✅ Pehle Yeh Aata Hai?
Pichle posts mein humne seekha ki har rational number ka decimal ya ruk jaata hai ya dohrata hai: Decimal Forms: Terminating aur Repeating aur Repeating Decimal ko p/q Mein Badalna. Aaj ka sawaal ulta hai: agar decimal na ruke aur na dohraye, to woh kaisa number hoga? Pehle teen chhote sawaal:
- 1/8 ko decimal mein likho.
- 1/3 ko decimal mein likho.
- 0.4545… (45 dohrata hai) ko p/q mein likho.
Answers dekho
- 1 ÷ 8 = 0.125. Yeh ruk jaata hai, yaani terminating.
- 1 ÷ 3 = 0.333…. Yahan 3 baar baar aata hai, yaani repeating.
- Block 2 digit ka hai, to 100x − x = 45, 99x = 45, x = 45/99 = 5/11 (9 se bhaag).
Decimal point ka matlab thoda dhundhla ho to: MN-03: Decimal Point ka asli matlab padho.
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🎬 Kahani: Tile ka Ajeeb Naap
Rohan ke ghar mein 1 meter × 1 meter ki ek square tile lagi hai. Usne dori se tile ka ek kona doosre saamne wale kone tak naapa aur calculator se pata kiya ki lambai 1.41421356… meter hai. Usne apni behen Gudiya se kaha, “Calculator to bas kuch digits dikhata hai. Isko fraction mein likh do, jaise 3/2 ya 7/5.”
Gudiya ne 1.4142 ko 14142/10000 likh diya. Rohan ne calculator mein aur digits maange: 1.41421356237… aur phir aur… Digits khatam hi nahi hote, aur koi pattern bhi nahi dikhta! 🤔
Yeh lambai (√2, Pythagoras se: 1² + 1² = 2) ek aise number hai jo kabhi rukta nahi aur kabhi dohrata nahi. Aise numbers ko irrational numbers kehte hain. Aaj hum seekhenge ki ye kya hote hain aur inhe pehchante kaise hain.
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📖 Concept: Irrational Number Kya Hota Hai?
Step 1: Rational yaad karo
Rational number woh hai jo p/q ke roop mein likha ja sake, jahan p aur q poore number (integers) hain aur q ≠ 0. Jaise 3/4, −5, 0.25, 0.333… (= 1/3).
Step 2: Irrational ki definition
Irrational number woh number hai jo p/q ke roop mein likha hi nahi ja sakta (p, q integers, q ≠ 0). Naam mein “ratio” chhupa hai: ir-ratio-nal = jo ratio na ban sake.
Step 3: Decimal se pehchaan
Pichle posts ka nateeja: rational ⇒ decimal terminating ya repeating. Ulta bhi sach hai, isliye:
WHY? Agar kisi number ka decimal ruk jaye ya dohraye, to guna-ghatao ki trick se hum use p/q bana lete hain, yaani woh rational hai. Isliye jo p/q nahi ban sakta, uska decimal in dono mein se kuch bhi nahi kar sakta.
Step 4: Kuch jaane-pehchane irrational numbers
| Number | Decimal ki jhalak |
|---|---|
| √2 | 1.41421356… |
| √3 | 1.7320508… |
| π (pie) | 3.14159265… |
| 0.1010010001… | 1, phir 1 zero, 1, 2 zero, 1, 3 zero… (zeros badhte jaate hain) |
WHY 0.1010010001… irrational hai jabki isme “pattern” hai? Pattern hai, par koi ek fixed block baar baar nahi dohrata. Repeating ka matlab hai ek hi block bina badle dohrana (jaise 36, 36, 36…). Yahan zeros ki ginti har baar badh jaati hai, isliye block hai hi nahi.
Step 5: Square root kab irrational?
√4 = 2, √9 = 3, √16 = 4. Yeh rational hain, kyunki 4, 9, 16 perfect square hain. Par √2, √3, √5, √6, √7, √10 jaise root irrational hain, kyunki 2, 3, 5, 6, 7, 10 perfect square nahi hain. Yaad rakho: √ ka nishan dikhna matlab irrational nahi. Dekhna padta hai ki andar ka number perfect square hai ya nahi.
Abhi ek baat dhyan se: √2 aur √3 irrational kyun hain, iska pura saboot agle post mein aayega. Aaj hum maan kar chalte hain aur pehchaan seekhte hain.
Step 6: π ka sach
π = (gol cheez ki paridhi) ÷ (uska vyas). Yeh irrational hai. Aapne 22/7 aur 3.14 padhe honge, par woh sirf paas-paas ki value hain, π ke barabar nahi. Kyunki 22/7 = 3.142857142857… (repeating, rational) hai aur π = 3.14159265… hai.
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Decimal kabhi dohrata kyun nahi?
1) Contradiction wali soch: maan lo √2 ka decimal kisi jagah se dohrane lagta. Tab pichle post ki guna-ghatao trick se √2 = p/q ban jaata. Par √2 irrational hai (saboot agle post mein). Matlab hamari “dohrane” wali maan-ni galat thi. Isliye irrational ka decimal na rukta hai na dohrata hai.
2) Calculator irrational dikha hi nahi sakta: calculator ke screen par kuch hi digits aate hain, jaise 1.414213562. Yeh ruka hua decimal hai, yaani rational number (1414213562/10⁹). Woh √2 ki sirf ek paas ki value hai. Asli √2 ke digits kabhi khatam nahi hote.
3) “Irrational” ka matlab “bekaar” nahi: yeh numbers real duniya mein roz milte hain: square ka kona (√2), gol cheez ki paridhi (π). Bas inhe fraction se naapna mumkin nahi.
— sirf charumam.com readers ke liye 🔐
✏️ Solved Examples
Example 1 (Easy): Batao rational ya irrational: (i) √49 (ii) √50 (iii) 3.14 (iv) 0.7070070007…
(i) √49 = 7, rational. (ii) 50 perfect square nahi hai (7² = 49, 8² = 64), to √50 irrational. (iii) 3.14 = 314/100, rational. (iv) 7 ke beech zeros 1, 2, 3… badhte jaate hain, koi block nahi dohrata, to irrational.
WHY? Root ke liye perfect square check karte hain. Decimal ke liye “ruka? dohraya?” check karte hain.
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Example 2 (Medium): x = 0.12112111211112… (12, 112, 1112, 11112… ek ke baad ek) aur y = 0.121212… mein se kaun irrational hai?
y mein “12” block baar baar dohrata hai, to y = 12/99 = 4/33, rational. x mein 1 ki ginti har baar badhti hai (2, 3, 4, 5…), koi fixed block nahi dohrata aur decimal ruka nahi, to x irrational hai.
WHY? Dono mein “pattern” dikhta hai, par rational ka pattern fixed block ka dohrana hota hai. x ka pattern badalta rehta hai.
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Example 3 (NCERT style): Kya π = 22/7 likhna bilkul sahi hai? Dono ke decimal dekh kar batao.
22/7 = 3.142857142857… (142857 dohrata hai), yaani rational. π = 3.14159265… (na ruka, na dohraya), yaani irrational. Rational aur irrational kabhi barabar nahi ho sakte, to π ≠ 22/7. Farq lagbhag 3.142857 − 3.141593 = 0.00126 hai, bahut chhota, isliye 22/7 ek achhi approximation hai.
WHY? Hisaab ke liye 22/7 ya 3.14 kaam aate hain, par “barabar” likhna galat hai. Hum “≈” likhte hain.
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Rohan ka jawab: tile ka kona-se-kona naap √2 meter hai. Yeh fraction nahi ban sakta, isliye Gudiya ka 14142/10000 sirf paas ki value tha. ✅
🚀 Aur Aage: NCERT ke bahar
1) Rational + irrational = irrational (extra): jaise 2 + √3. WHY? Agar 2 + √3 = p/q (rational) hota, to √3 = p/q − 2 bhi rational ban jaata. Par √3 irrational hai, to yeh ho nahi sakta.
2) Nonzero rational × irrational = irrational (extra): jaise 3√2, aur √2/4. WHY? Agar 3√2 rational r hota, to √2 = r/3 rational hota. Dhyan dein: nonzero zaroori hai, kyunki 0 × √2 = 0, jo rational hai.
3) Irrational + irrational ka koi pakka niyam nahi (extra): √2 + (3 − √2) = 3 (rational). √2 × √8 = √16 = 4 (rational). Par √2 + √2 = 2√2 irrational hai. Isliye aise sawaal mein jawab number dekh kar hi tay hota hai.
4) Cube root (extra): ∛8 = 2 (rational) hai, par ∛2 irrational hai. √ ki tarah ∛ mein bhi andar ka number perfect cube hona chahiye.
5) Aur bhi irrational numbers (extra): π ke saath e ≈ 2.71828… bhi irrational hai. Jab aap upar ke ≈ values yaad rakho, to samajh lo yeh sirf paas ki values hain.
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🌍 Real Life Mein
Square tile ya kamre ka kona: 3 m × 3 m ke kamre mein kona-se-kona dori 3√2 m (lagbhag 4.24 m) hoti hai. Mistri ise “4 meter 24 centimetre ke aas-paas” kehta hai, kyunki exact fraction nahi hota.
Gol cheez: 70 cm vyas wale pahiye ki paridhi π × 70 cm hoti hai. Dukaandar π ko 22/7 maan ke 220 cm bolta hai (aasan, kyunki 70 mein 7 kat jaata hai). Asli value 219.9 cm ke aas-paas hai.
A4 paper: A4 sheet ki lambai aur chaudai ka anupaat lagbhag 1.414 hota hai, jo √2 ke paas hai. Isliye aadhi karne par bhi sheet ka shape ek jaisa rehta hai.
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⚖️ Kab Use Karein / Kab Nahi
| Kab “irrational” kahein | Kab nahi |
|---|---|
| Decimal na rukta ho aur koi block baar-baar na dohrata ho | Decimal lamba ho par ruk jaata ho (jaise 3.14159) ya dohrata ho (jaise 0.2727…) |
| √ ke andar ka number perfect square na ho (√2, √7, √50) | √ ke andar perfect square ho (√25 = 5, √0.49 = 0.7) |
| π, ya π jaisi value (nonzero rational) × irrational | Calculator ki dikhi hui value ko exact maan kar “rational/irrational” tay karna |
— 📌 charumam.com
❌→✅ Worked Mistake
Sawaal: √25 aur 0.5050050005… mein se kaunsa irrational hai?
❌ Galat: “√25 mein root hai, to irrational. Aur 0.5050050005… mein pattern hai, to rational.”
✅ Sahi: √25 = 5, jo rational hai (25 perfect square hai). 0.5050050005… mein 5 ke beech zeros 1, 2, 3… badhte hain, koi fixed block nahi dohrata, to yeh irrational hai.
WHY? Root ka nishan hone se koi number irrational nahi banta. Aur “pattern dikhna” aur “block dohrana” alag baatein hain. Rational ke liye ek hi block baar baar aana zaroori hai.
— sirf charumam.com pe milenge!
📝 10 MCQs
Q1. Inme se irrational number kaun sa hai?
Solution dekho
Sahi jawab: (c)
- (a) galat: √81 = 9, rational.
- (b) galat: √0.09 = 0.3, rational.
- (c) sahi: 11 perfect square nahi hai (3² = 9, 4² = 16), to √11 irrational.
- (d) galat: 22/7 do poore numbers ka ratio hai, to rational.
Q2. Kaun sa decimal irrational number dikhata hai?
Solution dekho
Sahi jawab: (c)
- (a) galat: 0.3333… mein 3 dohrata hai, = 1/3, rational.
- (b) galat: 0.2525… mein 25 dohrata hai, = 25/99, rational.
- (c) sahi: 4 ke beech zeros badhte jaate hain, koi block nahi dohrata aur decimal ruka nahi, to irrational.
- (d) galat: 0.125 ruk jaata hai, = 1/8, rational.
Q3. Irrational number ki sahi definition kaun si hai?
Solution dekho
Sahi jawab: (b)
- (a) galat: Yeh to terminating decimal wale rational numbers ki pehchaan hai.
- (b) sahi: p/q ban hi na sakna hi irrational ki definition hai.
- (c) galat: −3 = −3/1 rational hai, to negative hona irrational hone ka matlab nahi.
- (d) galat: 1/2 poora number nahi hai, par rational hai.
Q4. π ke baare mein sahi kya hai?
Solution dekho
Sahi jawab: (c)
- (a) galat: 22/7 = 3.142857… repeating hai, π = 3.14159… alag hai.
- (b) galat: 3.14 ruka hua rational number hai, π nahi.
- (c) sahi: π ka decimal na rukta hai na dohrata hai, to irrational.
- (d) galat: 3.14159 sirf pehle kuch digits hain, π ke digits aage chalte rehte hain.
Q5. Irrational number ka decimal expansion hota hai:
Solution dekho
Sahi jawab: (c)
- (a) galat: Terminating wale rational hote hain.
- (b) galat: Non-terminating repeating bhi rational hote hain (p/q ban jaata hai).
- (c) sahi: Na rukna aur na dohrana, yahi irrational ki pehchaan hai.
- (d) galat: Yeh dono ek saath ho hi nahi sakte.
Q6. Inme se rational number kaun sa hai?
Solution dekho
Sahi jawab: (c)
- (a) galat: 5 perfect square nahi, √5 irrational.
- (b) galat: Zeros badhte jaate hain, irrational.
- (c) sahi: √(4/9) = 2/3, rational.
- (d) galat: π irrational hai.
Q7. 2 + √3 kaisa number hai? (extra)
Solution dekho
Sahi jawab: (b)
- (a) galat: Agar rational hota, to √3 = (2 + √3) − 2 bhi rational hota, jo ho nahi sakta.
- (b) sahi: Rational + irrational = irrational.
- (c) galat: Integer rational hota hai, par yeh to irrational hai.
- (d) galat: √3 ≈ 1.732, to 2 + √3 ≈ 3.732, 5 nahi.
Q8. √2 × √2 ki value aur type kya hai?
Solution dekho
Sahi jawab: (b)
- (a) galat: Value sahi hai par 2 = 2/1 rational hai.
- (b) sahi: √2 × √2 = 2 = 2/1, rational. Do irrational ka guna rational ho sakta hai.
- (c) galat: 4 nahi, kyunki √2 × √2 = (√2)² = 2.
- (d) galat: √4 = 2, jo rational hai.
Q9. Kaun sa number irrational NAHI hai?
Solution dekho
Sahi jawab: (c)
- (a) galat: Zeros badhte hain, irrational hai.
- (b) galat: 23 perfect square nahi, irrational.
- (c) sahi: Yeh ruka hua decimal hai (31415926/10⁷), rational hai. Isi liye yeh irrational nahi.
- (d) galat: π irrational hai, to π + 1 bhi irrational (rational + irrational).
Q10. Calculator par √5 = 2.236067977 aaya. Kya yeh √5 ki exact value hai?
Solution dekho
Sahi jawab: (b)
- (a) galat: Calculator limited digits dikhata hai, exact nahi.
- (b) sahi: 2.236067977 = 2236067977/10⁹, rational hai. √5 ka asli decimal aage chalta rehta hai.
- (c) galat: Is decimal mein koi block dohrata nahi dikhta.
- (d) galat: 5 perfect square nahi, to √5 irrational hai.
❓ FAQs (Bachchon Ke Asli Sawaal)
1. “Irrational” ka matlab kya pagal ya be-sar-pair number hai?
Nahi. “Irrational” mein “ratio” chhupa hai. Matlab woh number jo do poore numbers ke ratio p/q ke roop mein nahi likha ja sakta. Yeh bilkul theek-thaak, asli numbers hain.
2. Calculator √2 ke 10 digits dikhata hai, phir yeh irrational kaise?
Calculator ki screen chhoti hai, wo sirf kuch digits dikha sakta hai. Jo dikhta hai (1.414213562) ruka hua decimal hai, jo rational hota hai. Asli √2 ke digits kabhi khatam nahi hote, calculator bas uski paas ki value deta hai.
3. 0.1010010001… mein to pattern hai, phir yeh repeating kyun nahi?
Repeating ka matlab hai ek hi block baar-baar aaye, jaise 36, 36, 36. Yahan zeros ki ginti 1, 2, 3… badhti jaati hai, to koi ek block baar baar nahi aata. Isliye yeh non-repeating hai aur irrational hai.
4. Kya har non-terminating decimal irrational hota hai?
Nahi. 0.3333… non-terminating hai par repeating hai, aur 1/3 ke barabar rational hai. Irrational ke liye decimal non-terminating AUR non-repeating dono hona chahiye.
5. Hum π = 22/7 kyun padhte hain agar yeh sahi nahi?
Kyunki 22/7 = 3.142857… π ke kaafi paas hai (farq lagbhag 0.00126), aur hisaab mein aasan hai. Par yeh “approximation” hai, barabar nahi. Isliye likhte hain π ≈ 22/7.
6. Kya irrational numbers sach mein hote hain ya sirf kitaab mein?
Sach mein hote hain. 1 m ki square tile ka kona-se-kona naap √2 m hai aur gol cheez ki paridhi vyas ki π guna hoti hai. Yeh roz dikhne wali lambaiyan hain, bas inhe fraction se likh nahi sakte.
🧠 HOTS (Soch Wale Sawaal)
HOTS 1. Ek student kehta hai: “0.999… ka decimal kabhi rukta nahi, to yeh irrational hoga.” Kya woh sahi hai?
Solution dekho
Nahi. 0.999… mein 9 dohrata hai, to yeh repeating hai. Guna-ghatao se x = 0.999…, 10x − x = 9, x = 1. Yaani yeh 1 hai, rational. WHY? Sirf “na rukna” kaafi nahi, “na dohrana” bhi chahiye.
HOTS 2. x = 3 + √5 aur y = 3 − √5 ka jod aur guna nikaalo. Kya inka jod/guna rational hai?
Solution dekho
x + y = 6 aur x × y = 3² − (√5)² = 9 − 5 = 4. Dono rational hain, jabki x aur y khud irrational hain (3 + √5 rational + irrational hai). WHY? √5 jod mein +√5 − √5 se kat gaya, aur guna mein √5 × √5 = 5 ban gaya.
HOTS 3. Ek square ka area 2 cm² hai aur doosre square ka area 9/4 cm² hai. Kiski side rational hai?
Solution dekho
Pehle ki side √2 cm irrational hai. Doosre ki side √(9/4) = 3/2 cm rational hai. WHY? Side = √area, aur 9/4 = (3/2)² perfect square hai, jabki 2 kisi rational ka square nahi.
HOTS 4. Agar p ek nonzero rational number hai aur x irrational hai, to kya p/x irrational hoga? (extra)
Solution dekho
Haan. Maan lo p/x = r rational hota. p ≠ 0 hai to r ≠ 0, aur x = p/r rational ban jaata, jo ho nahi sakta. Jaise 1/√2 = √2/2 irrational hai. WHY? Contradiction se saaf hota hai.
🏆 Olympiad Corner
Oly 1. a = 0.123456789101112131415… (1, 2, 3, 4… natural numbers ek ke baad ek). Kya a irrational hai? Kaaran do.
Solution dekho
Haan, irrational. Agar a ka decimal kisi jagah se L digit ke block mein dohrata, aur us dohrate hisse mein L zeros ek saath aa jaate, to block poora 0 hota aur aage har digit 0 hota (yaani a terminating hota). Par a mein 10, 100, 1000, … likhe hain, to kitni bhi door jaakar zeros ki bahut lambi sequence milti hai, jiske baad phir non-zero digit aata hai. Isliye koi bhi dohrane wala block ban hi nahi sakta. WHY? Repeating decimal ek chhote block ko hamesha dohrata hai, par yahan zeros ki lambaiyan badhti jaati hain aur beech mein non-zero digit aate rehte hain.
Oly 2. (√3 + 1)(√3 − 1) ki value nikaalo. Kya ek rational number do irrational numbers se bana?
Solution dekho
(√3 + 1)(√3 − 1) = (√3)² − 1² = 3 − 1 = 2. Haan, 2 rational hai, aur √3 + 1 tathaa √3 − 1 dono irrational hain. WHY? (a + b)(a − b) = a² − b² wali pehchaan se √3 ka guna 3 ban gaya.
Oly 3. 1 se 50 tak ke natural numbers n ke liye √n mein kitne irrational aur kitne rational hain?
Solution dekho
Perfect squares 1, 4, 9, 16, 25, 36, 49 hain (1² se 7²), yaani 7 rational. Baaki 50 − 7 = 43 irrational. WHY? √n rational tab hota hai jab n perfect square ho.
Oly 4. a = √2 + 1 aur b = √2 − 1 lo. a × b aur a − b nikaalo. Aur batao a = 2.41421356… aur b = 0.41421356… ke decimal mein kya khaas hai?
Solution dekho
a × b = (√2)² − 1 = 2 − 1 = 1 aur a − b = 2. Dono rational. Kyunki a − b = 2 (poora number), to a aur b ke decimal point ke baad ke digits bilkul ek jaise hain (41421356…), bas poora ank 2 aur 0 ka farq hai. WHY? Agar do numbers ka farq poora number ho, to unka decimal hissa barabar hota hai.
💖 Jo Naap Mein Na Aaye, Uski Kadar Kam Nahi
Maths: irrational numbers ko fraction mein baandh nahi sakte, phir bhi woh asli hain aur tile ke kone se lekar pahiye ke ghere tak har jagah milte hain. Jo cheez naap mein poori na aaye, uska wajood kam nahi hota.
Life: kuch talents aur khoobiyan bhi “marks” ya “rank” ke fraction mein nahi samaati. Aapki mehnat, aapki lagan, aapka dhairya naapne ke liye number kam padte hain. Khud ko sirf ek score se mat tolo. 🌟
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🎯 Master Recap
- Irrational number woh hai jo p/q (p, q integers, q ≠ 0) ke roop mein likha na ja sake.
- Decimal: non-terminating aur non-repeating = irrational. Terminating ya repeating = rational.
- Pattern hona kaafi nahi: rational mein ek hi block baar-baar dohrata hai. Zeros/ones badhte jaayein to irrational.
- √n tabhi rational jab n perfect square. √2, √3, √5, √7, √10 irrational. √25 = 5 rational. (√2, √3 ka saboot agle post mein.)
- π irrational hai. 22/7 aur 3.14 sirf approximation hain.
- Extra: rational + irrational = irrational; nonzero rational × irrational = irrational. Irrational + irrational kuch bhi ho sakta hai.
- Aam galtiyan: calculator ka ruka decimal irrational maan lena, ya √ dikhte hi irrational kehna.
— charumam.com 🎯
🏋️ Practice Zone — 15 Questions
Q1. Easy √64 rational hai ya irrational?
Solution dekho
√64 = 8 = 8/1, to rational. WHY? 64 = 8² perfect square hai.
Q2. Easy √13 rational hai ya irrational?
Solution dekho
13 perfect square nahi (3² = 9, 4² = 16), to √13 irrational.
Q3. Easy 0.625 ko p/q mein likho aur batao rational hai ya nahi.
Solution dekho
0.625 = 625/1000 = 5/8. Ruka hua decimal hai, to rational.
Q4. Easy 0.5656… (56 dohrata hai) rational hai ya irrational? p/q bhi likho.
Solution dekho
Block 56 dohrata hai, to rational. 100x − x = 56, 99x = 56, x = 56/99.
Q5. Easy 0.2020020002… (2 ke beech zeros 1, 2, 3… badhte hain) rational hai ya irrational?
Solution dekho
Na ruka, na koi fixed block dohraya, to irrational. WHY? Zeros ki ginti badalti rehti hai.
Q6. Medium √9, √10, 2/3, 0.45, 0.1121231234…, π, √(16/25) mein se irrational numbers chuno.
Solution dekho
√9 = 3, 2/3, 0.45 = 9/20 aur √(16/25) = 4/5 rational hain. Irrational: √10, 0.1121231234… (1, 12, 123, 1234… badhta pattern) aur π. Kul 3.
Q7. Medium Kya 3.1416 rational hai? Isko p/q mein likho aur batao yeh π kyun nahi.
Solution dekho
3.1416 = 31416/10000 = 3927/1250 (8 se bhaag), rational. π ka decimal 3.14159265… aage chalta hai, to 3.1416 sirf π ki approximation hai. WHY? Ruka hua decimal hamesha rational hota hai.
Q8. Medium 5 + √7 rational hai ya irrational? Kaaran do. (extra)
Solution dekho
7 perfect square nahi, to √7 irrational. Agar 5 + √7 = p/q rational hota, to √7 = p/q − 5 rational hota, jo ho nahi sakta. To 5 + √7 irrational.
Q9. Medium 0.2020020002… mein 2 ke beech zeros badhte jaate hain (1, 2, 3 …). Decimal ke baad 10 digits 0.2020020002 hain. Uske agle 5 digits kya honge?
Solution dekho
Pattern: 2, 0, 2, 00, 2, 000, 2, aur ab 4 zeros ke baad 2 aayega. Agle 5 digits 00002 hain. WHY? Zeros ki ginti har baar 1 badhti hai: 1, 2, 3, ab 4.
Q10. Medium 22/7 aur π ke decimal likho aur dikhao ki yeh barabar nahi hain.
Solution dekho
22/7 = 3.142857142857… (142857 dohrata hai), rational. π = 3.14159265… (na ruke na dohraye), irrational. Rational ≠ irrational, to 22/7 ≠ π, farq lagbhag 0.00126.
Q11. NCERT / Extra (2 + √3) + (4 − √3) ka jod nikaalo. Kya yeh rational hai? (extra)
Solution dekho
2 + 4 + √3 − √3 = 6. Haan, rational. Do irrational numbers ka jod rational ban gaya kyunki √3 kat gaya.
Q12. NCERT / Extra √5 × √20 nikaalo. Kya yeh rational hai? (extra)
Solution dekho
√5 × √20 = √100 = 10, rational. √20 irrational hai (20 perfect square nahi), phir bhi guna rational nikla. WHY? Do roots ka guna ek hi root mein milta hai: √a × √b = √(ab).
Q13. NCERT / Extra 21 se 40 ke beech ke kin natural numbers n ke liye √n rational hai?
Solution dekho
Perfect squares: 5² = 25, 6² = 36. 7² = 49 (bahar). Yaani n = 25 aur n = 36. Baaki sab ke √n irrational hain.
Q14. NCERT / Extra 3√2 aur √2/4 kya irrational hain? (extra)
Solution dekho
Dono nonzero rational × irrational hain. Agar 3√2 = r rational hota, to √2 = r/3 rational hota, jo ho nahi sakta. Isi tarah √2/4 bhi. Dono irrational.
Q15. NCERT / Extra 0.123456789 aur 0.123456789101112… (natural numbers ek ke baad ek) mein kaunsa rational hai aur kaunsa irrational? (extra)
Solution dekho
Pehla 9 digits par ruk jaata hai, = 123456789/10⁹, rational. Doosra na rukta hai, na koi block dohrata hai (Olympiad 1 ki tarah), to irrational. WHY? Ruka hua decimal rational hota hai, chaahe kitna bhi lamba ho.
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Ab aap irrational numbers ko pehchaan sakte ho. Agle post mein hum dekhenge ki √2 aur √3 ke irrational hone ka pakka saboot kaise diya jaata hai. Milte hain √2 aur √3 Irrational Kyun Hain se.
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