✅ Pehle Yeh Aata Hai?
Pichle post Rational Numbers aur Unki Properties mein hum standard form aur additive inverse seekh chuke hain. Ab 1 minute mein yeh teen sawaal try kijiye:
- 7/3 ko mixed number (poora + hissa) mein likho.
- −5/2 ka additive inverse kya hai?
- −1/2 aur −3/4 mein kaun bada hai? (Abhi apna andaza lagao.)
Answers dekho
- 7 ÷ 3 = 2 baaki 1, to 7/3 = 2 1/3.
- Sign badlo: 5/2, kyunki −5/2 + 5/2 = 0.
- −1/2 = −2/4, aur −2/4 > −3/4, to −1/2 bada hai. Yeh isi post mein number line se pakka karenge.
Atak gaye? Class 8 ke posts dekh lijiye: Positive aur Negative Numbers (Number Line Par) aur Comparison of Rational Numbers.
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🎬 Kahani: Darzi ka Tape
Anaya ko kurta silwana hai. Darzi uncle ne tape nikala aur bole, “Beti, 5/2 meter kapda chahiye.” Anaya ne tape dekha: us par 0, 1, 2, 3… meter ke bade nishaan hain. “Uncle, 5/2 to kisi nishaan par likha hi nahi!”
Uncle hanse: “Arre, 2 aur 3 ke theek beech mein dekho.” Anaya ne dekha, 2 meter ke baad aadha meter aur chalo, wahi 5/2 hai.
Is post mein hum yahi seekhenge: kisi bhi rational number ko number line par sahi jagah par kaise rakhte hain, negative numbers kahan aate hain, aur number line se bada-chhota kaise pehchante hain. 😊
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📖 Concept: Rational Numbers aur Number Line
Step 1: Number line kya hai?
Ek seedhi line jis par hum ek point ko 0 maante hain, aur uske dahine (right) taraf barabar doori par 1, 2, 3 … rakhte hain. Baayen (left) taraf −1, −2, −3 … aate hain. Do bade nishaanon ke beech ki doori hamesha barabar hoti hai (isse hum “unit length” kehte hain).
WHY barabar doori? Taaki doori se hum numbers ka farak naap sakein. Agar nishaan kahin paas, kahin door hon to number line kisi kaam ki nahi.
Yaad rakho: number line par jo number jitna right mein, utna bada.
Step 2: p/q ko 0 aur 1 ke beech mark karna (0 < p < q)
Rule: 0 se 1 ke segment ko q barabar hisson mein baanto. 0 se shuru karke p-th nishaan par jao. Wahi p/q hai.
Example: 3/4. Segment ko 4 barabar hisson mein baanta, har hissa 1/4 ka hai. 3 hisse chal kar pahunche, to 3/4 mila.
WHY? Har hissa poore 1 ka 1/q hai. p hisse chalne par p × (1/q) = p/q door pahunchte hain.
Step 3: 1 se bade fraction (improper) ko mark karna
Pehle use mixed number mein badlo, phir dekho woh kin do integers ke beech hai. 7/3 = 2 1/3. To 7/3, 2 aur 3 ke beech hai: 2 se shuru karke 2 aur 3 ke segment ko 3 barabar hisson mein baanto, aur pehle nishaan par jao (kyunki extra hissa 1/3 hai).
WHY? 7/3 = 2 + 1/3. Pehle 2 tak jao, phir 1/3 aur aage.
Step 4: Negative rational numbers
−p/q, p/q ka mirror image hota hai, 0 ke doosri taraf, 0 se utni hi doori par. −7/3 ke liye: 7/3 jahan hai (0 se 2 1/3 right), uske barabar left mein jao, yaani −2 ke baad 1/3 aur left mein. Isliye −7/3, −3 aur −2 ke beech hai.
WHY mirror image? p/q aur −p/q ka sum 0 hai, yaani dono 0 se barabar doori par hain, bas ulti taraf.
Step 5: Number line se tulna (compare)
Jo number right mein, woh bada. Example: −1/2 aur −3/4. −1/2 = −2/4, aur −3/4 usse ek hissa aur left mein hai. To −3/4 < −1/2.
Yaad rakho: barabar fractions (jaise 1/2 aur 2/4) number line par ek hi point par aate hain, kyunki woh ek hi number hain. Compare karne ka calculation wala tareeka Class 8 ke Comparison of Rational Numbers post mein detail mein hai.
Step 6: 0 se doori (absolute value)
Kisi number ki 0 se doori ko uska absolute value kehte hain. |−3/4| = 3/4 aur |3/4| = 3/4. Doori kabhi negative nahi hoti.
| Number | Mixed form | Kin do integers ke beech? |
|---|---|---|
| 5/2 | 2 1/2 | 2 aur 3 |
| −5/2 | −2 1/2 | −3 aur −2 |
| 3/5 | — | 0 aur 1 |
| −3/5 | — | −1 aur 0 |
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🧠 Deep Concept: Negative number ke mixed form mein dhokha
−7/3 ko sahi se −2 1/3 padhna padega, yaani “−2 aur +1/3” nahi, balki “−2 se 1/3 aur left”. Bahut bachche −7/3 = −2 + 1/3 maan lete hain aur point ko −2 ke right mein rakh dete hain. Galat! Safe tareeka: pehle positive version 7/3 mark karo, phir uska mirror image lo.
Doosri gehri baat: number line par har rational number ka ek hi point hota hai, aur point ke liye uske kai “naam” (1/2, 2/4, 3/6…) ho sakte hain. Isliye “kaun sa number kahan hai” ka matlab hai standard form dekhna. Aur jab bhi q bada ho, mark karne ke liye segment ko q barabar hisson mein baantna padta hai. Isliye chhote denominator wale numbers mark karna aasaan hai.
— 📌 sirf charumam.com readers ke liye 🔐
✏️ Solved Examples
Example 1 (Easy): 5/6 ko number line par mark karo.
5/6, 0 aur 1 ke beech hai (5 < 6). 0 se 1 ke segment ko 6 barabar hisson mein baanto. 0 se shuru karke 5th nishaan par jao. Yeh 1 se sirf ek hissa (1/6) pehle hai.
WHY? Har hissa 1/6 hai, 5 hisse = 5/6 ✅. Check: 1 − 5/6 = 1/6, yaani 1 se 1/6 left mein.
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Example 2 (Medium): −11/4 kahan hoga? Kin do integers ke beech?
Pehle 11/4 = 2 3/4 (11 ÷ 4 = 2 baaki 3). To −11/4 = −2 3/4 = −2.75. Yeh −3 aur −2 ke beech hai.
Mark karne ke liye: −3 aur −2 ke segment ko 4 barabar hisson mein baanto. −2.75, −3 se 1/4 right mein hai (kyunki −3 + 1/4 = −11/4), yaani −3 ke baad pehle nishaan par.
WHY? −3 + 1/4 = −12/4 + 1/4 = −11/4 ✅.
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Example 3 (NCERT style): −5/6 aur −7/9 mein kaun bada hai?
Dono ko common denominator 18 par laao: −5/6 = −15/18 aur −7/9 = −14/18. Number line par −14/18, −15/18 se right mein hai (−14 > −15). To −7/9 > −5/6.
WHY? Negative numbers mein jo 0 ke paas hota hai wohi bada hota hai. −14/18, −15/18 se 0 ke zyada paas hai.
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Anaya ka sawaal: 5/2 = 2 1/2. Woh 2 aur 3 ke theek beech mein hai. Darzi uncle sahi the! 🎉
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🚀 Aur Aage: NCERT ke bahar
Yeh hisse extra / exam ke liye hain.
1) Do rational numbers ke beech ki doori
Doori = bada − chhota (hamesha positive). Example: −3/4 aur 5/6 ke beech: 5/6 − (−3/4) = 5/6 + 3/4 = 10/12 + 9/12 = 19/12. WHY? Number line par 0 se 5/6 tak ki doori 5/6 hai, aur −3/4 se 0 tak ki doori 3/4. Dono ko jodne se poori doori.
2) Ek hi line par thirds aur quarters
Agar ek hi number line par 1/3 aur 1/4 dono mark karne hain, to har unit ko 12 barabar hisson mein baanto (12 = 3 aur 4 ka LCM). Tab 1/3 = 4 hisse aur 1/4 = 3 hisse. WHY? Taaki har fraction ka position poore hisson ki ginti se mil jaye.
3) |x| = a wale sawaal
|x| = 5/7 ka matlab hai x ki 0 se doori 5/7 hai. Number line par 0 se 5/7 door do point hain: x = 5/7 ya x = −5/7.
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🏠 Real Life Mein Number Line
- 📏 Ruler aur tape: 2 1/2 cm ya 5/2 meter, sab number line ke points hain.
- 🌡️ Thermometer: −2.5 °C yaani −5/2, 0 ke neeche.
- 🛗 Lift: basement −1, −2 aur ground 0. Beech ke points bhi hote hain jaise −1.5.
- 🛣️ Highway ke milestones: har kilometer par bada nishaan, beech mein 1/2 km ke chhote nishaan.
- ⏱️ Race ka timeline: 0 second se shuru, har 1/4 second par reading.
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🧭 Kab Use Karein, Kab Nahi
| ✅ Number line kaam aati hai jab… | ❌ Dhyan rakho… |
|---|---|
| Bada-chhota dekhna ho (right = bada) | Bahut bade denominator (jaise 1/97) ko line par mark karna mushkil, wahan calculation wala tareeka better hai. |
| Do numbers ke beech ki doori nikaalni ho | Negative numbers mein “bada” ka matlab 0 ke paas hona hai, bade digits wala nahi (−2 > −5). |
| Number kin do integers ke beech hai, yeh dekhna ho | Mixed form se negative number mark karte waqt pehle positive version banao. |
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❌→✅ Worked Mistake
Sawaal: 1/4 ko 0 aur 1 ke beech mark karo.
❌ Galat: “4 ka matlab 4 nishaan. 0 aur 1 ke beech 4 nishaan bana diye aur pehle par 1/4 likh diya.”
Galti kya hai? 4 nishaan banane se segment 5 hisson mein bat jata hai, 4 mein nahi. Phir har hissa 1/5 ka hota hai, 1/4 ka nahi.
✅ Sahi: Segment ko 4 barabar hisson mein baanto. Iske liye 0 aur 1 ke beech sirf 3 nishaan chahiye. Pehla nishaan 1/4 hai.
Doosri galti: −3/4 ko 0 ke right mein rakh dena, ya −1 ke bhi left mein. −3/4 = −0.75, to woh 0 aur −1 ke beech hai, 0 ke left mein.
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📝 10 MCQs
Q1. 3/5 kin do integers ke beech hai?
Solution dekho
Sahi jawab: (b)
- (a) galat: 3/5 = 0.6, yeh 1 se chhota hai.
- (b) sahi: 0 < 3/5 < 1, kyunki numerator denominator se chhota hai.
- (c) galat: yeh negative numbers ki range hai.
- (d) galat: 3/5, 2 se bahut chhota hai.
Q2. 9/4 kin do integers ke beech hai?
Solution dekho
Sahi jawab: (d)
- (a) galat: 9/4 = 2.25 hai, 1 se bada.
- (b) galat: 9/4 = 2.25 hai, 2 se bada.
- (c) galat: 9/4, 3 se chhota hai (3 = 12/4).
- (d) sahi: 9/4 = 2 1/4, to 2 aur 3 ke beech.
Q3. −7/2 kin do integers ke beech hai?
Solution dekho
Sahi jawab: (a)
- (a) sahi: −7/2 = −3.5, jo −4 aur −3 ke beech hai.
- (b) galat: −3 aur −2 ke beech −2.5 aata (−5/2).
- (c) galat: 7/2 positive hota, yeh −7/2 nahi.
- (d) galat: −7/2 = −3.5, −4 se right mein hai.
Q4. 5/8 ko 0 aur 1 ke beech mark karne ke liye segment ko kitne barabar hisson mein baantein?
Solution dekho
Sahi jawab: (b)
- (a) galat: 5 hisson se 1/5 ke multiples milte hain.
- (b) sahi: denominator 8 hai, to 8 barabar hisse, aur 5th nishaan.
- (c) galat: 5 + 8 jodne ka koi rule nahi.
- (d) galat: 3 hisson se 1/3 ke multiples milte hain.
Q5. 5/7 ka 0 ke baare mein mirror image kya hai?
Solution dekho
Sahi jawab: (d)
- (a) galat: yeh reciprocal hai, mirror image nahi.
- (b) galat: khud wahi point, mirror nahi.
- (c) galat: sign badla par numbers bhi ulta kar diye.
- (d) sahi: 0 se barabar doori, ulti taraf: −5/7.
Q6. −1/2, −2/3, −3/4 mein sabse bada kaun hai?
Solution dekho
Sahi jawab: (c)
- (a) galat: −3/4 = −9/12, sabse chhota.
- (b) galat: −2/3 = −8/12, beech wala.
- (c) sahi: −1/2 = −6/12, 0 ke sabse paas, to sabse bada.
- (d) galat: teeno ke denominator 12 par values −9, −8, −6 alag hain.
Q7. 3/4 aur 9/12 number line par kahan hain?
Solution dekho
Sahi jawab: (b)
- (a) galat: 9/12 ko 3 se divide karo to 3/4 milta hai, alag nahi.
- (b) sahi: dono barabar number hain, isliye ek hi point.
- (c) galat: barabar numbers mein se koi left ya right nahi hota.
- (d) galat: upar jaisa hi: barabar numbers ek hi point par.
Q8. −7/5 ki 0 se doori kya hai?
Solution dekho
Sahi jawab: (c)
- (a) galat: doori negative nahi hoti.
- (b) galat: numerator-denominator ulta kar diya.
- (c) sahi: |−7/5| = 7/5.
- (d) galat: 0 se doori 0 tab hoti jab number khud 0 hota.
Q9. 1/2 aur −1/3 ke beech ki doori kya hai?
Solution dekho
Sahi jawab: (c)
- (a) galat: 1/2 − 1/3 galti se kiya, jabki −1/3 negative hai.
- (b) galat: tops aur bottoms alag-alag jod diye.
- (c) sahi: 1/2 − (−1/3) = 1/2 + 1/3 = 5/6.
- (d) galat: random value, calculation se match nahi.
Q10. −10/3 ke baare mein kaun sa kathan sahi hai?
Solution dekho
Sahi jawab: (a)
- (a) sahi: −10/3 = −3 1/3, to −3 se 1/3 aur left.
- (b) galat: right mein hota to number −3 aur −2 ke beech hota (−8/3).
- (c) galat: −3 = −9/3, −10/3 se alag.
- (d) galat: 3 door hota to number −6 hota.
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❓ FAQs (Bachchon Ke Asli Sawaal)
1. 0 aur 1 ke beech kitne rational numbers hain?
Bahut zyada. Yeh baat hum Post 4 (Density, jald aa raha hai) mein gehraai se dekhenge. Abhi itna yaad rakho: har bade denominator ke saath naye points mil jaate hain.
2. Kya 0 positive hai ya negative?
Dono nahi. 0 number line ka beech ka point hai, jahan se positive aur negative alag hote hain.
3. Mirror image ka kya matlab hai?
Maan lo 0 par ek aaina (mirror) lagaya hai. Right ka point 3/4 aaine mein left ka point −3/4 dikhta hai, 0 se barabar doori par.
4. Kya unit length koi bhi rakh sakte hain?
Haan, chhoti ya badi, par poori line par ek hi unit rehna chahiye. Agar kabhi chhota, kabhi bada hoga to doori ka sahi andaza nahi lagega.
5. −5 bada hai ya −2?
−2 bada hai, kyunki number line par −2, −5 ke right mein hai. Negative numbers mein bade digits ka matlab zyada chhota hota hai.
6. Kya har rational number number line par aata hai?
Haan, har rational number ka number line par ek point hota hai. Aur number line par aise points bhi hain jo rational nahi hain; unki baat aage ke posts mein hogi.
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🔥 HOTS (Sochne Wale Sawaal)
HOTS 1. Point A number line par −3/4 par hai. Point B, A ke right mein hai aur AB = 5/4. B kahan hai?
Solution dekho
B = −3/4 + 5/4 = 2/4 = 1/2. Check: 1/2 − (−3/4) = 1/2 + 3/4 = 5/4 ✅.
HOTS 2. 7/9, 8/11 aur 3/4 mein se kaun 1 ke sabse paas hai?
Solution dekho
1 se doori: 1 − 7/9 = 2/9 ≈ 0.222, 1 − 8/11 = 3/11 ≈ 0.273, 1 − 3/4 = 1/4 = 0.25. Sabse chhoti doori 2/9 hai, to 7/9 sabse paas hai.
HOTS 3. 2/5 aur uska reciprocal 5/2 mein se number line par kaun right mein hai? Kya yeh hamesha hoga jab 0 < x < 1?
Solution dekho
2/5 = 0.4 aur 5/2 = 2.5, to 5/2 right mein hai. Hamesha: agar 0 < x < 1, to 1/x > 1 > x, kyunki chhote positive number ka reciprocal bada hota hai.
HOTS 4. A = −1/2. B, A ke right mein hai aur AB = 2/3. C, B ke right mein hai aur BC = 3/4. C kahan hai?
Solution dekho
B = −1/2 + 2/3 = −3/6 + 4/6 = 1/6. C = 1/6 + 3/4 = 2/12 + 9/12 = 11/12. Seedha: −1/2 + 2/3 + 3/4 = 11/12 ✅.
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🏆 Olympiad Corner
Olympiad 1. −17/4 aur 29/6 ke beech kitne integers hain?
Solution dekho
−17/4 = −4.25 aur 29/6 = 4.83…. In dono ke beech ke integers: −4, −3, …, 4. Ginti: 4 negative + 0 + 4 positive = 9 integers.
Olympiad 2. −2 aur 2 ke beech (dono sahit) k/6 form ke kitne numbers hain, jahan k integer hai?
Solution dekho
−2 ≤ k/6 ≤ 2 → −12 ≤ k ≤ 12. k ke values: −12 se 12 tak = 12 + 1 + 12 = 25.
Olympiad 3. Ek mendhak 0 se shuru karta hai. Woh pehle 5/6 right kudta hai, phir 1/3 left, phir 5/6 right, phir 1/3 left, aise hi. 9 chhalang ke baad woh kahan hoga?
Solution dekho
9 chhalang mein 4 poore jode (right + left) aur ek aakhri right chhalang. Har jode ka net = 5/6 − 1/3 = 5/6 − 2/6 = 1/2. 4 jode = 2. Phir + 5/6: 2 + 5/6 = 17/6.
Olympiad 4. |x − 1/2| = 3/4 ho to x ke dono values ka product kya hai?
Solution dekho
x − 1/2 = 3/4 ya x − 1/2 = −3/4. To x = 5/4 ya x = −1/4. Product = (5/4)(−1/4) = −5/16.
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💖 CharuMam Motivation
Maths ka sabak: Number line sabko ek hi qataar mein rakhti hai: chhote bachche jaise numbers left mein, bade right mein, aur beech mein aadha, chauthai, teen-chauthai sab ke liye jagah hai. Har number ka apna ek point hai.
Life ka sabak: Kisi ka point “beech mein” hone se chhota nahi ho jata. 5/2 bhi 2 aur 3 ke beech mein hai, phir bhi uski apni pehchaan hai. Aap bhi jahan ho, wahan ki apni jagah hai. Bas apni raftaar se aage badhte raho. 🌟
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🟢 Master Recap
- Number line par jo right mein, woh bada.
- p/q (0 < p < q): 0–1 ko q barabar hisson mein baanto, p-th nishaan.
- Improper fraction: pehle mixed number banao, phir integer ke baad hissa mark karo.
- −x, x ka mirror image hai (0 se barabar doori).
- Barabar fractions ek hi point par aate hain.
- 0 se doori = absolute value, kabhi negative nahi. Do points ki doori = bada − chhota.
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🏋️ Practice Zone — 15 Questions
Q1. Easy 3/7 mark karne ke liye 0–1 ko kitne barabar hisson mein baantein aur kaun sa nishaan lein?
Solution dekho
7 barabar hisson mein baantein, aur 0 se 3rd nishaan lein, kyunki 3 × (1/7) = 3/7.
Q2. Easy 11/3 kin do integers ke beech hai?
Solution dekho
11 ÷ 3 = 3 baaki 2, to 11/3 = 3 2/3. Yeh 3 aur 4 ke beech hai.
Q3. Easy −17/5 ko mixed form mein likho aur batao woh kin do integers ke beech hai.
Solution dekho
17 ÷ 5 = 3 baaki 2, to 17/5 = 3 2/5. To −17/5 = −3 2/5, jo −4 aur −3 ke beech hai.
Q4. Easy 4/9 ka mirror image (0 ke baare mein) kya hai?
Solution dekho
−4/9, kyunki 4/9 + (−4/9) = 0, yaani dono 0 se barabar doori par.
Q5. Easy 2/5 aur 3/8 mein kaun bada hai?
Solution dekho
LCM 40: 2/5 = 16/40, 3/8 = 15/40. 16 > 15, to 2/5 bada hai.
Q6. Medium −5/8, 1/4, −3/4, 0, 7/8 ko chhote se bade ke order mein likho.
Solution dekho
Denominator 8: −5/8, 2/8, −6/8, 0, 7/8. Order: −6, −5, 0, 2, 7 yaani −3/4 < −5/8 < 0 < 1/4 < 7/8.
Q7. Medium −2/3 aur 3/4 ke beech ki doori kya hai?
Solution dekho
3/4 − (−2/3) = 3/4 + 2/3 = 9/12 + 8/12 = 17/12.
Q8. Medium 13/5 ko number line par kaise mark karoge?
Solution dekho
13 ÷ 5 = 2 baaki 3, to 13/5 = 2 3/5. 2 aur 3 ke segment ko 5 barabar hisson mein baanto, aur 2 se 3rd nishaan par jao.
Q9. Medium |−7/12| aur |7/12| ki tulna karo.
Solution dekho
|−7/12| = 7/12 aur |7/12| = 7/12. Dono barabar hain, kyunki dono 0 se barabar doori par hain.
Q10. Medium A = −1/3, B = 5/6. C, A ke left mein hai aur AC = AB. C kahan hai?
Solution dekho
AB = 5/6 − (−1/3) = 5/6 + 2/6 = 7/6. C = A − 7/6 = −2/6 − 7/6 = −9/6 = −3/2.
Q11. NCERT / Extra −4/9 aur −5/12 mein kaun bada hai?
Solution dekho
LCM 36: −4/9 = −16/36, −5/12 = −15/36. −15 > −16, to −5/12 bada hai.
Q12. NCERT / Extra −13/3 aur 13/3 ke beech kitne integers hain?
Solution dekho
13/3 = 4.33…. Integers: −4, −3, …, 4. Ginti: 4 + 1 + 4 = 9.
Q13. NCERT / Extra 3/2, −3/2, 4/3, −5/4 ko bade se chhote ke order mein likho.
Solution dekho
Decimal mein: 1.5, −1.5, 1.33…, −1.25. Bade se chhote: 3/2, 4/3, −5/4, −3/2.
Q14. NCERT / Extra Aisa rational number p/5 dhundo jo 2 aur 3 ke beech hai aur jiska numerator 3 ka multiple hai.
Solution dekho
2 < p/5 < 3 → 10 < p < 15, p ∈ {11, 12, 13, 14}. In mein 3 ka multiple sirf 12 hai. Number 12/5.
Q15. NCERT / Extra Ek number ki 0 se doori 9/7 hai aur woh −1 ke left mein hai. Number kya hai?
Solution dekho
0 se doori 9/7 wale do number: 9/7 aur −9/7. −1 ke left mein hone ke liye number −1 se chhota hona chahiye: −9/7 = −1.28… < −1 ✅. Answer −9/7.
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👉 Aage Kya?
Ab aap rational numbers ko number line par rakh sakte ho. Agle post mein sawaal uthega: do rational numbers ke beech aur kitne rational numbers chhupe hain? Yeh density ki kahani hai.
Koi doubt? CharuMam AI Tutor se poochho, ya comment mein likho! 💬
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