संख्या जगत में खेळकूड

la[;k&txr esa [ksy&dwn

ih-ds- Jhfuoklu

Hkkjr Kku foKku lfefr

uo tuokpu vkanksyu bl fdrkc dk izdk'ku Hkkjr Kku foKku lfefr us ^lj nksjkcth VkVk VªLV* ds lg;ksx ls fd;k gSA bl vkanksyu dk edln vke turk esa iBu&ikBu laLÑfr fodflr djuk gSA

la[;k&txr esa [ksy&dwn ih- ds- Jhfuoklu ¯gnh vuqokn ;ksxs'k vxzoky dkWih laiknd jk/s';ke eaxksyiqjh xzkfiQDl vHk; dqekj >k doj xkWMÚs nkl izFke laLdj.k tuojh 2008 lg;ksx jkf'k 30 #i;s eqæ.k lu 'kkbu vkWiQlsV ubZ fnYyh & 110 018

Sankhya-Jagat Men Khel-Kood P.K. Srinivasan Hindi Translation Yogesh Agarwal Copy Editor Radheshyam Mangolpuri Graphics Abhay Kumar Jha Cover Godfrey Das First Edition January 2008 Contributory Price Rs. 30.00 Printing Sun Shine Offset New Delhi - 110 018

Publication and Distribution Bharat Gyan Vigyan Samiti Basement of Y.W.A. Hostel No. II, G-Block, Saket , New Delhi - 110017 Phone : 011 - 26569943, Fax : 91 - 011 - 26569773 Email : [email protected], [email protected] BGVS JAN 2008 2K 3000 NJVA 0128/2008

leiZ.k ;g iqLrd mu dYiuk'khy ekrk&firkvksa vkSj deZfu"B vè;kidksa dks lefiZr gS tks ekuotkfr ds fgr ds gsrq] xf.kr esa es/k'khy Nk=kksa dks [kkstus vkSj laokjus esa layXu gSaA

Øef.kdk vkeq[k 1- vkvks esjs lkFk 2- la[;kvksa ds izdkj

7 9 17

le vkSj fo"ke

17

xq.kt vkSj xq.kd

18

vHkkT; vkSj feJ

21

oxZ vkSj v&oxZ

23

?ku vkSj v&?ku

25

f=kdks.kh; vkSj v&f=kdks.kh;

28

fiNyh vkSj vxyh la[;k,a

32

3- u;k jksekap

35

4- [kkst vfHk;ku

39

igyk nkSj

39

nwljk nkSj

42

rhljk nkSj

45

pkSFkk nkSj

51

ikapoka nkSj

55

NBk nkSj

59

5- vyfonk

62

6- la[;k&fof'k"Vrk,a % la{ksi esa

66

8

vkeq[k cPps LoHkko ls gh ftKklq gksrs gSaA os gj ubZ oLrq dks viuh dkSrwgy&Hkjh utjksa ls ns[krs gSaA os viuh ifjiDork vkSj {kerk ds vuq:i ges'kk oLrqvksa ds vkilh laca/ksa dks [kkstus ds fy, mRlqd jgrs gSa µ u dsoy HkkSfrd oLrqvksa ds chp] vfirq vewrZ fopkjksa ds chp HkhA bu laca/ksa esa fNis uD'kksa dks [kkstuk ekuo&eu dk ewyHkwr LoHkko gSA volj rFkk izksRlkgu feyus ij gj ckyd bl dk;Z esa vkuUn dk vuqHko djrk gS vkSj mlesa fNik [kkstdrkZ fodflr gksus yxrk gSA izkÑfrd la[;k ra=k esa bl izdkj ds vufxur vn~Hkqr iSVuZ fNis gSa] lh/s&lkns ls ysdj vfr&tfVy rd] vkSj vfr&ljy ls ysdj vfr&xgjs HkhA izkÑfrd la[;kvksa ls cPpksa dk ifjp; izkFkfed Ldwy esa gh gks tkrk gSA os bu la[;kvksa ds lkFk tksM+uk&?kVkuk] xq.kk&Hkkx djuk bR;kfn lh[krs gSaA bl iqLrd dk iz;kl ;gh gS fd cPps fdl izdkj la[;k&ra=k ds bl vn~Hkqr jksekapdkjh txr dh M~;kS<+h dks yka?kdj mlesa izos'k dj ldsa] og Hkh ek=k tksM+us&?kVkus o xq.kk&Hkkx djus dh viuh ;ksX;rk dk iz;ksx djds_ vkSj viuh varcksZ/&'kfDr ds }kjk bl ra=k dh ljy o lqUnj fof'k"Vrkvksa dh Lor% gh [kkst dj ldsaA bl [kkst esa cPpksa dks ftl vkuUn vkSj vkRefo'okl dh vuqHkwfr gksxh] mlds }kjk gh os xf.kr ds vè;;u ds izfr mRlkfgr gks ldsaxsA vk'kk gS fd lHkh dYiuk'khy ekrk&firk o #fp'khy vè;kid bl u, iz;kl dk Lokxr djsaxs vkSj bl fof/ dks iz;ksx esa ykdj blds ifj.kkeksa dh tkudkjh ys[kd dks vo'; nsaxsA bl iqLrd ds Hkkoh laLdj.kksa esa lq/kj ds fy, lHkh lq>koksa dk Lokxr gSA & ih-ds- Jhfuoklu 9

10

1 vkvks esjs lkFk

eSa

d{kk lkr esa i<+rk gwaA eq>s dgkfu;ka i<+uk vkSj [ksyuk cgqr vPNk yxrk gSA esjs pkpkth xf.kr ds izksiQslj gaSA os vDlj eq>ls dgrs gSa] ¶rqe Hkh ubZ [kkstksa ds }kjk xf.kr dk vlyh vkuUn ys ldrs gks] ;fn rqe fofHkUu izkÑfrd la[;kvksa dks Hkyh&Hkkafr tku yks] csf>>d mUgsa igpkuks] vkSj muds lkFk [ksyksA blds fy, rqEgsa dsoy lh/s&lkns tksM+&?kVko o xq.kk&Hkkx dk vPNk vH;kl gksuk pkfg,] ftldk Kku rqe NksVh d{kkvkas esa igys gh izkIr dj pqds gksA¸ dqN le; rd eSaus muds lq>ko dks cgqr xEHkhjrk ls ugha fy;kA ,d fnu pkpkth dks ,d fo|ky; ds xf.kr Dyc ds lnL;ksa dks lEcksf/r djus dk fuea=k.k feykA xf.kr Dyc dk ukedj.k egku~ xf.krK vkW;yj ds uke ij fd;k x;k FkkA eSa Hkh pkpkth ds lkFk x;kA eSa mUgsa NksVh mez esa cPpksa dks lEcksf/r djrs gq, lquuk pkgrk Fkk vkSj ns[kuk pkgrk Fkk fd cPps mudh ckrksa dks dSls xzg.k djrs gSaA pkpkth us cPpksa dks crk;k fd dSls os [kqn Hkh xf.kr'kkL=kh cudj xf.kr ds jksekap dk vuqHko dj ldrs gSaA mUgksaus cPpksa ls dgk % ¶vktdy cgqr&lh cM+h [kkstsa dh tk jgh gSaA lkjs fo'o eas vkSj thou ds lHkh {ks=kksa esa [kkstdrkZvksa dh cgqr vko';drk gSA vPNs [kkstdrkZ cuus ds fy, gesa cpiu ls gh [kkstdk;Z esa #fp ysuh gksxhA [kkstdk;Z ds fy, ,d mÙke {ks=k gS la[;k,a] D;ksafd la[;kvksa ij [kkst djus ds fy, /u [kpZ djus dh dksbZ t:jr ugha gS] vkSj lkFk gh ;g cgqr vkuUne;h Hkh gSA 'kk;n rqEgsa ftKklk gksxh fd ,slk dSls lEHko gSA ,d ckj ;fn dksbZ tksM+&?kVko o xq.kk&Hkkx esa egkjr gkfly dj ys] rks og Lo;a gh la[;ktxr ds vn~Hkqr lkSUn;Z dk vUos"k.k dj ldrk gSA 11

pwafd rqe lHkh xq.kk&Hkkx vkfn Hkyh&Hkkafr tkurs gks] rqe Hkh bl vUos"k.k ds fy, rS;kj gksA rqe ftruh vf/d [kkstus dh ps"Vk djksxs] mrus gh liQy [kkstdrkZ cu ldrs gksA la[;kvksa esa ekStwn fdlh uD'ks] vkÑfr ;k Lo:i] ;k ftls vaxzsth esa iSVuZ dgrs gSa] dks igpku ikuk gh la[;ktxr ds vUos"k.k dh dqath gSA ;g rqe lHkh ds fy, laHko gSA la[;ktxr dks rqe la[;kvksa ds ,d vn~Hkqr lkxj ds leku le>ks] ftlesa fNis vufxur cgqewY; jRu rqe tSls [kksft;ksa dh izrh{kk dj jgs gSaA fdlh iSVuZ dh [kkst la[;kvksa ds ijLij laca/ksa dks n'kkZrh gSA bu laca/ksa ds vè;;u ls rqe la[;ktxr ds lqUnj fl¼karksa o eksgd fof'k"Vrkvksa rd igqap ldrs gksA la[;kvksa ds vè;;u ds fy, eSa dqN pkVZ ysdj vk;k gwaA vc bl pkVZ dks ns[kksA

rqe lksprs gksxs fd bl pkVZ dh lHkh iafDr;ka ,d tSlh D;ksa gSa\ lHkh esa ,d ls nl rd dh la[;k,aa Øe ls fy[kh gSaA ijUrq è;ku nks fd dqN la[;k,a yky jax ls fy[kh xbZ gSa] tks rhu&rhu la[;kvksa ds Øeokj lewgksa dks n'kkZrh gSaA ,sls gj lewg dks ge frdVh dk uke ns ldrs gSaA ,d ls nl rd dh la[;kvksa esa ekStwn ,slh lkjh frdfV;ka bl pkVZ esa fn[kkbZ xbZ gSa] tSls% (1] 2] 3)] (2] 3] 4)] --- (8] 9]10)A vc ,d&,d frdVh dks ns[kksA gj frdVh dh dqN ;k lHkh la[;kvksa dk ;ksx ;k xq.kuiQy fudkyks] vkSj mudk vè;;u dj mlesa fNis uD'kksa dks [kkstus dk iz;kl djksA dHkh ,slk iSVuZ Lor% gh fn[kkbZ nsxk] vkSj dHkh mls [kkstus ds fy, FkksMk+ xgjkbZ esa tkuk gksxkA tks ckr lHkh lewgksa ij ykxw ugha gksrh] mls ek=k ,d fofp=krk le>uk pkfg,A 'kq: esa gesa ,slh fofp=krkvksa ij le; ugha xaokuk 12

pkfg,A gesa ,sls fl¼karksa dks [kkstuk gS tks lHkh lewgksa ij ykxw gksrs gksAa mnkgj.k ds fy,] igyh frdVh (1] 2] 3) ls 'kq: djds lc lewgksa dks ns[kksA bl frdVh ds fy, 1$3 ¾ 2x2 fu'p; gh lR; gS] ;kfu frdVh dh igyh o rhljh la[;kvksa dk ;ksx fcpyh la[;k ds oxZ ds cjkcj gSA ijUrq nwljh frdVh (2] 3] 4) esa 2$4 vkSj 3x3 cjkcj ugha gSaA blfy, ;g fu;e gj frdVh ij ykxw ugha gksrk] vkSj bls dsoy igyh frdVh dh fofp=krk gh ekuuk mfpr gksxk] ,d fl¼kar ughaA ijUrq vc bls ns[kks µ (1, 2, 3)1+3 = 2 x 2 (2,3, 4) 2+4 = 2 x 3 (3, 4, 5)3+5 = 2 x 4

vkfn

rqe ns[kksxs fd gj frdVh esa igyh o vkf[kjh la[;k dk ;ksx fcpyh la[;k dk nksxquk gSA ;g okLro esa ,d fl¼kar dk mnkgj.k gSA bls bl izdkj le>ksA eku yks fd rqEgkjsa iz/kukpk;Z rqEgsa ;g izek.ki=k nsa fd rqe dHkh&dHkh cM+h cqf¼ekuh fn[krs gksA D;k rqe ,slk izek.ki=k ikdj izlUu gkssxs\ D;k rqe ugha pkgksxs fd izek.ki=k dsoy bruk dgs fd rqe cgqr cqf¼eku gks] ;kfu ges'kk cqf¼eku] u fd fliQZ dHkh&dHkhA blfy, rqEgsa la[;kvksa esa fNis fl¼karksa dks [kkstuk pkfg,A ;fn dqN fofp=krk,a feysa rks dsoy mUgsa uksV dj vyx j[kks vkSj Hkwy tkvksA mUgsa ,slh gh dkSrwgy dh oLrq le>ksµ tSls] fdlh ds gkFk esa ikap ds LFkku ij Ng maxfy;kaA ;fn dksbZ iSVuZ ckj&ckj lkeus vk,] rks le>ks fd mlesa ,d fl¼kar dh xU/ vk jgh gSA ;kn j[kks fd ;g ek=k ,d vuqeku gSA 13

t:jh ugha fd gj iSVuZ ds ihNs dksbZ fl¼kar gksA fdlh iSVuZ dks fl¼kar rHkh ekuk tk,xk tc mls fl¼ fd;k tk ldsA dksbZ iSVuZ fdruh vf/d ckj lkeus vk;k] ;g mls fl¼ djus ds fy, dkiQh ughaA fdlh fl¼kar dks fl¼ djuk okdbZ dkiQh jksekapd dke gSA ysfdu bls ge ckn esa ns[ksaxsA igys ge bu fl¼karksa ds ckjs esa vuqeku yxk,axsA gesa ;g è;ku j[kuk gS fd gekjk vuqeku ,slk gks tks cgqr&ls la[;k lewgksa ij lgh cSBrk gks] vkSj mlds okLro esa ,d fl¼kar gksus dh laHkkouk izrhr gksrh gksA pyks] vc ge ns[krs gSa fd rqeesa ls fdrus twfu;j xf.krK cu ldus ds fy, rS;kj gSaA vc ,d ckj fiQj pkVZ dks ns[kksA rqe ns[k pqds gks fd gj frdVh esa igyh vkSj rhljh la[;k dk ;ksx chp dh la[;k dk nksxquk gSA ,sls gh vkSj iSVuZ Hkh [kkstksA¸ lHkh cPps fy[kus yxs ;k ,d&nwljs ls iQqliQqlk&dj dqN iwNus yxsA fiQj dqN us vius gkFk mBk,A eq>s Hkh gj frdVh esa la[;kvksa dk ;ksx fudkyus ij ,d iSVuZ fn[kkbZ fn;kA 1$2$3 ¾ 6 2$3$4 ¾ 9 3$4$5 ¾ 12 4$5$6 ¾ 15 bR;kfnA eSaus bl iSVuZ dks dkxt ij uksV fd;k % gj frdVh esa la[;kvkas dk ;ksxiQy rhu ls foHkkftr gksrk gSA ;g uksV eSaus pkpkth dks ns fn;kA mUgksaus mls i<+k vkSj esjh vksj ,d LohÑfrlwpd eqLdku ds lkFk ns[kk] ftlls eq>s xkSjo dk vuqHko gqvkA ftu cPpksa us vius gkFk mBk, Fks] mueas ls dqN dks pkpkth us viuh&viuh [kkst izLrqr djus ds fy, cqyk;kA eq>s cM+h [kq'kh gqbZ tc eSaus ns[kk fd mUgksaus Hkh ogh iSVuZ [kkstk Fkk] tks eSausA dqN vkSjksa us rhuksa la[;kvksa ds xq.kuiQy esa ,d iSVuZ ns[kk FkkA mUgksaus cksMZ ij fy[kkµ 14

1x2x3=6 2 x 3 x 4 = 24 3 x 4 x 5 = 60

bR;kfnA [kksft;ksa ls dgk x;k fd os viuh [kkst dks 'kCnksa esa crk,aA mUgksaus dqN bl izdkj dgk % rhu yxkrkj la[;kvksa dk xq.kuiQy Ng ls foHkkftr gksrk gSA Ng dh la[;k fdlh Hkh rhu yxkrkj izkÑfrd la[;kvksa ds xq.kuiQy dk ,d xq.kd gSA fdUgha Hkh rhu yxkrkj la[;kvksa dks xq.kk djks vkSj fiQj Ng ls Hkkx nks] rks og iw.kZr% foHkkftr gksrh gSA bu lHkh dk vFkZ yxHkx ,d gh FkkA bl dke esa cPpksa ds mRlkgiwoZd Hkkx ysus ls pkpkth cM+s izlUu gq,A og cPpksa dks mudh [kkstksa ds fy, c/kbZ ns gh jgs Fks] fd ckb± vksj cSBs cPpksa esa ls rhu us ,d lkFk vius gkFk [kM+s fd,A pkpkth us ml vksj ns[kk rks rhuksa us [kM+s gksdj dgk fd mUgksaus ,d iSVuZ [kkstk gS tks dgha vf/d jkspd yxrk gSA mUgsa cqyk;k x;k fd os vkdj iwjh d{kk dks viuh [kkst crk,a vkSj vius [kksts gq, iSVuZ dks cksMZ ij fy[kdj lHkh dks fn[kk,aA mUgksaus cksMZ ij bl izdkj fy[kk % 4 x 5 x 6 = 120

1 x 3 = 2 x 2-1 2 x 4 = 3 x 3-1 3 x 5 = 4 x 4-1 4 x 6 = 5 x 5-1

bR;kfn

1 x 3+1 = 2 x 2 2 x 4+1 = 3 x 3 3 x 5+1 = 4 x 4 4 x 6+1 = 5 x 5

bR;kfnA

15

muesa ls ,d dks bl [kkst dks 'kCnksa esa crkus ds fy, dgk x;kA mlus dgk % ¶;fn fdlh frdVh dh igyh vkSj rhljh la[;k dks xq.kk fd;k tk, rks xq.kuiQy chp dh la[;k ds oxZ ls ,d de gksrk gSA¸ eq>s mu cPpksa ls dqN bZ";kZ&lh gqbZ] D;ksafd eSa Lo;a ;g [kkst ugha dj ik;k FkkA pkpkth us mu cPpksa dh ihB FkiFkikbZ vkSj lHkh Nk=kksa ls iwNk] ¶crkvks] vc rd tks nks [kkstsa dh xb±] muesa ls fdlesa vf/d xgjkbZ gS\¸ lHkh tks'k ls fpYykdj cksys] ¶nwljh] nwljhA¸ pkpkth us crk;k] ¶fdlh [kkst esa ftruh xgjkbZ gksrh gS] mruh gh og [kkst vkuUnnk;d gksrh gSA¸ bl izdkj pkpkth us [kkst djus dk ,d ekgkSy&lk rS;kj dj fn;kA fIkQj pkpkth us dgk fd izkÑfrd la[;k,a dbZ izdkj dh gksrh gSaA vc vkSj ubZ [kkstsa djus ds fy, rqEgsa la[;kvksa ds bu fofHkUu izdkjksa dks tkuuk vkSj igpkuuk gksxkA pkpkth us mUgsa le vkSj fo"ke la[;kvksa] xq.kd (factor) o xq.kt (multiple)] oxZ vkSj ?ku la[;kvksa] rFkk vHkkT; (prime) o feJ (composite) la[;kvksa ds ckjs esa crk;k] vkSj muls bu lHkh ds mnkgj.k nsus dks dgkA pkpkth us mUgsa le>k;k fd dSls bu la[;kvksa dks igpkuk tk ldrk gSA mUgksaus cPpksa dks xf.kr dh 'kCnkoyh ds u, 'kCnksa ls ifjfpr djk;k] tSls vxyh la[;k (successor)] fiNyh la[;k (predecessor)] lEcfU/r la[;k (corresponding number)] bR;kfnA mUgksaus dgk fd bl 'kCnkoyh dh tkudkjh bl fo"k; ij ckrphr esa cgqr lgk;d gSA mUgksaus ;g Hkh dgk fd lHkh cPpksa dks 1 ls 100 ds chp bu lHkh izdkj dh lHkh la[;kvksa dh lwfp;ka cukuh pkfg,] vkSj ?ku la[;kvksa ds fy, ;g lwph 1000 rd tkuh pkfg,A tc pkpkth ;g lc dg jgs Fks] ,d cPps us viuk gkFk 16

mBk;kA pkpkth us #d dj mlls iwNk fd D;k mlus dksbZ vkSj xgjh [kkst dh gS\ og cksyk fd mlus ,d vuks[kh fof'k"Vrk [kksth gSA 1$2$3 ¾ 1 x 2 x 3 ijUrq 2$3$4 ≠ 2 x 3 x 4 mls c/kbZ nsus ds ckn pkpkth us dgk fd bl izdkj dh fof'k"Vrk feyus ij gesa mu lHkh la[;k lewgksa dks <wa<us dk iz;Ru djuk pkfg, ftu ij ;g fof'k"Vrk ykxw gksrh gksA bl ij ml cPps us dgk fd ;g fo'ks"krk vkSj fdlh Hkh la[;k lewg ij ykxw ugha gksrhA pkpkth us mlls dgk fd t:jh ugha fd la[;k lewg rhu Øeokj la[;kvksa dh frdVh gh gksA ;kfu ,slh la[;k,a [kkstks ftuds lHkh xq.kdksa dk ;ksx mu la[;kvksa ds cjkcj gksA cPps us dgk fd blesa rks cgqr le; yxsxkA vly esa og pkgrk Fkk fd pkpkth gh bldk mÙkj crk nsaA pkpkth us dgk fd bu la[;kvksa dks ns[kks % 1] 2] 4] 7] vkSj 14 A cPps us bu la[;kvksa dk ;ksxiQy fudkyk vkSj ik;k fd ml ;ksxiQy esa okdbZ og fof'k"Vrk ekStwn FkhA mlus dgk fd og ,slh vU; la[;k,a Lo;a [kkstuk pkgrk gSA pkpkth us mlds mRlkg dh iz'kalk dh vkSj vkxs cksys] ¶;fn rqe ,slh vU; la[;k,a tek dj yks] rks ;g fof'k"Vrk mu la[;kvksa ls lEcfU/r ,d fl¼kUr cu tkrk gSA ;g fdruh lqUnj ckr gSA¸ eq>s o vU; cPpksa dks Hkh pkpkth dh ;g ckr cM+h jkspd yxhA ,d cPps us mBdj iwNk % ,slh la[;k,a dkSu&lh gSa\ pkpkth us bl izdkj le>k;k % 1$2$3 ¾ 6] 1$2$4$7$14 ¾ 28 A vc ns[kks 1] 2 vkSj 3] ;s lHkh 6 ds xq.kd gSa] ;kfu 6 dks iw.kZr% foHkkftr djrs gSaA blh rjg 28 ds xq.kd gSa 1] 2] 4] 7 vkSj 14 A ,slh la[;kvksa] tSls 6 vkSj 28] dh fo'ks"krk ;g gS fd muds lHkh xq.kdksa dk ;ksx ml 17

la[;k ds cjkcj gksrk gSA ,d cPps us iwNk fd ,slh la[;kvksa dks dgrs D;k gaS\ pkpkth us mÙkj fn;k fd ,slh la[;kvksa dks vkn'kZ la[;k,a (perfect number) dgrs gSaA viuh ckrphr lekIr djrs le; pkpkth us mu lHkh cPpksa dks Lusgiw.kZ c/kbZ nh ftUgksaus bl nkSjku ubZ [kkstsa dh FkhaA mUgksaus ckdh cPpksa dks Hkh bl fo"k; ds izfr mRlkg o #fp fn[kkus ds fy, c/kbZ nhA mUgksaus dgk] ¶;fn rqe FkksM+k /S;Z j[kks] vkSj fo"k; esa #fp txk, j[kks] rks rqeesa gj dksbZ ,d twfu;j xf.krK cu dj fn[kk ldrk gSA¸ mUgksaus vius lkFk yk, gq, dqN pkVZ Dyc dks HksaVLo:i fn,A xks"Bh lekIr gks xbZA ?kj okil tkrs le; eSaus pkpkth dks crk;k fd eq>s xgjh [kkstsa djus okys cPpksa ls bZ";kZ gks jgh FkhA eSaus dgk fd vc eSa gj jfookj dks vkids ikl vk;k d:axk] vkSj vkids lkFk cSBdj ubZ [kkstsa djus dk iz;kl fd;k d:axkA ;g lqudj pkpkth izlUu gq, vkSj cksys] ¶rqEgkjh ;g ckr eq>s cgqr vPNh yxhA eSa vo'; rqEgkjs fy, dqN le; fu;r j[kawxkA¸ bl izdkj la[;ktxr esa esjk Hkze.k 'kq: gqvkA eSa pkpkth ds lkFk viuk okrkZyki v{kj'k% lcdks lqukrk gaw] ftlls os esjs vuqHko ls ykHk mBk ldsaA D;k rqe Hkh bl liQj esa esjs lg;k=kh cuksxs\

18

2 la[;kvksa ds izdkj ds lkFk esjh [kkst;k=kk dk ;g igyk jfookj FkkA og pkpkth pkgrs Fks fd eSa fofHkUu izdkj dh la[;kvksa ds ckjs esa vius Kku dks fiQj ls nksgjk ywa] rkfd mUgsa ns[krs gh igpku ldawA le vkSj fo"ke la[;k,a mUgksaus dqN daps mBk, vkSj eq>ls iwNk] ¶D;k fcuk fxus crk ldrs gks fd budh la[;k le gS ;k fo"ke\¸ eq>s le> esa ugha vk;k fd ,slk dSls lEHko gSA mudh fxurh djds eSa mÙkj ns ldrk Fkk] ;fn la[;k dk vkf[kjh] ;kfu bdkbZ okyk vad] nks ls foHkkftr gksrk gks rks le] ojuk fo"keA mUgksaus lq>ko fn;k fd eSa dapksa ds nks&nks ds tksM+s cukdj ns[kwa fd D;k gksrk gSA eSaus ,slk djuk 'kq: fd;k rks ik;k fd mudh la[;k rks le gS] D;ksafd tksM+s cukus ds ckn dksbZ dapk vdsyk ugha cpk FkkA fiQj mUgksaus eq>s dqN vkSj daps fn,A bl ckj eSaus tksM+s cuk, rks vUr esa ,d dapk vdsyk cp x;k] blfy, mudh la[;k fo"ke FkhA eSaus iwNk] ¶;fn dapksa dks fxudj] mudh la[;k ds bdkbZ vad dks ns[kdj] fd og nks ls foHkkftr gksrk gS ;k ugha] mÙkj fn;k tk,] rks D;k gkfu gS\¸ mUgksaus esjs iz'u dks ljkgk vkSj iwNk] ¶ge tks la[;k,a iz;ksx djrs gSa] mudk vk/kj (base) D;k gS\¸ eSaus dgk fd ge nl ds vk/kj okyh la[;k,a iz;ksx djrs gSa] ;kfu bdkbZ ds ckn vxyk vad ngkbZ (10 xquk) vkSj mlls vxyk lSdM+k (10x10) bR;kfn gksrk gSA mUgksaus dgk] ¶;fn ge 7 dks vk/kj cuk,a] rks 12 dks fdl izdkj fy[kksxs\¸ eSaus mÙkj fn;k] 15] D;ksafd bdkbZ ls vxys vad 19

dk eku vc 7 gS] 10 ughaA blfy, vk/kj 7 gksus ij 15 dk vFkZ gS 7$5 ¾ 12 A vc eq>s vpkud le> esa vk;k fd bdkbZ ds vad dks ns[kus dh vis{kk tksM+s cukuk D;ksa csgrj gSA 12 rks le la[;k gS] ij vk/kj 7 gksus ij bdkbZ dk vad 5 gks tkrk gS] tks fd 2 ls foHkkftr ugha gksrkA ;kfu ge xyr fu.kZ; ij igqaprsA tksM+s cukus dh fof/ ges'kk lgh mÙkj nsxh] vk/kj pkgs tks Hkh gksA pkpkth us dgk] gka ;g vo'; lgh gS fd ;fn la[;k nks ls foHkkftr gksrh gS rks og le gS] vU;Fkk fo"keA varr% mUgksaus eq>ls 100 rd dh lHkh le vkSj fo"ke la[;kvksa dh lwph cukus dks dgk ftlesa le vkSj fo"ke la[;k,a vyx&vyx ryksa ij gksaA ;g rks cgqr gh vklku dke FkkA eSaus 2 1

4 3

22 21

24 23

42 41

82

58

76

60 59

78 77

96 95

40 39

57

75 94

93

38

56

74

20 19

37

55

73 92

91

36

54

72

18 17

35

53

71 90

89

34

52

70

16 15

33

51

69 88

87

32

50

68

14 13

31

49

67 86

85

30

48

66

12 11

29

47

65 84

83

28

46

64

10 9

27

45

63

81

26

44

62

8 7

25

43

61

6 5

80 79

98 97

100 99

fuEu lwph cukbZµ tSlk fd ;g lwph n'kkZrh gS] nl ds vk/kj okyh iz.kkyh esa le la[;kvksa dk vafre vad gksrk gS 0] 2] 4] 6 ;k 8 vkSj fo"ke la[;kvksa dk 1] 3] 5] 7 ;k 9 A xq.kt vkSj xq.kd fiQj pkpkth us eq>ls iwNk] ¶D;k oLrqvksa dk dksbZ ,slk laxzg laHko gS ftls ,d drkj esa yxkuk lEHko u gks\¸ eSaus rqjUr dgk] ¶,slk dksbZ laxzg ugha gks ldrkA¸ fiQj mUgksaus eq>ls iwNk] ¶bl ckr dk foHkktu ds lanHkZ esa D;k vFkZ gS\¸ dqN fgpfdpkrs gq, eSaus dgk] ¶,d dh la[;k lHkh izkÑfrd la[;kvksa dh xq.kd gS vkSj gj 20

izkÑfrd la[;k Lo;a dk xq.kd gSA¸ mUgksaus ;gh ckr xq.kd (Factor) ds LFkku ij xq.kt (Multiple) 'kCn iz;ksx dj dgus dks dgkA eSaus dgk] ¶gj ,d izkÑfrd la[;k ,d dh xq.kt gS] vkSj Lo;a dh xq.kt Hkh gSA¸ mUgksaus eq>s vyekjh ls fudkydj ,d dVksjk fn;k] ftlesa lksMkokVj cksryksa ds cgqr ls s le> esa ugha vk;kA fiQj mUgksaus le>k;k fd
lajpuk cukbZµ ;kfu rhu drkjsa] vkSj ikap LrEHkA ¶bl lajpuk esa dqy fdrus s u\¸ ¶gka] le>kA bl izdkj 15 ;gka xq.kt gS] vkSj 3 o 5 mlds xq.kdA¸ ¶fcYdqy Bhd] 15 xq.kt gS 3 o 5 dkA 5 vkSj 3] nksuksa 21

mlds xq.kd gSaA¸ ¶tc dksbZ la[;k dqN vU; la[;kvksa dk xq.kuiQy gksrh gS rks xq.kuiQy dks xq.kt dgrs gSa] vkSj ftu la[;kvksa dks xq.kk djds mls izkIr fd;k x;k] os lc gSa ml la[;k ds xq.kdA¸ ¶D;k le la[;k,a xq.kt gksrh gS\¸ ¶gka] os lHkh 2 dh xq.kt gSaA¸ ¶vkSj fo"ke la[;k,a\¸ ¶gka] muesa Hkh cgqr&lh xq.kt gks ldrh gSa] tSls 21 xq.kt gS 7 dkA¸ ¶vly esa gj ,d la[;k de ls de 1 dh xq.kt rks vo'; gh gS] vFkkZr~ og la[;k Lo;a gh xq.kt Hkh vo'; gSA Bhd\¸ ¶gka] eSa le> x;kA¸ ¶,d ,slh la[;k crkvks] tks ,d le vkSj ,d fo"ke] nksuksa la[;kvksa dk xq.kd gksA¸ ¶ij la[;kvksa dks ns[ks fcuk dSls crkmQa\¸ ¶D;ksa ugha\¸ ¶vPNk] le>kA mÙkj gS la[;k] 1 Bhd\¸ ¶fcYdqy BhdA de ls de ,d la[;k ,slh vo'; gS] tks fdlh le] vkSj fdlh fo"ke la[;k] nksuksa dh xq.kd gks\ ,slk xq.kd dkSu&lk gks ldrk gS\¸ ¶og rks 1 dh la[;k gh gks ldrh gSA¸ ¶D;k rqe eq>s ,slh ,d le vkSj ,d fo"ke la[;k dk mnkgj.k ns ldrs gks\¸ ¶ns[ksa rks --- gka --- 2 vkSj 3A¸ ¶dksbZ vkSj mnkgj.k\¸ 22

¶3 vkSj 4 A¸ ¶D;k rqEgkjk eryc gS fd os nksuksa Øeokj la[;k,a gksuh pkfg,\¸ ¶;g t:jh ugha] ysfdu mÙkj nsus esa ;g vklku gSA¸ ¶cgqr vPNkA vc ge vxys gÝrs feysaxsA¸ eSa cksryksa ds dqN kA vHkkT; la[;k ds dsoy nks gh xq.kd gksrs gSa] vkSj feJ la[;kvksa ds nks ls vf/dA Bhd gS u] vkSj 1 vkSj 23

2 ds ckjs esa dSls fu.kZ; gks\¸ ¶gka] 1 vkSj 2 ds ckjs esa ;g fu.kZ; vk;rkdkj lajpuk cukdj ugha gks ldrkA ysfdu buds xq.kdksa dh la[;k dks ns[kdj ;g fu.kZ; fd;k tk ldrk gSA¸ ¶fcYdqyA 2 ds dsoy nks gh xq.kd gSa] 1 vkSj 2A blfy, ;g vHkkT; gSA ysfdu 1 dk rks dsoy ,d gh xq.kd gS] ;kfu 1A blfy, ;g u rks vHkkT; gqvk] vkSj u gh feJ! eSaus Bhd dgk u\¸ ¶rqeus fcYdqy Bhd dgkA D;k vc rqe dqN la[;k&igsfy;ka gy djuk pkgksxs\¸ ¶cM+h izlUurk lsA¸ ¶crkvks] lcls igyh vHkkT; la[;k gS\¸ ¶nksA¸ ¶BhdA vkSj lcls igyh fo"ke vHkkT; la[;k\¸ ¶rhuA¸ ¶cgqr vPNsA lcls igyh feJ la[;k dkSu&lh gS\¸ ¶pkjA ;g Hkh le la[;k gSA¸ ¶BhdA igyh fo"ke feJ la[;k dkSu&lh gS\¸ ¶tjk lkspus nhft,A gka] fey xbZ] og gS 9A Bhd gS u\¸ ¶gka] fcYdqy BhdA vc izkÑfrd la[;kvksa dh nks ryksa okyh ,d rkfydk rS;kj djks] ftlesa vHkkT; la[;k,a mQijh ry ij gksa] vkSj feJ la[;k,a fupys ry ijA¸ eSa rkfydk cukus esa yx x;kA ;g FkksM+k eqf'dy vuqHko FkkA le vkSj fo"ke la[;kvksa dh ,slh rkfydk cukuk vklku gS] D;ksafd ;s ,d ds ckn ,d vny&cny dj vkrh gSaA ysfdu vHkkT; vkSj feJ la[;kvksa ds lkFk ,slk ugha gSA eSaus rkfydk bl izdkj cukbZµ 24

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7 6

11 8

23 22

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26

44

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46

64

65

66

84

85

86

43 42

27 47

9 29

28

10

82

63 83

87

30

48

49

50

68

69 89

70

88

17 14

15

16

31

67 62

13 12

90

33 53

52

34

35

36

38

54

55

56

57

58

74

75

76

77 97

78

94

95

96

73 72

91

20

37 32

51 71

19 18

92

93

39 59

21 41

40 61 60

79

98

99

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81

100

---

¶,d vkSj lkS ds chp fdruh vHkkT; la[;k,a gSa\¸ ¶iPphlA¸ ¶vkSj feJ la[;k,a\¸ ¶lkS dks feyk dj\¸ ¶gka&gkaA¸ ¶ns[kk tk, rks --- 1 u rks vHkkT; gS] vkSj u feJA blfy, 1 ls 100 rd feJ la[;kvksa dh la[;k gksxh 99&25] ;kfu 74A¸

oxZ vkSj v&oxZ la[;k,a ¶vk;rkdkj lajpuk cukrs le; D;k dHkh ,slk gqvk fd drkjksa dh la[;k] vkSj gj drkj esa oLrqvksa dh la[;k] nksuksa cjkcj jgh gkas\¸ ¶gka] ,slk dbZ ckj gqvkA mnkgj.k ds fy, tSls 9 oLrq,a gksa] rc rhu drkjsa curh gSa] vkSj gj drkj eas rhu gh oLrq,a gksrh gSaA¸ ¶,slh la[;k,a ,d fo'ks"k izdkj dh vk;rkdkj lajpuk ls

lac¼ gksrh gSaA tkurs gks] ,sls vk;r dks D;k dgrs gSa\¸ ¶dgha oxZ rks ugha\¸ 25

¶gka] Bhd dgkA pwafd ,slh la[;k,a oxkZdkj lajpuk ls laca/ j[krh gSa] mUgsa oxZ la[;k,a dgrs gSaA D;k vk;rkdkj la[;k,a oxZ gks ldrh gSa\¸ ¶dsoy rHkh] tc drkjksa dh la[;k gj drkj eas oLrqvksa dh la[;k ds cjkcj gksA¸ ¶rks D;k izR;sd oxZ la[;k vk;rkdkj gS\¸ ¶gka] fu'p; ghA¸ ¶;kfu gj oxZ la[;k vk;rkdkj gS] ysfdu gj vk;rkdkj la[;k oxZ ughaA vPNk] oxZ la[;k ds xq.kdksa ds ckjs esa D;k dgksxs\¸ ¶9 ¾ 3×3] 4 ¾ 2×2] bR;kfnA ;kfu ;fn la[;k dks nks leku xq.kdksa ds xq.kuiQy ds :i esa fy[kk tk lds] rks og oxZ la[;k gS] Bhd\¸ ¶gka] BhdA vc ;g crkvks] D;k 1 oxZ la[;k gS\¸ ¶vc ;g dSls irk yxk,a] D;ksafd dsoy ,d oLrq dks oxkZdkj lajpuk esa dSls ltk,a\¸ ¶,d oxZ la[;k ds nks leku xq.kd gksrs gSaA vFkkZr~ og nks leku la[;kvksa dk xq.kuiQy gSA ysfdu 1 ds nks ls vf/d leku xq.kd gks ldrs gSa] D;ksafd ge 1 dks bl izdkj fy[k ldrs gSa% 1×1×1×--- bR;kfnA¸ ¶gka] fcYdqy] ,d gS 1×1 A bl izdkj ;g oxZ la[;k gqbZA Bhd\¸ ¶gkaA vc izkÑfrd la[;kvksa dks nks ryksa okyh ,d lwph cukvks] ftlesa oxZ la[;k,a mQijh ry ij gksa] vkSj v&oxZ fupys ry ijA¸ lwph bu izdkj cuhµ 26

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9 5 25

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44 64

61 81

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16 10

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37

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36 26

27

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65

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29 49

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89

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la[;kvksa dh bl izdkj nks ry okyh lwph cukuk eq>s vPNk yxk] D;ksafd nks Øeokj oxZ la[;kvksa ds chp fdruh v&oxZ la[;k,a iM+rh gSa] blesa Hkh ,d iSVuZ gSA ¶nks oxZ la[;kvksa ds chp iM+us okyh bu v&oxZ la[;kvksa dks rqe vUrjky Øe (interval sequence) Hkh dg ldrs gksA¸ ¶Bhd] gjsd vUrjky Øe esa v&oxksZa dh la[;k 2 ls foHkkftr gksrh gSA¸ ¶;g rks cM+k jkspd iSVuZ gSA¸ ¶fcYdqy BhdA¸ ?ku vkSj v&?ku vc pkpkth us ?ku la[;kvksa dh ppkZ 'kq: dhA mUgksaus eq>s dqN cksryksa ds
27

¶eku yks fd rqEgsa rhu ryksa okyk ?ku cukuk gks] rks dSls cukvksxs\¸ ¶;kfu ftlesa gj <sjh esa ry&mQij rhu
¶fcYdqy BhdA ,d oxZ la[;k nks leku xq.kdksa dk xq.kuiQy gSA ,d ?ku la[;k ds fy, fdrus leku xq.kdkssa dh t:jr gksxh\¸ ¶8 = 2×2×2, 27 = 3×3×3A gjsd ?ku la[;k rhu leku xq.kdksa dk xq.kuiQy gSA¸ ¶BhdA vc ,d ds ckjs esa D;k dgksxs\ og oxZ la[;k rks gS gh] D;k ?ku la[;k Hkh gS\¸ ¶gka] ,d dks ge fy[k ldrs gaS] 1×1×1] blfy, ;g ?ku la[;k Hkh gqbZA Bhd gS u\¸ ¶rqeus fcYdqy Bhd dgkA ;fn fdlh la[;k dks rhu leku xq.kdksa ds xq.kuiQy ds :i esa fy[kk tk ldrk gS] rks og vo'; gh ?ku la[;k gSA vc 8 dh la[;k 2 dk ?ku gSA D;k rqe tkurs gks 2 vkSj 8 dk D;k fj'rk gS\ ;kfu] 8 rks 2 dk ?ku gS] ij nks 8 dk D;k gS\¸ 28

¶2×2 ¾ 4 A 4 dh la[;k 2 dk oxZ gS] vkSj 2 gS 4 dk oxZewyA blh izdkj] D;ksafd 2×2×2 ¾ 8 A vr% 2 gqvk 8 dk ?kuewyA eSaus Bhd dgk u\¸ ¶,dne lgh dgkA vc 1 ls 100 rd dh la[;kvksa dh ,d f}ryh; lwph cukvks] ftlesa ?ku vkSj v&?ku la[;k,a mQijh o fupys ry ij n'kkZbZ xbZ gksaA¸ ftl izdkj eSaus oxZ vkSj v&oxZ la[;kvksa dh f}ryh; lwph cukbZ Fkh] mlh izdkj ;g lwph cukuk Hkh eq>s cM+k jkspd dk;Z yxkA ;g lwph uhps nh gqbZ gSA 1

8 2

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6

21

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41

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44 64

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65

81

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7 27

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bl lwph ds varjky Øe dkiQh yEcs gSaA bu varjky Øeksa esa v&?kuksa dh la[;k Øe'k% 6] 18] 36] 60] 90 bR;kfn gS] tks lHkh 6 ds xq.kd gSaA ¶vPNk] vc 1000 rd dh la[;kvksa esa dsoy ?ku la[;kvksa dh ,d lwph cukvksA crkvks] dSls cukvksxs\¸ ¶rhu leku xq.kdksa dk xq.kuiQy ysdj] xq.kd la[;k,a gksaxh 1] 2] 3--- bR;kfnA eSa ;g vklkuh ls dj ldrk gwaA¸ ¶cgqr [kwc] vc cukvks ?kuksa dh lwphA¸ ¶1] 8] 27] 64] 125] 216] 343] 512] 729] vkSj 1000A¸ ¶cgqr vPNkA vxys gÝrs ge f=kdks.kh; la[;kvksa ds ckjs esa tkusaxsA¸ ¶;g f=kdks.kh; la[;k,a D;k gksrh gSa\¸ ¶/hjt ds lkFk FkksM+h izrh{kk djks] vxys gÝrs rdA¸ 29

f=kdks.kh; vkSj v&f=kdks.kh; la[;k,a pkpkth us eq>ls dgk fd tks dqN os djus tk jgs Fks] eSa mls è;ku ls ns[kwaA mUgksaus dqN
vc mUgksaus eq>ls bl Øe esa vxyh nks lajpuk,a cukus dks dgkA igys eSaus mudh cukbZ gqbZ lajpukvksa dk è;kuwioZd vè;;u fd;kA ¶gj drkj esa us ds ckn bl Øe dh vxyh nks lajpuk,a eSaus bl izdkj cukb±µ

¶gjsd lajpuk dk vkdkj D;k gS\¸ ¶f=kdks.kh;A¸ ¶gka] bUgha ls f=kdks.kh; la[;k,a cuh gSaA vc fxu dj ns[kks fd gj lajpuk esa dqy fdrus kA 3] 6] 10] 15] 21] 28A¸ 30

¶vkSj f=kdks.kh; lajpuk,a cuk, fcuk D;k rqe bl lwph dh vxyh nks la[;k,a crk ldrs gks\¸ ¶igys f=kdks.k dh vkf[kjh drkj esa 2 tkrs gks! pyks] iQVkiQV rkfydk cukvksA¸ eq>s rkfydk cukus esa dksbZ ijs'kkuh ugha gqbZA D;ksafd tYn gh eSaus ik;k fd ;fn f=kdks.kh; la[;kvksa ds chp ds varjky Øe esa yEckbZ ,d ls 'kq: gksdj gj ckj ,d ls c<+ tkrh gSA ;kfu varjky 31

Øe dh yackbZ 1 ls 'kq: gksdj izkÑfrd la[;kvksa dh rjg] ;kfu --1] 2] 3] c<+rh tkrh gSA rkfydk bl izdkj cuh % 1

3 2

6 4

5 25 45

10 7

21 22

23

24

41

42

43

44

61

62

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65

81

82

83

84

85

26

27

46 66

47

86

8 28

9

15 11

12

13

14

16 36

17

18

19

20

37

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29

40

56

57

58 78

59

60

79

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99

100

29

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35 55

48

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71 91

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98

¶D;k vc rqe ,d ijh{kk ds fy, rS;kj gks\¸ ¶vo';A¸ ¶,d ds ckn vxyh dkSu&lh la[;k gS] tks oxZ Hkh gS vkSj f=kdks.kh; HkhA¸ ¶rkfydk dk vè;;u djus ds ckn eSaus dgk] 36 A¸ ¶fcYdqy BhdA ,d vkSj 100 ds chp fdruh f=kdks.kh; la[;k,a gSa\¸ ¶ckjgA¸ ¶lghA vc igyh f=kdks.kh; la[;k 1 gS] nwljh 3] rhljh 6] pkSFkh 10 bR;kfnA ;fn eSa rqels iwwNwa fd chloha f=kdks.kh; la[;k dkSu&lh gS] rks dSls crkvksxs\¸ ¶ns[krk gwaA ,d lwph cukdj ns[krk gwaA¸ Øekad f=kdks.kh; la[;k

bR;kfn ¶eq>s yxrk gS fd eq>s Øeokj tksM+uk pkfg,] tSls 1 esa 2 tksM+us ij 3 feyk] fiQj 3 esa 3 tksM+us ij 6 feyk] 6 esa 4 tksM+k rks 10] 10 esa 5 tksM+k rks 15] 15 esa 6 tksM+us ij 21] bR;kfnA eSa 32

;g Øe tkjh j[k ldrk gwa] tc rd fd chloha f=kdks.kh; la[;k rd u igqap tkmQaA D;k blls vkSj vklku rjhdk ugha gks ldrk\¸ pkpkth us eq>ls dgk fd eSa fdlh Hkh f=kdks.kh; la[;k ds nks f=kdks.k ywa] vkSj muesa ls ,d dks mYVk djds nwljs ds mQij j[kwa] tSlk fd uhps fn, fp=k esaA

¶,slk djus ls D;k cuk\¸ ¶;g rks ,d vk;rkdkj la[;k cu xbZA¸ ¶bu vk;rkdkj la[;kvksa esa rqEgsa dksbZ fo'ks"krk fn[kkbZ nsrh gS\ tjk xkSj ls ns[kks] gj vk;r esa fdruh drkjsa gSa] vkSj gj drkj esa fdruh oLrq,a\¸ ¶vksg] le>kA gj vk;rkdkj la[;k esa ,d drkj esa oLrqvkasa dh la[;k drkjksa dh la[;k ls ,d vf/d gS] ;kfu Øeokj izkÑfrd la[;k,a tSls (2]3)] (3]4)] (4]5) bR;kfnA D;k ,slh vk;rkdkj la[;kvksa dk dksbZ fo'ks"k uke gksrk gS\¸ ¶gka] bUgsa dgrs gSa vaMkdkj (oblong) la[;k,aaA D;k rqe ,slh la[;kvksa dh lwph cuk ldrs gks\¸ ¶buesa gj la[;k ,d f=kdks.kh; la[;k dks nksxquk djds cuh gS] --blfy, budh lwph bl izdkj gksxh % 6] 12] 20] bR;kfnA¸ ¶gka] rqeus fcYdqy Bhd dgkA¸ ¶vc eSa le>kA gj f=kdks.kh; la[;k nks Øeokj la[;kvksa ds xq.kuiQy dh vk/h gSA blfy, igyh f=kdks.kh; la[;k gksxh (1×2)@2 ¾1 A nwljh f=kdks.kh; la[;k gksxh (2×3)@2 ¾ 3] vkSj 33

blh rjg vkxs gh f=kdks.kh; la[;k,a vklkuh ls fudkyh tk ldrh gSaA blfy, 10 oha f=kdks.kh; la[;k gksxh] (10×11)@2 ¾55A Bhd gS u\¸ ¶gka] rqe fcYdqy Bhd le>sA vc crkvks] lcls igyh izkÑfrd la[;k dkSu&lh gS] tks oxZ Hkh gks] ?ku Hkh] vkSj f=kdks.kh; HkhA¸ ¶og rks 1 gh gS] gS u\¸ ¶cgqr vPNs] fcYdqy 1 gh gSA vc] ?ku la[;kvksa ds chp ds eè;kUrj Øe esa fdruh la[;k,a gSa] bl ij è;ku nksA¸ ¶vkidk eryc gS] 6] 18] 36] 60 vkSj 90 bR;kfn\¸ ¶gka] oghA rqe ns[k ldrs gks fd ;s lc 6 ds xq.kt gSaA vc ns[kks fd 6 dks fdu la[;kvksa ls xq.kk djus ij 6] 18] 36] 60] 90 bR;kfn la[;k,a feyrh gSaA¸ ¶vkidk rkRi;Z gS] 1] 3] 6] 10] 15 A vjs] bu la[;kvksa ds ;s xq.kd rks f=kdks.kh; la[;k,a fudysA vjs okg] vk'p;Z gSA¸ vxyh (successor) o fiNyh (predecessor) la[;k,a ¶vc rqEgkjs okrkZyki dh ljyrk ds fy, dqN vkSj 'kCnksa dk iz;ksx lh[kuk gksxkA D;k rqe izkÑfrd la[;k,a Øe ls crk ldrs gks\¸ ¶1] 2] 3] 4] 5] 6] 7] bR;kfnA¸ ¶vc fdlh la[;k dh ^fiNyh la[;k* (predessor) dk vFkZ gS mlls rqjUr igys vkus okyh la[;kA rks crkvks] og dkSu&lh la[;k gS ftldh fiNyh la[;k gS gh ugha\¸ ¶1 A¸ ¶fdlh la[;k dh ^vxyh la[;k* (successor) dk vFkZ gS mlds rqjUr ckn vkus okyh la[;kA dksbZ la[;k crkvks ftudh vxyh la[;k ekStwn gksA¸ 34

¶og rks gjsd la[;k dh gSA¸ ¶fdl la[;k dh vxyh la[;k ugha gS\¸ ¶,slh dksbZ la[;k ugha gSA¸ ¶;g rqeus dSls tkuk\¸ ¶eq>s dksbZ Hkh izkÑfrd la[;k nhft,] eSa mlesa 1 tksM+dj vxyh la[;k crk ldrk gwaA¸ ¶Bhd dgkA vc iw.kZ la[;k,a (whole numbers) Øe ls crkvksA¸ ¶0] 1] 2] 3] 4] 5] --- bR;kfnA¸ ¶D;k bl Øe esa 1 dh dksbZ fiNyh la[;k gS\¸ ¶gka] 1 ls igys 0 vkrk gSA¸ ¶rks D;k ge dg ldrs gaS fd dksbZ la[;k o mldh vxyh la[;k] rFkk dksbZ la[;k o mldh fiNyh la[;k] dh Øeokj la[;k,a gSa\¸ ¶gka] vo'; ge ,slk dg ldrs gSaA¸ ¶vc ,d drkj esa izkÑfrd la[;k,a fy[kks] vkSj mlds Bhd uhps nwljh drkj esa le la[;k,a fy[kksA¸ ¶1 2 3 4 5 6 7 8 9 10 --2 4 6 8 10 12 14 16 18 20 ---¸ ¶Bhd] vc crkvks] nwljh drkj dh la[;k,a igyh drkj dh la[;kvksa esa fdl izdkj ik ldrs gks\¸ ¶igyh drkj dh gj la[;k dks 2 ls xq.kk djdsA¸ ¶;kfu 1 ls lacaf/r (corresponding) le la[;k gqbZ 2 vkSj 2 ls lacaf/r le la[;k gqbZ 4A rks 7 ls lacaf/r le la[;k D;k gS\¸ ¶pkSngA¸ 35

¶'kkck'kA vc crkvks] le la[;k 18 ls lacaf/r izkÑfrd la[;k D;k gS\¸ ¶ukSA¸ pkpkth us esjh vcrd dh liQyrk ij eq>s c/kbZ nh] vkSj crk;k fd vxys gÝrs ls la[;ktxr dh bl ;k=kk dk dgha vf/d jksekapd nkSj 'kq: gksxkA ¶pkpkth] bls la[;ktxr dh ctk; la[;k&lkxj D;ksa ugha dgrs\¸ ¶;fn rqe pkgks rks og Hkh dg ldrs gksA¸

36

3 u;k jksekap

eSa

bl ckj pkpkth ls cgqr vk'kkfUor gksdj feykA mUgksaus eq>s ;kn fnyk;k fd bl ;k=kk esa esjs fy, tksM+&?kVko] xq.kk&Hkkx] oxZ o ?ku la[;k,a fudkyuk] dsoy bruk gh Kku dkiQh gksxkA mUgksaus eq>s lykg nh fd eSa la[;kvksa ds ijLij laca/ksa] vkSj muds chp tks iSVuZ mHkj dj vk,a] mu ij viuh utj yxkrkj cuk, j[kwaA mUgksaus eq>s ;g fo'okl Hkh fnyk;k fd bl ;k=kk esa ubZ [kkstsa djus esa fofHkUu izdkj dh la[;kvksa ds ckjs esa esjh tkudkjh fo'ks"k :i ls lgk;d gksxhA mUgksaus eq>s dqN i=kd (flash cards) fn[kk,A igys i=kd ij ;g rkfydk Fkh % d [k x ?k 3 4 7 13 8 1 9 9 5 7 12 36 9 2 && && ¶rkfydk dk è;ku ls vè;;u djksA igys ;g tkuus dh dksf'k'k djks fd LrEHk x dh la[;kvksa dk LrEHk d vkSj [k dh la[;kvksa ls D;k laca/ gSA fiQj mlds vk/kj ij LrEHk x dh vkf[kjh drkj esa mfpr la[;k HkjksA¸ ¶eq>s rks mQijh rkSj ij dqN ,slk iSVuZ fn[krk gS % 3$4 ¾ 7 8$1 ¾ 9 5$7 ¾ 12 37

blfy, LrEHk x esa vkf[kjh la[;k gksuh pkfg,] 9$2 ¾ 11 A D;k ge dg ldrs gSa fd LrEHk x LrEHk d vkSj [k dk ;ksx gS\¸ ¶'kkck'k! ,slk fcYdqy dg ldrs gSaA bls ge bl izdkj Hkh dg ldrs gSa % x ¾ d$[kA vc irk yxkvks fd ?k la[;kvksa dk d vkSj [k la[;kvksa ls D;k laca/ gS] vkSj fiQj LrEHk ?k esa fjDr LFkku dks HkjksA¸ ¶;gka iSVuZ ,dne lkiQ rks fn[kkbZ ugha nsrkA eq>s dbZ jkLrs vktekus gksaxsA tksM+ rks eSaus igys gh fudky j[kk gSA LrEHk ?k dk d vkSj [k ds ;ksx ls dksbZ fj'rk utj ugha vkrkA ?kVkus ls Hkh dqN iQk;nk gksrk ugha fn[kkbZ nsrkA LrEHk d vkSj [k dk xq.kuiQy vktekrk gwaA og dke djsxk D;k\¸ ¶,d rkfydk cukvks] ftlds ,d LrEHk esa d vkSj [k dk xq.kuiQy gks] vkSj nwljs esa LrEHk ?k dh la[;k,aA¸ rkfydk uhps nh gqbZ gSA d × [k ?k 12 13 8 9 35 36 ¶vksgA rks vc [kqyk budk HksnA nksukaas LrEHkksa dh rqyuk djus ls ;g fcYdqy lkiQ gS fd LrEHk ?k dh la[;k d vkSj [k ds xq.kuiQy dh vxyh la[;k gSA blfy, LrEHk ?k dh vkf[kjh drkj esa tks la[;k gksuh pkfg,] og gS 9×2$1 ¾ 19] Bhd gS u\¸ ¶'kkck'k] cgqr vPNkA rks ge dg ldrs gSa] ?k ¾ d × [k $ 1] Bhd\¸ ¶gka] fcYdqyA eSa [kqn Hkh ;gh dguk pkgrk FkkA ;g fcYdqy lgh gSA¸ 38

¶vc bl i=kd dks ns[kks] p N t 3 8 10 5 24 26 6 35 37 4 && && vkSj [kkst dj crkvks] fd p la[;kvksa dk N vkSj t la[;kvksa ls D;k laca/ gS] vkSj fiQj vkf[kjh drkj esa [kkyh LFkkuksa dks HkjksA¸ ¶;g iSVuZ [kkstus ds fy, D;k eSa p la[;k dks vius&vki eas tksM+wa ;k Lo;a ls xq.kk d:a\ ¶Bhd gS] rqe ;g nksuksa phtsa vktek ldrs gksA ysfdu gj vktekb'k dh rkfydk cukuk er HkwyukA¸ ¶igys eSa tksM+dj rkfydk cukrk gwaA¸ p $ p N 6 8 10 24 12 35 ;gka rks eq>s dksbZ iSVuZ utj ugha vkrkA vc p dks Lo;a ls xq.kk djds rkfydk cukrk gwaA p×p N 9 8 25 24 36 35 vc iSVuZ ,dne lkiQ gSA izR;sd N la[;k p la[;k ds oxZ ls fiNyh la[;k gSA blfy, ;fn p la[;k 4 gS] rks N la[;k gksxh 39

16&1 ¾ 15A¸ pkpkth esjs dke dks yxkrkj ns[k jgs Fks] vkSj tc mUgksaus eq>s 15 fy[krs gq, ns[kk rks izlUu gksdj mUgksaus esjs dk;Z dh ljkguk dhA ¶vc p vkSj t la[;kvksa dk ikjLifjd laca/ rqe vklkuh ls fudky ldrs gksA¸ eSaus p la[;kvksa ds oxksZa vkSj t la[;kvksa dh vksj è;ku fn;k] rks iSVuZ fn[kkbZ fn;kA ¶t la[;k,a p la[;kvksa ds oxksZa dh vxyh la[;k,a gSa] blfy, p la[;k 4 dh drkj esa t la[;k gksxhA 16$1 ¾ 17A¸ esjh bl liQyrk ls pkpkth ds psgjs ij izlUurk dh >yd lkiQ fn[kkbZ ns jgh FkhA mUgksaus dgk fd vxys gÝrs ls 'kq: gksus okys [kkst vfHk;ku ds fy, vc eSa ,dne rS;kj gks pqdk gwaA

40

4 [kkst vfHk;ku % igyk nkSj izkÑfrd la[;kvksa ds Øeokj tksM+s eSaus ns[kk fd pkpkth dh est ij cgqr ls pkVZ iM+s gq, FksA ;s oSls gh pkVZ Fks] tSls pkpkth us xf.kr&Dyc dks lEcksf/r djrs le; iz;ksx fd, FksA mUgksaus eq>ls xqykch pkVZ mBkdj mldk vè;;u djus ds fy, dgkA mUgksaus dgk] ¶rqEgsa Øeokj vkus okys la[;kvksa ds tksM+ksa dh vksj è;ku nsuk gSA¸ 1 1 1

2 2 2

3 3 3

4 4 4

5 5 5

6 6 6

7 7 7

8 8 8

9 9 9

10 10 10

¶budk vè;;u djds crkvks] rqEgsa D;k utj vkrk gS\¸ ¶eq>s bl pkVZ esa la[;kvksa ds Øeokj tksM+s utj vk jgs gSa] tSls (1]2)] (2]3)] (3]4) bR;kfnA eSa gj tksM+s dh la[;kvksa dk ;ksx fudkydj ns[krk gwaA 1$2 ¾ 3 4$5 ¾ 9 2$3 ¾ 5 5$6 ¾ 11 3$4 ¾ 7 6$7 ¾ 13] bR;kfnA ;s la[;k,a] 3] 5] 7] 9] 11 oxSjg] lHkh fo"ke la[;k,a gSaA le>kA gj Øeokj tksM+s dh la[;kvksa dk ;ksx ,d fo"ke la[;k gSA Bhd\¸ 41

¶fcYdqy lghA ysfdu D;k blds vykok rqe dksbZ vkSj fo'ks"krk Hkh <wa<+ ldrs gks\¸ ¶,d fo'ks"krk tksM+us ls feyh] rks ,d ?kVkus ls Hkh feyuh pkfg,A dksf'k'k djrk gwaA 2&1 ¾ 1 3&2 ¾ 1 4&3 ¾ 1 bR;kfnA nks Øeokj la[;kvkas dk varj ges'kk ,d gksrk gSA¸ ¶D;k rqEgkjs fopkj esa ;g ckr Hkh mruh gh jkspd ;k egRoiw.kZ gS] tks bu tksM+ksa dk ;ksx ysus ls feyh Fkh\¸ ¶ugha] ;g rks fcYdqy ekewyh ckr yxrh gSA¸ ¶blfy, ,slh fo'ks"krk dks ekewyh (trivial) fo'ks"krk dgrs gSaA¸ ¶vPNk] vc eSa xq.kk djds ns[krk gwaA** 1×2 ¾ 2 4×5 ¾ 20 2×3 ¾ 6 5×6 ¾ 30 3×4 ¾ 12 6×7 ¾ 42 bR;kfnA 2] 6] 12] 20] 30] 42 bR;kfn esa lHkh le la[;k,a gSaA ;kfu] nks Øeokj la[;kvksa dk xq.kuiQy lnk le gksrk gSA¸ ¶;g ifj.kke Hkh mruk gh vPNk gS] ftruk bu tksM+ksa ds ;ksx ls feyk FkkA¸ ¶D;k eSa blls Hkh xgjk ,d ifj.kke [kkstus dh dksf'k'k d:a\¸ ¶;fn rqe ,slk dj ikvks] rks cgqr vPNk gksxkA¸ ¶;fn eSa gj tksM+s esa igyh la[;k dk oxZ ywa] vkSj bls nwljh la[;k ls foHkkftr djds ,d rkfydk cukmQa] rks D;k gksxk\¸ 42

¶djds ns[kksA¸ ¶1@2 ¾ 0] 'ks"k 1 4@3 ¾ 1] 'ks"k 1 9@4 ¾ 2] 'ks"k 1 16@5 ¾ 3] 'ks"k 1 25@6 ¾ 4] 'ks"k 1 bR;kfnA vjs okg] ;g rks cgqr vn~Hkqr iSVuZ gSA fdlh la[;k ds oxZ dks ;fn ml la[;k dh vxyh la[;k ls foHkkftr fd;k tk,] rks 'ks"k ges'kk 1 gh cprk gSA¸ ¶vkSj HkkxiQy ds ckjs esa D;k dg ldrs gks\¸ ¶HkkxiQy] ftl la[;k dk oxZ fy;k x;k Fkk (;kfu foHkkftr la[;k ds oxZewy) dh fiNyh la[;k gSA¸ ¶vfr mÙkeA vPNk] vc ;fn rqe tksM+s dh nwljh la[;k ds oxZ dks igyh la[;k ls Hkkx nks] rks Hkh D;k ,slk gh iSVuZ feysxk\¸ ¶eSa vHkh djds ns[krk gwaA

igys tksM+s dks NksM+dj] ckdh lHkh tksM+s ,d tSlk iSVuZ n'kkZrs gSaA gj ckj HkkxiQy foHkktd ls nks vf/d gS] vkSj 'ks"kiQy ges'kk ,d gSA¸ ¶cgqr lqUnjA vc ge vxys gÝrs feyrs gSaA¸ 43

[kkst vfHk;ku % nwljk nkSj Øeokj la[;kvksa dh frdfV;ka bl gÝrs pkpkth us eq>ls yky pkVZ ysdj mldk vè;;u djus dks dgkA ;g ogh pkVZ Fkk ftldk iz;ksx pkpkth us vkW;yj xf.kr&Dyc ds le{k fd;k FkkA 1 1 1 1

2 2 2 2

3 3 3 3

4 4 4 4

5 5 5 5

6 6 6 6

7 7 7 7

8 8 8 8

9 9 9 9

10 10 10 10

¶;g ogh pkVZ gS u] tks vkius xf.kr&Dyc ds Nk=kksa dh lHkk esa fn[kk;k Fkk] tc eSa Hkh vkids lkFk Fkk\¸ ¶gkaA ;g ogh gSA D;k rqEgsa ;kn gS] rhu Øeokj la[;kvksa dh frdfV;kasa dh dkSu&lh fo'ks"krk,a ml fnu [kksth xbZ Fkha\¸ ¶gka] fcYdqyA eSa Hkwy dSls ldrk gwa! esjs efLr"d esa os vHkh Hkh rktk gSaA ;s jgha os fo'ks"krk,a % 1- gj frdVh dh igyh vkSj vkf[kjh la[;k dk ;ksx chp dh la[;k dk nksxquk gksrk gSA 2- gj frdVh dh igyh vkSj vkf[kjh la[;k dk xq.kuiQy chp dh la[;k ds oxZ ls ,d de gksrk gSA ;kuh] chp dh la[;k ds oxZ ls fiNyh la[;k ckdh nks la[;kvksa ds xq.kuiQy ds cjkcj gksrh gSA 3- gj frdVh dh rhuksa la[;kvksa dk ;ksx ges'kk 3 dk xq.kt gksrk gSA 4- frdVh dh rhukas la[;kvksa dk xq.kuiQy lnk 6 dk xq.kt gksrk gSA crkb,] eSaus Bhd dgk u\¸ ¶vfr lqUnjA rqeus frdfV;ksa dh fo'ks"krkvksa dks muds egRo 44

o jkspdrk ds vuq:i Øe ls crk;k gSA rqEgkjh bruh vPNh ;knnk'r ij eSa rqEgsa c/kbZ nsrk gwaA vPNk] vc dqN vkSj fo'ks"krk,a [kkstksA¸ ¶;fn ge frdVh dh igyh nks la[;kvksa ds xq.kuiQy dks vkf[kjh la[;k ls foHkkftr djsa] rks D;k iSVuZ mHkjsxk\ pyks] ns[krk gwaA 2@3 ¾ 0] 'ks"k 2 30@7 ¾ 4] 'ks"k 2 6@4 ¾ 4] 'ks"k 2 42@8 ¾ 5] 'ks"k 2 12@5 ¾ 2] 'ks"k 2 56@7 ¾ 6] 'ks"k 2 20@6 ¾ 3] 'ks"k 2 72@10 ¾ 7] 'ks"k 2 bR;kfnA vksg] gkaA le> x;kA tc frdVh dh igyh nks la[;kvksa ds xq.kuiQy dks vkf[kjh la[;k ls foHkkftr fd;k tkrk gS] rks ges'kk 2 'ks"k jgrk gSA¸ ¶fcYdqy BhdA vkxs pyksA¸ ¶vc ;fn eSa frdVh dh Nksj okyh] ;kfu igyh vkSj vkf[kjh] la[;kvksa ds xq.kuiQy dks fcpyh la[;k ls foHkkftr d:a\ D;k gksxk\ ns[krk gwa\ 3@2 ¾ 1] 'ks"k 1 24@5 ¾ 4] 'ks"k 4 8@3 ¾ 2] 'ks"k 2 35@6 ¾ 5] 'ks"k 5 15@4 ¾ 3] 'ks"k 3 47@7 ¾ 6] 'ks"k 6 bR;kfnA fdruk vn~Hkqr iSVuZ gSA ;fn frdVh dh Nksj okyh la[;kvksa ds xq.kuiQy dks eè; dh la[;k ls foHkkftr fd;k tk,] rks HkkxiQy vkSj 'ks"kiQy nksuksa lnk gh leku vkrs gSaA¸ ¶D;k HkkxiQy vkSj 'ks"kiQy dk eè; dh la[;k ls dksbZ fj'rk utj vkrk gS\¸ ¶vjs gkaA eSa ml rjiQ è;ku nsus gh okyk FkkA 'ks"kiQy vkSj HkkxiQy] nksuksa gh eè; dh la[;k ls fiNyh la[;k gSA¸ 45

¶c<+rs jgksA¸ ¶vc ;gh ns[kuk ckdh gS fd ;fn frdVh dh vkf[kjh nks la[;kvksa ds xq.kuiQy dks igyh la[;k ls foHkkftr fd;k tk,] rks D;k gksxkA D;k oSlk dh dksbZ iSVuZ fudysxk\ pyks] ns[krk gwaA

;fn igyh nks frdfV;ksa dks NksM+ nsa] rks ckdh esa ,d iSVuZ fn[kkbZ nsrk gSA 'ks"kiQy ges'kk 2 gS] vkSj HkkxiQy 6 ls 'kq: gksdj izkÑfrd la[;kvksa dh Ük`a[kyk ds leku pyrk gSA igyh nks frdfV;ksa dks NksM+dj] HkkxiQy ges'kk foHkktd ls 3 vf/d gSA¸

46

[kkst vfHk;ku % rhljk nkSj pkj Øeokj izkÑfrd la[;k,a (pkSdM+h) ¶vkt ds pkVZ esa rqEgsa D;k feyus dh mEehn gS\¸ ¶igys pkVZ esa izkÑfrd la[;kvksa ds tksM+s Fks] vkSj nwljs esa frdfV;kaA vxj ;gh Øe pkyw jgk] rks vkt ds pkVZ esa pkj yxkrkj la[;kvksa ds lewg gksus pkfg,A Bhd\¸ ¶vPNk] yks] ;g ns[kks vkt dk pkVZ % 1 1 1 1 1

2 2 2 2 2

3 3 3 3 3

4 4 4 4 4

5 5 5 5 5

6 6 6 6 6

7 7 7 7 7

8 8 8 8 8

9 9 9 9 9

10 10 10 10 10

11 11 11 11 11

vc tjk dksf'k'k djksA¸ ¶;gka eq>s tjk vf/d iz;Ru djuk iM+sxk] D;ksafd gj Øeokj lewg esa vc T;knk la[;k,a gSaA 'kk;n HkkX; esjk lkFk nsA pyks] gj lewg esa nksuksa okyh la[;kvksa dk ;ksx ysdj mudh rqyuk djrk gawA D;k blesa dksbZ iSVuZ fudysxk\ ,d rkfydk cuk dj ns[krk gawA Nksj okyh nks la[;kvksa dk ;ksx

chp dh nks la[;kvksa dk ;ksx

(1$4¾)5 (2$5¾)7 (3$6¾)9 (4$7¾)11

(2$3¾) (3$4¾) (4$5¾) (5$6¾)

5 7 9 11

bR;kfnA

bl iSVuZ esa ,d lknxh vkSj lqUnjrk gSA gj pkSdM+h esa Nksj okyh nks la[;kvksa dk ;ksx chp dh nks la[;kvksa ds ;ksx ds cjkcj gksrk gSA ;gh ugha] ;gka ,d vkSj iSVuZ Hkh gSA ;s lkjh fo"ke la[;k,a gSaA 47

vc ;fn eSa tksM+us ds LFkku ij xq.kk d:a] rks D;k feysxk\ fiQj ls rkfydk cukdj ns[krk gwaA Nksj okyh nks la[;kvksa dk xq.kuiQy (1×4=)4

chp dh nks la[;kvksa dk xq.kuiQy (2×3=) 6

(2×5=)10 (3×6=)18

(3×4=) 12 (4×5=) 20

(4×7=)28

(5×6=) 30

(5×8=)40

(7×6=) 42 bR;kfnA

gka] blesa Hkh ,d lqUnj iSVuZ gSA lkjs xq.kuiQy le la[;k,a gSaA blds vykok] gj pkSdM+h dh Nksj okyh la[;kvksa dk xq.kuiQy chp dh la[;kvksa ds xq.kuiQy ls ges'kk 2 de gSA¸ ¶BhdA vc lkjs xq.kuiQy dks Øe ls fy[kks] D;k blesa dksbZ jkspd ckr fn[kkbZ nsrh gS\¸ ¶;s xq.kuiQy Øe ls bl izdkj gSa% 4] 6] 10] 12] 18] 20] 28] 30] 40] 42] bR;kfnA¸ ¶viQlksl] blesa dksbZ iSVuZ [kkstuk rks tjk Vs<+k dke yxrk gSA D;ksafd la[;kvksa ds vUrj dHkh ?kVrs gSa] rks dHkh c<+rs gSaA ;fn Øeokj la[;kvksa ds vUrj ysa] rks ;s gksaxs% 2] 4] 2] 6] 2] 8] 2] 10] 2] bR;kfnA¸ ¶vjs D;k rqEgsa blesa dqN vn~Hkqr ugha yxrk\¸ ¶vjs gka] ;g rks fu'p; gh cM+k jkspd gSA ;fn bl Øe esa ,d NksM+dj gj ckj vkus okyh 2 dh la[;kvksa dks gVk nsa] rks ;g 4 ls 'kq: gksdj le la[;kvksa dh Ük`a[kyk cu tkrh gSA¸ ¶rks vc D;k djksxs\¸ ¶vHkh eSaus gj pkSdM+h dh la[;kvksa ds ;ksxiQy dk vè;;u ugha fd;k gSA¸ 48

¶Bhd gS] rks ;g Hkh djds ns[kksA¸ ¶bu pkSdfM+;ksa ds ;ksx bl izdkj gaS % 1$2$3$4¾10 2$3$4$5¾14 3$4$5$6¾18 4$5$6$7¾22 5$6$7$8¾26 bR;kfnA ;g iSVuZ vf/d jkspd rks ugha yxrk] fiQj Hkh gS rks ;g ,d iSVuZ ghA ;s ;ksxiQy 10 ls 'kq: gksus okyk ,d la[;kØe cukrs gSa] tks gj ckj 4 ds vUrj ls c<+rk tkrk gSA¸ ¶bl Øe dks bruh vklkuh ls njfdukj er djksA eSa rqEgsa blds ckjs esa ,d jkspd ckr crkrk gwaA vPNk] vc dqN cksryksa ds
¶budh cukoV dSlh yxrh gS\¸ ¶le>k] igys dks NksM+dj ckdh lcdh vkÑfr leyac ;k VSªihft;e (trapezium) tSlh gS] Bhd\¸ 49

¶fcYdqy BhdA blhfy, bu la[;kvksa dks leyacd la[;k,a (trapezoidal numbers) dgrs gSaA¸ ¶vkSj igyh la[;k dk D;k\¸ ¶og Hkh ,d leyacd gS] ysfdu ,d fo'ks"k izdkj dhA oSls gh tSls fd oxZ ,d fo'ks"k izdkj dk vk;r gSA¸ ¶;kfu f=kdks.kh; la[;k,a ,d fo'ks"k izdkj dh leyacd la[;k,a gaS] ;k fiQj ;ksa Hkh og ldrs gSa fd leyacd la[;k,a f=kdks.kh; la[;kvksa dks O;kidrk iznku djrh gSaA Bhd gS u\¸ ¶rqeus Bhd le>kA rqe cM+h rsth ls ,d twfu;j xf.krK curs tk jgs gksA D;k rqEgsa yxrk gS fd rqe bu pkSdfM+;ksa esa dqN vkSj iSVuZ [kkst ldrs gks\¸ ¶gka] bu pkSdfM+;ksa dh la[;kvksa ds xq.kuiQy dh vksj rks eSaus vHkh rd è;ku gh ugha fn;kA¸ ¶Bhd gS] rks mls Hkh ns[kksA¸ ¶buds xq.kuiQy gSa] 1×2×3×4= 24 2×3×4×5= 120 3×4×5×6= 360 4×5×6×7= 840 5×6×7×8=1680 6×7×8×9=3024 bR;kfnA

ysfdu buesa rks dksbZ iSVuZ fn[kkbZ ugha nsrkA¸ ¶bu xq.kuiQyksa dh vxyh o fiNyh la[;kvksa dh vksj è;ku nks] vkSj ns[kks fd dksbZ iSVuZ mHkjdj vkrk gS ;k ughaA¸ ¶gka&gka] ;g djds ns[krk gwaA budh fiNyh la[;k,a gSa 23] 119] 359] 839] 1679] 3023] bR;kfnA vc vxyh la[;kvksa dks 50

ns[krk gwa] ;s gSa 25] 121] 361] 841] 1681] 3025 oxSjgA buesa ls 25 vkSj 121 rks oxZ la[;k,a gSa] vkSj ckdh---A¸ ¶;gka est ij j[kh fdrkc esa oxZ la[;kvksa dh rkfydk nh gqbZ gS] tjk mlesa ns[kksA¸ ¶gka] fcYdqyA 361] 841] 1681] 3025] ;s lHkh oxZ la[;k,a gSaA ;kfu ;g iSVuZ tjk xgjk gSA gj pkSdM+h dh pkjksa la[;kvksa ds xq.kuiQy esa ,d tksM+us ij ges'kk oxZ la[;k feyrh gSA¸ ¶vfr mÙkeA rqEgsa bl [kkst esa vkuUn vk jgk gS] gS u\ pyks] vc ns[kks fd D;k rqe bu oxZ la[;kvksa dk pkSdfM+;ksa ls lh/k laca/ [kkst ldrs gks\ ;kfu dsoy pkSdM+h dks ns[kdj fcuk xq.kk fd, ;g tku ldrs gks fd xq.kuiQy esa ,d tksM+us ij fdl la[;k dk oxZ izkIr gksxk\ gj pkSdM+h dks è;ku ls ns[kks] vkSj irk yxkus dh dksf'k'k djksA¸ ¶vPNk] dksf'k'k djrk gawA 25¾52¾ (4$1)2] ;kfu 1 ls 4 okyh pkSdM+h ls feyk (4$1) dk oxZA 121¾112 ¾ (10$1)2] ;kfu 2 ls 5 dh pkSdM+h ls feyk (10$1) dk oxZA blh rjg vkSj vkxs c<+sa rks budh rkfydk cusxh % pkSdM+h oxZ la[;k ftldk oxZ 1 % 4 25 4$1 2 % 5 121 10$1 3 % 6 361 18$1 4 % 7 841 28$1 vc le> esa vk;kA gj pkSdM+h ls lEcfU/r oxZ ml pkSdM+h dh igyh vkSj vkf[kjh la[;k ds xq.kuiQy ls vxyh la[;k dk oxZ gSA ;g iz'u iwNus ds fy, /U;oknA vc eSa gj pkSdM+h dh Nksj okyh la[;kvksa] vkSj eè; dh la[;kvksa dk ;ksx ysdj rqyuk djuk pkgrk gwaA mudh ,d rkfydk cukrk gawA 51

pkjkas la[;kvksa dk ;ksx 10 14 18

eè; dh nks la[;kvksa dk ;ksx 5 7 9

Nksj dh nks la[;kvksa dk ;ksx 5 7 9 bR;kfnA

iSVuZ fcYdqy lh/k gSA pkSdM+h dh pkjksa la[;kvksa dk ;ksx eè; dh nks la[;kvksa ds ;ksx dk nksxquk gS] vkSj Nksj okyh nks la[;kvksa ds ;ksx dk Hkh nksxqukA¸ ¶cgqr vPNs] vkSj dqN\¸ ¶;fn pkSdM+h dh igyh vkSj rhljh la[;k ds ;ksx] vkSj nwljh o pkSFkh la[;k ds ;ksx dh rqyuk d:a] rks\ ,d rkfydk cukdj ns[krk gwa fd blls D;k mHkjrk gSA igyh o rhljh la[;k dk ;ksx 4 6 8 10

nwljh o pkSFkh la[;k dk ;ksx 6 8 10 12

;gka Hkh iSVuZ ,dne lh/k gS] ysfdu vf/d xgjk ughaA gj pkSdM+h esa igyh o rhljh la[;k dk ;ksx nwljh o pkSFkh la[;k ds ;ksx ls 2 de gSA pkpkth] esjk ,d iz'u gSA ge dsoy Øeokj la[;kvksa dks gh D;ksa ns[k jgs gaS\¸ ¶vPNk] rks D;k vc rqEgkjk eu bl dk;Z ls mpVus yxk gS\¸ ¶,slh ckr ugha gSA eq>s vkids fn, pkVksZa esa Hkh ,d iSVuZ utj vkus yxk gS] vkSj eSa vc vuqeku yxk ldrk gwa fd vxyk pkVZ dkSu&lk gksxkA esjk eryc gS fd vc blesa mruk vkuUn ugha jgk tks 'kq: esa FkkA¸ ¶vPNk] rks Bhd gS] vxys gÝrs dh eqykdkr esa rqEgsa dqN u;k feysxkA¸ 52

[kkst vfHk;ku % pkSFkk nkSj f=kdks.kh; la[;kvksa ds Øeokj tksMs+ ¶rks ;g jgk vkt ds [kkst vfHk;ku dk pkVZA bls ns[kdj crkvks fd D;k rqEgsa ,sls pkVZ dh vIks{kk Fkh\¸ 1 1 1

3 3 3

6 6 6

10 10 10

15 15 15

21 21 21

28 28 28

36 36 36

45 45 45

55 55 55

66 66 66

¶ugha] eSaus fcYdqy ugha lkspk Fkk fd vkt dk pkVZ ,slk gksxkA /U;oknA rks vki pkgrs gSa fd eSa bl pkVZ eas n'kkZ, f=kdks.kh; la[;kvksa ds Øeokj tksM+ksa dk vè;;u d:aA Bhd\¸ ¶gkaA fcYdqy BhdA pyks] 'kq: gks tkvksA¸ ¶igys eSa gj tksM+s dh la[;kvkas ds ;ksx dh vksj è;ku nsrk gwaA ;s ;ksx gS % 1 $ 3 ¾ 4 10 $ 15 ¾ 25 3 $ 6 ¾ 9 15 $ 21 ¾ 36 6 $ 10¾ 16 21 $ 28 ¾ 49 cgqr lqUnjA nks Øeokj f=kdks.kh; la[;kvkas dk ;ksx rks oxZ la[;k gSaA¸ ¶cgqr vPNsA vc ;fn eSa rqels nloas ,sls tksM+s dk ;ksx fcuk mls ns[ks fudkyus dks dgwa\ D;k dj ldrs gks\¸ ¶dksf'k'k djrk gwaA igys tksM+s dk ;ksx gS 2 dk oxZA nwljs dk gS 3 dk oxZ] vkSj rhljs dk gS 4 dk oxZA le>k] rks bl izdkj nlosa tksM+s dk ;ksx gksxk 11 dk oxZ] ;kfu 121A Bhd\¸ ¶'kkunkjA vkxs c<+ksA¸ ¶vc eSa bu tksM+kas ds xq.kuiQyksa dh vksj ns[krk gwaA ;s xq.kuiQy 53

gSa % 1×3=3 3 × 6 = 18

10 × 15= 150 15 × 21= 315

6 × 10 = 60

bR;kfnA

bl iSVuZ esa dqN [kkl ugha fn[krkA cl] lHkh xq.kuiQy 3 ls foHkkftr gksrs gSaA budks nks xq.kdksa esa rksM+dj ns[krk gwa] ftuesa ls ,d 3 gks] 'kk;n dqN u;k lkeus vk,A 3 ¾ 3×1 150 ¾ 50 × 3 18¾ 6 × 3 315 ¾ 105 × 3] bR;kfnA 60¾ 20 × 3 cl eSaus rks gkj eku yhA pkpkth] vc eSa rhu yxkrkj f=kdks.kh; la[;kvksa dks ns[kuk pkgrk gwaA¸ 1 1 1 1

3 3 3 3

6 6 6 6

10 10 10 10

15 15 15 15

21 21 21 21

28 28 28 28

36 36 36 36

45 45 45 45

55 55 55 55

66 66 66 66

¶Bhd gS] og gjk pkVZ ys yks] vkSj 'kq: gks tkvksA¸ ¶igys eSa bu frdfV;ksa ds Nksj okyh la[;kvksa ds ;ksx dh rqyuk gj frdVh dh chp okyh la[;k ls djrk gwaA bldh rkfydk bl izdkj cusxhA Nksj okyh la[;kvksa dk ;ksx (1$6 ¾) 7 (3$10 ¾) 13 (6$15 ¾) 21 (10$21 ¾) 31 (15$28¾) 43 54

eè; dh la[;k 3 6 10 15 21

(21$36¾) 57

28 oxSjgA

iSVuZ lkiQ rks fn[kkbZ ugha nsrkA pyks] gj ;ksxiQy dks frdVh dh eè; la[;k ls foHkkftr djrk gaw] 'kk;n dqN feysA 7 ÷ 3 ¾ 2] 'ks"k 1 31 ÷ 15 ¾ 2] 'ks"k 1 13 ÷ 6 ¾ 2] 'ks"k 1 43 ÷ 21 ¾ 2] 'ks"k 1 21 ÷ 10 ¾ 2] 'ks"k 1 57 ÷ 28 ¾ 2] 'ks"k 1 vc le> vk;kA gj frdVh esa Nksj dh la[;kvksa dk ;ksx chp dh la[;k ds nksxqus dh vxyh la[;k gSA¸ ¶;g rks rqeus deky dj fn[kk;kA vkSj vkxs c
eè; dh la[;k 3 6 10 15 21 bR;kfnA

gka] le> esa vk;kA Øeokj f=kdks.kh; la[;kvksa dh gj frdVh esa Nksj okyh la[;kvkas dk xq.kuiQy 3 ls foHkkftr gksrk gSA vc bu xquiQyksa dks nks lcls utnhdh xq.kdkas esa foHkkftr djds fy[krk gwaA xq.kuiQy 6 30 90 210 420

nks lcls utnhdh xq.kd 2×3 5×6 9 × 10 14 ×15 20 ×21 bR;kfnA 55

igyh ckr ;g fd gj ckj nksukas xq.kd Øeokj la[;k,a gSaA vc igys xq.kdksa dh Ük`a[kyk (2] 5] 9] 14] 20] ---) ij è;ku nsa rks blesa nks Øeokj la[;kvksa ds vUrj gSa% 3] 4] 5] 6 bR;kfn] tks 3 ls 'kq: gksdj izkÑfrd la[;kvksa dh Üka`[kyk gS A blds vykok] gj xq.kuiQy ds nks xq.kdksa esa ls ,d f=kdks.kh; la[;k gS A pkpkth] ge ges'kk Øeokj la[;kvksa ds lewgksa dk gh vè;;u D;ksa djrs gSa\¸ ¶;g t:jh ughaA vxyh ckj tc ge feysaxs] rks rqEgsa dqN vkSj u;k feysxkA¸

56

[kkst vfHk;ku % ikapoka nkSj fo"ke la[;kvksa dh Ük`a[kykvksa ds ;ksxiQy ¶bl fupys pkVZ dks ns[kksA dksbZ [kkl ckr fn[kkbZ nsrh gS\¸ 1 1 1 1 1

3 3 3 3 3

5 5 5 5 5

7 7 7 7 7

9 9 9 9 9

11 11 11 11 11

13 13 13 13 13

15 15 15 15 15

17 17 17 17 17

19 19 19 19 19

¶;s rks fo"ke la[;kvksa dh Ük`a[kyk,a gSaA vki pkgrs gSa fd eSa Øeokj vkus okyh fo"ke la[;kvksa dk vè;;u d:a\¸ ¶gka] ysfdu Ük`a[kyk dh yEckbZ dks ,d&,d dj c<+rs gq,] tSlk fd pkVZ esa fn[kk;k x;k gSA¸ ¶Bhd gS] eSa igys buds ;ksxiQyksa dk vè;;u djrk gwaA 1$3¾4 1$3$5¾9 1 $ 3 $ 5 $ 7 ¾ 16 1 $ 3 $ 5 $ 7 $ 9 ¾ 25 bR;kfn A

;gka Hkh ,d lVhd vkSj lqUnj iSVuZ ekStwn gSA ,d ls 'kq: gksus okyh fo"ke la[;kvksa dh gj Ük`a[kyk dk ;ksxiQy ,d oxZ la[;k gksrh gSA¸ ¶BhdA vc bl pkVZ dks mBkvksA blesa Hkh fo"ke la[;k,a gSaA ns[kks fd blesa D;k djuk gS\¸

1 1 1 1

3 3 3 3

5 5 5 5

7 7 7 7

9 9 9 9

11 11 11 11

13 13 13 13

15 15 15 15

17 17 17 17

19 19 19 19

57

¶pkpkth] vkius rks eq>s vpEHks esa Mky fn;kA ;g rks okLro esa cM+k jkspd [kkst vfHk;ku gSA Bhd gS] eSa gj drkj dh la[;kvksa dks ns[krk gwaA 1 ¾ 1 3 $ 5 ¾ 8 7 $ 9 $ 11 ¾ 27 13 $15 $ 17 $ 19 ¾ 64 21 $ 23 $ 25 $ 27 $ 29 ¾ 125 bR;kfnA egku vk'p;Z! ;s lHkh ;ksxiQy rks ?ku la[;k,a gSaA bldk vFkZ gS fd igyh fo"ke la[;k] ;kfu 1] ?ku la[;k gSA mlls vxyh nks fo"ke la[;kvksa dk ;ksx Hkh ?ku la[;k gSA mlls vxyh rhu fo"ke la[;kvksa dk ;ksx Hkh ?ku la[;k gSA vkSj ;g Øe blh izdkj tkjh jgrk gSA¸ ¶;kfu ge bUgsa lewgksa dh Ük`a[kyk dgsa rks\¸ ¶vkidk eryc gS fd igys lewg esa dsoy ,d fo"ke la[;k gS] vkSj og gS 1_ nwljs lewg esa vxyh nks fo"ke la[;k,a gSa] 3 vkSj 5] ftudk ;ksx gS 8_ rhljs lewg esa mlls vxyh rhu fo"ke la[;k,a gSa] 7] 9] vkSj 11] ;ksxiQy 27_ pkSFks lewg esa vxyh pkj fo"ke la[;k,a gSa] 13] 15] 17] vkSj 19] ;ksxiQy 64] bR;kfnA¸ ¶D;k rqe crk ldrs gks fd ,sls nlosa lewg esa vkus okyh igyh fo"ke la[;k dkSu&lh gS\¸ ¶vki pkgrs gSa fd bu lewgksa esa vkus igyh la[;kvksa dk vè;;u d:aA Bhd\¸ 58

¶gka] rqe Bhd le>sA vkxs c<+ksA¸ ¶eSa igys lewg dks NksM+ nsrk gwaA nwljs ls 'kq: djds ;fn rkfydk cukmQa] rks bl izdkj cusxh % lewg Øe&la[;k lewg dh igyh fo"ke la[;k 2 3 3 7 4 13 bR;kfnA nwljs LrEHk dh la[;k dks ;fn igys LrEHk dh lacaf/r la[;k ls foHkkftr d:a] rks feysxk %

;g rks okdbZ vk'p;Ztud gSA 'ks"kiQy ges'kk 1 vkrk gS] vkSj HkkxiQy foHkktd ls ,d deA vkSj foHkktd rks lewg dh Øe&la[;k gh gSA ftl la[;k dks foHkkftr fd;k tk jgk gS] og gS lewg dh igyh fo"ke la[;kA eSa le> x;kA pkpkth] vc eSa vkids iz'u dk mÙkj ns ldrk gawA nloka lewg 91 ls 'kq: gksxk] D;ksafd foHkktd gS 10] HkkxiQy 9] vkSj 'ks"kiQy 1] ;kfu lewg dh igyh fo"ke la[;k gksxh 10 × 9 $ 1 ¾ 91A¸ ¶HkbZ] ;g rks rqeus deky gh dj fn;kA vc crkvks fd nlosa lewg esa fo"ke la[;kvksa dk ;ksxiQy D;k gksxk\¸ ¶eSa nlosa lewg dh lkjh fo"ke la[;k,a] vkSj mudk ;ksx vkidks crkrk gwaA Bhd gS\]¸ ¶Bhd gS] crkvksA¸ 59

¶91 $ 93 $95 $ 97 $ 99 $ 101 $ 103 $ 105 $107 $ 109 ¾ 1000¸ ¶yktokcA¸ ¶rks pkpkth] D;k bl izdkj dk vkSj dksbZ pkVZ vHkh ckdh gS\¸ ¶;fn rqe ,slk ,d vkSj pkVZ pkgrs gks] rks ;g pkVZ mBk yksA¸ 1 1 1 1 1

3 3 3 3 3

5 5 5 5 5

7 7 7 7 7

9 9 9 9 9

11 11 11 11 11

13 13 13 13 13

15 15 15 15 15

17 17 17 17 17

19 19 19 19 19

21 21 21 21 21

¶vksg] le>kA vki pkgrs gSa fd eSa pkj Øeokj fo"ke la[;kvksa ds lewgksa dk vè;;u d:aA buds ;ksxiQy fudkydj mUgsa ns[krk gwawA 1 $ 3 $ 5 $ 7 ¾ 16 3 $ 5 $ 7 $ 9 ¾ 24 5 $ 7 $ 9 $ 11¾ 32 7 $ 9 $ 11 $13¾ 40 bR;kfnA le> x;kA pkj yxkrkj fo"ke la[;kvksa dk ;ksx 8 ls foHkkftr gksrk gSA Bhd\¸ ¶vc rqe cM+h tYnh iSVuZ dks igpkuus yxs gksA¸ ¶pkpkth] D;k vkids ikl oxZ vkSj ?ku la[;kvksa ds pkVZ Hkh gSa\¸ ¶vPNk] rqe ,slk D;ksa ugha djrs fd ,sls pkVZ [kqn cukvks] vkSj tSls pkgks mudk vè;;u djksA¸ ¶èkU;okn] eSa ,slk gh d:axk] vkSj viuh [kkstksa ds ckjs esa vkidks vxys gÝrs crykmaQxkA¸ 60

[kkst vfHk;ku % NBk nkSj oxZ la[;kvksa ds Øeokj tksM+s igys eSaus oxZ la[;kvksa dks ysdj muds nks pkVZ cuk,] ftuesa igyk uhps fn;k gqvk gS % 1 1 1

4 4 4

9 9 9

16 16 16

25 25 25

36 36 36

49 49 49

64 64 64

81 81 81

100 100 100

;g pkVZ oxZ la[;kvksa ds Øeokj tksM+ksa ds vè;;u dk U;kSrk nsrk gSsA ;fn bu tksM+ksa dh la[;kvksa dk ;ksxiQy fy;k tk,] rks mlesa dksbZ jkspd iSVuZ ugha fudyrk] flok; blds fd lHkh ;ksxiQy fo"ke la[;k,a gSaA nks Øeokj oxks± dk vUrj Hkh fo"ke la[;k gh gSA bu tksM+ksa ds ;ksxiQy lHkh ,d&nwljs ls fHkUu gSa] vr% esjh ftKklk bu ;ksxiQyksa dh Ük`a[kyk dh vksj c<+h] ;kfu 5] 13] 25] 41] 61] 85 vkfnA eSaus bl Ük`a[kyk ds gj Øeokj tksMs+ ds vUrj dks ns[kuk 'kq: fd;k] rks ;g feyk 8] 12] 16] 20] 24] --- bR;kfnA vc ;fn bl Ük`a[kyk ds tksM+ksa dk Hkh vUrj fy;k tk,] rks feyrk gS% 4] 4] 4] --- gS u ;g jkspd\ vc eSaus oxZ la[;kvksa ds gj tksM+s ds vUrj dh vksj è;ku fn;k] D;ksafd ;s Hkh ,d&nwljs ls fHkUu FksA 3] 5] 7] 9] 11] --vkfnA bl Ük`a[kyk ds tksM+ksa dk vUrj fy;k tk,] rks feyrk gSs 2] 2] 2] 2] --eSaus dqN u;k [kkst fudkyk FkkA oxks± ds tksM+ksa ds ;ksxiQy ds lkFk rks Ük`a[kyk dk vUrj nwljs pj.k esa tkdj cjkcj gqvk] ysfdu tc tksM+ksa ds vUrj dh Ük`a[kyk cukbZ xbZ rks igys pj.k esa gh 61

mldk vUrj leku gks x;kA esjs fy, ;g [kkst ,d u;k vkSj fHkUu vuqHko Fkk] vkSj blus esjs vè;;u ds fy, ,d ubZ jkg [kksy nhA eSaus tks nwljk pkVZ cuk;k] og uhps fn;k x;k gS % 1 1 1 1

4 4 4 4

9 9 9 9

16 16 16 16

25 25 25 25

36 36 36 36

64 64 64 64

49 49 49 49

81 81 81 81

100 100 100 100

;g pkVZ nkor nsrk gS oxZ la[;kvksa dh Øeokj frdfV;ksa ds vè;;u dhA gj frdVh dh Nksj okyh la[;kvksa dk ;ksxiQy fudkydj eSaus tc mldh rqyuk eè; dh la[;k ls dh] rks ,d cM+k xgjk iSVuZ lkeus vk;kA frdVh dh Nksj okyh la[;kvksa dk ;ksx

eè; dh la[;k

(1 $ 9 ¾ ) 10 (4 $ 16 ¾) 20 (9 $ 25 ¾) 34 (16$ 36 ¾) 52 ;fn igys LrEHk dh la[;k dks nwljs LrEHk foHkkftr djsa] rks feyk

4 9 16 25 dh la[;k ls

bR;kfnA ;kfu] oxZ la[;kvksa dh Øeokj frdfV;ksa dh Nksj okyh la[;kvksa dk ;ksxiQy eè; dh la[;k ds nksxqus ls nks vfèkd gSA eSaus tks rhljk pkVZ cuk;k] og ?ku la[;kvksa ij vkèkkfjr FkkA 62

1 1 1 1

8 8 8 8

27 27 27 27

64 64 64 64

125 125 125 125

216 216 216 216

343 343 343 343

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729 729 729 729

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;g pkVZ gekjk è;ku vkdf"kZr djrk gS ?ku la[;kvksa dh Ük`a[kykvksa ij tks 1 ls 'kq: gksrh gS] vkSj mudh yEckbZ Øe'k% c<+rh tkrh gSA ;kfu igyh Ük`a[kyk esa dsoy ,d ?ku la[;k (1)] nwljh esa igyh nks ?ku la[;k,a (1] 8)] fiQj igyh rhu (1] 8] 25)] fiQj pkj (1] 8] 25] 64)] bR;kfnA eSaus bu Ük`a[kykvksa ds ;ksxiQy fudkys] rks eSa pfdr jg x;k] fd os lHkh oxZ la[;k,a fudyhaA 1 $ 8 ¾ 9 1 $ 8 $ 27 ¾ 36 1 $ 8 $ 27 $ 64 ¾ 100 bR;kfnA ,d ls 'kq: gksus okyh ?ku la[;kvksa dh izR;sd Ük`a[kyk dk ;ksxiQy ,d oxZ la[;k gSA ;fn Ük`a[kyk ds LFkku ij lewg 'kCn dk iz;ksx djsa rks ;ksa Hkh dg ldrs gSa % igyh nks ?ku la[;kvksa ds lewg dk ;ksxiQy 3 dk oxZ gSA igyh rhu ?ku la[;kvksa ds lewg dk ;ksxiQy 6 dk oxZ gSA igyh pkj ?ku la[;kvksa ds lewg dk ;ksxiQy 10 dk oxZ gSA eq>s rqjUr ;g vkHkkl gqvk fd 3] 6] 10] ;s lHkh f=kdks.kh; la[;k,a gSaA blls eq>s viuh [kkst dks ,d u, izdkj ls O;Dr djus dk ekxZ feykA igyh nks ?ku la[;kvksa dk ;ksxiQy nwljh f=kdks.kh; la[;k dk oxZ gSA igyh rhu ?ku la[;kvksa dk ;ksxiQy rhljh f=kdks.kh; la[;k dk oxZ gSA bR;kfnA 63

5 vyfonk

o

xZ vkSj ?ku la[;kvksa dh viuh [kkstksa dks ysdj eSa pkpkth ls feyus x;kA og esjs dk;Z dks ns[kdj cgqr izlUu gq, vkSj eq>s xys yxk;kA esjs lkFk&lkFk pkpkth dks Hkh bl liQyrk ij xoZ dk vuqHko gks jgk FkkA mUgksaus eq>s ^la[;kvksa ds lkFk vkeksn&izeksn* uked iqLrd HksaV esa nhA mUgksaus crk;k fd vxys Ng eghus og fons'kksa esa O;k[;ku nsus ds fy, nkSjs ij tk jgs gSa vkSj okil vkdj esjs lkFk lIrkgkar dh eqykdkrsa fiQj ls 'kq: djsaxsA eq>s FkksM+k nq[k gqvk fd brus gÝrksa rd eSa muls ugha fey ldwaxkA eSaus muls iwNk fd dbZ fo"k;ksa] tSls le la[;k,a] vHkkT; la[;k,a] feJ la[;k,a bR;kfn ij dksbZ pkVZ D;ksa ugha gSA mUgksaus dgk] ¶rqe vc ,d twfu;j xf.krK cu pqds gks vkSj bl [kkst vfHk;ku dks fcuk fdlh lgk;rk ds viuh Lo;a dh {kerk vkSj Kku ds vk/kj ij tkjh j[k ldrs gksA blfy, rqEgsa pkfg, fd Lo;a] tSlk pkgks oSlk u, pkVZ rS;kj djks] tks mu la[;kvksa ij vk/kfjr gksa ftuij [kkst djus esa rqEgkjh #fp gSA esjh 'kqHkdkeuk,a rqEgkjs lkFk gSaA¸ pkpkth dh ckrsa lqudj eSa vius&vki dks lkrosa vkleku ij eglwl dj jgk FkkA eSaus mudh lykg igys u ekuus ds fy, {kek ekaxhA ¶dksbZ [ksn u djks] ,slk cgqrksa ds lkFk gksrk gS¸] mUgksaus dgkA pyus ds igys mUgksaus eq>ls cSaxuh pkVZ ysus dks dgkA 1$2 ¾ 3 4$5$6 ¾ 7$8 9$10$11$12 ¾ 13$14$15 16$17$18$19$20 ¾ 21$22$23$24 64

vkSj mUgksaus eq>ls bl izdkj nloka lehdj.k crkus dks dgk] ysfdu fcuk chp ds lehdj.kksa dks fy[ksA vHkh rd tks eSa djrk vk;k Fkk] ;g mlls ,dne fHkUu Fkk] vkSj blfy, dkiQh pqukSrhiw.kZA eSa bl pqukSrh esa liQy gksus dk vkuUn vo'; ikuk pkgrk FkkA ges'kk dh rjg eSaus ,d fu;fer :i ls bl iz'u dk vè;;u 'kq: fd;kA lehdj.k Øe&la[;k 1 2 3 4

ckb± vksj dh igyh la[;k 1 4 9 16 bR;kfnA

;g lkiQ Fkk fd gj lehdj.k ,d oxZ la[;k ls izkjaHk gks jgk FkkA igyk lehdj.k 1 ds oxZ ls izkjEHk gqvkA nwljk lehdj.k 2 ds oxZ ls izkjEHk gqvk] vkSj blh izdkj vkxs HkhA blfy, nloka lehdj.k 10 ds oxZ] ;kfu 100 ls 'kq: gksuk pkfg,A ysfdu ;g pkpkth ds iz'u dk vkaf'kd mÙkj gh FkkA vc eq>s ;g irk djuk Fkk fd lehdj.k dh ckbZa vksj fdruh la[;k,a gksuh pkfg,] vkSj nkfguh vksj fdruhA lehdj.k Øe la[;k 1 2 3 4

ckbZa vksj la[;kvksa dh la[;k 2 3 4 5

nkfguh vksj la[;kvksa dh la[;k 1 2 3 4 bR;kfnA

bl izdkj igys lehdj.k esa ckbZa vksj nks la[;k,a gSa] vkSj nkfguh vksj ,dA nwljs esa ckbZ vksj rhu] vkSj nkfguh vksj nks] vkSj ;g iSVuZ tkjh jgrk gSA blfy, nlosa lehdj.k esa ckbZa vksj 11 la[;k,a gksuh pkfg,] vkSj nkfguh vksj 10 A 65

bu fu"d"kZ ij igqaprs gh] eSaus nloka lehdj.k fy[k fn;kA 100$101$102$103$104$105$106$107$108$109$110 ¾ 111$112$113$114$115$116$117$118$119$120 pkpkth izlUu gksdj [kM+s gks x, vkSj eq>ls gkFk feykdj mUgksaus esjs dk;Z dks ljkgkA mUgksaus dgk fd xf.kr ds {ks=k esa ubZ [kkst djus ds fy, mfpr iz'u mBkus dh ;ksX;rk gksuk lcls vf/d vko';d gSA mUgksaus eq>ls iwNk fd bl pkVZ esa fn[kk, iSVuZ ls lacaf/r vkSj dkSu&ls iz'u mBk, tk ldrs gSaA eSaus dgk % 1- fdlh lehdj.k esa nkfguh vksj dh igyh la[;k dkSu&lh gksxh\ 2- fdlh lehdj.k dh ckbZa vksj dh la[;kvksa dk ;ksxiQy D;k gS\ 3- fdlh lehdj.k dh nkfguh vksj dh la[;kvksa dk ;ksxiQy D;k gS\ 4- fdlh lehdj.k dh nkfguh vksj dh vfUre la[;k dkSu&lh gS\ 5- fdlh lehdj.k dh ckbZa vksj dh vfUre la[;k dkSu&lh gS\ 6- D;k le la[;kvksa ;k fdlh la[;k ds xq.ktksa dks ysdj ge ,sls gh lehdj.k cuk ldrs gSa\ ¶ns[kk] fu'p; gh rqe ,sls iz'u mBkus vkSj Lo;a muds mÙkj [kkstus dh {kerk vkSj cqf¼eÙkk j[krs gksA 'kks/dk;Z djus ds fy, blh dh vko';drk gSA eSa ?kks"k.kk djrk gwa fd rqe vc ,d uUgsa 'kks/d xf.krK cu x, gksA¸ ¶pkpkth] uUgsa xf.krK ls ,d iw.kZ ofj"B xf.krK cuus esa fdruk le; yxsxk\¸ 66

¶tc fdlh esa vius vuqekuksa dks lgh ;k xyr fl¼ djus dh {kerk vkSj dq'kyrk fodflr gksus yxs rks le>uk pkfg, fd ;g ofj"B xf.krK cuus dh jkg ij py iM+k gSA vkf[kjdkj] ;s lHkh fof'k"Vrk,a tks rqeus [kksth gaS] bUgsa vHkh dsoy ,d equkflc vuqeku gh ekuk tk ldrk gS] ;kfu ;s lR; izrhr rks gksrh gSa] ij fu'p;iwoZd ugha dgk tk ldrkA okil ykSVus ds ckn 'kh?kz gh eSa rqEgsa xf.kr ds vuqekuksa dks fl¼ djus ds fo"k; esa crkmQaxkA¸ ¶Bhd gS] pkpkthA u, vkuUn rks eSa [kkstrk gh jgrk gwa] ij bu dqN gÝrksa esa vkius thou ds ,d vuwBs vkuUn ls esjk ifjp; djk;k gSA eSa fdl izdkj vkidks /U;okn nwaA¸ ;g dgrs esjh vka[ksa Hkj vkbZaA eSaus pkpkth dks iz.kke fd;k vkSj muls fonk ysdj py iM+kµ vc vdsys gh la[;k&txr ds bl [ksy&dwn Hkjs [kkst&vfHk;ku ij vkxs c<+us ds fy,A

67

6 la[;k&fof'k"Vrk,a % la{ksi esa 12345-

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1268

nks Øeokj la[;kvksa dk ;ksxiQy ,d fo"ke la[;k gksrh gSA nks Øeokj la[;kvksa dk vUrj lnk ,d gksrk gSA nks Øeokj la[;kvksa dk xq.kuiQy le la[;k gksrh gSA fdlh la[;k ds oxZ dks mldh vxyh la[;k ls foHkkftr djus ij 'ks"kiQy ,d feyrk gSA (1] 2) dks NksM+dj] Øeokj la[;kvksa ds gj tksM+s esa] cM+h la[;k ds oxZ dks NksVh ls foHkkftr djus ij 'ks"kiQy ,d feyrk gSA rhu Øeokj la[;kvksa dh fVdVh esa igyh vkSj vkf[kjh la[;k dk ;ksx eè; dh la[;k dk nksxquk gksrk gSA frdVh dh Nksj dh la[;kvksa dk xq.kuiQy eè; dh la[;k ds oxZ ls ,d de] ;kfu mldh fiNyh la[;k gksrh gSA rhu Øeokj la[;kvksa dk ;ksxiQy rhu dk xq.kt gS] ;kfu rhu ls iw.kZr% foHkkftr gksrk gSA rhu Øeokj la[;kvksa dk xq.kuiQy Ng dk xq.kt gSA frdVh dh igyh nks la[;kvksa ds xq.kuiQy dks rhljh la[;k ls foHkkftr djus ij 'ks"kiQy 2 feyrk gSA rhu Øeokj la[;kvksa dh frdVh dh Nksj okyh la[;kvksa ds xq.kuiQy dks ;fn chp okyh la[;k ls foHkkftr fd;k tk,] rks HkkxiQy vkSj 'ks"kiQy ,d leku gksrk gS] vkSj ;g la[;k chp dh la[;k dh fiNyh la[;k gksrh gSA 3 ;k mlls cM+h la[;k ls 'kq: gksus okyh fdlh Hkh frdVh esa] ;fn vfUre nks la[;kvksa ds xq.kuiQy dks igyh la[;k ls

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foHkkftr fd;k tk,] rks HkkxiQy foHkktd ls 3 vf/d gksrk gSA fdlh Hkh pkj Øeokj la[;kvksa ds lewg (pkSdM+h) esa chp dh nks la[;kvksa dk ;ksx Nksj okyh la[;kvksa ds ;ksx ds cjkcj gksrk gSA fdlh Hkh pkSdM+h esa Nksj okyh la[;kvksa dk xq.kuiQy eè; dh nks la[;kvksa ds xq.kuiQy ls 2 de gksrk gSA nks ;k mlls vf/d fdlh la[;k ls 'kq: gksus okyh fdlh Hkh yEckbZ dh Øeokj Ük`a[kyk ds ;ksxiQy dks leyacd la[;k dgrs gSaA gj f=kdks.kh; la[;k ,d leyacd la[;k gS] ijUrq gj leyacd la[;k f=kdks.kh; ughaA fdlh pkSdM+h dh pkjksa la[;kvksa dk xq.kuiQy] pkSdM+h dh Nksj okyh la[;kvksa ds xq.kuiQy dh vxyh la[;k ds oxZ ls ,d de gksrk gSA fdlh pkSdM+h dh pkjksa la[;kvksa dk ;ksx eè; dh nks la[;kvksa ds ;ksx dk nksxquk gksrk gS] vkSj Nksj dh nks la[;kvksa ds ;ksx dk Hkh nks xqukA pkSdM+h dh igyh vkSj rhljh la[;k dk ;ksx nwljh vkSj pkSFkh la[;k ds ;ksx ls nks de gksrk gSA pkSdM+h dh pkjksa la[;kvksa dk ;ksxiQy 8 dk xq.kt gksrk gSA nks Øeokj f=kdks.kh; la[;kvksa dk ;ksx lnk ,d oxZ la[;k gksrh gSA rhu Øeokj f=kdks.kh; la[;kvksa esa Nksj okyh nks la[;kvksa dk ;ksx chp dh la[;k ds nksxqus dh vxyh la[;k gksrh gSA rhu Øeokj f=kdks.kh; la[;kvksa esa Nksj okyh nks la[;kvksa dk xq.kuiQy 3 dk ,d xq.kt gS] ;kfu og 3 ls iw.kZr% foHkkftr gksrk gSA 69

23- fdlh izkÑfrd la[;k ^d* ds fy, (d) (d&1) $ 1 ls izkjaHk djds ^d* Øeokj fo"ke la[;kvksa dk ;ksxiQy ^d* ds ?ku ds cjkcj gksrk gSA mnkgj.k ds fy,] ;fn d¾ 3] rc 7$9$11¾27] tks fd 3 dk ?ku gSA 24- ,d ls 'kq: gksus okyh fo"ke la[;kvksa dh fdlh Hkh Øeokj Ük`a[kyk dk ;ksx ,d oxZ la[;k gksrh gSA mnkgj.k ds fy, 1$3$5$7 ¾ 16] ;kfu 4 dk oxZA 25- fo"ke la[;kvksa ds Øeokj tksM+ksa ds ;ksxiQyksa dh ;fn ,d Ük`a[kyk cukbZ tk, rks bl Ük`a[kyk ds inksa dk lkekU; vUrj nwljh voLFkk esa 4 gksrk gSA 26- fo"ke la[;kvksa ds Øeokj tksM+ksa ds vUrjksa dh ;fn ,d Ük`a[kyk cukbZ tk,] rks bl Ük`a[kyk ds inksa dk lkekU; vUrj igyh voLFkk esa 2 gksrk gSA 27- rhu Øeokj oxZ la[;kvksa dh frdVh esa Nksj okyh la[;kvksa dk ;ksxiQy eè; dh la[;k ds nksxqus ls 2 vf/d gksrk gSA 28- ,d ls 'kq: gksdj ^d* Øeokj ?ku la[;kvksa dk ;ksxiQy] f=kdks.kh; la[;k dh Ük`a[kyk esa ^d* LFkku ij vkus okyh la[;k dk oxZ gksrk gSA 29- (d&1)×2 ls izkjEHk gksus okyh izkÑfrd la[;kvksa dh Ük`a[kyk esa igyh ^d* la[;kvksa dk ;ksx] mlls vxyh (d&1) la[;kvksa ds ;ksx ds cjkcj gksrk gSA ❐

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