Base 2, Base 5: Number Badlo | Class 8 Maths Book 1 Ch 3

📘 Class 8 · Ganita Prakash Book 1Ch 3 · A Story of NumbersPost 5 / 8⏱ 25–30 min

Base 2, Base 5: Number Badlo 🔄

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:

Base-5 ke landmark?
2⁵ = ?
17 ÷ 5 = ? shesh ?

Answers dekho

1, 5, 25, 125…
32
3, shesh 2

Teeno sahi? Badhiya, aage chaliye! Koi galat hua? Koi baat nahi, Post 4 — Number System Mein Base Kya Hai? dobara padh lijiye.

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🎬 Kahani: Robot Bittu Ka Ajeeb Jawab

Meenu ne science fair ke robot Bittu se poochha: “Meri umar 13 hai. Tum ise kaise likhoge?” Screen pe aaya: 1101. 🤖

Golu hans pada: “Bittu pagal hai! 13 ko ek hazaar ek sau ek bol raha hai!”

Par Bittu bilkul sahi tha. Woh sirf do ank (0, 1) jaanta hai. Is post mein hum Bittu ki bhasha (base-2), aur kisi bhi base mein aana-jaana seekhenge. 🔄

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📖 Concept: Ek Number, Kai Kapde

💡 Is post ki ek badi baat: Number wahi rehta hai, sirf kapde (roop) badalte hain. Base-n mein har jagah ka ank batata hai ki us landmark (n^k) ke kitne group hain, aur har ank 0 se (n − 1) tak.

Step 1: Base-n ka place value roop

Post 4 — Number System Mein Base Kya Hai? mein chinh dohraaye the (■■ ▲ ●●●). Ab chinh ki jagah ginti likho, aur position se landmark batao:

$58 = 2\times25 + 1\times5 + 3\times1 \;\Rightarrow\; 213_5$

Chhota “5” = base. Daayen se: ikaai (5⁰), 5 wala (5¹), 25 wala (5²). WHY ank 0–4 hi? 5 group ho gaye to agla landmark ban jaata (Post 4 — Number System Mein Base Kya Hai?).

Step 2: Tareeka A: Landmark se (bade se chhota)

58 ko base-2 mein: landmark 1, 2, 4, 8, 16, 32, 64.

58 − 32 = 26 → 26 − 16 = 10 → 10 − 8 = 2 → 2 − 2 = 0. Kaun aaye: 32, 16, 8, 2.

32	16	8	4	2	1
1	1	1	0	1	0

→ 111010₂. WHY kaam karta? Har kadam pe sabse bada landmark nikaal rahe: Misr wala tareeka hi, bas ank likh diye.

Step 3: Tareeka B: Baar-baar bhaag (shesh neeche se upar)

58 ÷ 2 = 29 shesh 0; 29 ÷ 2 = 14 shesh 1; 14 ÷ 2 = 7 shesh 0; 7 ÷ 2 = 3 shesh 1; 3 ÷ 2 = 1 shesh 1; 1 ÷ 2 = 0 shesh 1.

Shesh neeche se upar: 111010₂ ✅ (Tareeka A jaisa hi!)

WHY kaam karta? 58 cheezon ko 2-2 ki jodi mein baanto: jo akela bacha (shesh) = ikaai ka ank. Jodiyon (29) ko phir jodi mein: shesh = 2 wala ank. Har bhaag ek landmark “upar” le jaata hai.

Step 4: Waapas base-10 mein: khol ke jodo

$1101_2 = 1\times8 + 1\times4 + 0\times2 + 1\times1 = 13$

Bittu sahi tha! WHY? Har ank × uske landmark = utne cheezein; sab jodo = poora number. Yahi har conversion ka check hai.

Step 5: Kab kaunsa tareeka?

Sthiti	Behtar	Kyun
Chhota number, landmark yaad (2 ki powers)	A	Do-teen ghataav mein ho jaata
Bada number ya ajeeb base (7, 8)	B	Landmark yaad nahi rakhne padte
Base-n se base-10	Khol ke jodo	Seedha landmark × ank

🪜 Atak Gaye? Aise Socho

Pehle base ke landmark likho: base-8 → 1, 8, 64, 512.
Number se bada pehla landmark dhundho; usse ek chhota se shuru karo.
Tareeka B mein har line pe: bhaagfal neeche, shesh daayen.
Shesh hamesha base se chhota hona chahiye (base-2 → 0 ya 1).
Jawab milne ke baad khol ke jodo: wahi number aaya? ✅

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🔐 Deep Concept: Number Ek, Roop Anek: Base Sirf “Kapde” Hai

13 bakriyaan 13 hi hain, chahe aap “13”, “1101₂”, “23₅” ya “XIII” likho. Number = maatra; roop = likhne ka tareeka (Post 1 — Ginti Ki Shuruaat Kaise Hui? ka number vs numeral).

Ek jaadu: kisi bhi base mein “10” = base khud! 10₂ = 2, 10₅ = 5, 10₈ = 8. Kyun? “1 group, 0 akela” = ek poora group = base. Isliye har base walon ke liye apna base “das” jaisa lagta hai. 😄

Chhota base = lamba roop: 58 = 111010₂ (6 ank), 213₅ (3 ank), 58₁₀ (2 ank). Kam ank ke choice → zyada jagah chahiye. Computer ko lamba roop chalta hai kyunki usse bas on/off pehchaanna hai.

Unifying idea: conversion = dobara group karna. Tareeka A upar se group banaata, Tareeka B neeche se. Dono ek hi grouping tak pahunchte. 🎯

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✏️ Solved Examples

Example 1 (Easy): 200 ko base-5 mein likho (Tareeka B).

200 ÷ 5 = 40 shesh 0; 40 ÷ 5 = 8 shesh 0; 8 ÷ 5 = 1 shesh 3; 1 ÷ 5 = 0 shesh 1.

Neeche se upar: 1300₅.

Check: 1 × 125 + 3 × 25 + 0 + 0 = 125 + 75 = 200 ✅

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Example 2 (Medium): 100 ko base-8 mein likho (Tareeka A).

Landmark 1, 8, 64. 100 = 1 × 64 + 36; 36 = 4 × 8 + 4 → 144₈.

Check: 64 + 32 + 4 = 100 ✅. Ank 1, 4, 4 sab 8 se chhote ✅

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Example 3 (NCERT style): 10110₂ aur 243₅ ko base-10 mein. Kaun bada?

10110₂ = 16 + 4 + 2 = 22. 243₅ = 2 × 25 + 4 × 5 + 3 = 73. 243₅ bada.

Check: 22 ÷ 2 baar-baar: 0, 1, 1, 0, 1 → 10110 ✅

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Bittu ka jawab: 13 = 8 + 4 + 1 = 1101₂. Golu ne maafi maangi, aur apni umar (14) Bittu ki bhasha mein khud nikaali: 1110₂! 🤖👏

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🏠 Base Conversion Kahan Kaam Aata Hai?

💻 Computer: har photo, gaana, message andar 0-1 (base-2) mein; screen pe hum base-10 dekhte.
🎨 Rang ke code: website pe #FF8800 jaise rang base-16 (hexadecimal) mein: FF = 255.
⏰ Ghadi: 150 minute = 2 ghante 30 minute = base-60 mein “2, 30”.
🥚 Ande ka hisaab: 150 ande = 1 gross (144) + 6 = base-12 jaisa grouping.
💡 Ungliyon pe 31 tak: har ungli on/off = base-2 mein ek haath se 0–31!

Har baar “itne ke group” sochte ho, aap base badal rahe ho.

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🧭 Kab Use Karein, Kab Nahi

Situation	Kya karein	Kyun
Base-10 → base-n, bada number	✅ Baar-baar bhaag	Shesh neeche se upar
Base-10 → base-2, chhota	✅ Landmark (2 ki powers)	32, 16, 8…
Base-n → base-10	✅ Ank × landmark, jodo	Hamesha check bhi
Shesh ≥ base aaya	❌ Bhaag galat	Dobara karo
Shesh upar se neeche padha	❌ Ulta jawab	Neeche se upar!

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❌→✅ Worked Mistake

Sawaal: 11 ko base-2 mein likho.

11 ÷ 2 = 5 shesh 1; 5 ÷ 2 = 2 shesh 1; 2 ÷ 2 = 1 shesh 0; 1 ÷ 2 = 0 shesh 1.

❌ Galat: upar se neeche padha: “1101₂”. Woh to 13 hai!

✅ Sahi: neeche se upar: 1011₂. Check: 8 + 0 + 2 + 1 = 11 ✅

Doosri common galti: base-5 mein ank 5 likh dena (jaise 153₅). Base-5 mein ank sirf 0–4!

🧠 Rule of thumb: Shesh neeche se upar padho, aur hamesha khol ke check karo. Base-n mein ank n se chhote.

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📝 10 MCQs

Q1. 1101₂ = ?

(a) 11

(b) 13

(c) 1,101

(d) 9

Solution dekho

Sahi jawab: (b)

(a) galat: Yeh 1011₂ hai: ank ulte padhe.
(b) sahi: Khol ke: 8 + 4 + 0 + 1 = 13 ✅
(c) galat: Base-10 jaisa padh liya, galat.
(d) galat: 4 wala 1 bhool gaye: 8 + 1 = 9.

Q2. 20 base-2 mein?

(a) 10100

(b) 10010

(c) 11000

(d) 10200

Solution dekho

Sahi jawab: (a)

(a) sahi: 16 + 4 = 20 → 10100₂ ✅
(b) galat: 16 + 2 = 18 hai, 20 nahi.
(c) galat: 16 + 8 = 24 hai, 20 nahi.
(d) galat: Base-2 mein ank 2 nahi aata.

Q3. Base-5 mein kaunsa ank nahi aa sakta?

(a) 0

(b) 3

(c) 4

(d) 5

Solution dekho

Sahi jawab: (d)

(a) galat: 0 to aata hai: jagah khaali.
(b) galat: 3 base se chhota hai, chalta.
(c) galat: 4 sabse bada ank hai base-5 mein.
(d) sahi: 5 = base, ank 0–4 hi ✅

Q4. 44₅ = ?

(a) 44

(b) 24

(c) 20

(d) 9

Solution dekho

Sahi jawab: (b)

(a) galat: Base-10 maan liya, base-5 hai.
(b) sahi: Khol ke: 4 × 5 + 4 = 24 ✅
(c) galat: Ikaai ka 4 jodna bhool gaye.
(d) galat: Ankon ka jod kar diya, galat.

Q5. Kisi bhi base mein “10” = ?

(a) 10

(b) Base khud

(c) 1

(d) 0

Solution dekho

Sahi jawab: (b)

(a) galat: Sirf base-10 mein 10 = das.
(b) sahi: 1 group + 0 akela = base ✅
(c) galat: “1” ek ikaai hai, “10” nahi.
(d) galat: 10 khaali nahi, ek poora group.

Q6. Baar-baar bhaag mein shesh kaise padhein?

(a) Upar se neeche

(b) Neeche se upar

(c) Beech se

(d) Koi bhi

Solution dekho

Sahi jawab: (b)

(a) galat: Upar wala shesh ikaai hai, daayen jaata.
(b) sahi: Aakhri shesh = sabse bada landmark ✅
(c) galat: Beech se padhne ka koi niyam nahi.
(d) galat: Kram badla to number badal gaya.

Q7. 100 base-8 mein?

(a) 144

(b) 124

(c) 104

(d) 1100

Solution dekho

Sahi jawab: (a)

(a) sahi: 64 + 32 + 4 = 100 → 144₈ ✅
(b) galat: 64 + 16 + 4 = 84 hota hai.
(c) galat: 64 + 0 + 4 = 68 hota hai.
(d) galat: Base-2 jaisa likh diya.

Q8. Chhota base, number ka roop?

(a) Chhota

(b) Lamba

(c) Same

(d) Pata nahi

Solution dekho

Sahi jawab: (b)

(a) galat: Kam ank ke choice = zyada jagah.
(b) sahi: Chhota base → zyada ank → lamba ✅
(c) galat: Lambaai base ke saath badalti hai.
(d) galat: Pattern saaf: 2 mein sabse lamba.

Q9. 1300₅ = ?

(a) 200

(b) 1,300

(c) 175

(d) 325

Solution dekho

Sahi jawab: (a)

(a) sahi: 1 × 125 + 3 × 25 = 200 ✅
(b) galat: Base-10 maan liya, galat.
(c) galat: 3 × 25 ko 50 maana, galat.
(d) galat: 1 ko 250 maana: 125 hai.

Q10. 31 ko base-2 mein kitne ank?

(a) 4

(b) 5

(c) 6

(d) 31

Solution dekho

Sahi jawab: (b)

(a) galat: 4 ank max 15 tak jaate hain.
(b) sahi: 31 = 11111₂, yaani 5 ank ✅
(c) galat: 6 ank 32 se shuru hote hain.
(d) galat: Tally jaisa soch liya, galat.

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❓ FAQs (Bachchon Ke Asli Sawaal)

1. Base-16 mein 10 se bade ank kaise?

Akshar: A = 10, B = 11 … F = 15. FF₁₆ = 15 × 16 + 15 = 255.

2. Kya fractions bhi base-2 mein?

Haan: 0.1₂ = 1/2, 0.01₂ = 1/4. Par yeh Class 8 se aage ka maza hai.

3. Base-2 ko binary kyun kehte?

“Bi” = do. Do ank 0 aur 1. Computer mein bijli on/off.

4. Ek base se doosre (5 → 2) seedha?

Aasaan raasta: pehle base-10 mein lao, phir doosre mein. Kuch khaas jodiyan (2 ↔ 8 ↔ 16) mein shortcut hota hai.

5. Chhota “5” likhna zaroori?

Haan, warna 213 ko log base-10 padh lenge. Base likhna roop ka “label” hai.

6. Ungliyon pe 1,023 tak?

Do haath = 10 ungli = 10 binary ank → 2¹⁰ − 1 = 1,023 tak!

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🏋️ Practice Zone (12 Sawaal)

Q1. Easy 9 ko base-2 mein.

Solution dekho

8 + 1 → 1001₂. Check: 8 + 0 + 0 + 1 = 9 ✅

Q2. Easy 111₂ = ?

Solution dekho

4 + 2 + 1 = 7. Check: 7 ÷ 2 shesh 1, 1, 1 ✅

Q3. Easy 32₅ = ?

Solution dekho

3 × 5 + 2 = 17. Check: 17 ÷ 5 = 3 shesh 2 ✅

Q4. Easy Base-8 ke ank kaun se?

Solution dekho

0–7. WHY: 8 = agla landmark. Check: 8 = 10₈ ✅

Q5. Medium 45 ko base-2 mein (Tareeka B).

Solution dekho

Shesh 1, 0, 1, 1, 0, 1 → 101101₂. Check: 32 + 8 + 4 + 1 = 45 ✅

Q6. Medium 90 ko base-5 mein.

Solution dekho

90 = 3 × 25 + 3 × 5 + 0 → 330₅. Check: 75 + 15 = 90 ✅

Q7. Medium 63 ko base-8 mein.

Solution dekho

7 × 8 + 7 → 77₈. Check: 56 + 7 = 63 ✅

Q8. Medium 1000₂, 1000₅, 1000₈ base-10 mein?

Solution dekho

8, 125, 512. WHY: “1000” = base³. Check: 2³, 5³, 8³ ✅

Q9. NCERT style Base-3 mein 50.

Solution dekho

50 ÷ 3 → shesh 2, 1, 2, 1 → 1212₃. Check: 27 + 18 + 3 + 2 = 50 ✅

Q10. NCERT style 101₂ + 11₂ base-2 mein hi jodo.

Solution dekho

1 + 1 = 10 (0 likho, 1 carry), 0 + 1 + 1 = 10, 1 + 1 = 10 → 1000₂. Check: 5 + 3 = 8 ✅

Q11. NCERT style Kis base mein 121 = 49?

Solution dekho

n² + 2n + 1 = (n + 1)² = 49 → n + 1 = 7 → base 6. Check: 36 + 12 + 1 = 49 ✅

Q12. NCERT style Ek haath ki 5 ungliyon pe binary: sabse bada number? Teen ungli (pehli, teesri, paanchvi) uthaayi to?

Solution dekho

11111₂ = 31. 10101₂ = 16 + 4 + 1 = 21. Check: 21 ÷ 2 baar-baar: 1, 0, 1, 0, 1 ✅

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🧠 HOTS (Sochne Wale Sawaal)

Q1. Kya koi number base-2 mein aur base-10 mein ek jaisa dikhta hai (2 se bada)?

Solution dekho

Sirf 0 aur 1. 2 se bade number ke liye base-10 ka ank ≥ 2 chahiye, jo base-2 mein nahi. WHY: ank set alag.

Q2. Kisi base mein 100 = 81. Base?

Solution dekho

n² = 81 → 9. Check: 100₉ = 81 ✅

Q3. Base-2 mein even number ki pehchaan?

Solution dekho

Aakhri ank 0. WHY: ikaai ka ank = 2 se bhaag ka shesh (Tareeka B ki pehli line). Check: 10₂ = 2, 110₂ = 6 ✅

Q4. Base-5 mein kisi number ke peeche 0 lagao to kya hota?

Solution dekho

5 guna! 23₅ = 13, 230₅ = 65. WHY: har ank ek landmark upar.

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🏆 Olympiad Corner

Q1. Sabse chhota base jisme 2026 ke 4 ank hon?

Solution dekho

4 ank ke liye n³ ≤ 2026 < n⁴. n = 6: 6⁴ = 1,296 ≤ 2026, to 5 ank. n = 7: 343 ≤ 2026 < 2,401 → 4 ank → base 7.

Q2. Base-2 mein 2¹⁰ − 1 kaisa?

Solution dekho

1111111111 (das 1).

Q3. 234₅ + 1 = ? (base-5 mein)

Solution dekho

240₅ (4 + 1 = 10₅).

Q4. Ek number base-4 mein 33 aur base-n mein 21. n?

Solution dekho

33₄ = 15 = 2n + 1 → n = 7.

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💗 CharuMam Motivation

Golu ne Bittu ko pagal kaha, par asal mein Bittu ek alag “bhasha” bol raha tha. Maths mein bhi jab koi alag lage, pehle uska “base” samjho! 🤝

Aaj aapne woh seekha jo har programmer roz use karta hai. Agli baar computer chalaao to socho: andar sab 0-1 ka khel hai! 💻

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🟢 Master Recap

💡 Chaabi: Number wahi rehta hai, sirf kapde (roop) badalte hain. Base-n mein har jagah ka ank batata hai ki us landmark (n^k) ke kitne group hain, aur har ank 0 se (n − 1) tak.
Base-n roop: ank × n ki power, daayen se n⁰, n¹, n²…
Ank 0 se (n − 1).
Tareeka A: sabse bada landmark nikaalte jao.
Tareeka B: baar-baar bhaag, shesh neeche se upar.
Waapas: ank × landmark, jodo (yahi check).
Kisi bhi base mein 10 = base.
13 = 1101₂: Bittu sahi tha!

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👉 Aage Kya?

Romans hisaab kaise karte the? Ek board, kuch goliyaan, aur base-10 ka jaadu: agle post mein abacus aur decimal system, aur Misr ke system ki kami. 🧮

← Pichla: Post 4 — Number System Mein Base Kya Hai?📚 Chapter 3 ke saare posts Agla: Post 6 — Abacus Aur Decimal System →

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

✅ Pehle Yeh Aata Hai?

🎬 Kahani: Robot Bittu Ka Ajeeb Jawab

📖 Concept: Ek Number, Kai Kapde

Step 1: Base-n ka place value roop

Step 2: Tareeka A: Landmark se (bade se chhota)

Step 3: Tareeka B: Baar-baar bhaag (shesh neeche se upar)

Step 4: Waapas base-10 mein: khol ke jodo

Step 5: Kab kaunsa tareeka?

🪜 Atak Gaye? Aise Socho

🔐 Deep Concept: Number Ek, Roop Anek: Base Sirf “Kapde” Hai

✏️ Solved Examples

🏠 Base Conversion Kahan Kaam Aata Hai?

🧭 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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