✅ Pehle Yeh Aata Hai?
Pichle post Irrational Numbers Kya Hain mein humne maana tha ki √2 aur √3 irrational hain. Aaj hum iska pakka saboot dekhenge. Rational numbers ki purani baat ke liye Rational Numbers aur Unki Properties bhi dekh sakte ho. Pehle teen chhote sawaal:
- Agar p ek odd number hai (jaise 7), to p² odd hoga ya even?
- 12/30 ko sabse chhote roop (lowest terms) mein likho.
- Kya 2 ek perfect square hai?
Answers dekho
- 7² = 49, jo odd hai. Odd × odd hamesha odd hota hai.
- 12 aur 30 dono 6 se katte hain: 12/30 = 2/5. Ab 2 aur 5 mein koi common factor nahi bacha.
- 1² = 1 aur 2² = 4, to 2 in dono ke beech hai. Nahi, 2 perfect square nahi hai.
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🎬 Kahani: Gudiya ka Fraction Dhoondo Challenge
Rohan ke ghar mein 1 m × 1 m ki tile hai. Uska kona-se-kona dori ki lambai Pythagoras se √(1² + 1²) = √2 meter aati hai. Gudiya ne chunauti li: “Main √2 ke barabar ek fraction dhoondh ke dikhaungi!”
Usne 7/5 try kiya: (7/5)² = 49/25 = 1.96. Bahut paas, par 2 nahi. Phir 17/12 try kiya: (17/12)² = 289/144 ≈ 2.0069. Phir paas, par phir 2 nahi! 😮 Wo har baar paas pahunchti, par 2 tak kabhi nahi.
Kya Gudiya kabhi jeet sakti hai? Aaj ka jawab ek hi shabd mein: Nahi! Aur hum yeh ek aise tarike se saabit karenge jo sirf “dekh ke” nahi, balki tark (logic) se pakka hota hai. 💪
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📖 Concept: Contradiction se Saboot
Step 1: Saboot ka tarika (contradiction)
Hum ulta maante hain: “maan lo √2 rational hai”. Phir tark chalate hain. Agar aakhir mein koi aisi baat nikle jo ho nahi sakti, to hamari maan-ni galat thi, yaani √2 rational nahi hai. WHY? Sahi tark se kabhi galat nateeja nahi nikalta. To galti sirf shuru ki maan-ni mein ho sakti hai.
Step 2: Do zaroori auzaar
Auzaar A (lowest terms): har fraction ko chhota kar sakte hain, to hum p/q aisa lete hain jisme p aur q ka koi common factor 1 ke alawa nahi (inhe co-prime kehte hain). Jaise 12/30 ko 2/5 bana liya.
Auzaar B: agar p² even hai, to p bhi even hai. WHY? Agar p odd hota, p = 2k + 1 likho. Tab p² = 4k² + 4k + 1 = 2(2k² + 2k) + 1, jo odd hai. To p² even tabhi hoga jab p even ho.
Step 3: √2 ka saboot
Isliye hamari maan-ni galat thi. √2 irrational hai. ✅
Step 4: √3 ka saboot
Yahan Auzaar B ki jagah naya auzaar chahiye. Auzaar C: agar p², 3 se divisible hai, to p bhi 3 se divisible hai. WHY? Agar p = 3k + 1 hota, p² = 9k² + 6k + 1 = 3(3k² + 2k) + 1, yaani remainder 1. Agar p = 3k + 2 hota, p² = 9k² + 12k + 4 = 3(3k² + 4k + 1) + 1, phir remainder 1. Dono mein p² 3 se divisible nahi. To p² divisible tabhi hoga jab p = 3k ho.
Gudiya ki chunauti ka jawab mil gaya: koi bhi p/q √2 ke barabar ho hi nahi sakta. 7/5 aur 17/12 sirf paas aate hain.
Irrational numbers ki pehchaan (decimal se) ki poori detail ke liye yeh post padho: Irrational Numbers Kya Hain.
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Saboot yahi tarika √4 par kyun fail hota hai?
1) Test karo: √4 = p/q maano. Square: p² = 4q². Iska matlab p = 2q (kyunki p, q positive). Yahan p = 2q aaya, par yeh q ke even hone par majboor nahi karta. Jaise q = 1, p = 2 chalta hai: 2/1 co-prime hai. Koi contradiction nahi, kyunki √4 = 2 sach mein rational hai.
2) √2 mein kya alag tha? p² = 2q² ne p ko even banaya, phir q² = 2m² ne q ko bhi even bana diya. Yeh “do baar 2 nikalna” co-prime ke saath takrata hai. Perfect square mein yeh chain nahi banti.
3) Co-prime maan-na jaayaz kyun hai? Agar koi p/q likha ja sakta hai, to use chhota karke co-prime bana sakte hain. To “co-prime p/q” maan-na koi chhoot nahi, balki sabse sakht shart hai.
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✏️ Solved Examples
Example 1 (Easy): Gudiya ne 3/2 aur 10/7 try kiye. Dono ke square nikaalo aur batao kya koi √2 ke barabar hai.
(3/2)² = 9/4 = 2.25. (10/7)² = 100/49 ≈ 2.0408. Dono 2 nahi hain, to koi √2 ke barabar nahi.
WHY? Fraction √2 ke barabar tabhi hota jab uska square bilkul 2 aata. Saboot ne dikhaya ki aisa koi fraction hai hi nahi.
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Example 2 (Medium): Saabit karo ki 4 − √3 irrational hai.
Maan lo 4 − √3 = r, jahan r rational hai. Tab √3 = 4 − r. Rational − rational = rational, to √3 rational ban gaya. Par √3 irrational hai (abhi saabit kiya). Contradiction! Isliye 4 − √3 irrational hai.
WHY? Agar yeh rational hota, to √3 bhi rational ho jaata. Ek ulti baat se doosri ulti baat nikli, to maan-ni galat thi.
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Example 3 (NCERT style): Saabit karo ki √5 irrational hai.
Maan lo √5 = p/q, p, q co-prime, q ≠ 0. Square: p² = 5q². To p², 5 se divisible hai. 5 prime hai, to p bhi 5 se divisible hai (p ke prime factors wahi hote hain jo p² ke). Likho p = 5m. Tab 25m² = 5q², yaani q² = 5m², to q bhi 5 ka multiple. Dono mein 5 common, par co-prime the. Contradiction! √5 irrational hai.
WHY prime zaroori? 5 prime hai, isliye 5 ka p² ko baantna p ko baantne par majboor karta hai. Yeh baat 4 jaise number par sach nahi hai (aage dekho).
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Gudiya ka jawab: koi fraction √2 ke barabar nahi hai, wo sirf paas pahunch sakti hai. 7/5 aur 17/12 achhe approximations hain. ✅
🚀 Aur Aage: NCERT ke bahar
1) Har prime p ke liye √p irrational (extra): √2, √3, √5, √7, √11, √13 sab irrational. WHY? Saboot wahi chalta hai: p² = (prime) × q² se p us prime ka multiple, phir q bhi. Prime hona hi chaabi hai.
2) Prime nahi to sambhal ke (extra): “p², 4 se divisible ⇒ p, 4 se divisible” galat hai. Jaise p = 6: 6² = 36, 4 se divisible, par 6 nahi. Isliye √4 ya √9 par saboot nahi chalta, aur chalna bhi nahi chahiye (woh rational hain).
3) √6 jaise numbers (extra): √6 = p/q se p² = 6q², to p² even, p even, p = 2m. Phir 4m² = 6q², yaani 2m² = 3q². Ab 3q² even hai, to q² even, to q even. Dono even, contradiction. To √6 irrational. Is tarah ke aur sawaal Olympiad corner mein hain.
4) √2 + √3 irrational hai (extra): maan lo √2 + √3 = r (rational, r > 0). Tab √3 = r − √2. Square: 3 = r² − 2r√2 + 2, to √2 = (r² − 1)/(2r), jo rational hai. Par √2 irrational hai. Contradiction! WHY? Yahan humne √3 ko akela chhod ke √2 ko akela kar liya.
5) 1/√2 irrational (extra): 1/√2 = √2/2. Agar yeh rational r hota, to √2 = 2r rational hota. Isliye irrational.
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🌍 Real Life Mein
Kamre ka kona: 3 m × 3 m ke kamre mein kona-se-kona dori 3√2 m ≈ 4.24 m hoti hai. Mistri 4.24 m maan ke kaam chalata hai, par exact kabhi nahi bata sakta. Ab aap jaante ho ki isme galti mistri ki nahi, √2 ke irrational hone ki hai.
Calculator aur computer: screen par √2 ke sirf kuch digits aate hain, yaani ek ruka hua rational number. Isliye sensitive hisaab mein engineers “approximation ki seema” likhte hain.
Sochne ka tarika: “Agar aisa ho to kya galat hoga?” wali soch detective aur judge bhi use karte hain. Alibi ke jhoot ko pakadne ke liye bhi contradiction hi kaam aata hai. 🕵️
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⚖️ Kab Use Karein / Kab Nahi
| Kab contradiction wala saboot | Kab nahi |
|---|---|
| “Yeh number rational nahi hai” saabit karna ho (√2, √3, √5, √7) | Number sach mein rational ho (√4, √49): saboot chalega hi nahi |
| √n mein n prime ho, to p² ⇒ p ka fact seedha lagta hai | n composite ho, to pehle prime factors dekho (√6 ke liye alag dhyan) |
| Doosre numbers (4 − √3, 3√5) ko irrational dikhana ho, jab ek irrational pehle se maloom ho | Calculator ke digits dekh ke “saboot” ka dava karna (yeh saboot nahi hota) |
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❌→✅ Worked Mistake
Sawaal: √2 ke saboot mein p² = 2q² se aage kya likhein?
❌ Galat: “p² = 2q² to p = 2q.”
✅ Sahi: p² = 2q² se sirf itna pata chalta hai ki p² even hai, to p even hai. Phir p = 2m likho aur daalo: 4m² = 2q², yaani q² = 2m².
WHY? Agar p = 2q hota, to p² = 4q² hota, jo 2q² nahi hai (q ≠ 0). Square-root dono taraf lene par factor √2 aata, 2 nahi. Isliye p = 2q galat hai.
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📝 10 MCQs
Q1. √2 irrational saabit karne ke liye kaun sa tarika use hota hai?
Solution dekho
Sahi jawab: (b)
- (a) galat: Calculator sirf kuch digits deta hai, saboot nahi.
- (b) sahi: Pehle “√2 = p/q” maante hain, phir kuch aisa nikalta hai jo ho nahi sakta.
- (c) galat: Ek-do udaharan se saboot nahi bante.
- (d) galat: Andaaze se pakki baat nahi hoti.
Q2. Agar p² even hai, to:
Solution dekho
Sahi jawab: (d)
- (a) galat: Odd p ka square odd hota hai (p = 2k + 1 ⇒ p² odd).
- (b) galat: p = 4, 6 bhi chalte hain, jo prime nahi hain.
- (c) galat: p = 2 sirf ek udaharan hai, hamesha nahi. p = 4 par bhi p² = 16 even hai.
- (d) sahi: p² even tabhi hota hai jab p even ho.
Q3. √2 ke saboot mein contradiction kis baat se aata hai?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Dono even matlab common factor 2, par humne co-prime maana tha.
- (b) galat: Humein dono even mile, odd nahi.
- (c) galat: q ≠ 0 shuru se maana hua hai, aur nikalta bhi nahi.
- (d) galat: p = q se to √2 = 1 hota, jo galat hai, par yeh saboot ka step nahi hai.
Q4. Inme se kaun sa irrational hai?
Solution dekho
Sahi jawab: (c)
- (a) galat: √36 = 6, rational.
- (b) galat: √(9/16) = 3/4, rational.
- (c) sahi: 20 perfect square nahi (4² = 16, 5² = 25), to √20 irrational.
- (d) galat: √0.04 = 0.2 = 1/5, rational.
Q5. √3 ke saboot mein kaun sa fact kaam aata hai?
Solution dekho
Sahi jawab: (d)
- (a) galat: p² = 3p sirf p = 3 par sach hai, har p ke liye nahi.
- (b) galat: p = 3 lo: 3² = 9, par 3, 9 se divisible nahi.
- (c) galat: p = 6 multiple of 3 hai par even.
- (d) sahi: p = 3k+1 ya 3k+2 hota to p² ka remainder 1 aata, isliye p = 3k.
Q6. (17/12)² kitna hai (lagbhag)?
Solution dekho
Sahi jawab: (b)
- (a) galat: 289/144 ≠ 2, kyunki 2 × 144 = 288.
- (b) sahi: 289/144 = 2.00694…, yaani 2 se thoda zyada.
- (c) galat: 1.96 to (7/5)² hai.
- (d) galat: 2.5 bahut door hai.
Q7. √7 irrational kyun hai?
Solution dekho
Sahi jawab: (c)
- (a) galat: Odd hona irrational hone ka kaaran nahi, 9 odd hai par √9 = 3.
- (b) galat: Calculator saboot nahi hai.
- (c) sahi: Yahi saboot hai: dono ka common factor 7 co-prime se takrata hai.
- (d) galat: Bada hona kaaran nahi, 16 > 2 par √16 = 4.
Q8. 2√3 kaisa number hai?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Nonzero rational × irrational = irrational.
- (b) galat: Agar 2√3 = r rational hota, to √3 = r/2 rational hota, jo nahi ho sakta.
- (c) galat: Integer rational hote hain, par 2√3 ≈ 3.464 aisa nahi.
- (d) galat: Natural number bhi rational hote hain.
Q9. √2 = p/q ko square karne par kya milta hai?
Solution dekho
Sahi jawab: (d)
- (a) galat: q² ki jagah q likh diya, par q ka bhi square aata hai.
- (b) galat: Ulta ho gaya: 2 = p²/q² se p² = 2q², q² = 2p² nahi.
- (c) galat: Isse p seedha nahi milta, p² hi milta hai.
- (d) sahi: 2 = p²/q² ko q² se guna karne par p² = 2q².
Q10. √4 par yahi saboot kyun fail hota hai?
Solution dekho
Sahi jawab: (b)
- (a) galat: 4 even hai, odd nahi.
- (b) sahi: p = 2q hone se koi contradiction nahi, kyunki 2/1 co-prime hai.
- (c) galat: 4 = 2 × 2, prime nahi hai.
- (d) galat: q ≠ 0 bana rehta hai.
❓ FAQs (Bachchon Ke Asli Sawaal)
1. Calculator √2 ke 10 digits deta hai, phir saboot ki kya zaroorat?
Calculator ruke hue decimal dikhata hai, jo hamesha rational hote hain. Digits kitne bhi hon, wo √2 ka sach nahi hota. Saboot ek tark hai jo saare digits ke baare mein ek saath baat karta hai.
2. Ulta maan ke saboot dena theek kyun hai?
Agar sahi tark se hum galat nateeje par pahunchte hain, to galti shuru ki maan-ni mein hi hogi. Yeh bilkul waise hai jaise “agar yeh aadmi shaher mein tha, to wo ek hi samay par do jagah tha” se pata chalta hai ki wo shaher mein nahi tha.
3. p/q ko co-prime kyun maan liya? Agar na ho to?
Koi bhi fraction chhota karke co-prime bana sakte hain (jaise 12/30 = 2/5). To “√2 koi p/q hai” ka matlab hi hai “√2 koi co-prime p/q hai”. Isme koi chhoot nahi li gayi.
4. Kya √2 ke irrational hone ke aur bhi saboot hain?
Haan, kai tarah ke saboot hote hain. Humne sabse seedha waala dekha, jo NCERT ke dhaanche mein bhi aata hai. Aap aage aur seekhoge to naye tarike bhi milenge.
5. √2 irrational hai to number line par uski jagah kahan hai?
Zaroor hai! Wo 1.414… par hai, aur use compass aur Pythagoras se banaya ja sakta hai. Iski poori detail agle post “Square Root Spiral” (jald aa raha hai) mein aayegi.
6. Kya √4 par bhi saboot try karke dekh sakte hain?
Haan, aur karna chahiye! p² = 4q² se p = 2q milta hai, par koi contradiction nahi aata. Isse samajh aata hai ki saboot kahan “perfect square nahi hai” ki baat pakadta hai.
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🧠 HOTS (Soch Wale Sawaal)
HOTS 1. √5 irrational hai (maan lo). Saabit karo ki 3 − 2√5 irrational hai.
Solution dekho
Maan lo 3 − 2√5 = r (rational). Tab 2√5 = 3 − r, to √5 = (3 − r)/2, jo rational hai. Par √5 irrational hai. Contradiction! To 3 − 2√5 irrational hai. WHY? Rational ko rational se jodne-ghatane aur 2 se baantne par rational hi rehta hai.
HOTS 2. Ek student kehta hai: “√2 = 1.4142 = 14142/10000, to √2 rational hai.” Galti kahan hai?
Solution dekho
1.4142 sirf √2 ki paas ki value hai. (1.4142)² = 1.99996164, yaani 2 nahi; actual √2 = 1.41421356… hai. Saboot kehta hai ki koi bhi fraction √2 ke bilkul barabar nahi ho sakta. WHY? Approximation aur barabar mein farq hota hai.
HOTS 3. Do irrational numbers batao jinka jod rational ho, aur do aur jinka guna rational ho.
Solution dekho
√2 aur −√2: jod = 0 (rational). √2 aur √8: guna = √16 = 4 (rational). WHY? √2 ka −√2 se cut gaya aur √2 × √8 ek perfect square ban gaya.
HOTS 4. Saabit karo ki 1/√3 irrational hai.
Solution dekho
1/√3 = √3/3. Agar yeh rational r hota, to √3 = 3r rational hota, jo nahi ho sakta. WHY? Nonzero rational se guna karne par rationality nahi badalti.
🏆 Olympiad Corner
Oly 1. Saabit karo ki √3 + √5 irrational hai (√3, √5 irrational maan lo).
Solution dekho
Maan lo √3 + √5 = r rational (r > 0). Tab √5 = r − √3. Square: 5 = r² − 2r√3 + 3, to √3 = (r² − 2)/(2r), rational. Contradiction. WHY? Square karke √3 akela rakh diya.
Oly 2. Saabit karo ki √6 irrational hai.
Solution dekho
Maan lo √6 = p/q (co-prime). p² = 6q², to p even, p = 2m. 4m² = 6q², yaani 2m² = 3q². 3q² even hai, to q² even, to q even. Dono even, contradiction. WHY? 6 ke factor 2 se hi pehla kadam chala, aur doosre kadam par bhi 2 hi aaya.
Oly 3. (√2 + √3)/2 √2 aur √3 ke beech ka number hai. Kya yeh irrational hai?
Solution dekho
Haan. Agar (√2 + √3)/2 = r rational hota, to √2 + √3 = 2r rational hota. Par √2 + √3 irrational hai (Aur Aage mein saabit hua). Aur yeh dono ka average hai, to √2 aur √3 ke beech aata hai. WHY? Do numbers ka average unke bilkul beech mein hota hai.
Oly 4. √12, √18, √20, √36, √45 mein se kitne rational hain?
Solution dekho
√36 = 6 rational hai. √12 = 2√3, √18 = 3√2, √20 = 2√5, √45 = 3√5 sab irrational. Yaani sirf 1 rational, 4 irrational. WHY? Nonzero rational × irrational = irrational.
💖 Jo Dikhta Nahi, Use Bhi Saabit Kiya Ja Sakta Hai
Maths: √2 ke saare digits koi kabhi dekh nahi sakta, phir bhi ek chhote se tark se hum pakka jaan lete hain ki wo fraction nahi hai. Soch ki taaqat dikhne wali cheez se bhi badi hoti hai.
Life: kai baar mehnat ka phal turant nahi dikhta. Par agar aapki neev saaf soch aur lagan par hai, to aap bharosa rakh sakte ho ki nateeja aayega. Aaj ke chhote kadam kal ki badi jeet ka saboot hote hain. 🌟
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🎯 Master Recap
- Saboot ka tarika: ulta maano (√2 = p/q, co-prime), tark chalao, contradiction nikaalo.
- Auzaar B: p² even ⇒ p even. Auzaar C: p², 3 se divisible ⇒ p, 3 se divisible. (WHY: odd/remainder check.)
- √2: p² = 2q² ⇒ p = 2m ⇒ q² = 2m² ⇒ q even. Dono even, co-prime se takrate hain.
- √3: p² = 3q² ⇒ p = 3m ⇒ q² = 3m² ⇒ q multiple of 3. Contradiction.
- Extra: har prime p ke liye √p irrational. 4 jaise composite par fact (p² ⇒ p) galat hai.
- Extra: 4 − √3, 3 − 2√5, 2√3, 1/√3 jaise numbers irrational. √2 + √3 aur √3 + √5 bhi irrational.
- Aam galtiyan: p² = 2q² se p = 2q likhna, ya calculator ke digits ko saboot samajhna.
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🏋️ Practice Zone — 15 Questions
Q1. Easy Kya √2 = 3/2 ho sakta hai? Square karke dekho.
Solution dekho
(3/2)² = 9/4 = 2.25, jo 2 nahi hai. To nahi. WHY? √2 ka square 2 hona chahiye.
Q2. Easy Khali jagah bharo: p² even hai to p ______ hai.
Solution dekho
Even. WHY? Odd p ka square hamesha odd hota hai.
Q3. Easy p = 7 odd hai. p² kya hai aur odd hai ya even?
Solution dekho
7² = 49, odd. WHY? p = 2k + 1 ka square 2(2k² + 2k) + 1 hota hai.
Q4. Easy √11 rational hai ya irrational? Kaaran do.
Solution dekho
11 prime hai aur perfect square nahi (3² = 9, 4² = 16), to √11 irrational hai.
Q5. Easy 12/30 ko lowest terms mein likho.
Solution dekho
12 aur 30 ka sabse bada common factor 6 hai. 12/30 = 2/5.
Q6. Medium p = 12, 14, 21 ke liye p² ko 3 se divide karke dekho. Kab p² divisible hai?
Solution dekho
12² = 144 = 3 × 48 (divisible, aur 12 bhi 3 ka multiple). 14² = 196 = 3 × 65 + 1 (divisible nahi, 14 bhi nahi). 21² = 441 = 3 × 147 (divisible, 21 bhi 3 ka multiple). Dekha: p² divisible tabhi jab p divisible.
Q7. Medium Saabit karo ki 5 + √2 irrational hai.
Solution dekho
Maan lo 5 + √2 = r rational. Tab √2 = r − 5 rational ho jaata hai, jo nahi ho sakta. To 5 + √2 irrational hai.
Q8. Medium Saabit karo ki 7√3 irrational hai.
Solution dekho
Maan lo 7√3 = r rational. Tab √3 = r/7 rational hota, par √3 irrational hai. Contradiction, to 7√3 irrational hai.
Q9. Medium √3 ke saboot mein p = 3m daalne par kya milta hai? Aage kya natija nikalta hai?
Solution dekho
p² = 3q² mein p = 3m: 9m² = 3q², to q² = 3m². Isliye q², 3 se divisible, to q bhi 3 ka multiple. p aur q dono mein 3 common, jo co-prime ke khilaf hai.
Q10. Medium √2/3 rational hai ya irrational?
Solution dekho
Maan lo √2/3 = r rational. Tab √2 = 3r rational ho jaata hai, jo nahi ho sakta. To irrational hai.
Q11. NCERT / Extra Saabit karo ki √7 irrational hai.
Solution dekho
Maan lo √7 = p/q (co-prime). p² = 7q², to p², 7 se divisible. 7 prime hai, to p bhi 7 ka multiple, p = 7m. 49m² = 7q², q² = 7m², to q bhi 7 ka multiple. Contradiction! √7 irrational.
Q12. NCERT / Extra Saabit karo ki 1/√2 irrational hai. (extra)
Solution dekho
1/√2 = √2/2. Agar yeh rational r hota, to √2 = 2r rational hota. Contradiction. To 1/√2 irrational hai.
Q13. NCERT / Extra Dikhao ki “p², 4 se divisible ⇒ p, 4 se divisible” galat hai. (extra)
Solution dekho
p = 6 lo: 6² = 36 = 4 × 9, 4 se divisible. Par 6, 4 se divisible nahi. To baat galat hai. WHY? Yeh fact sirf prime ke liye pakka hota hai.
Q14. NCERT / Extra 1 se 30 tak n ke liye √n mein kitne rational aur kitne irrational hain? (extra)
Solution dekho
Perfect squares: 1, 4, 9, 16, 25 yaani 5 rational. Baaki 30 − 5 = 25 irrational.
Q15. NCERT / Extra Saabit karo ki √2 + √5 irrational hai. (extra)
Solution dekho
Maan lo √2 + √5 = r rational (r > 0). Tab √5 = r − √2. Square: 5 = r² − 2r√2 + 2, to √2 = (r² − 3)/(2r), rational. Par √2 irrational hai. Contradiction! √2 + √5 irrational.
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👉 Aage Kya?
Ab aapko √2 aur √3 ke irrational hone ka pakka saboot aa gaya. Agle post mein hum dekhenge ki √2, √3, √5 jaise numbers ko number line par kaise dikhate hain, ek mazedaar “Square Root Spiral” ke saath.
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