✅ Pehle Yeh Aata Hai?
Yeh post shuru karne se pehle, in teen sawaalon ko 1 minute mein khud try kijiye:
- 1.4 × 1.4 kitna hota hai?
- Kaun sa number khud se guna hoke 49 deta hai?
- Kya 0.5 ko fraction mein likh sakte hain?
Answers dekho
- 1.96
- 7 (7 × 7 = 49), isliye √49 = 7
- Haan, ½
Teeno sahi? Waah! Decimal ka guna kamzor hai to ek baar practice kar lijiye, is post mein bahut kaam aayega. Pichhle do post: Post 1 aur Post 2.
— 📌 charumam.com
🎬 Kahani: Rohan Ka Calculator
Rohan ne Post 1 padh ke socha, “1 m side ki vedi ka diagonal kitna lamba hai? Seedha calculator se √2 nikaal leta hoon.”
Calculator ne dikhaaya: 1.414213562. Rohan khush: “Bas! Yahi exact answer hai.”
Priya ne 1.414213562 × 1.414213562 kiya… aaya 1.99999999… ke aas-paas, poora 2 nahi! 😲
“Shayad aur digits chahiye?” Rohan ne bada calculator liya, aur digits aaye, phir bhi poora 2 nahi. Kya √2 ka decimal kabhi khatam hoga? Aaj ka sawaal yahi hai! 🕵️
— 📌 charumam.com
📖 Concept: √2: Woh Number Jiska Square 2 Hai
Step 1: Isosceles right triangle ka hypotenuse
Jis right triangle ki do sides barabar hon, use isosceles right triangle kehte hain. 90° ke saamne wali sabse lambi side hypotenuse (karn) hai.
Barabar sides 1 hon, to yeh triangle 1 side wale square ka aadha hai, aur hypotenuse us square ka diagonal. Post 1 se: diagonal pe bana square = 2 × 1 = 2 sq. units. To c × c = 2, yaani c = √2.
WHY √? √2 (“root 2”) ka matlab: woh positive number jo khud se guna hokar 2 de.
Step 2: √2 ko gher ke pakdo (bounds)
| Check | Square | Nateeja |
|---|---|---|
| 1 aur 2 | 1² = 1, 2² = 4 | 1 < √2 < 2 |
| 1.4 aur 1.5 | 1.96, 2.25 | 1.4 < √2 < 1.5 |
| 1.41 aur 1.42 | 1.9881, 2.0164 | 1.41 < √2 < 1.42 |
| 1.414 aur 1.415 | 1.999396, 2.002225 | 1.414 < √2 < 1.415 |
WHY yeh tareeka? √2 ko seedha nahi pakad sakte, par har baar ek digit aur pakki kar sakte hain: ek number ka square 2 se kam, agle ka 2 se zyada, to √2 dono ke beech.
Step 3: Decimal kabhi khatam kyun nahi hota?
Maano kisi khatam hone wale decimal ka aakhri (non-zero) digit d hai. Uska square karo to aakhri digit d × d ka aakhri digit hoga (jaise 1.4² = 1.96, aakhri 6 = 4 × 4 ka). d = 1 se 9 ke liye d × d kabhi 0 pe khatam nahi hota (1, 4, 9, 6, 5, 6, 9, 4, 1). To square mein decimal ke baad hamesha kuch bachega, woh poora 2 kabhi nahi banega.
Isliye √2 ka decimal non-terminating hai: 1.41421356… chalta hi rehta hai, aur koi pattern repeat bhi nahi karta.
Step 4: √2 fraction bhi nahi ban sakta
Maano √2 = m/n (m, n poore numbers). To m² = 2n². Ab prime factors mein 2 kitni baar aata hai, gino:
- m² mein: m mein jitni baar 2, uska double, yaani even baar.
- 2n² mein: n² mein even baar, upar se ek extra 2, yaani odd baar.
Ek hi number mein 2 even baar bhi aur odd baar bhi, yeh namumkin hai! To √2 = m/n ho hi nahi sakta. Yeh saboot Euclid ki kitaab Elements (c. 300 BCE) mein hai.
WHY zaroori? Yeh dikhata hai ki number line pe aise numbers bhi hain jo kisi fraction se nahi likhe ja sakte.
Step 5: General formula: c² = 2a²
Barabar sides a hon, to hypotenuse pe bana square = 2 × a², yaani c² = 2a² aur c = a√2.
WHY kaam ka? Kisi bhi isosceles right triangle ka hypotenuse turant: a = 5 → c = 5√2 ≈ 7.07.
— 📌 charumam.com
🔐 Deep Concept: √2 Ki Lambaai Haath Mein Pakdo
√2 ka decimal kabhi khatam nahi hota, phir bhi uski lambaai bilkul pakki hai! 1 cm side ka square banao aur diagonal naapo, wahi √2 cm hai. Number “adhoora” lagta hai, par line poori hai. 📏
Isliye Baudhayana ko √2 ka decimal jaanne ki zaroorat hi nahi thi; unhone diagonal ko rassi se naap ke vedi bana li.
Fir bhi Śulba-Sūtra mein √2 ka ek bahut achha anumaan diya gaya hai jo 1.4142156… deta hai, kai decimal tak sahi! 🔐
sirf charumam.com readers ke liye 🔐
✏️ Solved Examples
Example 1 (Easy): Isosceles right triangle ki barabar sides 12 hain. Hypotenuse kis do poore numbers ke beech hai?
c² = 2 × 12² = 2 × 144 = 288. 16² = 256 aur 17² = 289.
256 < 288 < 289, to 16 < c < 17 (17 ke bahut paas, c ≈ 16.97).
WHY sahi hai? c = 12√2 ≈ 12 × 1.414 ≈ 16.97, jo 16 aur 17 ke beech ✅
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Example 2 (Medium): Isosceles right triangle ka hypotenuse √72 hai. Barabar sides kitni hain?
c² = 72 = 2a² → a² = 36 → a = 6.
WHY sahi hai? 6² + 6² = 36 + 36 = 72 ✅
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Example 3 (NCERT style): Barabar sides 3 hon to hypotenuse batao aur use ek decimal tak gher ke dikhao.
c² = 2 × 9 = 18, to c = √18.
4.2² = 17.64, 4.3² = 18.49. To 4.2 < √18 < 4.3 (c ≈ 4.24).
WHY sahi hai? 17.64 < 18 < 18.49 ✅
📌 charumam.com
Rohan ka naya formula: “Isosceles right triangle? Hypotenuse = side × √2, aur √2 ≈ 1.414!” 🎉
— 📌 charumam.com
🏠 Real Life Mein
- 📐 Set square: Geometry box ka 45°–45°–90° set square ek isosceles right triangle hai. Uski lambi side = chhoti side × √2.
- 📄 A4 kaagaz: A4 sheet ki lambaai ÷ chaudaai ≈ 1.414 (√2) hoti hai. Isliye aadha modne pe bhi same shape (A5) milti hai!
- 🏏 Square maidaan: Kone se kone tak daudna = side × 1.414, side ke saath-saath ghoomne (2 × side) se chhota raasta.
- 📺 Screen size: TV ka size diagonal se naapte hain; square screen mein diagonal = side × √2.
— 📌 charumam.com
🧭 Kab Use Karein, Kab Nahi
| Situation | Kya karein | Kyun |
|---|---|---|
| Do sides barabar + 90° ka kona | ✅ c = a√2 | c² = 2a² |
| Square ka diagonal chahiye | ✅ Side × √2 | Diagonal = isosceles right triangle ka hypotenuse |
| √2 ka exact decimal likhna hai | ❌ Mumkin nahi | Non-terminating, kabhi khatam nahi |
| Exam mein approx value chahiye | ✅ √2 ≈ 1.414 | 1.414² = 1.999396 ≈ 2 |
— 📌 charumam.com
❌→✅ Worked Mistake
Sawaal: 10 cm side ke square ka diagonal?
❌ Galat: “Diagonal = side + side = 20 cm” ya “diagonal = 2 × side = 20 cm”.
Galti kya hai? Triangle ki koi side baaki do sides ke jod jitni nahi ho sakti. Aur 20 × 20 = 400 = 4 × 100, jabki diagonal square ka area sirf 2 × 100 = 200 hona chahiye ❌
✅ Sahi:
- d² = 2 × 10² = 200.
- d = 10√2 ≈ 10 × 1.414 = 14.14 cm.
Yaad rakho: Diagonal hamesha side se lamba, par 2 × side se chhota.
sirf charumam.com pe milenge!
📝 10 MCQs
Q1. √2 ka matlab?
Solution dekho
Sahi jawab: (b)
- (a) galat: Aadha to 1 hai, 1 × 1 = 1.
- (b) sahi: √ = jiska square diya gaya number ho ✅
- (c) galat: 2 × 2 = 4 = 2².
- (d) galat: 2 + 2 = 4.
Q2. √2 kin do decimals ke beech hai?
Solution dekho
Sahi jawab: (b)
- (a) galat: 1.4² = 1.96, abhi bhi 2 se kam, to √2 > 1.4.
- (b) sahi: 1.96 < 2 < 2.25 ✅
- (c) galat: 1.5² = 2.25 pehle hi 2 se zyada.
- (d) galat: 1.9² = 3.61, bahut bada.
Q3. 1.41² kitna hai?
Solution dekho
Sahi jawab: (a)
- (a) sahi: 141 × 141 = 19881, to 1.9881 ✅
- (b) galat: Yeh 1.42² hai.
- (c) galat: Yeh 1.4² hai.
- (d) galat: Poora 2 kisi decimal ka square nahi.
Q4. Barabar sides 5 wale isosceles right triangle ka c²?
Solution dekho
Sahi jawab: (c)
- (a) galat: 10 = 2 × 5, square nahi kiya.
- (b) galat: 25 = ek side ka square.
- (c) sahi: 2 × 25 = 50 ✅
- (d) galat: 100 = 4 × 25.
Q5. Hypotenuse √50 hai (isosceles right). Barabar side?
Solution dekho
Sahi jawab: (a)
- (a) sahi: 2a² = 50 → a² = 25 → a = 5 ✅
- (b) galat: 25 to a² hai, side nahi.
- (c) galat: 50 to c² hai.
- (d) galat: 10² + 10² = 200, 50 nahi.
Q6. √2 ka decimal…
Solution dekho
Sahi jawab: (c)
- (a) galat: 1.414² = 1.999396, 2 nahi.
- (b) galat: Koi repeat pattern nahi.
- (c) sahi: Non-terminating ✅
- (d) galat: 2 decimal pe khatam nahi.
Q7. √2 = m/n kyun nahi ho sakta?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Prime factor ginti ka virodh ✅
- (b) galat: m ke size se fark nahi padta.
- (c) galat: n zero ho hi nahi sakta (bhaag nahi hota).
- (d) galat: √2 positive hai.
Q8. √2 ka saboot ki woh fraction nahi, kiski kitaab mein hai?
Solution dekho
Sahi jawab: (b)
- (a) galat: Baudhayana ne vedi ke niyam diye.
- (b) sahi: Elements, c. 300 BCE ✅
- (c) galat: Fermat 1600s mein the, alag sawaal.
- (d) galat: Wiles ne 1994 mein Fermat ka theorem prove kiya.
Q9. 12 side ke isosceles right triangle ka hypotenuse kin poore numbers ke beech?
Solution dekho
Sahi jawab: (c)
- (a) galat: 13² = 169, 288 se bahut kam.
- (b) galat: 16² = 256, abhi bhi 288 se kam.
- (c) sahi: 256 < 288 < 289 ✅
- (d) galat: Yeh to 2 × 12 ke paas hai, bahut bada.
Q10. A4 kaagaz ki lambaai ÷ chaudaai lagbhag?
Solution dekho
Sahi jawab: (b)
- (a) galat: A4 square nahi hai.
- (b) sahi: √2 ≈ 1.414 ✅
- (c) galat: 2 hota to bahut patla.
- (d) galat: 1.5 approx sahi nahi, √2 hai.
— 📌 charumam.com
❓ FAQs (Bachchon Ke Asli Sawaal)
1. √2 ko calculator exact kyun nahi dikhata?
Calculator sirf 8–10 digit dikha sakta hai, aur √2 ka decimal kabhi khatam nahi hota. Isliye calculator ka jawab bhi ek anumaan hi hai.
2. √2 ≈ 1.414 likhna theek hai?
Haan, jab tak sawaal mein “approx” ya decimal maanga ho. Warna 5√2 jaisa exact roop likho.
3. Hypotenuse kaun si side hoti hai?
90° ke kone ke saamne wali side, jo hamesha sabse lambi hoti hai.
4. Kya √3, √5 bhi aise hi hain?
Haan, jo numbers perfect square nahi (jaise 2, 3, 5, 6, 7, 8…), unke √ bhi fraction mein nahi likhe ja sakte.
5. Non-terminating ka matlab?
Decimal ke baad digits kabhi khatam nahi hote.
6. √2 aur 2 ka aadha same hain?
Bilkul nahi! 2 ka aadha 1 hai. √2 ≈ 1.414, kyunki 1.414 × 1.414 ≈ 2.
— 📌 charumam.com
🏋️ Practice Zone (12 Sawaal)
Q1. Easy Barabar sides 4 wale isosceles right triangle ka c² aur c (approx)?
Solution dekho
c² = 32, c = 4√2 ≈ 5.66.
Q2. Easy 1.42² kitna hai, aur kya yeh 2 se zyada hai?
Solution dekho
2.0164, haan zyada.
Q3. Easy 8 cm square ka diagonal (approx)?
Solution dekho
8 × 1.414 ≈ 11.31 cm.
Q4. Easy √49 aur √50 mein kaun sa poora number hai?
Solution dekho
√49 = 7 poora; √50 ≈ 7.07 nahi.
Q5. NCERT style Barabar sides 6: hypotenuse ko ek decimal tak ghero.
Solution dekho
c² = 72. 8.4² = 70.56, 8.5² = 72.25. 8.4 < c < 8.5.
Q6. NCERT style Barabar sides 8: hypotenuse ghero.
Solution dekho
c² = 128. 11.3² = 127.69, 11.4² = 129.96. 11.3 < c < 11.4.
Q7. NCERT style Barabar sides 9: hypotenuse ghero.
Solution dekho
c² = 162. 12.7² = 161.29, 12.8² = 163.84. 12.7 < c < 12.8.
Q8. NCERT style Isosceles right triangle ka hypotenuse 10 hai. Baaki sides?
Solution dekho
2a² = 100 → a² = 50 → a = √50 ≈ 7.07.
Q9. Medium Hypotenuse √98 hai. Barabar sides?
Solution dekho
a² = 49, a = 7.
Q10. Medium √2 ko 1.4142 lekar 1.4142² nikaalo. 2 se kitna kam?
Solution dekho
1.99996164, yaani 0.00003836 kam.
Q11. Medium Barabar sides 1.5 hon to c²?
Solution dekho
2 × 2.25 = 4.5.
Q12. Medium Kya 7/5 = √2 ho sakta hai? Check karo.
Solution dekho
(7/5)² = 49/25 = 1.96 ≠ 2. Nahi.
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🧠 HOTS (Sochne Wale Sawaal)
Q1. Dikhao ki 1.4142135 bhi √2 nahi hai.
Solution dekho
Aakhri digit 5, to square ka aakhri digit 5 (kyunki 5 × 5 = 25). Square mein decimal ke baad kuch bachega, poora 2 nahi. ✅
Q2. Kya √8 ko a√2 roop mein likh sakte hain?
Solution dekho
8 = 4 × 2, to √8 = 2√2 ✅ (2√2 × 2√2 = 4 × 2 = 8).
Q3. Ek square ka diagonal 2 hai. Area kya hai, aur side kya hai?
Solution dekho
Area = d²/2 = 2. Side = √2. Mazedaar: diagonal 2, side √2!
Q4. Golu: “√2 × √2 = √4 = 2, to √2 to bas 2 ka aadha hai.” Galti?
Solution dekho
√2 × √2 = 2 sahi hai, par iska matlab √2 + √2 = 2 nahi. 2 ka aadha 1 hai, √2 ≈ 1.414.
sirf charumam.com pe milenge!
🏆 Olympiad Corner
Q1. Kya √2 ka decimal kabhi repeat hone lagta hai (jaise 1/3 = 0.333…)?
Solution dekho
Nahi. Repeating decimal hamesha fraction ban jaata hai, aur √2 fraction nahi hai. Isliye na khatam, na repeat.
Q2. 1.4, 1.41, 1.414, 1.4142… mein agla digit kaise dhoondhoge?
Solution dekho
1.4142x ke liye x = 0 se 9 tak square karo, jo sabse bada 2 se kam ho. 1.41421² = 1.9999899… < 2, 1.41422² > 2. Agla digit 1.
Q3. Ek square ka area 50 hai. Diagonal?
Solution dekho
d² = 2 × 50 = 100, d = 10. (Side √50 adhoora, diagonal poora!)
Q4. (√2 + 1)(√2 − 1) kitna hai?
Solution dekho
√2 × √2 − 1 × 1 = 2 − 1 = 1.
sirf charumam.com readers ke liye 🔐
💗 CharuMam Motivation
Rohan ko laga calculator sab jaanta hai. Par ganit ne sikhaaya: kuch sach calculator se bade hote hain. √2 ka decimal kabhi khatam nahi hota, phir bhi hum usse vedi bana lete hain! 🌌
Har sawaal ka jawab “poora number” nahi hota. Adhoorepan se mat daro, use samjho. Wahi asli ganitagya ki pehchaan hai! ✨
— 📌 charumam.com
🟢 Master Recap
- Isosceles right triangle: do sides barabar + 90°. Sabse lambi side = hypotenuse.
- Barabar sides 1 → c² = 2 → c = √2.
- Bounds: 1.414 < √2 < 1.415; √2 = 1.41421356…
- √2 ka decimal non-terminating hai (aakhri digit ka square 0 nahi deta).
- √2 fraction nahi (m² = 2n² mein 2 ki ginti even vs odd), Euclid ka saboot.
- General: c² = 2a², c = a√2.
— charumam.com 🎯
👉 Aage Kya?
Ab tak dono sides barabar thi. Agar sides alag-alag hon, jaise 3 aur 4? Agle post mein aata hai chapter ka hero: Baudhayana–Pythagoras Theorem, a² + b² = c², saboot ke saath! 🦸
Koi doubt? CharuMam AI Tutor se poochho, ya comment mein likho! 💬