The World of Numbers (Numbers ki Duniya): Class 9 Maths Chapter 3 Hinglish mein

Notches ginne se real number line tak

Ganita Manjari Part 1 · Hinglish notes, simulations aur practice · Official NCERT chapter PDF ↗

Published 12 Oct 2026 · Updated 12 Oct 2026

110 min
1 simulation
47 practice questions
40 MCQs

Is chapter mein aap kya seekhenge

Samjhao ki number system counting numbers se integers, rational numbers aur real numbers tak kaise bada hua, aur har step ki zaroorat kyun padi
Zero aur signed numbers ke liye Brahmagupta ke rules lagao, aur samjhao ki do negative numbers ka product positive kyun hota hai
Rational numbers ko add, subtract, multiply aur divide karo, do fractions barabar hain ya nahi yeh test karo, aur distributive law use karo
Number line par rational numbers locate karo, absolute values aur distances nikalo, aur kinhi bhi do diye gaye rational numbers ke beech aur rational numbers dhoondho
Contradiction se prove karo ki 2 ka square root irrational hai, aur number line par 2 aur 3 ke square roots jaisi lengths construct karo
Pehle se batao ki kisi fraction ka decimal terminating hoga ya repeating, aur repeating decimals ko fractions mein badlo
2 ke square root aur pi jaise irrational numbers ko unke non-terminating, non-repeating decimals se pehchano

NCERT Mathematics Class 9, Chapter 3, pp. 41–67 par based original companion text. Yeh NCERT se juda nahi hai aur NCERT ne ise endorse nahi kiya hai.

3.1 Mathematics ki Shuruaat: Insaan ko Ginne ki Zaroorat

Socho, ek birthday party hai aur paper plates ka ek stack rakha hai. Kya har bachche ke liye kaafi plates hain? Iske liye tumhe yeh jaanna zaroori nahi ki kitne bachche aaye. Har bachche ko ek plate do. Agar plates khatam ho gayin aur koi abhi bhi intezaar kar raha hai, to plates kam hain. Agar kuch plates bach gayin, to tumhare paas extra hain.

Ek collection ki har cheez ko doosre collection ki theek ek cheez ke saath jodna one-to-one correspondence kehlata hai. Isse hum do collections ke size compare kar lete hain, bina ek bhi number word use kiye. Kisi ke numeral likhne se bahut pehle log isi par bharosa karte the. Lakdi par ek notch, rassi mein ek gaanth ya thaile mein ek patthar, har jaanwar, har din ya anaaj ki har tokri ki jagah le sakta tha.

Kisi samay logon ne dekha ki paanch bakriyan, paanch ungliyan aur paanch kankad, sab ek doosre se match karte hain. Inmein jo common hai woh ek idea hai: number paanch. Ginne ke liye jo numbers hum use karte hain, 1, 2, 3, 4 waghairah, woh natural numbers (प्राकृत संख्याएँ) hain. In sabke collection ko $ℕ = {1, 2, 3, 4, \dots}$ likhte hain. Teen dots batate hain ki list kabhi khatam nahi hoti: har natural number ke baad ek aisa number aata hai jo usse 1 bada hai.

Definition

Natural numbers

Ginne wale numbers $1, 2, 3, 4, \dots$ In sabke collection ko $ℕ$ likhte hain.

ExampleIs book mein kitne pages hain, ya tumhari class mein kitne students hain.

3.1.1 Haddi par Likha Itihaas

Aage padhne se pehle Fig. 3.1 wali puzzle try karo. Uske chaar cards ki counts ek asli cheez se aayi hain: haddi ka bana lagbhag 10 cm lamba ek handle. Ise twentieth century ke beech mein Ishango mein khod kar nikala gaya tha, jo Democratic Republic of the Congo mein Lake Edward ke kinare hai, Nile ke ek source ke paas. Ise 20,000 se bhi zyada saal pehle taraasha gaya tha, aur yeh un sabse purani cheezon mein se ek hai jo numbers record karti lagti hain.

4/410 se 20 tak ke baaki sab numbers chhote factors mein toot jaate hain. Yeh chaar nahi tootte: yahi 10 aur 20 ke beech ke primes hain.

Fig. 3.1Ishango bone ke ek column ki counts, ek puzzle ki tarah dobara banayi gayi.NCERT Fig. 3.1 · p. 42

Chaar cards par chhote taraashe gaye notches ka ek-ek group hai, do rows mein. Pehle step mein groups gine nahi gaye hain aur har ek ke neeche question mark hai. Doosre step mein counts dikhti hain: 11, 13, 17 aur 19. Teesre step mein 10 se 20 tak numbered cells ki ek strip aati hai aur dashed arrows har count ko uske cell tak le jaate hain, jo shaded ho jaata hai. Aakhri step mein bina shade wale numbers ke neeche unhe chhote factors mein todne ka tareeka dikhta hai, jaise 10 ke neeche 2 × 5 aur 12 ke neeche 3 × 4, jabki shaded cells 11, 13, 17 aur 19 ka aisa koi split nahi hai: yeh 10 aur 20 ke beech ke primes hain.

11, 13, 17 aur 19 ko chhodkar 10 se 20 tak ka har number chhote whole numbers ke product mein toot jaata hai, isliye yahi chaar us range ke prime numbers hain. Haddi par notches ke do aur columns hain. Unmein se ek mein numbers apne doubles ke bagal mein baithe lagte hain, jaise 3 aur 6 ya 5 aur 10. Kya carver sach mein primes ke baare mein jaanta tha? Yeh koi nahi keh sakta.

Isse bhi purani ek cheez Lebombo Mountains ki ek gufa se mili hai, jahan South Africa aur Eswatini milte hain: baboon ki ek chhoti haddi jis par 29 cuts hain, aur jo kam se kam 35,000 saal purani hai. Moon ko ek full Moon se agle full Moon tak lagbhag 29½ din lagte hain, isliye ho sakta hai yeh haddi ek calendar rahi ho. Aisi cheezein batati hain ki log likhna seekhne se bahut pehle se ginti rakhte the, aur shayad patterns bhi notice karte the.

3.1.2 Indian Context: Trade aur Astronomy

Harappa, Mohenjo-daro, Dholavira aur Lothal ki khudaai mein patthar ke saikdon chhote, chikne cubes mile hain. Yeh weights hain, aur ek strict series mein aate hain. Sabse chhote se shuru karo to har weight pichhle wale se double bhaari hai: 1, 2, 4, 8, 16, 32 aur 64 units. Isse bhaari weights ten ke multiples mein aage badhte hain. Weights ka ek hi exact set poore region mein chalta tha. Isliye door-door ke shehron ke traders, aur samundar paar Mesopotamia ke merchants bhi, is baat par agree kar sakte the ki kitni cotton ya kitne beads ka len-den hua.

Vedas ke rachne walon ko bhi bade numbers bahut pasand the. Kabhi-kabhaar naya word banane ki jagah unhone ten ki har power ko apna alag naam diya, jaisa table mein dikhta hai. Yajurveda ki list parārdha, yaani $10^{12}$ , ek million million, par khatam hoti hai.

Table 3.1Yajurveda mein ten ki powers ke kuch naam

Naam	Value	Naam	Value
eka	$1$	prayuta	$10^{6}$
daśa	$10$	arbuda	$10^{7}$
śata	$10^{2}$	samudra	$10^{9}$
sahasra	$10^{3}$	anta	$10^{11}$
ayuta	$10^{4}$	parārdha	$10^{12}$

Ten ki har power ka naam ho, to numbers ko likhne ka bhi ek tareeka soojhta hai: digit ki position batayegi ki woh ten ki kaunsi power gin raha hai. Tab wahi 5 paanch, pachaas ya paanch sau ban sakta hai. Ise place value kehte hain. Iska Indian roop, sirf das digits ke saath, baad mein Arabic texts ke zariye west tak pahuncha aur wahi system ban gaya jo tum roz likhte ho. Lekin ise ek aur cheez chahiye: yeh dikhane ka tareeka ki kisi position mein kuch nahi hai. Wahin se agla section shuru hota hai.

Quick check

3 mein se 1

Ek teacher har student ko ek worksheet deti hain aur dekhti hain ki teen worksheets bach gayin. Bina kisi ko gine woh kya keh sakti hain?

Worksheets se teen students zyada hain.Students se teen worksheets zyada hain.Theek teen students hain.Kuch nahi, jab tak woh dono ko gin na lein.

Answer dekho

Answer (B) Students se teen worksheets zyada hain.

Har student ko theek ek worksheet mili, yaani students kuch worksheets ke saath one-to-one match hue, aur teen worksheets ka koi partner nahi tha. Ginne ki zaroorat hi nahi: yahi one-to-one correspondence ki taakat hai.

Practice · 4 questions

Practice 3.1: answers ke saath

NumericalEasyNCERT Exercise Set 3.1, Q1
Purane India ke astronomers poore din aur raat ko 60 ghaṭikās mein baantte the. Time ka hisaab rakhne ka ek tareeka tha copper ki ek chhoti katori, jiske tale mein chhed hota tha aur jo paani ke bartan mein tairti thi: woh har ghaṭikā mein ek baar bharkar doob jaati thi. Katori har 2 ghante mein 5 baar doobti hai. Subah 6 baje sunrise se shaam 6 baje sunset tak woh kitni baar doobti hai?
Solution dikhao
6 am se 6 pm tak 12 ghante hain, yaani 2 ghante ke $12 \div 2 = 6$ blocks. Har block mein katori 5 baar doobti hai, isliye woh $6 \times 5 = 30$ baar doobti hai. Yeh poore din ki 60 ghaṭikās ka aadha hai, jaisa 24 mein se 12 ghanton ke liye hona chahiye.
Short answerEasyNCERT Exercise Set 3.1, Q2
Ek taraashi gayi stick par 23, 29, 31 aur 37 notches ke groups hain. Ek dost kehta hai ki carver ne bas odd numbers chun liye. Aisa test dhoondho jismein 23, 29, 31 aur 37 sab pass hon lekin chhode gaye odd numbers 25, 27, 33 aur 35 sab fail hon. Phir 37 ke baad ke agle teen numbers batao jo tumhara test pass karte hain.
Solution dikhao
AnswerChaaron taraashe gaye numbers prime hain aur chhode gaye numbers nahi; agle teen primes 41, 43 aur 47 hain.
Har number ko do chhote whole numbers ke product mein todne ki koshish karo. Chhode gaye numbers toot jaate hain: $25 = 5 \times 5$ , $27 = 3 \times 9$ , $33 = 3 \times 11$ aur $35 = 5 \times 7$ . 23, 29, 31 aur 37 mein se koi nahi tootta, isliye har ek prime hai: carver ne 23 se 37 tak ke theek primes hi chune.
37 ke baad 38, 39 aur 40 composite hain ( $2 \times 19$ , $3 \times 13$ , $2 \times 20$ ), isliye agle primes 41, 43 aur 47 hain. (42, 44, 45 aur 46 sab composite hain.)
3
Short answerMediumNCERT Exercise Set 3.1, Q3
Do counting numbers ko add karne par hamesha counting number hi milta hai, isliye hum kehte hain ki $ℕ$ addition ke under closed hai. Kya $ℕ$ in operations mein se har ek ke under closed hai? Har jawab ke liye ek example do.
1. multiplication
  closedclosed nahi
  Answer dekho
  Answer (A) closed
  Do counting numbers ka product counting number hota hai, jaise $6 \times 9 = 54$ .
2. subtraction
  closedclosed nahi
  Answer dekho
  Answer (B) closed nahi
  $4 - 9 = - 5$ aur $7 - 7 = 0$ natural numbers nahi hain, aur yeh dikhane ke liye ki $ℕ$ closed nahi hai, ek example hi kaafi hai.
3. division
  closedclosed nahi
  Answer dekho
  Answer (B) closed nahi
  $3 \div 4 = \frac{3}{4}$ natural number nahi hai.
4
ActivityMediumNCERT Exercise Set 3.1, Q4
Indian markets mein kai sabzi wale bina calculator ke darjan ginte hain. Angootha usi haath ki baaki chaar ungliyon ke segments (creases ke beech ke hisse) par tap karta hai, har item ke liye ek tap.
1. Angoothe ke liye segments khatam hone tak ek haath par kitne taps ho sakte hain?
  Answer dekho
  Answer 12
  4 ungliyan × 3 segments = 12 taps: theek ek darjan.
2. Jab bhi ek haath ek darjan poora karta hai, seller doosre haath ki ek ungli mod leta hai, angoothe samet. Dono haath milkar sabse badi kitni count rakh sakte hain?
  Answer dekho
  Answer 60
  Paanch ungliyan 5 darjan record karti hain: $5 \times 12 = 60$ .
3. Seller ne 40 kele gin liye hain. Tally wale haath par kitni ungliyan mudi hain, aur counting wale haath par angoothe ne kitne segments tap kiye hain?
  Solution dikhao
  Answer3 ungliyan mudi hain aur 4 segments tap hue hain, kyunki $40 = 3 \times 12 + 4$ .
  $40 = 3 \times 12 + 4$ : teen poore darjan (teen mudi ungliyan) aur 4 aur (chaar taps). Dono haath asal mein 40 ko base 12 mein likh rahe hain, ek haath par darjan aur doosre par units.
4. Roz ki zindagi ki do aisi cheezein batao jo aaj bhi 12-12 ya 60-60 mein gini jaati hain.
  Solution dikhao
  AnswerJaise: darjan ke hisaab se bikne wale ande aur kele, saal mein 12 mahine, ek ghante mein 60 minute aur ek minute mein 60 seconds.
  Twelve kaam ka number hai kyunki yeh halves, thirds, quarters aur sixths mein barabar bant jaata hai, aur 60 aur bhi zyada barabar hisson mein bantta hai. Yahi ek wajah hai ki darjan, ghante, minute aur degrees aaj bhi 12 aur 60 use karte hain.

3.2 Śhūnya ki Kranti: Jab Kuch Nahi Bhi Kuch Ban Gaya

Agar tumhare number system mein zero na ho, to tum do sau paanch kaise likhoge? Tum 2 aur 5 likhkar beech mein gayab tens ke liye khaali jagah chhod sakte ho. Lekin khaali jagah aasaani se nazar se chhoot jaati hai, aur 25, 205 aur 250 ek jaise dikhne lagte hain. Place-value numbers likhne wale shuruaati logon ke saamne theek yahi problem thi.

Doosri civilisations ke saamne bhi yahi problem aayi. Babylonian scribes ne aakhir mein khaali jagah mark karne ke liye clay mein wedges ki ek khaas jodi dabani shuru ki, aur Mayan astronomers ne shell jaisa ek sign banaya. Yeh signs kaam ke the, lekin sirf jagah ko khaali rakhte the. Kisi ne 7 plus khaali, ya khaali times 3 jaisa sum nahi likha. Fig. 3.2 khaali column se ek aise number tak ka safar dikhata hai jisse calculation ho sake, aur yeh safar India mein poora hua.

4/4Zero ek digit aur ek number ban jaata hai, jise hum add, subtract aur multiply kar sakte hain.

Fig. 3.2Khaali jagah se number zero tak.NCERT p. 43

Hundreds, tens aur ones label wale teen boxes mein do kankad, kuch nahi aur paanch kankad hain. Pehle step mein hum sirf boxes dekhte hain aur yeh sawaal ki number kaise likhein. Doosre step mein ise 2, ek khaali dashed box, phir 5 ki tarah likha gaya hai, aur sawaal hai ki iska matlab 25 hai, 205 ya 250. Teesre step mein ek solid dot, yaani bindu, khaali jagah bhar deta hai. Aakhri step mein dot digit 0 ban gaya hai, jisse 205 milta hai, aur teen chhote sums dikhate hain ki zero ek number ki tarah kaam karta hai: 5 − 5 = 0, 7 + 0 = 7 aur 9 × 0 = 0.

3.2.1 Philosophy se Mathematics tak: Śhūnyatā ka Concept

Sanskrit mein śhūnya (शून्य) ka matlab hai khaali, aur Hindi mein aaj bhi yeh zero ke liye roz ka word hai. Digit ka naam banne se bahut pehle iske peeche ka idea philosophy mein important tha. Buddhist teachers, Upanishads aur Patañjali ke Yoga Sūtras, sab śhūnyatā (शून्यता), yaani khaalipan, ki baat karte hain. Meditate karne wale ke liye yeh kuch khona nahi, balki pahunchne ki ek manzil hai: bechain vichaaron se mukt ek shaant mann. Kai historians sochte hain ki jo culture khaalipan ko itna keemti maanta tha, uske liye kuch nahi ko apna naam aur apna symbol dena natural tha.

Jo astronomers lambe numbers verse mein likhte the, woh digit 0 ke liye aise kai words use karte the: śhūnya, kha aur ākāśa (khaali aur aakaash). Piṅgala ne Sanskrit metre par apni purani study mein śhūnya ko ginti ke ek tareeke mein working mark ki tarah use kiya tha. Lagbhag 499 CE mein Āryabhaṭa ne place value ko har jagah pichhli jagah se das guna bataya, jo tabhi kaam karta hai jab khaali jagahein dikhayi ja sakein. Phir Brahmagupta ne aakhri step liya aur zero ko uske apne rules diye.

3.2.2 Bakhśhālī Manuscript aur Brahmagupta ke Rules

1881 mein Bakhśhālī gaon ke paas, jo ab Pakistan mein hai, khudaai karte ek kisaan ko bhojpatra (birch bark) par likhi ek purani manuscript mili. Yeh arithmetic ke sawalon se bhari hai, aur khaali jagah ko ek solid dot, yaani bindu, se mark karti hai. Carbon dating batati hai ki iske kuch hisse shuruaati centuries CE tak jaate hain.

Kya tum 7 mein ek dot add kar sakte ho, ya 5 ko usse multiply kar sakte ho? Aise sawalon ka jawab Brahmagupta ne diya, jo aaj ke Rajasthan ke Bhinmal ke ek astronomer the. Unki Brāhmasphuṭasiddhānta, jo 628 CE mein poori hui, sabse purani jaani-maani book hai jo zero ko apne rules wala ek number maanti hai. Unka starting point roz ka ek subtraction tha: $7 - 7$ , $5 - 5$ ya koi bhi number minus khud wahi, zero bachata hai, isliye $a - a = 0$ . Baaki rules neeche hain.

PropertyZero ke liye Brahmagupta ke rules

Har number $a$ ke liye: $a + 0 = a$ , $a - 0 = a$ aur $a \times 0 = 0$ .

Dekho list mein kya nahi hai: zero se divide karna. Brahmagupta ne yeh case bhi handle karne ki koshish ki, lekin unke jawab baaki arithmetic ke saath fit nahi baithte. Aaj hum kehte hain ki zero se division defined nahi hai. Neeche wali puzzle dikhati hai kyun.

Quick check

3 mein se 1

Placeholder aur number zero mein kya fark hai?

Koi fark nahi: yeh ek hi idea ke do naam hain.Placeholder sirf khaali jagah mark karta hai; zero ko add, subtract aur multiply bhi kiya ja sakta hai.Placeholder ek number hai, jabki zero sirf ek symbol hai.Placeholders Brahmagupta se aaye, aur zero Babylonians se.

Answer dekho

Answer (B) Placeholder sirf khaali jagah mark karta hai; zero ko add, subtract aur multiply bhi kiya ja sakta hai.

Babylonian aur Mayan marks sirf yeh dikhate the ki likhe hue number mein koi jagah khaali hai. Brahmagupta ne zero ko

a + 0 = a

aur

a \times 0 = 0

jaise rules diye, jisse woh calculation karne layak number ban gaya. Aakhri option mein history ulti hai.

3.3 Integers: Horizon ko Badhana

Meena ke purse mein ₹4 hain. Ek dukaan par, jo uski family ko jaanti hai, woh ₹6 ki notebook kharidti hai aur baaki paise kal dene ka vaada karti hai. Ab uske paas kitne paise hain? Zero nahi: uske paas kuch nahi se bhi kam hai, kyunki use ₹2 dene hain. Natural numbers aur zero ise describe nahi kar sakte, kyunki unmein $4 - 6$ ka koi jawab nahi hai.

Brahmagupta ne theek market ki yahi bhasha use ki. Jo amount tumhare paas hai use unhone dhana (धन) kaha, yaani daulat, aur jo amount tumhe dena hai use ṛiṇa (ऋण), yaani karz. Meena ki 4 ki daulat 2 ka karz ban gayi hai, jise hum $- 2$ likhte hain. To $4 - 6 = - 2$ .

Karz ko number line par zero ke left mein apni jagah milti hai, natural numbers ki mirror images ki tarah (Fig. 3.3). Counting numbers, zero aur karz milkar integers (पूर्णांक) banate hain. Mathematicians is collection ko $ℤ = {\dots, - 3, - 2, - 1, 0, 1, 2, 3, \dots}$ likhte hain.Letter $ℤ$ Zahlen se aaya hai, jo German mein numbers ke liye word hai. Counting numbers mein 0 jod dein to unhe whole numbers bhi kehte hain.

4/4Brahmagupta ne left side ko karz (ṛiṇa) aur right side ko daulat (dhana) ki tarah padha.

Fig. 3.3Integers: natural numbers, zero aur unki mirror images.NCERT Fig. 3.2 · p. 45

Pehle step mein ek ray 1 se shuru hoti hai aur counting numbers 1 se 6 tak blue dots ki tarah dikhati hai. Doosre step mein shuruaat mein zero ek open circle ki tarah jud jaata hai. Teesre step mein negative numbers −1 se −6 tak left mein orange dots ki tarah aate hain, aur dashed arches 3 ko −3 se aur 5 ko −5 se jodte hain, yeh dikhane ke liye ki har pair zero se same distance par hai. Aakhri step mein line ki negative side ek dashed orange band mein hai jis par ṛiṇa (karz) likha hai, aur positive side ek solid blue band mein jis par dhana (daulat) likha hai; −4 par owe ₹4 aur 4 par have ₹4 likha hai.

Har integer $a$ ka ek partner $- a$ hota hai jo zero ki doosri taraf utni hi distance par hai, aur $a + (- a) = 0$ . Hum $- a$ ko $a$ ka additive inverse kehte hain. Jaise, 7 ka additive inverse $- 7$ hai, aur $- 3$ ka additive inverse 3 hai, kyunki $(- 3) + 3 = 0$ .

3.3.1 Integers ka Arithmetic

Brahmagupta ne daulat aur karz ke saath calculation ke rules bhi bataye. Table unhe modern symbols mein deti hai, jahan $a$ aur $b$ positive numbers hain.

Table 3.2Daulat aur karz ke liye Brahmagupta ke rules

Rule	Symbols mein	Example
Do daulat milkar daulat banti hain	$a + b$ positive hai	$6 + 9 = 15$
Do karz milkar karz bante hain	$(- a) + (- b) = - (a + b)$	$(- 6) + (- 9) = - 15$
Zero hataane se kuch nahi badalta	$a - 0 = a, (- a) - 0 = - a$	$(- 8) - 0 = - 8$
Daulat mein se badi daulat ghataao to karz bachta hai	agar $b > a$ , to $a - b = - (b - a)$	$4 - 6 = - 2$
Karz times daulat karz hai	$(- a) \times b = - a b$	$(- 7) \times 2 = - 14$
Karz times karz daulat hai	$(- a) \times (- b) = a b$	$(- 7) \times (- 2) = 14$

Inmein se zyadatar rules paison ki aam samajh se match karte hain. Agar tumhe ek dost ko ₹6 aur doosre ko ₹9 dene hain, to kul milakar tumhe ₹15 dene hain. Aakhri rule hairaan karne wala hai. Do karz multiply hokar daulat kyun dein? Chalo ek pattern dhoondhte hain, jaise Fig. 3.4 mein.

3/33-3 badhte rehne ke liye agle jawab $(- 3) \times (- 1) = 3$ aur $(- 3) \times (- 2) = 6$ hone chahiye.

Fig. 3.4Negative times negative positive kyun hai: pattern ko follow karo.NCERT p. 46

Left mein products ki rows hain: pehle step mein (−3) × 3 = −9, (−3) × 2 = −6 aur (−3) × 1 = −3, jinke jawab neeche number line par mark hain aur +3 label wali chhalaangon se jude hain. Doosre step mein row (−3) × 0 = 0 judti hai aur chhalaangein 0 tak pahunchti hain. Aakhri step mein blue mein do aur rows aati hain, (−3) × (−1) = 3 aur (−3) × (−2) = 6, chhalaangein 3 aur 6 tak jaati hain, aur ek note kehta hai ki pattern se (−3) × (−1) = 3 hona hi padega.

List ko upar se neeche padho. Har baar jab doosra number 1 kam hota hai, jawab 3 badh jaata hai. Jab pattern $(- 3) \times 0 = 0$ tak pahunchta hai, to aage chalne ka ek hi tareeka hai: $(- 3) \times (- 1) = 3$ , phir $(- 3) \times (- 2) = 6$ . Isliye agar arithmetic ke rules consistent rehne hain, to negative times negative positive hi hona chahiye.

Example

Ek cold-storage room

Apples ke ek cold-storage room ka temperature cooling on hone ke baad har ghante 3 °C lagaatar girta hai. Dopahar 12 baje thermometer 0 °C dikhata hai.

(a) 4 pm par woh kya dikhayega? (b) Agar temperature pehle se isi rate se gir raha tha, to 10 am par usne kya dikhaya tha?

Pehle khud try karo. Phir solution ek-ek step karke kholo.

Quick check

4 mein se 1

(- 8) + (- 5) - (- 6)

nikalo.

Answer dekho

Answer -7

(- 8) + (- 5) = - 13

, aur

- 6

subtract karne ka matlab 6 add karna hai:

- 13 + 6 = - 7

Practice · 4 questions

Practice 3.2: answers ke saath

NumericalEasyNCERT Exercise Set 3.2, Q1
Thoothukudi ke paas ek pearl diver sea level se 12 m neeche hai. Woh 5 m upar aati hai, phir 9 m aur neeche gota lagaati hai. Ise integers wali calculation ki tarah likho aur uski final position nikalo.
m
Solution dikhao
Shuruaat: $- 12$ m. 5 m upar aane par 5 add hota hai, 9 m gota lagaane par 9 subtract hota hai: $- 12 + 5 - 9 = - 7 - 9 = - 16$ . Woh sea level se 16 m neeche hai.
NumericalEasyNCERT Exercise Set 3.2, Q2
Mahilaon ka ek self-help group ek account book rakhta hai. Is mahine shuruaat mein group ko bank ko ₹2,400 dene the, usne achaar bechkar ₹3,150 kamaaye, apne stall ka ₹1,200 rent diya aur ek supplier se ₹325 ka refund mila. Group ki final position ke liye ek integer expression likho aur use solve karo. Kya group ko ab bhi paise dene hain?
₹
Solution dikhao
$- 2400 + 3150 - 1200 + 325 = 750 - 1200 + 325 = - 450 + 325 = - 125$ . Result negative hai, isliye group ko ab bhi ₹125 dene hain.
3
NumericalEasyNCERT Exercise Set 3.2, Q3
Brahmagupta ke rules se inhe solve karo.
1. $(- 14) \times 3$
  Answer dekho
  Answer -42
  Karz times daulat karz hai.
2. $(- 6) \times (- 11)$
  Answer dekho
  Answer 66
  Karz times karz daulat hai.
3. $- 9 - (- 17)$
  Answer dekho
  Answer 8
  $- 17$ subtract karne ka matlab 17 add karna hai: $- 9 + 17 = 8$ .
4. $(- 48) \div (- 6)$
  Answer dekho
  Answer 8
  $(- 6) \times 8 = - 48$ , isliye $(- 48) \div (- 6) = 8$ .
5. $(- 2) \times (- 3) \times (- 5)$
  Answer dekho
  Answer -30
  $(- 2) \times (- 3) = 6$ , phir $6 \times (- 5) = - 30$ .
Short answerMediumNCERT Exercise Set 3.2, Q4
Ek quiz mein har galat jawab ke $- 2$ points milte hain. Riya ka score 9 hai. Phir quizmaster ko pata chalta hai ki uska ek jawab galti se galat mark ho gaya tha, aur woh uske record se woh $- 2$ hata deta hai. Is situation se samjhao ki negative number ko hataane ka wahi asar hota hai jo uske positive partner ko add karne ka.
Solution dikhao
Answer $- 2$ hataane ka matlab hai $9 - (- 2)$ , aur uska score badhkar 11 ho jaata hai, jo $9 + 2$ hai.
Riya ke 9 points mein galat penalty $- 2$ pehle se shaamil hai. Us penalty ko uske record se hataana calculation $9 - (- 2)$ hai.
Penalty ke bina uska score 2 points zyada hai, yaani 11. To $9 - (- 2) = 11 = 9 + 2$ : 2 ka karz hataane ka wahi asar hai jo 2 ki daulat jodne ka.

3.4 Khaali Jagah Bharna: Fractions aur Rational Numbers

Chaar basic operations mein se division hi aisa hai jise integers hamesha handle nahi kar paate. $12 \div 4 = 3$ $ℤ$ ke andar rehta hai, lekin 2 m ke ek phatte ko 3 shelves mein barabar baantna, ya 1 km ko 4 relay runners mein, aisa nahi karta. Jawab, $\frac{2}{3}$ m aur $\frac{1}{4}$ km, fractions hain. Inhe apne numbers mein jod lene se division tab hamesha ho sakta hai jab divisor zero na ho.

Karz bhi fractions mein aate hain: aadhe rupaye ka nuksaan $- \frac{1}{2}$ hai, aur zero se dhaai degree neeche girna $- \frac{5}{2}$ hai. Har positive fraction ki aisi ek mirror image hoti hai, aur dono milkar zero bante hain, jaise $\frac{2}{3} + (- \frac{2}{3}) = 0$ : yeh fraction ka additive inverse hai. Minus sign aage, upar ya neeche kahin bhi ho sakta hai, kyunki upar ya neeche kisi ka bhi sign badalne se quotient ka sign badal jaata hai: $- \frac{2}{3} = \frac{- 2}{3} = \frac{2}{- 3}$ .

Ab tak mile saare numbers par ek nazar daalo: counting numbers, zero, karz aur dono signs wale fractions. Inmein se har ek ko ek integer divided by doosra integer likha ja sakta hai. Is tarah ke numbers rational numbers (परिमेय संख्याएँ) hain. Naam ratio se aaya hai, aur symbol $ℚ$ quotient se.

Definition

Rational number

Koi bhi number jo do integers ke quotient $\frac{p}{q}$ ke barabar ho, jahan denominator $q$ zero nahi hai.

Example $\frac{5}{8}$ , $- \frac{7}{3}$ , $4 = \frac{4}{1}$ aur $0 = \frac{0}{9}$ .

Is definition ke baare mein aksar teen sawaal aate hain.

Kya $- 12$ jaisa integer rational hai? Haan: $- 12 = \frac{- 12}{1}$ , integers ka ek quotient. Yahi trick har integer ke liye kaam karti hai, isliye $ℤ$ , aur uske saath $ℕ$ bhi, poori tarah $ℚ$ ke andar hai.
Kya $\frac{3}{4}$ aur $\frac{15}{20}$ alag rational numbers hain? Nahi. Upar aur neeche ko ek hi non-zero integer se scale karne par usi number ka ek aur naam milta hai: $\frac{3}{4} = \frac{6}{8} = \frac{15}{20} = \frac{300}{400}$ . Aise fractions ko equivalent kehte hain.
Hum kaunsa naam use karte hain? Aam taur par sabse simple wala, yaani lowest terms mein. Har common factor cancel karo, taaki upar aur neeche dono ko divide karne wala akela positive whole number 1 ho (yaani woh co-prime hon), aur denominator ko positive bana do. Jaise, $\frac{- 18}{24}$ ban jaata hai $- \frac{3}{4}$ . Number line par number mark karte waqt hum yahi naam use karte hain.

Fractions ke saath calculation ke yeh rules kam se kam Brahmagupta tak jaate hain, jo inhe daulat ki tarah karz ke liye bhi aasaani se use karte the. Aaj ke symbols mein, jahan $a$ , $b$ , $c$ , $d$ integers hain aur koi denominator zero nahi hai:

Do fractions barabar hain ya nahi, yeh test karna

\frac{a}{b} = \frac{c}{d} sirf tab jab a d = b c

Same denominator ke saath add aur subtract karna

\frac{a}{b} + \frac{c}{b} = \frac{a + c}{b}, \frac{a}{b} - \frac{c}{b} = \frac{a - c}{b}

Multiply aur divide karna

\frac{a}{b} \times \frac{c}{d} = \frac{a c}{b d}, \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{a d}{b c} (c \neq 0)

Alag-alag denominators wale fractions ko add ya subtract karne ke liye pehle dono ko ek common denominator ke saath dobara likho. Fig. 3.5 dikhata hai ki yeh kyun kaam karta hai: pieces ginne se pehle hum dono wholes ko same size ke pieces mein kaatte hain.

3/3Ab pieces match karte hain: $8 + 3 = 11$ twelfths, isliye $\frac{2}{3} + \frac{1}{4} = \frac{11}{12}$ .

Fig. 3.5Fractions add karne ke liye pehle dono wholes ko same size ke pieces mein kaato.NCERT p. 48

Barabar length ki do bars. Pehle step mein upar wali bar 3 barabar parts mein kati hai jismein 2 shaded hain, label two-thirds, aur neeche wali bar 4 barabar parts mein jismein 1 part hatched hai, label one-quarter; pieces ke size alag-alag hain. Doosre step mein dashed lines dono bars ko twelfths mein kaat deti hain, aur labels badalkar eight-twelfths aur three-twelfths ho jaate hain. Aakhri step mein barah barabar parts wali ek teesri bar mein 8 shaded aur 3 hatched pieces saath aa jaate hain, aur ek part khaali bachta hai; label eleven-twelfths hai, aur neeche sum likha hai.

Do rational numbers ka sum hamesha rational kyun hota hai? Formula $\frac{a}{b} + \frac{c}{d} = \frac{a d + b c}{b d}$ dekho. Iska upar aur neeche integers ko multiply aur add karke bane hain, isliye woh integers hain, aur $b d \neq 0$ . Aisa hi check subtraction, multiplication aur non-zero number se division ke liye bhi chalta hai. Isliye $ℚ$ chaaron operations ke under closed hai, bas hum kabhi 0 se divide na karein. Jaane-pehchane laws bhi lagu rehte hain: $x + y = y + x$ aur $x y = y x$ (commutative), aur $x (y + z) = x y + x z$ (distributive).

Example

Rational numbers ke saath calculation

(a) Kya $\frac{- 14}{21}$ aur $\frac{10}{- 15}$ barabar hain? (b) $- \frac{5}{6} + \frac{7}{10}$ nikalo. (c) $(- \frac{9}{10}) \div \frac{3}{4}$ nikalo.

Pehle khud try karo. Phir solution ek-ek step karke kholo.

Practice · 8 questions

Practice 3.3: answers ke saath

1
Short answerEasyNCERT Exercise Set 3.3, Q1
Test $\frac{a}{b} = \frac{c}{d}$ sirf tab jab $a d = b c$ use karke tay karo ki har pair ke do rational numbers barabar hain ya nahi.
1. $\frac{3}{4}$ aur $\frac{9}{12}$
  barabarbarabar nahi
  Answer dekho
  Answer (A) barabar
  $3 \times 12 = 36$ aur $4 \times 9 = 36$ : products match karte hain, isliye yeh barabar hain.
2. $\frac{7}{5}$ aur $\frac{21}{16}$
  barabarbarabar nahi
  Answer dekho
  Answer (B) barabar nahi
  $7 \times 16 = 112$ lekin $5 \times 21 = 105$ , isliye yeh barabar nahi hain. ( $\frac{7}{5}$ ke barabar $\frac{21}{15}$ hai.)
3. $- \frac{4}{9}$ aur $\frac{12}{- 27}$
  barabarbarabar nahi
  Answer dekho
  Answer (A) barabar
  $(- 4) \times (- 27) = 108$ aur $9 \times 12 = 108$ , isliye yeh barabar hain.
4. $- 2$ aur $\frac{14}{- 7}$
  barabarbarabar nahi
  Answer dekho
  Answer (A) barabar
  $- 2 = \frac{- 2}{1}$ likho. Phir $(- 2) \times (- 7) = 14$ aur $1 \times 14 = 14$ , isliye yeh barabar hain.
2
NumericalEasyNCERT Exercise Set 3.3, Q2
Har sum lowest terms mein nikalo.
1. $\frac{3}{8} + \frac{5}{12}$
  Answer dekho
  Answer 0.7916666666666666
  $\frac{9}{24} + \frac{10}{24} = \frac{19}{24}$ .
2. $\frac{5}{6} + \frac{4}{9}$
  Answer dekho
  Answer 1.2777777777777777
  $\frac{15}{18} + \frac{8}{18} = \frac{23}{18}$ .
3. $- \frac{5}{6} + \frac{7}{18}$
  Answer dekho
  Answer -0.4444444444444444
  $- \frac{15}{18} + \frac{7}{18} = - \frac{8}{18} = - \frac{4}{9}$ .
3
NumericalEasyNCERT Exercise Set 3.3, Q3
Har difference lowest terms mein nikalo.
1. $\frac{7}{10} - \frac{2}{5}$
  Answer dekho
  Answer 0.3
  $\frac{7}{10} - \frac{4}{10} = \frac{3}{10}$ .
2. $\frac{9}{4} - \frac{5}{6}$
  Answer dekho
  Answer 1.4166666666666667
  $\frac{27}{12} - \frac{10}{12} = \frac{17}{12}$ .
3. $- \frac{3}{8} - (- \frac{5}{6})$
  Answer dekho
  Answer 0.4583333333333333
  $- \frac{9}{24} + \frac{20}{24} = \frac{11}{24}$ .
4
NumericalEasyNCERT Exercise Set 3.3, Q4
Har product lowest terms mein nikalo.
1. $\frac{4}{9} \times \frac{15}{8}$
  Answer dekho
  Answer 0.8333333333333334
  $\frac{4 \times 15}{9 \times 8} = \frac{60}{72} = \frac{5}{6}$ .
2. $\frac{5}{12} \times \frac{9}{10}$
  Answer dekho
  Answer 0.375
  $\frac{45}{120} = \frac{3}{8}$ .
3. $- \frac{6}{7} \times \frac{14}{15}$
  Answer dekho
  Answer -0.8
  $- \frac{84}{105} = - \frac{4}{5}$ .
5
NumericalEasyNCERT Exercise Set 3.3, Q5
Har quotient lowest terms mein nikalo.
1. $\frac{4}{9} \div \frac{2}{3}$
  Answer dekho
  Answer 0.6666666666666666
  $\frac{4}{9} \times \frac{3}{2} = \frac{12}{18} = \frac{2}{3}$ .
2. $\frac{5}{12} \div \frac{15}{8}$
  Answer dekho
  Answer 0.2222222222222222
  $\frac{5}{12} \times \frac{8}{15} = \frac{40}{180} = \frac{2}{9}$ .
3. $- \frac{6}{7} \div \frac{9}{14}$
  Answer dekho
  Answer -1.3333333333333333
  $- \frac{6}{7} \times \frac{14}{9} = - \frac{84}{63} = - \frac{4}{3}$ .
Short answerEasyNCERT Exercise Set 3.3, Q6
Dono sides solve karke dikhao ki $(\frac{2}{3} + \frac{1}{6}) \times \frac{9}{5} = \frac{2}{3} \times \frac{9}{5} + \frac{1}{6} \times \frac{9}{5}$ .
Solution dikhao
AnswerDono sides $\frac{3}{2}$ ke barabar hain.
Left side: $\frac{2}{3} + \frac{1}{6} = \frac{4}{6} + \frac{1}{6} = \frac{5}{6}$ , aur $\frac{5}{6} \times \frac{9}{5} = \frac{45}{30} = \frac{3}{2}$ .
Right side: $\frac{2}{3} \times \frac{9}{5} = \frac{18}{15} = \frac{6}{5}$ aur $\frac{1}{6} \times \frac{9}{5} = \frac{9}{30} = \frac{3}{10}$ . Inka sum $\frac{12}{10} + \frac{3}{10} = \frac{15}{10} = \frac{3}{2}$ hai.
Dono sides $\frac{3}{2}$ ke barabar hain, jaisa distributive law vaada karta hai.
NumericalMediumNCERT Exercise Set 3.3, Q7
Distributive property se $\frac{5}{8} (\frac{4}{5} - \frac{2}{3})$ ko simplify karo.
Solution dikhao
$\frac{5}{8} (\frac{4}{5} - \frac{2}{3}) = \frac{5}{8} \times \frac{4}{5} - \frac{5}{8} \times \frac{2}{3} = \frac{20}{40} - \frac{10}{24} = \frac{1}{2} - \frac{5}{12} = \frac{1}{12}$ .
Pehle brackets solve karke check karo: $\frac{4}{5} - \frac{2}{3} = \frac{2}{15}$ , aur $\frac{5}{8} \times \frac{2}{15} = \frac{10}{120} = \frac{1}{12}$ .
8
Short answerHardNCERT Exercise Set 3.3, Q8
Fractions wale equations.
1. Woh rational number $x$ nikalo jiske liye $\frac{2}{3} (x + \frac{3}{4}) = \frac{1}{2} x + 1$ .
  Solution dikhao
  Distribute karo: $\frac{2}{3} x + \frac{2}{3} \times \frac{3}{4} = \frac{2}{3} x + \frac{1}{2}$ .
  To $\frac{2}{3} x + \frac{1}{2} = \frac{1}{2} x + 1$ . Dono sides se $\frac{1}{2} x$ aur $\frac{1}{2}$ subtract karo: $(\frac{2}{3} - \frac{1}{2}) x = \frac{1}{2}$ , yaani $\frac{1}{6} x = \frac{1}{2}$ , isliye $x = 3$ .
  Check: $\frac{2}{3} (3 + \frac{3}{4}) = \frac{2}{3} \times \frac{15}{4} = \frac{5}{2}$ , aur $\frac{1}{2} \times 3 + 1 = \frac{5}{2}$ .
2. Ab $\frac{3}{2} (x - \frac{1}{3}) + \frac{1}{2} = \frac{3}{2} x$ try karo. Kya hota hai, aur isse $x$ ke baare mein kya pata chalta hai?
  Solution dikhao
  AnswerDono sides ek hi expression hain, isliye har rational number $x$ chalega.
  Left side $\frac{3}{2} x - \frac{3}{2} \times \frac{1}{3} + \frac{1}{2} = \frac{3}{2} x - \frac{1}{2} + \frac{1}{2} = \frac{3}{2} x$ hai, bilkul right side jaisa. Yeh equation har $x$ ke liye sach hai: yeh ek identity hai, ek jawab wali puzzle nahi.

3.4.1 Number Line par Rational Numbers ko Dikhana

Number line par $\frac{7}{3}$ kahan aayega? Integer line ke kisi tick par nahi. Woh ticks origin 0 se ek-ek unit ke poore steps mark karte hain: positive integers ke liye uske right mein, negative integers ke liye left mein. Fraction do ticks ke beech ke gap mein rehta hai, aur uska denominator batata hai ki us gap ko kitna baarik kaatna hai.

$\frac{p}{q}$ ko instructions ka ek set samjho. Denominator $q$ har step ka size batata hai: ek unit ko $q$ barabar pieces mein baanta gaya. Numerator $p$ batata hai ki 0 se kitne steps lene hain, aur uska sign batata hai ki kis taraf chalna hai: positive integers ki taraf ya negative integers ki taraf. Fig. 3.6 $\frac{7}{3}$ aur $- \frac{5}{6}$ ke liye in instructions ko follow karta hai.

3/3 $- 1$ se $0$ tak ki unit ko 6 parts mein kaato aur 0 ke left mein 5 steps lo.

Fig. 3.6

\frac{p}{q}

ko rakhne ke liye har unit ko

q

barabar parts mein kaato aur 0 se unmein se

p

gino.NCERT Fig. 3.5 aur 3.6 · p. 50

−2 se 3 tak ki number line jis par integers mark hain. Doosre step mein 0 se 3 tak ki har unit thirds mein kat jaati hai, aur one-third ki saat blue chhalaangein 0 se 2 ke thoda aage ek dot tak le jaati hain, jiska label seven-thirds hai, yaani 2 aur one-third. Aakhri step mein −1 se 0 tak ki unit sixths mein kat jaati hai, aur one-sixth ki paanch dashed orange chhalaangein 0 se left mein ek dot tak le jaati hain, jiska label negative five-sixths hai, −1 ke thoda right mein.

1 se bade fractions ke liye ek shortcut hai: pehle poori units alag kar lo. $\frac{7}{3}$ 2 poori units aur ek-third aur hai, isliye seedhe 2 par jump karo aur sirf agli unit ko thirds mein kaato.

Rational number ki absolute value

Kabhi-kabhi sirf yeh matter karta hai ki number zero se kitna door hai, yeh nahi ki woh kis side hai. 3 km ki auto-rickshaw ride ka kiraya same hota hai, chahe woh east jaaye ya west. Direction ke bina yeh size number ki absolute value (निरपेक्ष मान) hai, jise do khadi lines ke beech likhte hain: $∣ x ∣$ .

Definition

Absolute value

Koi number 0 se kitna door hai, chahe woh kisi bhi side ho. Symbols mein, $∣ x ∣ = x$ jab $x \geq 0$ , aur $∣ x ∣ = - x$ jab $x < 0$ .

Example $∣ - 2.5 ∣ = 2.5$ aur $∣ \frac{4}{7} ∣ = \frac{4}{7}$ .

Kyunki distances kabhi negative nahi hoti, $∣ x ∣ \geq 0$ hamesha, aur $∣ x ∣ = 0$ sirf $x = 0$ par. Yahi lines kinhi bhi do numbers ke beech ka gap bhi naapti hain: $a$ aur $b$ ek doosre se $∣ a - b ∣$ door hain. Isse fark nahi padta ki tum kisme se kise subtract karte ho, kyunki lines sign hata deti hain. Fig. 3.7 ise ek aise pair par check karta hai jo 0 ke dono taraf hai.

4/5 $b$ se $a$ tak ki distance $∣ a - b ∣ = ∣ \frac{5}{2} + \frac{3}{2} ∣ = 4$ hai. Yahan upar ke dono pieces milkar yahi bante hain.

Fig. 3.7Absolute value 0 se distance naapti hai;

∣ a - b ∣

a

aur

b

ke beech ki distance naapta hai.NCERT Fig. 3.8 · p. 52

−3 se 3 tak ki number line, jis par point b negative three-halves par aur point a five-halves par hai. Pehle step mein sirf dono points unki values ke saath dikhte hain. Doosre step mein line ke upar b se 0 tak ek dashed bracket dikhata hai ki b ki absolute value three-halves hai. Teesre step mein line ke upar 0 se a tak ek solid bracket dikhata hai ki a ki absolute value five-halves hai. Chauthe step mein line ke neeche ek lamba green bracket b ko a se jodta hai aur unke beech ki distance deta hai: a minus b ki absolute value, jo 4 hai, yaani upar ke dono pieces ka jod. Aakhri step mein usi bracket par b minus a likha hai, jiski absolute value bhi 4 hai.

Example 1NCERT Example 1 · p. 51

Absolute values aur distance

$∣ - \frac{7}{4} ∣$ , $∣ \frac{1}{2} ∣$ aur $∣ 0 ∣$ nikalo. Phir number line par $- \frac{7}{4}$ aur $\frac{1}{2}$ ke beech ki distance nikalo.

Pehle khud try karo. Phir solution ek-ek step karke kholo.

Quick check

4 mein se 1

- \frac{13}{4}

kin do integers ke beech hai?

- 3

aur

- 2

- 4

aur

- 3

3

aur

4

- 13

aur

- 4

Answer dekho

Answer (B) $- 4$ aur $- 3$

- \frac{13}{4} = - 3 \frac{1}{4} = - 3.25

, jo

- 3

se ek quarter unit left mein hai, isliye yeh

- 4

aur

- 3

ke beech hai.

- 3

aur

- 2

chunne wala bhool jaata hai ki aur left jaane par number chhota hota hai.

3.4.2 Rational Numbers ki Density

1 ke theek baad kaunsa integer aata hai? Aasaan hai: 2. Ab yeh try karo: 1 ke theek baad kaunsa rational number aata hai? Tum $1.1$ kaho, to koi $1.01$ bol sakta hai; tum $1.0001$ kaho, to $1.00001$ usse bhi paas hai. Koi agla rational number nahi hota. Do rational numbers kitne bhi paas hon, unke beech hamesha ek aur fit ho jaata hai.

Beech mein ek number ghusaane ka sabse simple tareeka hai theek aadhe raaste jaana, yaani average lena. $\frac{1}{3}$ aur $\frac{1}{2}$ ke theek beech mein $\frac{1}{2} (\frac{1}{3} + \frac{1}{2}) = \frac{5}{12}$ hai; $\frac{1}{3}$ aur $\frac{5}{12}$ ke theek beech mein $\frac{3}{8}$ hai. Fig. 3.8 zoom karta hi rehta hai, aur yeh process kabhi rukna zaroori nahi.

3/4 $\frac{1}{3}$ aur $\frac{5}{12}$ ke theek beech mein $\frac{3}{8}$ hai.

Fig. 3.8Do rational numbers ke beech hum kitna bhi zoom karein, unka average ek naya rational number hota hai.NCERT Fig. 3.9 · p. 52

Teen number lines ek ke upar ek. Pehle step mein upar wali line 0 se 1 tak hai, jis par one-third aur one-half mark hain aur unke beech ka gap highlight hai. Doosre step mein ek funnel us gap ko kheenchkar beech wali line bana deta hai, one-third se one-half tak, aur uska midpoint five-twelfths ek orange dot ki tarah dikhta hai. Teesre step mein one-third se five-twelfths tak ka gap kheenchkar neeche wali line banti hai, jiska midpoint three-eighths hai. Aakhri step mein one-third se three-eighths tak ka gap highlight hai aur ek note agle midpoints deta hai, seventeen forty-eighths aur phir eleven thirty-seconds, jo dikhata hai ki yeh process kabhi khatam nahi hota.

TheoremRational numbers dense hain

Agar $a$ aur $b$ rational numbers hain aur $a < b$ , to $\frac{a + b}{2}$ ek rational number hai aur $a < \frac{a + b}{2} < b$ .

Yeh rational hai. $a + b$ rational hai kyunki $ℚ$ addition ke under closed hai, aur ise non-zero rational number 2 se divide karne par yeh rational hi rehta hai.

Yeh $a$ aur $b$ ke beech hai. Ise dono ends se compare karo:

\frac{a + b}{2} - a = \frac{b - a}{2} > 0 aur b - \frac{a + b}{2} = \frac{b - a}{2} > 0

Dono differences positive hain kyunki $b > a$ . Isliye average $a$ se bada aur $b$ se chhota hai; asal mein yeh theek beech mein hai. Is result ko baar-baar use karne se $a$ aur $b$ ke beech infinitely many rational numbers milte hain.

Average hi akela tareeka nahi hai. Dono numbers ko ek bade common denominator ke saath likhne se ek saath kai numbers dikh jaate hain. Jaise $\frac{1}{3} = \frac{20}{60}$ aur $\frac{1}{2} = \frac{30}{60}$ , isliye $\frac{21}{60}, \frac{22}{60}, \dots, \frac{29}{60}$ unke beech nau aur rational numbers hain.

Example

Do fractions ke beech chaar rational numbers

$\frac{3}{5}$ aur $\frac{2}{3}$ ke beech chaar rational numbers dhoondho.

Pehle khud try karo. Phir solution ek-ek step karke kholo.

Ghusaate raho, fractions kabhi khatam nahi honge. Kya iska matlab hai ki line ke har point par ek fraction baitha hai? Is idea ko ek khaas point par test karo: unit square ke diagonal ki length, jise 0 se line ke saath bichhaya gaya ho. Section 3.5 dikhata hai ki kya hota hai.

Quick check

3 mein se 1

\frac{2}{7}

aur

\frac{3}{7}

ke beech kitne rational numbers hain?

ek bhi nahi, kyunki denominator 7 wala koi fraction unke beech fit nahi hotatheek ektheek chhahinfinitely many

Answer dekho

Answer (D) infinitely many

Doosre denominators wale fractions fit ho jaate hain, jaise

\frac{5}{14}

\frac{9}{28}

ya inmein se kinhi do ka average, aur hum hamesha average lete reh sakte hain. Pehle option ne sirf sevenths ko dekha.

Practice · 6 questions

Practice 3.4: answers ke saath

DiagramEasyNCERT Exercise Set 3.4, Q1
$- 2$ se $3$ tak ek number line banao aur us par $\frac{3}{5}$ , $- \frac{4}{3}$ aur $2 \frac{1}{4}$ rakho.
Solution dikhao
Answer $\frac{3}{5}$ 0 aur 1 ke beech hai, $- \frac{4}{3}$ $- 2$ aur $- 1$ ke beech, aur $2 \frac{1}{4}$ 2 aur 3 ke beech.
$\frac{3}{5}$ : 0 se 1 tak ki unit ko 5 barabar parts mein kaato aur 0 ke right mein unmein se 3 lo.
$- \frac{4}{3} = - 1 \frac{1}{3}$ : $- 2$ se $- 1$ tak ki unit ko 3 barabar parts mein kaato aur $- 1$ se ek part left wala point mark karo.
$2 \frac{1}{4}$ : 2 se 3 tak ki unit ko 4 barabar parts mein kaato aur 2 se ek part right wala point mark karo.
Short answerEasyNCERT Exercise Set 3.4, Q2
Teen aise rational numbers batao jo $- \frac{2}{3}$ se bade lekin $\frac{1}{6}$ se chhote hon.
Solution dikhao
AnswerJaise $- \frac{1}{2}$ , $- \frac{1}{6}$ aur $0$ .
Denominator 6 ke saath ends $- \frac{4}{6}$ aur $\frac{1}{6}$ hain. $- 3$ se $0$ tak koi bhi numerator chalega: $- \frac{3}{6} = - \frac{1}{2}$ , $- \frac{2}{6} = - \frac{1}{3}$ , $- \frac{1}{6}$ aur $\frac{0}{6} = 0$ . Koi bhi teen chuno.
NumericalEasyNCERT Exercise Set 3.4, Q3
$(- \frac{5}{6}) + \frac{3}{8} - (- \frac{1}{4})$ ko simplify karo.
Solution dikhao
$- \frac{5}{6} + \frac{3}{8} + \frac{1}{4} = - \frac{20}{24} + \frac{9}{24} + \frac{6}{24} = \frac{- 20 + 15}{24} = - \frac{5}{24}$ .
NumericalMediumNCERT Exercise Set 3.4, Q4
Ek cyclist ghat road par ek jaisi speed se $2 \frac{3}{4}$ ghante mein $16 \frac{1}{2}$ km chalati hai. Woh har ghante mein kitne kilometre tay karti hai?
km
Solution dikhao
$16 \frac{1}{2} = \frac{33}{2}$ aur $2 \frac{3}{4} = \frac{11}{4}$ . Ek ghante mein distance $= \frac{33}{2} \div \frac{11}{4} = \frac{33}{2} \times \frac{4}{11} = \frac{132}{22} = 6$ . Woh har ghante 6 km chalati hai.
Short answerMediumNCERT Exercise Set 3.4, Q5
School ki ek sprint mein do runners ka time 12.47 s aur 12.48 s aata hai. Teen aise times likho jo in dono ke strictly beech mein hon, aur samjhao ki har ek rational number kyun hai.
Solution dikhao
AnswerJaise 12.471 s, 12.475 s aur 12.479 s.
Ends ko 12.470 aur 12.480 likho. 12.471, 12.472, …, 12.479 mein se koi bhi inke beech hai, jaise 12.471, 12.475 aur 12.479. Har ek terminating decimal hai, isliye woh 1000 par ek fraction hai aur isliye rational hai (jaise $12.475 = \frac{12475}{1000} = \frac{499}{40}$ ).
Short answerHardNCERT Exercise Set 3.4, Q6
$\frac{a}{b}$ aur $\frac{c}{d}$ (jahan $b, d > 0$ ) ke beech fraction dhoondhne ka ek aur tareeka hai: upar wale add karo aur neeche wale add karo, aur $\frac{a + c}{b + d}$ pao. Ise $\frac{1}{3}$ aur $\frac{1}{2}$ par try karo. Kya tumhara jawab unke beech hai? Apne do aur pairs par try karo.
Solution dikhao
Answer $\frac{1 + 1}{3 + 2} = \frac{2}{5} = 0.4$ , jo $0.33 \dots$ aur $0.5$ ke beech hai.
$\frac{2}{5} = 0.4$ aur $\frac{1}{3} < 0.4 < \frac{1}{2}$ , isliye yahan yeh kaam karta hai. Is fraction ko mediant kehte hain.
Yeh positive denominator wale kinhi bhi do fractions ke liye kaam karta hai jahan $\frac{a}{b} < \frac{c}{d}$ . Jaise $\frac{3}{5}$ aur $\frac{2}{3}$ se $\frac{5}{8} = 0.625$ milta hai, jo $0.6$ aur $0.66 \dots$ ke beech hai; aur $\frac{1}{4}$ aur $\frac{3}{4}$ se $\frac{4}{8} = \frac{1}{2}$ milta hai. Average ke ulat, mediant aam taur par theek beech mein nahi hota.

3.5 Irrational Numbers

Ek priest-architect ke paas ek square fire altar hai, aur use aisa doosra altar banane ko kaha jaata hai jiska area theek double ho. Naye square ki side kitni lambi honi chahiye? Aise sawalon se Śulbasūtras bhare hain, jo rassi aur khoontiyon se Vedic altars ka layout banane ki manuals hain. Lagbhag 800 BCE ka Baudhāyana ka Śulbasūtra iska ek saaf jawab deta hai, aur uske andar ek naye tarah ka number chhupa hai.

Fig. 3.9 dikhata hai ki ise kaise banayein. Double square ki side pehle square ka diagonal $d$ hai. Agar pehle square ki side 1 unit hai, to Baudhāyana–Pythagoras theorem se $d^{2} = 1^{2} + 1^{2} = 2$ . Isliye $d$ woh positive number hai jiska square 2 hai, jise $\sqrt{2}$ likhte hain.

5/5Area 2 wale square mein $d^{2} = 2$ , isliye $d = \sqrt{2} \approx 1.414$ .

Fig. 3.9Square ko double karna, jaise altar banane wale karte the: nayi side purane square ka diagonal hai.NCERT Fig. 3.10 · p. 53

Dashed lines mein bani unit squares ki two-by-two grid, jismein upar-left wala square shaded hai aur har side par 1 likha hai. Pehle step mein note kehta hai side 1, area 1. Doosre step mein shaded square ka diagonal draw hota hai aur use d kaha jaata hai. Teesre step mein ek tircha square banta hai jiski chaaron sides chaar unit squares ke diagonals hain, aur har ek par d likha hai. Chauthe step mein tirche square ke andar ke chaar triangles shaded hote hain: har ek aadha unit square hai, isliye uska area 4 halves hai, yaani 2, pehle square ka double. Aakhri step mein note kehta hai d times d barabar 2, isliye d, 2 ka square root hai, lagbhag 1.414.

Squares $1.96$ , lagbhag $2.0069$ aur lagbhag $1.9988$ aate hain: 2 ke paas aur paas, lekin kabhi uske barabar nahi. Asal mein koi bhi fraction square hokar theek 2 nahi deta, aur agla section ise prove karta hai. Isliye $\sqrt{2}$ irrational number (अपरिमेय संख्या) ka hamara pehla example hai: number line par iski ek pakki jagah hai, phir bhi yeh do integers ka quotient nahi hai.

Definition

Irrational number

Aisa number jiski number line par jagah hai lekin jo do integers ka quotient nahi hai: koi integers $p$ aur $q \neq 0$ aise nahi hain ki $\frac{p}{q}$ uske barabar ho.

Example $\sqrt{2}$ , $\sqrt{3}$ aur $π$ .

3.5.1 $\sqrt{2}$ ki Irrationality ka Proof

Koi kaise pakka keh sakta hai ki koi bhi fraction square hokar theek 2 nahi deta? Fractions infinitely many hain, isliye unhe ek-ek karke check karna kabhi khatam nahi hoga. Hamein ek alag tarah ka argument chahiye, jaisa koi detective use karta. Agar chor garden ke raaste bhaaga hota, to geeli mitti mein pairon ke nishaan hote. Koi nishaan nahi hain. To chor us raaste nahi gaya. Yahi proof by contradiction hai: kuch maan lo, use follow karke ek impossible nateeje tak pahuncho, aur conclude karo ki jo maana tha woh galat tha.

$\sqrt{2}$ ke liye hum maante hain ki yeh ek fraction hai aur nishaan dhoondhte hain. Parampara ke hisaab se is proof ka credit Hippasus ko diya jaata hai, jo ancient Greece mein Pythagoras ke followers mein se ek the; wahan irrational lengths ki khoj Baudhāyana ke kuch sadiyon baad ek jhatke ki tarah aayi. Pehle hamein even aur odd numbers ke baare mein ek chhota sa fact chahiye.

LemmaEven squares even numbers se aate hain

Agar $p$ ek integer hai aur $p^{2}$ even hai, to $p$ even hai.

Theorem

\sqrt{2}

irrational hai

Aise koi integers $p$ aur $q$ , jahan $q \neq 0$ , nahi hain ki $\frac{p}{q} = \sqrt{2}$ .

Ulta maan lo. Maan lo kisi integers $p$ aur $q$ ke liye, jahan $q \neq 0$ , $\sqrt{2} = \frac{p}{q}$ . Pehle saare common factors cancel kar do, taaki fraction lowest terms mein ho: 1 se bada koi whole number $p$ aur $q$ dono ko divide nahi karta.

p^{2} = 2 q^{2}

To $p$ even hai. Right-hand side $2 q^{2}$ , 2 times ek integer hai, isliye $p^{2}$ even hai. Lemma se, $p$ khud even hai: kisi integer $k$ ke liye $p = 2 k$ .

q^{2} = 2 k^{2}

To $q$ bhi even hai. Ab $q^{2}$ , 2 times ek integer hai, isliye woh even hai, aur lemma se phir $q$ even hai.

Contradiction. $p$ aur $q$ dono even hain, isliye 2 ek common factor hai. Lekin humne $\frac{p}{q}$ aisa chuna tha ki 1 se bada koi whole number dono ko divide na kare. Dono baatein sach nahi ho sakti. Assumption ke baad ka har step sahi tha, isliye assumption hi galat tha: $\sqrt{2}$ ko $\frac{p}{q}$ nahi likha ja sakta. Yeh irrational hai.

Aisa proof ek hi baar mein har fraction ke liye sawaal ka faisla kar deta hai, unke liye bhi jinhe koi kabhi nahi likhega. Examples check karne ke mukable reasoning ki yahi taakat hai.

3.5.2 Length $\sqrt{n}$ ka Construction

Fig. 3.10 wale construction mein sirf do moves chahiye, jo baar-baar use hote hain. Khada karo: number line ke kisi point par line ke right angle par length 1 ka segment khada karo. Ghumao: us segment ke top ko 0 se jodo aur compass ki nok 0 par rakhkar is tirchi length ko ghumakar line par le aao. Baudhāyana–Pythagoras theorem tumhe batata hai ki tircha segment theek kitna lamba hai, jabki wahan ruler ka koi mark nahi hai.

0 se 1 tak ke unit segment se shuru karo. Ek baar khada-karo-aur-ghumao ise line ke saath length $\sqrt{1 + 1} = \sqrt{2}$ ka segment bana deta hai, isliye uska end point $\sqrt{2}$ mark karta hai. Us point par doosri baar khada-karo-aur-ghumao karne se $\sqrt{2 + 1} = \sqrt{3}$ milta hai. Dono dekhne ke liye figure ke steps mein aage badho.

5/5 $U$ ko ghumakar line par lao: arc point $\sqrt{3}$ mark karta hai.

Fig. 3.10Right triangles aur compass number line par

\sqrt{2}

aur

\sqrt{3}

ko theek sahi jagah rakhte hain.NCERT Fig. 3.11 · p. 56

0 se 2 tak ki number line, jismein origin O 0 par hai aur pehle step mein 0 se 1 tak ka unit segment highlight hai. Doosre step mein 1 par length 1 ka segment seedha khada hai, right-angle mark ke saath, aur O se uske top T tak ke blue tirche segment par root 2 likha hai. Teesre step mein O centre wala ek dashed compass arc T ko ghumakar number line par laata hai, jahan ek dot par root 2 likha hai, 1 se 2 ke aadhe raaste se thoda pehle. Chauthe step mein root 2 par length 1 ka doosra segment khada hota hai, aur O se uske top U tak ke green tirche segment par root 3 likha hai. Aakhri step mein doosra arc U ko ghumakar root 3 label wale dot tak laata hai, jo pehle wale se thoda right mein hai.

Har round length ke square mein 1 jodta hai, isliye wahi do moves $\sqrt{4} = 2$ , phir $\sqrt{5}$ , $\sqrt{6}$ waghairah dete hain. Kabhi-kabhi shortcut hota hai: legs 2 aur 1 wale right triangle ka hypotenuse $\sqrt{2^{2} + 1^{2}} = \sqrt{5}$ hota hai. Aise triangles dhoondhne ke liye simulation ka squared paper use karo.

SimulationSquared paper par lengths

Ek square grid jismein origin par ek fixed point O hai aur ek point B hai jise tum kisi bhi grid point par drag kar sakte ho. Ek segment O ko B se jodta hai, uske right triangle ke horizontal aur vertical legs draw hote hain, aur ek panel OB ki exact length dikhata hai, jaise root 5 ya 5, uski decimal value ke saath.

Fig. 3.11

B

ko kisi bhi grid point par drag karo. Panel

O B

ki exact length deta hai aur uska right triangle draw karta hai. Kuch lengths simplified dikhti hain:

\sqrt{8}

2 \sqrt{2}

ki tarah dikhta hai.

Try karo

$B$ ko $(2, 1)$ par drag karo. Panel $O B$ ki kya exact length dikhata hai?
Aisa grid point dhoondho jisse $O B = \sqrt{10}$ ho, aur ek aisa jisse $O B$ theek 5 ho.
$O B = \sqrt{3}$ banane ki koshish karo. Kya koi grid point aisa kar sakta hai?

3.5.3 Pi (π) ki Kahani aur Madhava ki Infinite Series

Ise ek steel ke glass aur ek dhaage ke saath try karo. Dhaage ko rim ke chaaron taraf ek baar lapeto, mark karo, aur length naapo; phir rim ki chaudai seedhe aar-paar naapo. Pehli ko doosri se divide karo. Glass kisi bhi size ka ho, tumhe 3 se thoda zyada milega. Har circle ke liye circumference ÷ diameter ek hi number hota hai, jise $π$ (पाई) kehte hain.

499 CE mein Āryabhaṭa ne iska ek nuskha verse mein diya: 100 mein 4 jodo, 8 se multiply karo, aur 62,000 jodo; yeh lagbhag us circle ki circumference hai jiska diameter 20,000 hai. Isse $π \approx \frac{62832}{20000} = 3.1416$ milta hai. Unhone is value ko āsanna, yaani paas aata hua, kaha: kareeb, lekin exact nahi. Unka saavdhaan rehna sahi tha. 1761 mein Johann Lambert ne prove kiya ki $π$ irrational hai, isliye koi bhi fraction, chahe uska numerator aur denominator kitna bhi bada ho, iske barabar nahi hai.

Saṅgamagrāma ke Mādhava Āryabhaṭa ke aath sau se zyada saal baad Kerala mein rahe, aur jin teachers ne unka kaam aage badhaya woh Kerala school kehlate hain. Unhone ek aisa expression dhoondha jo sirf odd numbers ki madad se $π$ ko exact deta hai:

π ke liye Mādhava ki series

π = 4 (1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} - \dots)

Pattern padhna aasaan hai: denominators mein odd numbers, aur signs baari-baari se badalte hain. Kabhi khatam na hone wala sum kabhi poora nahi kiya ja sakta, isliye uski value alag tarah se define hoti hai. Terms ko ek-ek karke jodo aur running total rakho, jise partial sum kehte hain. Yeh totals jis number par aakar tik jaate hain, wahi poore sum ki value hai. Fig. 3.12 dikhata hai ki totals $π$ ke upar-neeche jhoolte hain, har baar chhoti jhool ke saath, aur khud $π$ ke paas aate jaate hain.

4/5Aath terms ke baad total lagbhag $3.02$ hai; jhool chhoti hoti ja rahi hai.

Fig. 3.12Ek-ek term jodne par running totals

π

ke aas-paas zigzag karte hain, har baar chhoti jhool ke saath.NCERT p. 56

Neeche 2.5 se 4 tak ki ek number line hai, jismein pi, lagbhag 3.14, se ek dashed vertical line upar jaati hai. Mādhava ki series ka har running total apni row par ek dot hai, figure mein upar se neeche. Pehle step mein ek dot hai, 4 par. Doosre step mein doosra dot 2.67 par aata hai, pi se neeche. Teesre step mein totals 3.47 aur 2.90 judte hain. Chauthe step mein zigzag aath totals tak pahunchta hai, 3.34, 2.98, 3.28 aur 3.02, jo pi line ke dono taraf kam se kam jhoolte hain. Bhare hue dots pi se upar hain aur khaali dots neeche. Aakhri step mein ek note kehta hai ki 1000 terms ke baad total 3.1406 hai, aur ab bhi paas aa raha hai.

3.6 Real Numbers: Decimals aur Cyclic Patterns

Number line ke har point ko ek address diya ja sakta hai: uska decimal expansion. Integer part ek unit chunta hai. Point ke baad ka pehla digit us unit ke das tenths mein se ek chunta hai, agla digit uske andar ke das hundredths mein se ek, aur aise hi aage, point ke aur paas pahunchte hue. Neeche wala zoom tool aisa address ek-ek digit karke banata hai.

InteractiveNumber-line zoom

Ek interactive number line. Tum 13/8, 2/7, negative 7/6, root 2 ya pi chunte ho, ya apna fraction, decimal ya square root type karte ho. Line pehle −5 se 5 tak ke integers dikhati hai aur number ek dot ki tarah mark hota hai. Zoom in ka har press, ya highlight kiye gaye gap par tap, dot wale gap ko kheenchkar das parts mein kati ek nayi line bana deta hai, jo purani line ke neeche lagti hai. Ek readout decimal ko digit-by-digit banata hai, cards window, tick spacing aur, fractions ke liye, long-division remainder dikhate hain, aur ek message batata hai ki dot kab theek tick par aata hai, kab picture repeat hone lagti hai, ya ki woh kabhi nahi hogi.

Fig. 3.13Ek number chuno, phir zoom in karo. Har zoom highlight kiye gaye piece ko das hisson mein kaatta hai aur ek aur decimal digit dikhata hai. Fraction ke liye card long-division ka remainder dikhata hai.

Try karke dekho

Teen tarah ke address

Pehle guess karo

Zoom karne se pehle socho: kis number ke liye kuch zooms ke baad dot theek tick par aa jaayega?

\frac{2}{7}

\sqrt{2}

\frac{13}{8}

$\frac{13}{8}$ chuno aur tab tak zoom in karo jab tak dot theek tick par na baith jaaye. Kitne zooms lagte hain?
$\frac{2}{7}$ chuno aur remainder card dekhte hue saat baar zoom in karo. Kis zoom par remainder kisi aisi value par wapas aata hai jo tum pehle dekh chuke ho?
$\sqrt{2}$ chuno aur das baar zoom in karo. Kya dot kabhi tick par aata hai? Kya picture kabhi repeat hoti hai?

$\frac{13}{8} = 1.625$ : teen zooms (tenths, hundredths, thousandths) ke baad dot theek tick 1.625 par baithta hai, aur division remainder 0 par khatam hota hai.

$\frac{2}{7}$ ke liye zoom 1 par remainder 2 hai aur zoom 7 par wapas 2 aata hai. Uske baad har picture chhah zoom pehle wali picture jaisi hoti hai, isliye digits $285714$ hamesha repeat hote hain.

$\sqrt{2}$ ke liye dot kabhi tick par nahi aata aur koi picture repeat nahi hoti: $1.414213562 \dots$ bina kisi repeating block ke chalta rehta hai.

Rational points aur irrational points, dono ko is tarah ke address milte hain. Line ke saare points milkar, yaani saare rational aur saare irrational numbers, real numbers (वास्तविक संख्याएँ) hain, jinhe $ℝ$ likhte hain. Experiment dono tarah ke numbers ko alag pehchanne ka ek test bhi sujhata hai: rational number ka address ya to ruk jaata hai ya loop mein ghoomta hai, jabki irrational number ka address dono mein se kuch nahi karta. Agle teen sections batate hain kyun.

3.6.1 Rational Decimals: Terminating aur Repeating

Zoom tool mein har zoom long division ka ek step hai, aur remainder tay karta hai ki agli window mein dot kahan baithega. Agar remainder 0 aa jaaye, to dot tick par aa jaata hai aur address khatam ho jaata hai: fraction ka terminating decimal (सांत दशमलव) hai. Agar 0 kabhi na aaye, to address hamesha chalta rehta hai, digits ke ek repeat hote block ke saath: yeh repeating decimal (असांत आवर्ती दशमलव) hai. Block ke upar ek bar repeat ko mark karti hai: $0.2 \overline{3} = 0.2333 \dots$ aur $0. \overline{27} = 0.2727 \dots$

Example 2NCERT Example 2 · p. 57

Ruk jaane wala decimal

$\frac{7}{16}$ ko decimal mein likho.

Pehle khud try karo. Phir solution ek-ek step karke kholo.

Example 3NCERT Example 3 · p. 57

Repeat hone wala decimal

$\frac{5}{27}$ ko decimal mein likho.

Pehle khud try karo. Phir solution ek-ek step karke kholo.

Fig. 3.14 $\frac{2}{7}$ ki madad se long division ko ek wheel ke chaaron taraf ki sair bana deta hai. Har possible remainder wheel par ek stop hai. Agla stop sirf us stop par depend karta hai jahan tum ho: 10 se multiply karo, 7 se divide karo, aur jo bache woh rakho. 7 se divide karne par zyada se zyada chhah non-zero remainders bachte hain. Isliye jo sair beech wale 0 tak kabhi nahi pahunchti, use saat steps ke andar kisi aise stop par aana hi padega jahan woh pehle ja chuki hai. Uske baad woh sirf apna hi raasta dohra sakti hai.

4/5Remainder 2 wapas aa gaya hai, isliye wahi chhah digits repeat hote hain: $\frac{2}{7} = 0. \overline{285714}$ .

Fig. 3.147 se long division remainders ke ek loop mein ghoomta hai. Tum kahan se shuru karte ho, isse digits tay hote hain.NCERT p. 58

Ek ring mein lage chhah circles mein non-zero remainders 2, 6, 4, 5, 1 aur 3 hain; beech mein ek dashed circle mein 0 hai, jis par stop likha hai. Pehle step mein ek panel kehta hai ki 7 se divide karne par sirf 0 se 6 tak ke remainders bach sakte hain. Doosre step mein ek ring remainder 2 ko 2 divided by 7 ki shuruaat mark karti hai, aur teen blue arrows 2 se 6, 6 se 4, 4 se 5 jaate hain, jin par digits 2, 8 aur 5 likhe hain, jisse 0.285 milta hai. Teesre step mein baaki teen arrows ring poori karte hain, 5 se 1, 1 se 3 aur wapas 2 tak, digits 7, 1 aur 4 ke saath, jisse 0.285714 milta hai. Chauthe step mein remainder 2 shaded hai kyunki woh wapas aa gaya hai, isliye digits 285714 repeat hote hain. Aakhri step mein start ring remainder 1 par chali jaati hai: wahan se usi loop mein ghoomne par 1 divided by 7 milta hai, 0.142857 repeating.

Yahi argument kisi bhi denominator $q$ ke liye kaam karta hai. Zyada se zyada $q - 1$ non-zero remainders hote hain, isliye $\frac{p}{q}$ ka repeating block zyada se zyada $q - 1$ digits lamba hota hai. Zoom tool bhi yahi dikhata hai: remainder tay karta hai ki har window mein dot kahan baithega, isliye remainder repeat hone ka matlab hai picture repeat hona.

Decimal ke type ka andaaza lagana

Koi bhi division karne se pehle jawab ka andaaza lagane ka ek tareeka yeh hai. Terminating decimal asal mein ten ki kisi power par ek fraction hai, jaise $0.175 = \frac{175}{1000}$ . Ten ki power ke prime factors sirf 2 aur 5 hote hain, kyunki $10 = 2 \times 5$ . To sawaal yeh hai ki kya denominator ko ten ki power mein badla ja sakta hai. $\frac{7}{40}$ ke liye haan: $40 = 2^{3} \times 5$ , aur upar aur neeche ko $5^{2}$ se multiply karne par $2^{3} \times 5^{3} = 1000$ milta hai.

\frac{7}{40} = \frac{7}{2^{3} \times 5} = \frac{7 \times 5^{2}}{2^{3} \times 5^{3}} = \frac{175}{1000} = 0.175

TheoremDecimal kab terminate hota hai?

Maan lo $\frac{p}{q}$ lowest terms mein ek rational number hai jahan $q > 0$ . Iska decimal expansion theek tab terminate hota hai jab $q$ ke prime factors sirf 2 aur 5 hon, yaani jab kinhi whole numbers $m$ aur $n$ ke liye $q = 2^{m} \times 5^{n}$ ho.

Agar $q = 2^{m} 5^{n}$ , to decimal terminate hota hai. Maan lo $m \geq n$ (doosra case 2 aur 5 ki jagah badalkar same hai). Upar aur neeche ko $5^{m - n}$ se multiply karne par denominator $2^{m} 5^{m} = 10^{m}$ ban jaata hai, aur $10^{m}$ par fraction aisa decimal hai jismein point ke baad zyada se zyada $m$ digits hain.

Agar decimal terminate hota hai, to $q$ ka koi aur prime factor nahi hai. Point ke baad $k$ digits wala decimal kisi integer $N$ ke liye $\frac{N}{10^{k}}$ ke barabar hai. $\frac{N}{10^{k}}$ ko lowest terms mein laane mein sirf factors cancel hote hain, isliye reduced denominator $10^{k} = 2^{k} 5^{k}$ ko divide karta hai aur usmein 2 aur 5 ke alawa koi prime nahi ho sakta. Kisi number ka lowest-terms form unique hota hai, isliye woh reduced denominator $q$ hi hai.

Example

Divide karne se pehle andaaza

Long division ke bina tay karo ki inmein se kinke decimals terminating hain: $\frac{9}{80}$ , $\frac{11}{24}$ , $\frac{21}{35}$ aur $\frac{9}{75}$ .

Pehle khud try karo. Phir solution ek-ek step karke kholo.

Decimals ko $\frac{p}{q}$ form mein badalna

Terminating decimal pehle se hi ten ki power par ek fraction hai, jise hum phir simplify karte hain.

Example 4NCERT Example 4 · p. 59

Taraazu ki ek reading

Ek dukaan ka digital taraazu $2.64$ kg pyaaz dikhata hai. Is mass ko kilogram ke fraction ki tarah lowest terms mein likho.

Pehle khud try karo. Phir solution ek-ek step karke kholo.

Repeating decimal kabhi rukta nahi, isliye hum use seedhe ten ki power par nahi rakh sakte. Trick yeh hai ki number ki do aisi copies banao jinki kabhi khatam na hone wali tails bilkul match karein, aur unhe subtract karo: tails cancel ho jaati hain. $0. \overline{7}$ jaisa decimal, jiska repeating block point ke theek baad shuru hota hai, pure repeating decimal hai.

Example 5NCERT Example 5 · p. 59

Ninths ka use

$0. \overline{7}$ ko $\frac{p}{q}$ form mein likho.

Pehle khud try karo. Phir solution ek-ek step karke kholo.

Example 6NCERT Example 6 · p. 59

Poore block ko shift karna

$2. \overline{63}$ ko $\frac{p}{q}$ form mein likho.

Pehle khud try karo. Phir solution ek-ek step karke kholo.

Jab repeating block se pehle kuch digits aate hain, jaise $0.3 \overline{1}$ mein, to decimal general repeating decimal hai. Pehle multiply karke un digits ko point ke aage le aao; tab block point ke theek baad shuru hota hai, aur hum pehle ki tarah aage badhte hain. Fig. 3.15 dono copies ko line up hote dikhata hai.

5/5 $90 x = 28$ , isliye $x = \frac{28}{90} = \frac{14}{45}$ .

Fig. 3.15Tab tak multiply karo jab tak do copies ki tail bilkul same na ho jaaye, phir subtract karo.NCERT p. 59

Digits columns mein ek ke neeche ek lage hain, aur decimal points ek column mein hain. Pehle step mein x barabar 0.31111 waghairah, jahan 1s hamesha repeat hote hain. Doosre step mein 10x barabar 3.1111 waghairah, aur repeating 1s ab point ke theek baad shuru hote hain; woh shaded hain. Teesre step mein 100x barabar 31.1111 waghairah, aur 1s ki wahi shaded tail theek neeche hai. Chauthe step mein ek line kheenchi jaati hai aur note subtract karne ko kehta hai: 100x minus 10x barabar 31 minus 3, jo 28 hai, kyunki tails cancel ho jaati hain. Aakhri step mein isse 90x barabar 28 milta hai, isliye x, 28 over 90 hai, jo 14 over 45 hai.

Example 7NCERT Example 7 · p. 59

Pehla digit alag karna

$0.3 \overline{1}$ ko $\frac{p}{q}$ form mein likho.

Pehle khud try karo. Phir solution ek-ek step karke kholo.

Example 8NCERT Example 8 · p. 60

Calculator ne kisse divide kiya?

Arjun apne calculator par 1 ko ek whole number se divide karta hai, aur screen 0.0416666667 dikhati hai. Calculator ne aakhri digit round kar diya hai; 6s asal mein hamesha chalte rehte hain. Woh whole number dhoondho.

Pehle khud try karo. Phir solution ek-ek step karke kholo.

Example 9NCERT Example 9 · p. 60

Ek digit ke baad do-digit ka block

$3.1 \overline{27}$ ko $\frac{p}{q}$ form mein likho.

Pehle khud try karo. Phir solution ek-ek step karke kholo.

Table 3.3Repeating decimal

x

ko fraction mein badalna

Decimal ka type	$x$ ko kisse multiply karo	Phir
Pure repeating: $n$ digits ka block (jaise $2. \overline{63}$ )	$10^{n}$	$x$ subtract karo aur solve karo
General repeating: $n$ digits ke block se pehle $m$ digits (jaise $0.041 \overline{6}$ )	$10^{m}$ , aur alag se $10^{m + n}$	in dono ko subtract karo aur solve karo

Quick check

4 mein se 1

Inmein se kaunse fractions ke decimals terminating hain?

Saare sahi options chuno

Answer dekho

Answer

(A) $\frac{3}{64}$
(C) $\frac{14}{35}$
(E) $\frac{13}{125}$

64 = 2^{6}

aur

125 = 5^{3}

\frac{14}{35}

reduce hokar

\frac{2}{5}

banta hai. Lekin

\frac{7}{30}

aur

\frac{5}{18}

lowest terms mein hain aur unke denominator mein factor 3 hai, isliye woh repeat hote hain.

3.6.2 Cyclic Numbers ka Jaadu

Fig. 3.14 wale wheel par wapas jao. 7 se divide karna 1 se 6 tak ke har non-zero remainder par ek bade loop mein jaata hai. Numerator sirf yeh tay karta hai ki tum loop par kahan se shuru karte ho. Isliye chhah fractions $\frac{1}{7}, \frac{2}{7}, \dots, \frac{6}{7}$ ke wahi chhah repeating digits hote hain, bas har ek alag jagah se shuru hota hai. Table inhe us order mein dikhati hai jismein loop unke starting remainders tak pahunchta hai.

Table 3.4Sevenths, remainder loop ke order mein

Fraction	Kis remainder se shuru	Decimal
$\frac{1}{7}$	1	$0. \overline{142857}$
$\frac{3}{7}$	3	$0. \overline{428571}$
$\frac{2}{7}$	2	$0. \overline{285714}$
$\frac{6}{7}$	6	$0. \overline{857142}$
$\frac{4}{7}$	4	$0. \overline{571428}$
$\frac{5}{7}$	5	$0. \overline{714285}$

Har row apne upar wali row hai, bas uska pehla digit end mein chala gaya hai. Kyunki $\frac{k}{7} = k \times \frac{1}{7}$ , $\frac{k}{7}$ ka block $k \times 142857$ hai. Isliye 142857 ko 1 se 6 tak ke kisi bhi whole number se multiply karne par sirf uske digits ghoomte hain. Is property wale number ko cyclic number (चक्रीय संख्या) kehte hain.

142857 \times 3 = 428571, 142857 \times 5 = 714285, 142857 \times 7 = 999999

Aakhri product ek bonus hai. Yeh kehta hai ki $7 \times 0. \overline{142857} = 0. \overline{999999}$ , aur bilkul $7 \times \frac{1}{7} = 1$ . Is par hum Ek number ke do decimals mein wapas aayenge.

3.6.3 Irrational Decimals: Chaos aur Infinity

Irrational number ka terminating decimal nahi ho sakta, kyunki tab woh ten ki power par ek fraction ban jaayega. Uska repeating decimal bhi nahi ho sakta, kyunki har repeating decimal fraction mein badal jaata hai, jaisa humne abhi dekha. Isliye uske digits bina kisi repeat hote block ke hamesha chalte rehte hain: woh non-terminating aur non-repeating hain.

\sqrt{3} = 1.7320508075688772935 \dots π = 3.1415926535897932384626 \dots

Koi decimal repeating block ke bina bhi ek pattern follow kar sakta hai. $0.12112111211112 \dots$ mein har 2 ke baad pichhli baar se ek 1 zyada aata hai. Tum agle digits aasaani se likh sakte ho, phir bhi koi fixed block repeat nahi hota, isliye yeh number irrational hai. Rukna-ya-repeat hona matlab rational; dono mein se kuch nahi matlab irrational.

Ek number ke do decimals

Ek puzzle suno. Tum jaante ho ki $\frac{1}{3} = 0.333 \dots$ , hamesha 3s ke saath. Dono sides ko 3 se multiply karo. Left side 1 ban jaati hai aur right side $0.999 \dots$ To $0. \overline{9}$ 1 se thoda neeche nahi hai; yeh 1 hi hai. Inke beech koi gap nahi hai jismein koi aur number fit ho sake.

Yahi trick har terminating decimal ko ek judwaan deti hai jo kabhi na khatam hone wale 9s par khatam hota hai: $3.8 = 3.7 \overline{9}$ aur $0.25 = 0.24 \overline{9}$ . Isliye ek number ke do decimal address ho sakte hain. Riwaaz ke hisaab se hum woh use karte hain jo ruk jaata hai.

3.7 Conclusion: Kabhi Khatam Na Hone Wala Safar

Is chapter mein numbers ki har family ek aise sawaal ka jawab dene ke liye aayi jiska jawab purane numbers nahi de sakte the.

Kitne? Counting numbers, $ℕ$ .
Khaali jagah kaise dikhayein, aur jab kuch na bache to kya bachta hai? Zero.
$4 - 6$ kya hai? Negative integers, jinse $ℤ$ poora hota hai.
$2 \div 3$ kya hai? Fractions, jinse $ℚ$ milta hai.
Unit square ka diagonal kitna lamba hai? Irrational numbers, jo real numbers $ℝ$ ko poora karte hain.

Fig. 3.16 in steps ko ek hi number line par dobara dikhata hai.

5/5 $ℝ$ har chhed bhar deta hai: line ka har point ek real number hai.

Fig. 3.16Number line bharti jaati hai:

ℕ \subset ℤ \subset ℚ \subset ℝ

.NCERT Fig. 3.13 · p. 63

−4 se 4 tak ki ek number line. Pehle step mein blue dots natural numbers 1 se 4 tak mark karte hain. Doosre step mein orange squares 0 aur negative integers jodte hain. Teesre step mein 6 tak ke denominators wale fractions par chhote green ticks ki bheed aa jaati hai, jisse lagta hai ki rational numbers har jagah hain. Chauthe step mein khaali rings negative root 7, root 2 aur root 6 mark karte hain, jo points bilkul bhi rational nahi hain. Aakhri step mein ek moti shaded band poori line ko dhak leti hai: real numbers, bina kisi gap ke, aur upar chain N inside Z inside Q inside R dikhayi gayi hai.

Har family agli family ke andar baithi hai: $ℕ \subset ℤ \subset ℚ \subset ℝ$ , jahan $\subset$ ka matlab hai ke andar hai. Irrational numbers $ℝ$ ke andar hain lekin $ℚ$ ke bahar. Table batati hai ki har step ne kya joda.

Table 3.5Real numbers ki family

Numbers	Symbol	Kya jodte hain	Examples
Natural numbers	$ℕ$	ginti	1, 7, 2026
Integers	$ℤ$	zero aur negatives	$- 15$ , 0, 8
Rational numbers	$ℚ$	saare quotients $\frac{p}{q}$ ; ruk jaane wale ya repeat hone wale decimals	$- \frac{3}{4}$ , 0.125, $0. \overline{6}$
Irrational numbers	$𝕀$ (kuch books mein)	aise decimals jo na kabhi rukte hain na repeat hote hain	$\sqrt{3}$ , $- \sqrt{7}$ , $π$
Real numbers	$ℝ$	number line ka har point	upar ke sab

Timeline

Numbers ki kahani ke milestones

35,000+ saal pehle
Lebombo bone
29 notches, shayad Moon ke ek cycle ke din ginte hue.
20,000+ saal pehle
Ishango bone
11, 13, 17 aur 19 ke groups: 10 aur 20 ke beech ke primes.
lagbhag 2600–1900 BCE
Indus Valley ke weights
Standard weights jo double hote hain, phir tens mein badhte hain.
Vedic kaal
Ten ki powers ke naam
Parārdha, $10^{12}$ , tak.
lagbhag 800 BCE
Baudhāyana ka Śulbasūtra
Square ko double karne se $\sqrt{2}$ tak pahunchte hain.
Shuruaati centuries CE
Bakhśhālī manuscript
Ek dot khaali jagah mark karta hai.
499 CE
Āryabhaṭa
$π \approx 3.1416$ , jise āsanna, yaani approximate, kaha gaya.
628 CE
Brahmagupta
Zero aur negative numbers ke liye rules.
14th century
Mādhava
$π$ ke liye ek infinite series.
1761
Johann Lambert
Proof ki $π$ irrational hai.

Is chapter ka pattern batata hai ki kya karna chahiye. Har baar jab kisi equation ka jawab nahi tha, ek naya number declare kiya gaya: $x + 6 = 4$ se $- 2$ aaya, $3 x = 2$ se $\frac{2}{3}$ , aur $x^{2} = 2$ se $\sqrt{2}$ . $x^{2} = - 1$ ke liye mathematicians ne ek naya number $i$ declare kiya jiska square $- 1$ hai. Real line par iski koi jagah nahi hai, isliye yeh usse right angle par door point karta hai, aur numbers phailkar ek poora plane bhar dete hain. Yeh complex numbers, jo tum Class 11 mein padhoge, ghoomne ya kaampne wali kisi bhi cheez ke liye natural bhasha nikalte hain.

Quick check

3 mein se 1

Har number ko us sabse chhoti family se match karo jismein woh aata hai.

Left mein koi item tap karo, phir right mein uska match.

Answer dekho

Answer

$- 9$ : integers $ℤ$
$12$ : natural numbers $ℕ$
$- \frac{7}{3}$ : rational numbers $ℚ$
$\sqrt{10}$ : irrational numbers

12 ek counting number hai.

- 9

integer hai lekin natural number nahi.

- \frac{7}{3}

integers ka quotient hai lekin integer nahi.

\sqrt{10}

integers ka quotient bilkul nahi hai, kyunki 10 perfect square nahi hai.

Recap

Summary aur key points

Ginti one-to-one correspondence se nikli; natural numbers $ℕ = {1, 2, 3, \dots}$ hain (Section 3.1).
Lebombo aur Ishango bones jaisi notches wali haddiyan, Indus Valley ke weights aur ten ki powers ke Vedic naam dikhate hain ki number ke ideas kitne purane hain (Section 3.1.1, Section 3.1.2).
Zero ek placeholder ki tarah shuru hua aur Brahmagupta ke rules $a + 0 = a$ , $a - 0 = a$ aur $a \times 0 = 0$ ke saath ek number ban gaya; zero se division defined nahi hai (Section 3.2.2).
Integers $ℤ$ zero aur negative numbers jodte hain; Brahmagupta ke daulat aur karz se sign rules milte hain, jinmein $(- a) \times (- b) = a b$ bhi hai (Section 3.3.1).
Rational number ko $\frac{p}{q}$ likha ja sakta hai, jahan $p$ , $q$ integers hain aur $q \neq 0$ ; $ℚ$ $+$ , $-$ , $\times$ aur non-zero numbers se division ke under closed hai (Section 3.4).
Number line par $\frac{p}{q}$ rakhne ke liye har unit ko $q$ parts mein kaato aur 0 se unmein se $p$ gino; $∣ x ∣$ 0 se distance hai aur $∣ a - b ∣$ , $a$ aur $b$ ke beech ki distance (Section 3.4.1).
Rational numbers dense hain: average $\frac{a + b}{2}$ hamesha $a$ aur $b$ ke beech hota hai (Section 3.4.2).
$\sqrt{2}$ aur $π$ jaise numbers irrational hain; $\sqrt{2}$ ka irrational hona contradiction se prove hota hai (Section 3.5.1).
Right triangles aur compass number line par $\sqrt{2}, \sqrt{3}, \dots, \sqrt{n}$ construct karte hain (Section 3.5.2); Mādhava ki kabhi khatam na hone wali series $π$ ko exact deti hai (Section 3.5.3).
Rational number ka decimal ya to terminate hota hai ya repeat; lowest terms mein yeh theek tab terminate hota hai jab denominator $2^{m} 5^{n}$ ho (Section 3.6.1).
Repeating decimals ko 10 ki powers se multiply karke aur subtract karke fractions mein badla jaata hai; saath hi $0. \overline{9} = 1$ (Decimals ko $\tfrac{p}{q}$ form mein badalna, Ek number ke do decimals).
$\frac{1}{7}$ ka block, 142857, ek cyclic number hai kyunki 7 se division har non-zero remainder se hokar loop karta hai (Section 3.6.2).
Irrational decimals na kabhi khatam hote hain na kabhi repeat; rational aur irrational numbers milkar real numbers $ℝ$ banate hain (Section 3.6.3, Section 3.7).

Flash cards · 12

Key terms

1 / 12

Chapter ke end ki exercises

Neeche ke sawaal chapter ke har idea ki practice karwaate hain. Question 16 yahan bana square-root spiral use karta hai.

Fig. 3.17Ek square-root spiral: har bahar wali side 1 unit lambi hai.NCERT Fig. 3.14 · p. 66

Aath right triangles ek common corner O ke chaaron taraf clockwise ghoomte hue jude hain aur ek spiral banate hain. Pehle triangle ki do sides ki length 1 hai. Har naya triangle pichhle triangle ki sabse lambi side ko apni ek chhoti side ki tarah use karta hai, aur uski doosri chhoti side, spiral ke bahar ki taraf, 1 unit lambi hai. Jahan har bahar wali side pichhli lambi side se milti hai wahan right-angle marks hain, aur triangles par 1 se 8 tak number likhe hain.

Apna NCERT exercise question dhoondo

Hamare practice questions ncert.ai ke liye likhe gaye hain aur NCERT exercises ko follow karte hain. Apni textbook ke question se match karne wala question dhoondo; uska answer uske saath hai.

Tumhari NCERT textbook mein	Yahan practice question
NCERT Exercise Set 3.1, Q1	Practice 3.1 · Q1
NCERT Exercise Set 3.1, Q2	Practice 3.1 · Q2
NCERT Exercise Set 3.1, Q3	Practice 3.1 · Q3
NCERT Exercise Set 3.1, Q4	Practice 3.1 · Q4
NCERT Exercise Set 3.2, Q1	Practice 3.2 · Q1
NCERT Exercise Set 3.2, Q2	Practice 3.2 · Q2
NCERT Exercise Set 3.2, Q3	Practice 3.2 · Q3
NCERT Exercise Set 3.2, Q4	Practice 3.2 · Q4
NCERT Exercise Set 3.3, Q1	Practice 3.3 · Q1
NCERT Exercise Set 3.3, Q2	Practice 3.3 · Q2
NCERT Exercise Set 3.3, Q3	Practice 3.3 · Q3
NCERT Exercise Set 3.3, Q4	Practice 3.3 · Q4
NCERT Exercise Set 3.3, Q5	Practice 3.3 · Q5
NCERT Exercise Set 3.3, Q6	Practice 3.3 · Q6
NCERT Exercise Set 3.3, Q7	Practice 3.3 · Q7
NCERT Exercise Set 3.3, Q8	Practice 3.3 · Q8
NCERT Exercise Set 3.4, Q1	Practice 3.4 · Q1
NCERT Exercise Set 3.4, Q2	Practice 3.4 · Q2
NCERT Exercise Set 3.4, Q3	Practice 3.4 · Q3
NCERT Exercise Set 3.4, Q4	Practice 3.4 · Q4
NCERT Exercise Set 3.4, Q5	Practice 3.4 · Q5
NCERT Exercise Set 3.4, Q6	Practice 3.4 · Q6
NCERT Exercise Set 3.5, Q1	Practice 3.5 · Q1
NCERT Exercise Set 3.5, Q2	Practice 3.5 · Q2
NCERT Exercise Set 3.5, Q3	Practice 3.5 · Q3
NCERT Exercise Set 3.5, Q4	Practice 3.5 · Q4
NCERT Exercise Set 3.5, Q5	Practice 3.5 · Q5
NCERT End-of-chapter Exercises, Q1	Chapter ke end ki exercises · Q1
NCERT End-of-chapter Exercises, Q2	Chapter ke end ki exercises · Q2
NCERT End-of-chapter Exercises, Q3	Chapter ke end ki exercises · Q3
NCERT End-of-chapter Exercises, Q4	Chapter ke end ki exercises · Q4
NCERT End-of-chapter Exercises, Q5	Chapter ke end ki exercises · Q5
NCERT End-of-chapter Exercises, Q6	Chapter ke end ki exercises · Q6
NCERT End-of-chapter Exercises, Q7	Chapter ke end ki exercises · Q7
NCERT End-of-chapter Exercises, Q8	Chapter ke end ki exercises · Q8
NCERT End-of-chapter Exercises, Q9	Chapter ke end ki exercises · Q9
NCERT End-of-chapter Exercises, Q10	Chapter ke end ki exercises · Q10
NCERT End-of-chapter Exercises, Q11	Chapter ke end ki exercises · Q11
NCERT End-of-chapter Exercises, Q12	Chapter ke end ki exercises · Q12
NCERT End-of-chapter Exercises, Q13	Chapter ke end ki exercises · Q13
NCERT End-of-chapter Exercises, Q14	Chapter ke end ki exercises · Q14
NCERT End-of-chapter Exercises, Q15	Chapter ke end ki exercises · Q15
NCERT End-of-chapter Exercises, Q16	Chapter ke end ki exercises · Q16

Quick check: MCQs

3.1.2 Indian Context: Trade aur Astronomy3 questions
3.2.2 Bakhśhālī Manuscript aur Brahmagupta ke Rules3 questions
3.3.1 Integers ka Arithmetic4 questions
3.4 Khaali Jagah Bharna: Fractions aur Rational Numbers4 questions
Rational number ki absolute value4 questions
3.4.2 Rational Numbers ki Density3 questions
3.5.1 $\sqrt{2}$ ki Irrationality ka Proof4 questions
3.5.3 Pi (π) ki Kahani aur Madhava ki Infinite Series4 questions
Decimals ko $\tfrac{p}{q}$ form mein badalna4 questions
Ek number ke do decimals4 questions
3.7 Conclusion: Kabhi Khatam Na Hone Wala Safar3 questions

Glossary · 14

Key terms (English + Hindi)

absolute valueनिरपेक्ष मानnirpeksh maan: Number line par kisi number ki 0 se distance, jise $∣ x ∣$ likhte hain; yeh kabhi negative nahi hoti.Chapter mein pehli baar yahan aaya →
additive inverseयोज्य प्रतिलोमyojya pratilom: Woh number $- a$ jo $a$ mein jodne par 0 deta hai.Chapter mein pehli baar yahan aaya →

closedसंवृतsamvrit: Numbers ka collection kisi operation ke under closed hota hai jab uske kinhi bhi do members par woh operation karne se hamesha collection ka hi member milta hai.Chapter mein pehli baar yahan aaya →
cyclic numberचक्रीय संख्याchakriya sankhya: 142857 jaisa number jise 1, 2, 3, … se ek limit tak multiply karne par sirf uske digits ghoomte hain.Chapter mein pehli baar yahan aaya →

integerपूर्णांकpurnank: Natural number, zero, ya kisi natural number ka negative. Is collection ko $ℤ$ likhte hain.Chapter mein pehli baar yahan aaya →
irrational numberअपरिमेय संख्याaparimeya sankhya: Aisa real number jo integers ka ratio $\frac{p}{q}$ , jahan $q \neq 0$ , nahi hai, jaise $\sqrt{2}$ ya $π$ ; iska decimal na rukta hai na repeat hota hai.Chapter mein pehli baar yahan aaya →

lowest termsसरलतम रूपsaraltam roop: Fraction $\frac{p}{q}$ lowest terms mein hai jab $p$ aur $q$ dono ko divide karne wala akela positive whole number 1 ho, aur $q > 0$ .Chapter mein pehli baar yahan aaya →

natural numberप्राकृत संख्याprakrit sankhya: Ginne wala number: $1, 2, 3, \dots$ Is collection ko $ℕ$ likhte hain.Chapter mein pehli baar yahan aaya →

one-to-one correspondenceएकैकी संगतताekaiki sangatata: Ek collection ki har cheez ko doosre collection ki theek ek cheez ke saath jodna, taaki dono collections ko bina gine compare kiya ja sake.Chapter mein pehli baar yahan aaya →

proof by contradictionविरोधाभास द्वारा उपपत्तिvirodhabhas dwara upapatti: Aisa proof jo statement ka ulta maankar sahi reasoning se kisi impossible baat tak pahunchta hai, isliye assumption galat hi hona chahiye.Chapter mein pehli baar yahan aaya →

rational numberपरिमेय संख्याparimeya sankhya: Aisa number jise $\frac{p}{q}$ likha ja sake, jahan $p$ , $q$ integers hain aur $q \neq 0$ . Is collection ko $ℚ$ likhte hain.Chapter mein pehli baar yahan aaya →
real numberवास्तविक संख्याvastavik sankhya: Koi bhi rational ya irrational number; number line ka har point. Is collection ko $ℝ$ likhte hain.Chapter mein pehli baar yahan aaya →
repeating decimalअसांत आवर्ती दशमलवasaant aavarti dashamlav: Aisa decimal jo kabhi rukta nahi lekin jismein digits ka ek block hamesha repeat hota hai, jaise $0. \overline{185}$ .Chapter mein pehli baar yahan aaya →

terminating decimalसांत दशमलवsaant dashamlav: Aisa decimal jo finite digits ke baad ruk jaata hai, jaise $0.4375$ .Chapter mein pehli baar yahan aaya →

The World of Numbers (Numbers ki Duniya): Class 9 Maths Chapter 3 Hinglish mein

Is chapter mein aap kya seekhenge

3.1 Mathematics ki Shuruaat: Insaan ko Ginne ki Zaroorat

3.1.1 Haddi par Likha Itihaas

3.1.2 Indian Context: Trade aur Astronomy

Practice 3.1: answers ke saath

3.2 Śhūnya ki Kranti: Jab Kuch Nahi Bhi Kuch Ban Gaya

3.2.1 Philosophy se Mathematics tak: Śhūnyatā ka Concept

3.2.2 Bakhśhālī Manuscript aur Brahmagupta ke Rules

3.3 Integers: Horizon ko Badhana

3.3.1 Integers ka Arithmetic

Practice 3.2: answers ke saath

3.4 Khaali Jagah Bharna: Fractions aur Rational Numbers

Practice 3.3: answers ke saath

3.4.1 Number Line par Rational Numbers ko Dikhana

Rational number ki absolute value

3.4.2 Rational Numbers ki Density

Practice 3.4: answers ke saath

3.5 Irrational Numbers

3.5.1 $\sqrt{2}$ ki Irrationality ka Proof

3.5.2 Length $\sqrt{n}$ ka Construction

3.5.3 Pi (π) ki Kahani aur Madhava ki Infinite Series

3.6 Real Numbers: Decimals aur Cyclic Patterns

3.6.1 Rational Decimals: Terminating aur Repeating

Decimal ke type ka andaaza lagana

Decimals ko $\frac{p}{q}$ form mein badalna

3.6.2 Cyclic Numbers ka Jaadu

3.6.3 Irrational Decimals: Chaos aur Infinity

Ek number ke do decimals

Practice 3.5: answers ke saath

3.7 Conclusion: Kabhi Khatam Na Hone Wala Safar

Summary aur key points

Chapter ke end ki exercises

Practice questions answers ke saath (NCERT “Chapter ke end ki exercises”)

Apna NCERT exercise question dhoondo

Quick check: MCQs

Key terms (English + Hindi)

Tum chapter ke end tak pahunch gaye

3.1 Mathematics ki Shuruaat: Insaan ko Ginne ki Zaroorat

3.1.1 Haddi par Likha Itihaas

3.1.2 Indian Context: Trade aur Astronomy

3.2 Śhūnya ki Kranti: Jab Kuch Nahi Bhi Kuch Ban Gaya

3.2.1 Philosophy se Mathematics tak: Śhūnyatā ka Concept

3.2.2 Bakhśhālī Manuscript aur Brahmagupta ke Rules

3.3 Integers: Horizon ko Badhana

3.3.1 Integers ka Arithmetic

3.4 Khaali Jagah Bharna: Fractions aur Rational Numbers

3.4.1 Number Line par Rational Numbers ko Dikhana

Rational number ki absolute value

3.4.2 Rational Numbers ki Density

3.5 Irrational Numbers

3.5.1 2 ki Irrationality ka Proof

3.5.2 Length n ka Construction

3.5.3 Pi (π) ki Kahani aur Madhava ki Infinite Series

3.6 Real Numbers: Decimals aur Cyclic Patterns

3.6.1 Rational Decimals: Terminating aur Repeating

Decimal ke type ka andaaza lagana

Decimals ko pq form mein badalna

3.6.2 Cyclic Numbers ka Jaadu

3.6.3 Irrational Decimals: Chaos aur Infinity

Ek number ke do decimals

3.7 Conclusion: Kabhi Khatam Na Hone Wala Safar

Chapter ke end ki exercises

Apna NCERT exercise question dhoondo

Quick check: MCQs

Tum chapter ke end tak pahunch gaye

3.5.1 $\sqrt{2}$ ki Irrationality ka Proof

3.5.2 Length $\sqrt{n}$ ka Construction

Decimals ko $\frac{p}{q}$ form mein badalna