✅ Pehle Yeh Aata Hai?
Pichle post Decimal Forms: Terminating aur Repeating mein humne dekha ki p/q ka decimal ya ruk jaata hai ya dohrata hai. Ab ulta safar: dohrate decimal se wapas p/q tak. Pehle teen chhote sawaal:
- 100 × 0.27 kitna hota hai?
- 27.27 − 0.27 kitna hota hai?
- 36/99 ko sabse chhote form mein likho.
Answers dekho
- 100 se guna karne par decimal point 2 jagah daayen khisakta hai: 0.27 → 27.
- 27.27 − 0.27: decimal ke baad ke dono hisse (.27) barabar hain, kat jaate hain. Bachta hai 27.
- 36 aur 99 ka HCF 9 hai (36 = 9 × 4, 99 = 9 × 11). To 36/99 = 4/11. 4/11.
Decimal point khisakne ka matlab thoda dhundhla ho to: MN-03: Decimal Point ka asli matlab padho.
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🎬 Kahani: Dadi ka Ajeeb Calculator
Meena ke calculator par kuch dabane ke baad screen par aaya 0.6666666…. Usne dadi ko dikhaya. Dadi bole, “Beta, yeh to mere do-tihaayi (2/3) wali baat hai!” Meena hairan: “Dadi, yeh to ek lamba decimal hai, fraction kaise ho gaya?”
Dadi ne kaha, “Jo number khud ko dohrata hai, uski ek chaal hoti hai. Us chaal ko pakad lo, to woh seedha fraction ban jaata hai.” 🤔
Is post mein hum wahi chaal seekhenge: ek chhoti si guna aur ghatao ki trick, jisse har dohrata decimal ek pakka p/q fraction ban jaata hai. Matlab har repeating decimal ek rational number hai!
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📖 Concept: Dohrate Decimal ko Fraction Banana
Step 1: Chaal kya hai?
Dohrate decimal mein ek block baar baar aata hai. Agar hum number ko 10, 100 ya 1000 se guna karein, to decimal point khisak jaata hai, par dohrane wala hissa wahi ka wahi rehta hai. Phir dono numbers ko ghata do, to lamba “…” wala hissa kat jaata hai aur bachta hai ek seedha number.
Step 2: Sirf ek digit dohrata hai
Example: x = 0.7 (yaani 0.777…)
WHY 10 se guna? Block mein 1 digit hai, to decimal point 1 jagah khisakna hai taaki dono number ke “…” wale hisse bilkul ek jaise ho jayein. Tabhi ghatane par woh kat paate hain.
Step 3: Do digit ka block
Example: x = 0.36 (0.363636…)
WHY? 100x − x = 99x. Aur 4/11 ko bhaag karke dekho: 4 ÷ 11 = 0.3636… ✅.
Step 4: Shortcut nikalti hai
Jab decimal point ke turant baad se block dohrata hai (pure repeating), to fraction = block ÷ (utne 9, jitne block mein digit). Jaise 0.36 = 36/99, 0.045 = 45/999.
Step 5: Pehle kuch digit nahi dohrate (mixed)
Example: x = 0.83 (0.8333…)
WHY do baar guna? Pehle guna se dohrana point ke turant baad aata hai, doosre guna se ek block aage. In dono ke “…” ek jaise hain, isliye ghatane par kat jaate hain. Check: 5 ÷ 6 = 0.8333… ✅.
Step 6: Poora ank bhi ho
Example: x = 2.45 (2.4545…)
Yaad rakho: Hamesha ek hi chaal: x likho → 10 ki sahi power se guna → ghatao → x nikaalo → chhota karo.
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Ghatane ki trick sahi kyun hai?
1) Infinite number ko ghatana jaayaz hai kyunki “…” wale hisse bilkul barabar hote hain. 100x = 36.3636… aur x = 0.3636… dono mein decimal point ke baad bilkul ek hi sequence hai (3, 6, 3, 6, …). Digit-by-digit ghatane par har jagah 0 aata hai, to bachta hai sirf 36.
2) Isse ek bada nateeja nikalta hai: har dohrate decimal ka fraction ban sakta hai, kyunki akhir mein hum (poori sankhya) ÷ (9, 99, 999… ya 90, 900… jaisi poori sankhya) likhte hain. Poori sankhya ÷ poori sankhya = rational. Pichle post ka nateeja aur yeh milkar kehte hain: rational ⇔ terminating ya repeating decimal.
3) Sabse chaukane wali baat: 0.9 = 1. x = 0.999…, 10x = 9.999…, 9x = 9, x = 1. 0.999… 1 se “thoda sa kam” nahi hai: anant 9 lagane par difference 0.000…1 jaisa kuch bachta hi nahi. Doosra sabut: 1/3 = 0.333…, to 3 × (1/3) = 0.999…, aur 3 × (1/3) = 1 bhi hai.
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✏️ Solved Examples
Example 1 (Easy): 0.5 ko p/q mein likho.
x = 0.555…, 10x = 5.555…, 10x − x = 5, 9x = 5, to x = 5/9.
WHY? 1 digit ka block hai, isliye 10 se guna. Check: 5 ÷ 9 = 0.555… ✅.
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Example 2 (Medium): 0.16 ko p/q mein likho.
x = 0.1666…. 10x = 1.666…, 100x = 16.666…. Ghatao: 100x − 10x = 16 − 1 = 15, to 90x = 15, x = 15/90 = 1/6.
WHY? “1” nahi dohrata, “6” dohrata hai. Isliye do baar guna. Check: 1 ÷ 6 = 0.1666… ✅.
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Example 3 (NCERT style): 0.045 ko p/q mein likho.
x = 0.045045…. Block 3 digit ka, to 1000x = 45.045045…. 1000x − x = 45, 999x = 45, x = 45/999 = 5/111 (9 se bhaag).
WHY? Block mein 3 digit, isliye 1000 se guna (10 ki teesri power). Check: 5 ÷ 111 = 0.045045… ✅.
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Meena ka jawab: 0.666… mein block 6 hai, to x = 6/9 = 2/3. Dadi sahi thi! ✅
🚀 Aur Aage: NCERT ke bahar
1) Mixed decimal ka seedha shortcut (extra): Fraction = (non-repeating aur repeating dono milakar bana poora number − sirf non-repeating wala number) ÷ (repeating digits ke utne 9, uske baad non-repeating digits ke utne 0). Jaise 0.83: (83 − 8) / 90 = 75/90 = 5/6. Aur 0.63: (63 − 6) / 90 = 57/90 = 19/30.
WHY chalta hai? Yeh wahi guna-ghatao hai, bas short mein likha hua.
2) Ek number ke do roop (extra): 0.49 = (49 − 4)/90 = 45/90 = 1/2 = 0.5. Terminating number ko hum dohrate 9 ke saath bhi likh sakte hain. Kitaabon mein hamesha terminating roop (0.5) hi likhte hain.
3) Jodna-ghatana (extra): 0.27 + 0.72 = 27/99 + 72/99 = 99/99 = 1. Pehle fraction banao, phir jodo; decimal ko seedha jodna khatarnak hai.
4) Pehchaan (extra): agar chhote form mein denominator 9, 99 ya 999 ka factor ho aur 2, 5 na ho, to decimal point ke turant baad se dohrayega (pure repeating).
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🌍 Real Life Mein
Doodh ka batwara: 2 litre doodh 3 bachchon mein barabar baanto, to har ek ko 0.666… litre milta hai. Agar aap 0.67 maan lo, to teen bachchon ka 2.01 litre ho jaata hai. Sahi jawab 2/3 litre hai, jisme koi gadbad nahi.
Calculator ki ginti: calculator limited digit dikhata hai. 0.333333 × 3 = 0.999999, par asli 1/3 × 3 = 1 poora milta hai. Exact kaam (hisaab, naap-tol, bijli ka bill) mein fraction rakhna safe rehta hai.
Maths ka maza yeh hai ki “kabhi na rukne wala” number bhi ek chhote se fraction mein band ho jaata hai.
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⚖️ Kab Use Karein / Kab Nahi
| Kab use karein | Kab nahi |
|---|---|
| Decimal mein ek block baar-baar dohra raha ho (0.333…, 0.1212…, 0.45666…) | Decimal bina kisi pattern ke chalta ho (jaise √2 ka decimal). Wahan guna-ghatao se “…” nahi katega. |
| Exact fraction chahiye ho (jodna, ghatana, compare) | Sirf approximate value chahiye (jaise 0.33 ya 0.667 likhna ho) |
| Rational hone ka saboot dena ho | Decimal pehle se terminating ho (seedha 10, 100, 1000 ke denominator mein likho) |
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❌→✅ Worked Mistake
Sawaal: 0.36 ko p/q mein badlo.
❌ Galat: “0.3636… = 36/100 = 9/25.” Yeh galat hai kyunki 9/25 = 0.36, jo terminating hai aur 0.3636… se alag hai.
❌ Doosri galti: 10 se guna: 10x = 3.636…. 10x − x = 3.2727…, “…” nahi katta.
✅ Sahi: block 2 digit ka hai, to 100 se guna: 100x = 36.3636…, 100x − x = 36, 99x = 36, x = 36/99 = 4/11. Check: 4 ÷ 11 = 0.3636… ✅.
WHY? Multiplier block ke digit ginkar chuno (1 digit → 10, 2 digit → 100, 3 digit → 1000). Tabhi dono “…” bilkul ek jaise bante hain.
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📝 10 MCQs
Q1. 0.4 ka fraction kya hai?
Solution dekho
Sahi jawab: (b)
- (a) galat: 4/10 = 0.4 terminating hai, 0.444… nahi.
- (b) sahi: 1 digit ka block, to x = 4/9 (10x − x = 4). Check: 4 ÷ 9 = 0.444…
- (c) galat: 4/99 = 0.0404…, block alag hai.
- (d) galat: 40/99 = 0.4040…, jo 0.444… nahi.
Q2. 0.18 = ?
Solution dekho
Sahi jawab: (b)
- (a) galat: 9/50 = 0.18, terminating hai.
- (b) sahi: 18/99 = 2/11 (9 se bhaag). Check: 2 ÷ 11 = 0.1818…
- (c) galat: 18/9 = 2, jo 1 se bada hai, jabki 0.1818… chhota hai.
- (d) galat: 1/18 = 0.0555…, alag.
Q3. x = 0.257 ko fraction banane ke liye x ko kis se guna karein?
Solution dekho
Sahi jawab: (c)
- (a) galat: 10 se guna karne par sirf 1 digit khisakta hai, block 3 digit ka hai.
- (b) galat: 100 se 2 digit khisakte hain, 3 chahiye.
- (c) sahi: block mein 3 digit, to 10³ = 1000 se guna: 1000x − x = 257, x = 257/999.
- (d) galat: 999 se guna karne par decimal point ka khisakna nahi hota, yeh ghatane ke baad aane wala number hai.
Q4. 0.9 (yaani 0.999…) barabar hai:
Solution dekho
Sahi jawab: (c)
- (a) galat: 0.9 sirf ek 9 hai, 0.999… anant 9 hain.
- (b) galat: 99/100 = 0.99, sirf do 9.
- (c) sahi: 10x − x = 9, 9x = 9, x = 1. Yaani 0.999… aur 1 ek hi number hain.
- (d) galat: anant 9 lagane par “thoda kam” bachta hi nahi, difference 0 hota hai.
Q5. 0.13 (0.1333…) = ?
Solution dekho
Sahi jawab: (b)
- (a) galat: 13/99 = 0.1313…, block 13 hai, par yahan sirf 3 dohrata hai.
- (b) sahi: 10x = 1.333…, 100x = 13.333…, 90x = 12, x = 12/90 = 2/15. Check: 2 ÷ 15 = 0.1333…
- (c) galat: 13/90 = 0.1444…. Ghatane par 13 − 1 = 12 aata hai, 13 nahi.
- (d) galat: 1/13 = 0.076923…, bilkul alag.
Q6. 2.3 (2.333…) = ?
Solution dekho
Sahi jawab: (b)
- (a) galat: 23/9 = 2.555…, 2.333… nahi.
- (b) sahi: x = 2.333…, 10x = 23.333…, 9x = 21, x = 21/9 = 7/3. Check: 7 ÷ 3 = 2.333…
- (c) galat: 23/10 = 2.3 terminating hai.
- (d) galat: 2/3 = 0.666… hai, poora ank 2 kho gaya.
Q7. 0.27 + 0.72 = ?
Solution dekho
Sahi jawab: (b)
- (a) galat: 0.99 sirf do 9 hain, jabki jod anant 9 (0.999…) hai.
- (b) sahi: 27/99 + 72/99 = 99/99 = 1.
- (c) galat: 99/100 = 0.99, anant 9 nahi.
- (d) galat: 0.111… = 1/9 hai, jod isse bahut bada hai.
Q8. 0.504 ka sabse chhota fraction:
Solution dekho
Sahi jawab: (b)
- (a) galat: 63/125 = 504/1000 = 0.504, terminating, 0.504504… nahi.
- (b) sahi: 3 digit ka block, to 504/999. Dono ko 9 se bhaagne par 56/111.
- (c) galat: 504/99 = 5.09…, bahut bada.
- (d) galat: 504/9999 = 0.0504…, galat denominator (9 ki sankhya block ke digit jitni honi chahiye).
Q9. 0.36 ke fraction ka chhote form mein denominator kya hai?
Solution dekho
Sahi jawab: (c)
- (a) galat: 99 tab hota jab 36/99 chhota na kiya jaye.
- (b) galat: 100 ka sawal hi nahi, kyunki 36/100 terminating hota hai.
- (c) sahi: 36/99 = 4/11 (9 se bhaag), denominator 11.
- (d) galat: 9 denominator se 4/9 = 0.444… aata, 0.3636… nahi.
Q10. 0.63 (0.6333…) = ?
Solution dekho
Sahi jawab: (b)
- (a) galat: 7/11 = 0.6363…, block 63 hai, par yahan sirf 3 dohrata hai.
- (b) sahi: (63 − 6)/90 = 57/90 = 19/30. Check: 19 ÷ 30 = 0.6333…
- (c) galat: 7/10 = 0.7, terminating.
- (d) galat: 19/33 = 0.5757…, galat.
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❓ FAQs (Bachchon Ke Asli Sawaal)
1. Block ke digit ginkar hi 10 ki power kyun chunte hain?
Kyunki hume dono numbers ke decimal point ke baad ka hissa bilkul ek jaisa chahiye. 2 digit ka block ho aur hum 10 se guna karein, to x aur 10x ke “…” aapas mein mel nahi khate (0.3636… aur 3.636…) aur ghatane par kuch nahi katta. 100 se guna karne par 36.3636… aur 0.3636… ke “…” bilkul ek jaise ban jaate hain.
2. Anant digit wale numbers ko ghatana kya sach mein theek hai?
Haan, kyunki hum digit-by-digit ghata rahe hain aur har jagah digit barabar hain, to har jagah 0 aata hai. Isme koi “aakhri digit” ki tension nahi hai, kyunki koi aakhri digit hai hi nahi.
3. 0.999… = 1 sach mein? Mujhe yakeen nahi hota.
Sochiye 1 aur 0.999… mein farq kitna hai? 0.1? Nahi, 0.9 bhi 0.999… se chhota hai. 0.01? 0.99 bhi chhota hai. Jitna chhota farq batayenge, anant 9 us se bhi chhota farq bacha dete hain. Isliye farq 0 hi hai. Aur 9x = 9 wali trick aur 1/3 × 3 wala sabut dono yahi kehte hain.
4. Kya √2 jaise non-repeating decimal par bhi yeh trick chal sakti hai?
Nahi. Is trick ke liye “block dohrata hai” hona zaroori hai, tabhi guna karke “…” ko ek jaisa banaya jaa sakta hai. Non-repeating decimal mein koi block hi nahi, to ghatane se kuch nahi katta. Isliye woh p/q nahi ban sakte aur irrational kehlate hain. Yeh agle posts ka topic hai.
5. Denominator hamesha 9, 99, 999 aur 90, 900 kyun aata hai?
10ⁿx − x = (10ⁿ − 1)x, aur 10ⁿ − 1 ek 9-9-9… wali sankhya hai (10 − 1 = 9, 100 − 1 = 99). Mixed case mein ek extra 10 ya 100 se guna aata hai, isliye 9 ke peeche 0 lag jaate hain.
6. Mixed decimal mein do baar guna kyun karte hain?
Pehle guna se non-repeating hissa decimal point se pehle aa jaata hai aur dohrana point ke turant baad se shuru hota hai. Doosra guna ek block aage khisakta hai. In do number ke “…” hi ek jaise hote hain, jinhe ghata kar hum “…” hata dete hain.
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🧠 HOTS (Soch Wale Sawaal)
HOTS 1. 0.12 + 0.21 ka fraction chhote form mein batao.
Solution dekho
12/99 + 21/99 = 33/99 = 1/3. WHY? Dono ka denominator 99 hai, to seedha numerator jode. Check: 0.1212… + 0.2121… = 0.3333… ✅.
HOTS 2. 1/7 = 0.142857 hai. 0.285714 ka fraction kya hoga? (Bina guna-ghatao ke socho.)
Solution dekho
285714 = 2 × 142857. Aur 0.142857 = 1/7, to 0.285714 = 2/7. Guna-ghatao se bhi: 285714/999999 = 2/7 ✅. WHY? Dono ka block ek hi digits ka chakkar hai, bas 2 guna bada.
HOTS 3. 0.7 aur 0.77 mein kaun bada hai aur kitna?
Solution dekho
0.7 = 7/9, 0.77 = 77/100. 7/9 − 77/100 = (700 − 693)/900 = 7/900. Yaani 0.7 bada hai (0.7777… vs 0.77). WHY? 0.777… mein 0.77 ke baad bhi 7 aate hain, to woh thoda aur bada hai.
HOTS 4. 0.ab = 4/33 ho, to do digit ka number ab kya hai?
Solution dekho
0.ab = ab/99. 4/33 = (4 × 3)/(33 × 3) = 12/99. To ab = 12. Check: 4 ÷ 33 = 0.1212… ✅. WHY? Denominator 99 banane ke liye 33 ko 3 se guna.
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🏆 Olympiad Corner
Oly 1. 0.1 + 0.01 + 0.001 ka fraction nikaalo.
Solution dekho
1/9 + 1/99 + 1/999. Common denominator 10989 (9 = 3², 99 = 3² × 11, 999 = 3³ × 37, LCM = 27 × 11 × 37 = 10989). 1/9 = 1221/10989, 1/99 = 111/10989, 1/999 = 11/10989. Jod = 1343/10989. 1343 = 17 × 79 aur 10989 mein sirf 3, 11, 37 hain, to aur chhota nahi hota. WHY? Pehle sabko fraction banao, phir jodo.
Oly 2. 0.123 ko p/q mein likho.
Solution dekho
x = 0.12323…. 10x = 1.2323…, 1000x = 123.2323…. 1000x − 10x = 123 − 1 = 122, 990x = 122, x = 122/990 = 61/495. Check: 61 ÷ 495 = 0.12323… ✅. WHY? “1” nahi dohrata, “23” (2 digit) dohrata hai, isliye 10 aur 1000 se guna.
Oly 3. 1 se 9 tak kin digits a ke liye 0.a chhote form mein denominator 3 wala fraction hota hai?
Solution dekho
0.a = a/9. a = 3 par 3/9 = 1/3, a = 6 par 6/9 = 2/3. Baaki digits par denominator 9 ya 1 aata hai (a = 9 par 1). Isliye a = 3 aur a = 6. WHY? a/9 mein 3 tabhi chhota hota hai jab a, 3 ka multiple ho par 9 ka nahi.
Oly 4. 0.3 × 0.6 ka decimal roop kya hoga?
Solution dekho
1/3 × 2/3 = 2/9 = 0.2. WHY? Pehle fraction banane se guna aasaan ho jaata hai; decimal ko seedha guna karna mushkil hai.
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💖 CharuMam Motivation
Maths: Jo number khatam hi nahi hota, uski bhi ek chaal hoti hai. Chaal pakadne par woh chhote se fraction mein band ho jaata hai.
Life: Kuch aadatein, kuch galtiyan bhi “dohrati” rehti hain. Agar aap unki chaal (pattern) pehchaan lo, to unhe todna aasaan ho jaata hai. Pehle pattern dekho, phir upay socho. 🌟
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🎯 Master Recap
- Dohrate decimal ko x maano. Block ke digit ginkar x ko 10, 100, 1000 se guna karo.
- Dono ko ghatao, “…” kat jaata hai. x nikaalo aur chhota karo.
- Pure repeating: fraction = block / (utne 9). Jaise 0.36 = 36/99 = 4/11.
- Mixed: do baar guna (10 ki power) aur ghatao. Jaise 0.83 = 75/90 = 5/6.
- 0.9 = 1. Har repeating decimal rational hota hai.
- Aam galti: 0.3636… ko 36/100 likhna, ya galat power se guna karna.
— charumam.com 🎯
🏋️ Practice Zone — 15 Questions
Q1. Easy 0.2 ko p/q mein badlo.
Solution dekho
x = 0.222…, 10x = 2.222…, 9x = 2, x = 2/9 ✅.
Q2. Easy 0.8 ko p/q mein badlo.
Solution dekho
10x − x = 8, 9x = 8, x = 8/9 ✅. Check: 8 ÷ 9 = 0.888…
Q3. Easy 0.15 ko chhote form mein p/q likho.
Solution dekho
Block 2 digit ka, 100x − x = 15, 99x = 15, x = 15/99 = 5/33 (3 se bhaag) ✅.
Q4. Easy 0.63 ko p/q mein badlo.
Solution dekho
99x = 63, x = 63/99 = 7/11 (9 se bhaag) ✅. Check: 7 ÷ 11 = 0.6363…
Q5. Easy 0.81 ko p/q mein badlo.
Solution dekho
99x = 81, x = 81/99 = 9/11 (9 se bhaag) ✅. Check: 9 ÷ 11 = 0.8181…
Q6. Medium 0.234 ko p/q mein badlo.
Solution dekho
Block 3 digit ka, 1000x − x = 234, 999x = 234, x = 234/999 = 26/111 (9 se bhaag) ✅.
Q7. Medium 0.27 (0.2777…) ko p/q mein badlo.
Solution dekho
10x = 2.777…, 100x = 27.777…. 100x − 10x = 27 − 2 = 25, 90x = 25, x = 25/90 = 5/18 ✅. Check: 5 ÷ 18 = 0.2777…
Q8. Medium 1.6 (1.666…) ko p/q mein badlo.
Solution dekho
10x = 16.666…, 10x − x = 15, 9x = 15, x = 15/9 = 5/3 ✅. Check: 5 ÷ 3 = 1.666…
Q9. Medium 3.18 (3.1818…) ko p/q mein badlo.
Solution dekho
100x = 318.1818…, 100x − x = 315, 99x = 315, x = 315/99 = 35/11 (9 se bhaag) ✅. Check: 35 ÷ 11 = 3.1818…
Q10. Medium 0.45 (0.4555…) ko p/q mein badlo.
Solution dekho
10x = 4.555…, 100x = 45.555…. 100x − 10x = 45 − 4 = 41, 90x = 41, x = 41/90 ✅. Check: 41 ÷ 90 = 0.4555…
Q11. NCERT / Extra 0.037 ko p/q mein badlo. Chhota form dekhkar batao ki denominator kis sankhya ka ghan (cube) hai.
Solution dekho
1000x − x = 37, 999x = 37, x = 37/999. 999 = 27 × 37, to x = 1/27. 27 = 3³, ghan hai ✅. Check: 1 ÷ 27 = 0.037037…
Q12. NCERT / Extra (Extra) 0.4 + 0.5 ka fraction aur decimal batao.
Solution dekho
4/9 + 5/9 = 9/9 = 1. Decimal mein 0.444… + 0.555… = 0.999… = 1 ✅. WHY? Yeh 0.999… = 1 ka ek aur saboot hai.
Q13. NCERT / Extra 0.123 (0.12333…) ko p/q mein badlo.
Solution dekho
Do guna: 100x = 12.333…, 1000x = 123.333…. 1000x − 100x = 123 − 12 = 111, 900x = 111, x = 111/900 = 37/300 (3 se bhaag) ✅. Check: 37 ÷ 300 = 0.12333…
Q14. NCERT / Extra (Extra) 0.6 ÷ 0.3 kitna hota hai?
Solution dekho
0.6 = 6/9 = 2/3 aur 0.3 = 1/3. (2/3) ÷ (1/3) = 2/3 × 3/1 = 2 ✅. WHY? Fraction banane se bhaag aasaan ho jaata hai.
Q15. NCERT / Extra (Extra) 7/33 ko decimal mein likho (bina long division ke).
Solution dekho
33 ko 99 banane ke liye 3 se guna: 7/33 = 21/99. 21/99 = 0.21 = 0.212121… ✅. WHY? Denominator 99 ka matlab 2 digit ka block, numerator wahi block.
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👉 Aage Kya?
Ab aap kisi bhi dohrate decimal ko p/q mein badal sakte ho. Agle post mein sawaal ulta hoga: kya sach mein aise numbers bhi hote hain jo kabhi p/q nahi ban sakte? Milte hain Irrational Numbers se.
Koi doubt? CharuMam AI Tutor se poochho, ya comment mein likho! 💬
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