💬

✅ Pehle Yeh Aata Hai?

🎬 Kahani: Misr Ke Mandir Ki Deewar

📖 Concept: Group Ke Group: Base Ka Idea

Step 1: Misr ke landmark: 10-10 ka group

Step 2: Likhna: har landmark kitni baar?

Step 3: Base 10 hi kyun? Base 5 bhi!

Step 4: Base-n mein koi chinh n se kam baar

Step 5: Guna aasaan kyun?

🪜 Atak Gaye? Aise Socho

🔐 Deep Concept: Ek Hi Number Ki Powers Kyun? (Base Ka Jaadu)

✏️ Solved Examples

🏠 Base Ka Idea Roz Ki Zindagi Mein

🧭 Kab Use Karein, Kab Nahi

❌→✅ Worked Mistake

📝 10 MCQs

❓ FAQs (Bachchon Ke Asli Sawaal)

🏋️ Practice Zone (12 Sawaal)

🧠 HOTS (Sochne Wale Sawaal)

🏆 Olympiad Corner

💗 CharuMam Motivation

🟢 Master Recap

👉 Aage Kya?

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