Vastavik Sankhyayen Class 10 Notes shuru karne se pehle ek baat samajh len — Class 10 Maths ka pehla chapter dekhkar kai students sochte hain ki ye to bachpan wali cheez hai — HCF, LCM to hum Class 6 se padh rahe hain. Lekin board exam me yahi chapter students ko fasata hai, kyunki yahan sirf answer nahi, tareeka bhi manga jaata hai. Euclid division lemma se HCF nikalna, ye sabit karna ki root 2 irrational hai, aur ye batana ki kisi sankhya ka decimal expansion khatam hoga ya nahi rukega — ye sab aise sawaal hain jinme ek step galat hua to poora number kat jaata hai. Vastavik Sankhyayen Class 10 Notes is article me aapko poora Chapter 1 aasan Hindi me milega — har theorem seedhi bhasha me, har example step-by-step solve karke, aur board exam ke liye important points ke saath.
Pehle ye samajh len ki vastavik sankhyayen (real numbers) kya hoti hain. Jo sankhyayen hum roz istemal karte hain — jaise 2, -5, 3/4, 0.5, root 2, pi — ye sab vastavik sankhyayen hain. Inme parimey (rational) sankhyayen bhi hain aur aparimey (irrational) sankhyayen bhi. Rational sankhya use kehte hain jise p/q ke roop me likha ja sake, jahan p aur q poornank (integers) hon aur q kabhi zero na ho. Jaise 3/4, -2/5, 7. Irrational sankhya use kehte hain jise p/q ke roop me nahi likha ja sakta, jaise root 2, root 3, pi. Is chapter ka poora khel inhi do tarah ki sankhyayon ke aas-paas ghoomta hai.
Vastavik Sankhyayen Class 10 Notes — Euclid Division Lemma
Vastavik sankhyayen class 10 notes ki shuruaat Euclid division lemma se hoti hai. Ye lemma bahut seedha hai: kisi bhi do dhanaatmak poornank a aur b ke liye, aise unique poornank q aur r milte hain ki a = bq + r, jahan 0 se lekar r, b se chhota hota hai (0 ≤ r < b). Yahan a ko bhajya (dividend), b ko bhajak (divisor), q ko bhagfal (quotient) aur r ko sheshfal (remainder) kehte hain.
Seedhe shabdon me: kisi bhi sankhya ko doosri sankhya se bhag dene par jo bhagfal aur sheshfal milta hai, wo isi formula me bandha hota hai. Jaise 29 ko 6 se bhag dene par bhagfal 4 aur sheshfal 5 milta hai, kyunki 29 = 6 × 4 + 5. Yahan r = 5, jo b = 6 se chhota hai. Agar sheshfal bhajak se bada ho jaye, to samajh len ki bhag galat hua hai.
Euclid division lemma ka sabse bada kaam HCF nikalna hai. Euclid division algorithm ke hisaab se, do sankhyayon ka HCF nikalne ke liye badi sankhya ko chhoti se bhag den, phir jo sheshfal mile use naya bhajak banakar dobara bhag den. Ye process tab tak chalti hai jab tak sheshfal zero na aa jaye. Aakhri bhajak hi HCF hota hai.
HCF Nikalna — Euclid Division Algorithm Se Solved Example
Example 1: 306 aur 657 ka HCF nikaliye.
Step 1: 657 = 306 × 2 + 45 (sheshfal 45) Step 2: 306 = 45 × 6 + 36 (sheshfal 36) Step 3: 45 = 36 × 1 + 9 (sheshfal 9) Step 4: 36 = 9 × 4 + 0 (sheshfal 0)
Jab sheshfal zero aa gaya, to aakhri bhajak 9 hi HCF hai. Isliye HCF (306, 657) = 9.
Example 2: 24 aur 36 ka HCF nikaliye.
Step 1: 36 = 24 × 1 + 12 Step 2: 24 = 12 × 2 + 0
Isliye HCF (24, 36) = 12.
Board exam tip: Euclid algorithm wale sawaal me har step likhna zaroori hai. Sirf answer likhne par poore marks nahi milte. Har division ko a = bq + r ke roop me likhen, taaki examiner ko aapka tareeka saaf dikhe.
HCF Aur LCM Ka Relation — Sabse Important Formula
Do dhanaatmak poornankon ke liye ek bahut important relation hai: HCF × LCM = dono sankhyayon ka gunaphal. Matlab HCF (a, b) × LCM (a, b) = a × b. Ye formula sirf do sankhyayon ke liye lagu hota hai, teen sankhyayon ke liye nahi — ye baat board exam me aksar poochhi jaati hai aur kai students yahin galti kar dete hain.
Example 3: 24 aur 36 ka LCM is formula se nikaliye.
Hum jaan chuke hain ki HCF (24, 36) = 12. Ab formula lagayen: LCM = (24 × 36) / HCF = 864 / 12 = 72. Check: 24 × 36 = 864 aur 12 × 72 = 864. Dono barabar hain, isliye jawab sahi hai.
Example 4: 306 aur 657 ka LCM nikaliye.
HCF = 9 (upar nikala tha). LCM = (306 × 657) / 9 = 306 × 73 = 22338. Check: 306 × 73 = 22338, sahi hai. Isliye LCM (306, 657) = 22338.
Example 5: Prime factorisation se HCF aur LCM.
24 = 2 × 2 × 2 × 3 = 2³ × 3 36 = 2 × 2 × 3 × 3 = 2² × 3² HCF = common prime factors ki sabse chhoti power ka gunaphal = 2² × 3 = 12. LCM = sabhi prime factors ki sabse badi power ka gunaphal = 2³ × 3² = 8 × 9 = 72. Dono tareekon se jawab ek hi aata hai — Euclid algorithm se bhi 12 aur 72 hi mile the. Exam me jo tareeka poocha jaye, wahi istemal karen.
Fundamental Theorem Of Arithmetic
Is theorem ke hisaab se har composite sankhya (1 se badi sankhya jo prime nahi hai) ko prime sankhyayon ke gunaphal ke roop me likha ja sakta hai, aur ye tareeka unique hota hai (factors ka kram chhodkar). Jaise 24 = 2³ × 3, aur iske alawa koi doosra prime gunaphal 24 ke liye nahi ho sakta. Isi theorem ki wajah se HCF aur LCM nikalne ka prime factorisation wala tareeka kaam karta hai.
Example 6: 140 ka prime factorisation kijiye. 140 = 2 × 70 = 2 × 2 × 35 = 2 × 2 × 5 × 7 = 2² × 5 × 7.
Example 7: 156 ka prime factorisation kijiye. 156 = 2 × 78 = 2 × 2 × 39 = 2 × 2 × 3 × 13 = 2² × 3 × 13.
Prime factorisation karte waqt hamesha sabse chhoti prime sankhya se shuru karen — pehle 2 se check karen, phir 3 se, phir 5 se. Ye aadat galtiyan kam karti hai.
Irrational Numbers — Root 2 Irrational Hai, Ye Kaise Sabit Karen
Ye chapter ka sabse important proof hai aur board exam me aksar 3 ya 5 marks ke liye poocha jaata hai. Is proof ka tareeka “virodh se” (contradiction) hai — hum pehle maan lete hain ki root 2 rational hai, phir dikhate hain ki isse ek galat baat nikalti hai, isliye humara maan-na galat tha.
Proof: Root 2 ek irrational sankhya hai.
Maan lijiye root 2 ek rational sankhya hai. To ise p/q ke roop me likha ja sakta hai, jahan p aur q poornank hain, q zero nahi hai, aur p aur q me 1 ke alawa koi common factor nahi hai (matlab p/q apne sabse saral roop me hai).
Ab root 2 = p/q. Dono taraf square karne par: 2 = p²/q², ya p² = 2q². Iska matlab p², 2 se divisible hai. Aur agar kisi sankhya ka square 2 se divisible hai, to wo sankhya khud bhi 2 se divisible hoti hai. Isliye p, 2 se divisible hai. Maan lijiye p = 2r, jahan r ek poornank hai.
Ab p² = 2q² me p = 2r rakhne par: (2r)² = 2q², ya 4r² = 2q², ya q² = 2r². Iska matlab q² bhi 2 se divisible hai, isliye q bhi 2 se divisible hai.
Lekin ye hamari shuruaati baat se takraata hai — humne mana tha ki p aur q me 1 ke alawa koi common factor nahi hai, lekin ab dono 2 se divisible nikal rahe hain. Ye virodh hai. Isliye hamara maan-na galat tha ki root 2 rational hai. Atah, root 2 ek irrational sankhya hai.
Isi tareeke se aap ye bhi sabit kar sakte hain:
- Root 3 irrational hai (wahi proof, 2 ki jagah 3 lekar).
- Root 5 irrational hai.
- 2 + root 3 irrational hai (pehle maan len ki ye rational hai, to root 3 = (rational) − 2 bhi rational ho jayega, jo galat hai).
Board exam tip: Proof likhte waqt har step ka reason likhen — “kyunki p², 2 se divisible hai, isliye p bhi 2 se divisible hai” — ye line chhodne par marks kat sakte hain.
Decimal Expansion — Kab Khatam Hoga, Kab Nahi Rukega
Kisi rational sankhya p/q ka decimal expansion kab terminating (khatam hone wala) hoga aur kab non-terminating repeating (na rukne wala, dohraane wala) hoga, ye q ke prime factors se tay hota hai.
Niyam: p/q ko sabse saral roop me likhen. Agar q ke prime factors sirf 2 aur 5 hon, to decimal expansion terminating hoga. Agar q me 2 aur 5 ke alawa koi aur prime factor ho, to decimal expansion non-terminating repeating hoga.
Example 8: 3/8 ka decimal expansion kaisa hoga? 8 = 2³. Sirf 2 hai, isliye terminating hoga. Sach me 3/8 = 0.375, jo khatam ho jaata hai.
Example 9: 13/25 ka decimal expansion kaisa hoga? 25 = 5². Sirf 5 hai, isliye terminating hoga. Sach me 13/25 = 0.52.
Example 10: 17/8 ka decimal expansion kaisa hoga? 8 = 2³, isliye terminating hoga. 17/8 = 2.125.
Example 11: 1/7 ka decimal expansion kaisa hoga? 7 ka prime factor 7 hai, jo na 2 hai na 5, isliye non-terminating repeating hoga. Sach me 1/7 = 0.142857142857…, jisme 142857 baar-baar dohraata hai.
Example 12: 7/6 ka decimal expansion kaisa hoga? 6 = 2 × 3. Isme 3 hai, isliye non-terminating repeating hoga. Sach me 7/6 = 1.1666…
Ek aur zaroori baat: Har terminating decimal ek rational sankhya hoti hai, aur har non-terminating repeating decimal bhi rational hoti hai. Sirf non-terminating non-repeating decimal hi irrational hoti hai, jaise pi = 3.14159… jo kabhi khatam nahi hota aur kabhi dohraata bhi nahi hai.
Table Box — Vastavik Sankhyayen Chapter 1 Ek Nazar Me
| Vivaran | Jaankari |
|---|---|
| Topic | Vastavik Sankhyayen (Real Numbers) — Chapter 1 |
| Class/Course | Class 10 Maths |
| Isme kya-kya cover hai | Euclid division lemma, HCF-LCM, prime factorisation, irrational numbers ke proof, decimal expansion |
| Kis board/exam ke liye hai | CBSE aur state boards, NCERT syllabus, Class 10 board exam |
| Sabse important formula | HCF x LCM = dono sankhyayon ka gunaphal (sirf 2 sankhyayon ke liye) |
| Sabse important proof | Root 2 irrational hai (virodh se proof) |
Aur Solved Examples — Euclid Algorithm Aur HCF-LCM Check
Example 13: 135 aur 225 ka HCF Euclid algorithm se nikaliye, phir LCM nikaliye aur check kijiye.
Step 1: 225 = 135 x 1 + 90 (sheshfal 90) Step 2: 135 = 90 x 1 + 45 (sheshfal 45) Step 3: 90 = 45 x 2 + 0 (sheshfal 0)
Jab sheshfal zero aa gaya, to aakhri bhajak 45 hi HCF hai. Isliye HCF (135, 225) = 45.
Ab LCM ke liye formula lagayen: LCM = (135 x 225) / HCF = (135 x 225) / 45. 225 / 45 = 5, isliye LCM = 135 x 5 = 675.
Check: HCF x LCM = 45 x 675 = 30375. Aur 135 x 225 = 30375. Dono barabar hain, isliye jawab sahi hai. Exam me ye check aakhri line me zaroor likhen — isse examiner ko bhi bharosa hota hai aur aapki apni galti bhi pakdi jaati hai.
Example 14: 48 aur 180 ka HCF aur LCM nikaliye, dono tareekon se check kijiye.
Euclid algorithm se: Step 1: 180 = 48 x 3 + 36 (sheshfal 36, kyunki 48 x 3 = 144 aur 180 – 144 = 36) Step 2: 48 = 36 x 1 + 12 (sheshfal 12) Step 3: 36 = 12 x 3 + 0 (sheshfal 0)
Isliye HCF (48, 180) = 12.
LCM ke liye formula: LCM = (48 x 180) / 12 = 48 x 15 = 720, kyunki 180 / 12 = 15. Check: 12 x 720 = 8640 aur 48 x 180 = 8640. Dono barabar hain.
Prime factorisation se bhi check kar len: 48 = 2 x 2 x 2 x 2 x 3 = 2 ki power 4 x 3. 180 = 2 x 2 x 3 x 3 x 5 = 2 ki power 2 x 3 ki power 2 x 5. HCF = common factors ki sabse chhoti power ka gunaphal = 2 ki power 2 x 3 = 12. LCM = sabhi factors ki sabse badi power ka gunaphal = 2 ki power 4 x 3 ki power 2 x 5 = 16 x 9 x 5 = 720. Dono tareekon se ek hi jawab aaya, isliye ye pakka sahi hai.
Common Mistakes — Is Chapter Me Students Kahan Galti Karte Hain
Galti 1 — HCF x LCM wala formula teen sankhyayon par laga dena. Ye formula sirf do sankhyayon ke liye hai. Teen sankhyayon par lagayenge to jawab galat aayega, chahe calculation sahi ho.
Galti 2 — Aakhri sheshfal ko HCF samajh lena. Euclid algorithm me HCF aakhri bhajak hota hai, aakhri sheshfal nahi. Sheshfal to zero hota hai — zero HCF nahi ho sakta. Har step me ye dekhen ki bhajak kaun hai aur sheshfal kaun.
Galti 3 — Decimal expansion ka niyam bina saral roop ke laga dena. Niyam lagane se pehle p/q ko sabse saral roop me laana zaroori hai. Jaise 6/15 ko dekhkar lagega ki q = 15 me factor 3 hai, isliye non-terminating hoga — lekin 6/15 = 2/5 hai, aur 2/5 = 0.4, jo terminating hai. Saral roop na laana is chapter ki sabse chhupi hui galti hai.
Galti 4 — Root 2 ke proof me reason chhod dena. Sirf equations likhne par poore marks nahi milte. Har step par ye likhen ki kyun — jaise “p ki power 2, 2 se divisible hai, isliye p bhi 2 se divisible hai” — ye reason hi proof ki jaan hai.
Galti 5 — 1 ko prime samajh lena. 1 na prime hai na composite. Prime factorisation me 1 ka koi prime factor nahi hota, isliye use factor list me mat joden.
Galti 6 — LCM ka jawab check kiye bina chhod dena. LCM hamesha dono sankhyayon se bada ya barabar hota hai. Agar aapka LCM kisi sankhya se chhota aa raha hai, to calculation me galti pakki hai — turant dobara check karen.
Quick Revision Summary — Poora Chapter Ek Nazar Me
- Euclid division lemma: a = bq + r, jahan 0 se lekar r, b se chhota hota hai.
- HCF nikalne ke liye Euclid algorithm: badi sankhya ko chhoti se bhag den, sheshfal ko naya bhajak banayen, jab tak sheshfal zero na aa jaye. Aakhri bhajak hi HCF hai.
- HCF x LCM = dono sankhyayon ka gunaphal (sirf 2 sankhyayon ke liye).
- Prime factorisation: HCF ke liye common factors ki sabse chhoti power, LCM ke liye sabhi factors ki sabse badi power.
- Root 2, root 3 aur root 5 irrational hain — proof virodh se hota hai.
- Decimal expansion: q ke prime factors sirf 2 aur 5 hon to terminating, warna non-terminating repeating. Dono hi rational hote hain.
- Irrational ka decimal na khatam hota hai, na dohraata hai.
Vastavik Sankhyayen — Important Points Board Exam Ke Liye
- HCF × LCM = dono sankhyayon ka gunaphal (sirf 2 sankhyayon ke liye).
- Euclid division lemma: a = bq + r, jahan 0 ≤ r < b.
- Agar p ek prime sankhya hai aur p, a² ko divide karta hai, to p, a ko bhi divide karta hai. Yehi baat root 2 ke proof me kaam aati hai.
- Rational sankhya ka decimal expansion ya to terminating hota hai ya non-terminating repeating. Dono hi rational hote hain.
- Irrational sankhya ka decimal expansion non-terminating non-repeating hota hai.
- 1 na prime hai na composite. Prime factorisation me 1 ka koi prime factor nahi hota.
- Do consecutive positive integers hamesha co-prime hote hain, matlab unka HCF 1 hota hai. Jaise HCF (15, 16) = 1.
- HCF hamesha dono sankhyayon se chhota ya barabar hota hai, aur LCM hamesha dono se bada ya barabar hota hai. Agar aapka LCM sankhyayon se chhota aa raha hai, to calculation me galti hai.
Practice Questions — Khud Se Solve Karen
1. Euclid division algorithm se 4052 aur 12576 ka HCF nikaliye. (Hint: answer 4 aana chahiye — khud check karen: 12576 = 4052 × 3 + 420; 4052 = 420 × 9 + 272; 420 = 272 × 1 + 148; 272 = 148 × 1 + 124; 148 = 124 × 1 + 24; 124 = 24 × 5 + 4; 24 = 4 × 6 + 0. Aakhri bhajak 4 hai.) 2. Sabit kijiye ki root 5 irrational hai. 3. Batayen ki 17/20 ka decimal expansion terminating hoga ya non-terminating repeating, aur kyun. (Hint: 20 = 2² × 5, isliye terminating.) 4. Agar HCF (a, b) = 18 aur a × b = 3240 hai, to LCM (a, b) kitna hoga? (Hint: LCM = 3240 / 18 = 180.) 5. 96 aur 404 ka HCF prime factorisation se nikaliye aur Euclid algorithm se bhi check karen. (Hint: 96 = 2⁵ × 3, 404 = 2² × 101, HCF = 4.)
FAQ — Vastavik Sankhyayen Class 10
Q1. HCF aur LCM me kya farak hai? A. HCF wo sabse badi sankhya hai jo dono sankhyayon ko poora-poora divide kar de, aur LCM wo sabse chhoti sankhya hai jo dono sankhyayon se poori-poori divide ho jaye. Jaise 24 aur 36 ka HCF 12 hai aur LCM 72 hai.
Q2. Kya HCF × LCM wala formula teen sankhyayon par bhi lagta hai? A. Nahi. Ye formula sirf do sankhyayon ke liye hai. Teen sankhyayon ke liye HCF × LCM unke gunaphal ke barabar nahi hota.
Q3. Root 2 irrational kyun hai? A. Kyunki ise p/q ke roop me nahi likha ja sakta. Proof me virodh se dikhaya jaata hai ki agar root 2 rational hota, to p aur q dono 2 se divisible hote, jo is baat se takraata hai ki p/q sabse saral roop me hai.
Q4. Decimal expansion terminating kab hota hai? A. Jab p/q ke sabse saral roop me q ke prime factors sirf 2 aur 5 hon, tab decimal expansion terminating hota hai. Warna non-terminating repeating hota hai.
Q5. Kya har irrational sankhya ka decimal kabhi khatam nahi hota? A. Sahi hai — irrational sankhya ka decimal na kabhi khatam hota hai aur na kabhi ek pattern me dohraata hai. Jaise root 2 aur pi.
Aur bhi padhein: Quadratic Equations Class 10 Notes aur Arithmetic Progression Class 10 Notes.
Official Links
- NCERT Class 10 Maths Chapter 1 (Real Numbers) — ncert.nic.in
- CBSE Official Website — cbse.gov.in
- NCERT Books aur Solutions — ncert.nic.in