Kerala Panineeyam


kn{]IcWw

1. knhn`mKw

"kn' F ]Zn\v kmam\yamb Aw "tN' FmWtm. ckX{{]]Zmfn NneXv Xn tNcptm AhbpsS hw apXemb KpW amdntmIpp. apNneXv Xn tNcptm KpW am{Xa, ]ZmwXsbpw amdpp. thsd NneXv Xn F{XXs tNmepw bmsXmcwinepw amddw hcmsX AXmXns kvYnXnbns Ccnpp. CXpt]mse Ac, Asn hymIcWimkv{X{]Imcap h, Xn tNcptmgpw HmtcmXcw am Dmbnpw CmsXbpw hcpXmWv. BhI kwKXnIsfn hnhcnp `mKn\mWv hymIcWn "kn{]IcWw' Fp ]dbpXv.

h Xnep tbmKns et`Za\pkcnv ]mWn\n apXemb kwkvIrXhymIcWImm hn`PnnpXpt]mse, aebmfnepw kn kmam\ys ]Zaykn, ]Zmkn, D`bkn Fns\ aqpXcambn XncnmhpXmWv. {]IrXn{]Xyb tNnmWtm ]Zw DmIpXv. As\ Hcp ]Zn A`hnnp B cwi tNcptm am{XapmIp hhnImcw ]Zayknp hnjbw; cp ]Z Xn tNcptm am{XapmIpXv ]Zmknp hnjbw; ]Zayknbnepw ]Zmknbnepw hcpXv D`bknp hnjbw. DZmlcWw:

		{]IrXn	{]Xybw
]Zaykn acn	=	acw+	C
]Zmkn s]mq	=	s]m+ ]q
D`bkn aWnbdbn	=	aWn+Ad+	C

C\n knbn Xn tNcp hfpsS kzchyRvP\t`Za\pkcnv, kzc Xn tNcptm DXv "kzckn', kzcw hyRvP\tmSp tNcptm DXv "kzchyRvP\kn', hyRvP\w kzctmSp tNcptm DXv "hyRvP\kzckn', hyRvP\ Xn tNcptm DXv "hyRvP\kn' Cs\ knkmam\ys \mepXcambnncnmw:

DZm: kzckn  ag+ A= agb
kzchyRvP\kn Xmac+Ipfw= Xmacpfw
hyRvP\kzckn I+C= In
hyRvP\kn s\ev+aWn= s\Wn

kn hcptm hv DmImhp hnImcf\pkcnp t\mpXmbm "tem]kn', "BKakn', "ZznXzkn, "BtZikn' Cs\ \membnpw knkmam\ys XcwXncnmhpXmWv.

knphfn Hv CmsXbmIpXv "tem]w' knptm aqmasXmpwIqSn hptNcpXv "BKaw ', knphbn GsXnepw Hv CcnpXv "ZznXzw'; Hns kvYm\n asmmbnocpXv "BtZiw'

DZm : tem]w AX= AXv + A
BKaw agphn= agp + C
ZznXzw AhnsStmbn = AhnsS + t]mbn
BtZiw Fqdv = F + \pdv

Cs\ Hmtcm D]m[nt`Za\pkcnv ]eXcw hn`mKfpw sNmhpXmsWnepw HSphn ]d hn`mKsbmWv kuIcym ChnsS kzocInncnpXv. AXn "ZznXzkn' FXn\v, knp hfn HpXsbmWv BKaambn hcpXv Fp am{Xta BKaknsbm t`Zapp. kqvan AXpw BKaknbmbns hnNmcnmhpXmWv.

knbn Dmbnocp kIe hhnImcpw DmcWkuIcyamWv ASnkvYm\sapXv BhI hnImcfpsS kzcq]w t\mnbm Ffpn Adnbmw. hyRvP\v kzckwtbmKtmSpIqSnbmsX kv]jvSamb DmcWw kw`hnmXn\m cp hyRvP\ Xn tNmepw Acw Hp am{Xambnt hcpIbpp. Hcp DmcWnsbmWv B cp hyRvP\fpw ASnhcpXv Fw. kzcsmw {]tXyIwXs kv]amb DmcWw DXpsImv kzckwtbmKfnemWv hhnImcw A[nIambn BhiysSpXv. CuhI XzIqSn ChnsS {KlntXmIpp.

1. tem]kn

kzcnap]p tem]npw
kwhrXw hyamIbm;
AXns\ kzcambnt
hIhbvt knbn.

GsXnepw Hcp kzcw ]cambn hm kwhrtXmImcw tem]npw. hyRvP\w am{Xambnpcnp \ndpnbm kv]jvtSmmcWw kw`hnmXn\m "ImSpv', "\mSpv' apXemb ]Zfn Ahkm\n ImWp kwhrXw SImcns DmcWw kv]jvSammam{Xw DcnpXmWv. B kvYnXnv B hI kwhrXn\pta Hcp kzcw hcpXmbm B kzctmSp tNpwsImv Dcnmepw SImcn\p kv]jvSXbpmIpsancns kwhrXs Atm tNt Bhiyw hcpn. AXn\m kzcw ]camIpnSsmw kwhrXw DXmbns hnNmcntXpan.

DZm: XWpv + Dpv = XWppv.
Imddpv + ASnpp = ImddSnpp.

sNbvXp sNpp Fpsm
cmJymXmyapImchpw
tem]npw kZriwt]mse
kwhrtXm]PvRamIbm.

sNbvXp, sNpp apXemb IrXnIfpsS Ahkm\nep DImchpw kzcw]cambn hcptm tem]npw. IrXnIfpsS Ahkm\n ImWp Cu DImcnsbpw BZys kzcq]w hyRvP\fpsS DmcWkv]jvSXbvpthn tNp kwhrXambnpXsbmWv IrXnIsfmw hmIymhkm\n hcpXpsImv hmIyns _e]pjvSnpthn B kwhrXs hnhrXamn DcnpIbmWv sNbvXphcpXv. IrXywin Akm[mcWamb _ew ImWntnhcptm Cu amXncn hnhrtXmmcWwsImpt]mepw aXnbmImsX AXns\ ZoLamn DcnpIIqSn ]Xnhpv. "hp icabw' Fpw addpap {]tbmKw t\mpI. GsXmw kvYefn IrXnIptijw Ahntijw ImWnm tNp Jne[mXpcq]tfm ]pcpj{]Xybtfm Dmbnhcppthm BhI kvYefn Cu kwhrXw hmIymhkm\nembvIbm hmIy_en\pthn AXns\ hnhrXamn Dcnt Bhiyw hcpn. As\bp kvYefn tem]hpw hcpw. CXmWv "kZriwt]mse' Fp ]dnpXv. Jne[mXpcq]tfm ]pcpj{]Xybtfm ]camIptm Cu tem]w \nXyambnpw \n]mX ]cfmbm hnIev]ambnpw BWv hcpXv.

DZm:	\nXyw:	Ip+ C = In;	ImWpp+ Dv = ImWppv
		Ip+ B = Im;	ImWpp+ G = ImWpt.
	hnIev]w:	Ip+Hm = Ipthm, Itm

KZynsemw IrXnI hmIymhkm\n hcpXn\m ta]d Xcw i_vZfmsX addp i_vZ IrXnIptijw hcpn. AXn\m AhnsS knXs BhiysSpn. ]ZynemIs, "CXp' F \]pwkIcq]tmSpIqSn "sNpnXohoc' F amXncnbn {]tbmKnpIbmWp thXv. "l \opztlm! Fpw Nne {]tbmK ImWppv

A Cbvytem]
ambn t]mbnpsamt]m.

A, C Cu i_vZfpsS Ayamb AImchpw, Bbn, t]mbn ChbpsS Ayamb CImchpw kzcw ]camIptm kZriwt]mse tem]npw.

DZm: A+ Fv= Asv; C+ Fv= Csv
Bbn+ Fv= Bsbv; t]mbn+Fv= t]msbv

t]scfpsS Ahkm\nep AImcw tem]nv "]qWpnrXn' Fpw addpw {]tbmKnpXp icnb. BhI kvYefn "]pWpsmcnrXn Fp thXmWv. AXnse "Hcp' FXv "A' Fp t]sc{]Xybn\p ]Icw hcpXpamWv.

{]m\m{InbmwKn
ymImchpans\.

{InbmwKw Fm hn\sbw. AXns hIt`Zfn Hmb \Sphn\sbn\p k_lpam\amb t{]cWbpw Aapv. AXns HSphnes AImchpw tem]npw.

DZm: AdnI+ Aatcizc!= AdnIatcizc!
hcnI+ FtSm= hcnsItSm

AImcw ep]vXambvmWpw
thdnpw ]seSfn.

thsd ]e kvYefnepw knbn AImcw tem]npImWpw.

DZm: ]e+ FSfn = ]seSfn.

As, BsX, DsS, DusS,
Chbyw Ifpamw;
{lkzw \omw _enmbv;
Xmw {]m[m\ylm\nbn.

\ntbmPI{]Xybamb "As' FXnsbpw adphn\sbamb "BsX' FXnsbpw kw_nIm{]Xybamb "DsS" FXnsbpw KXnbmb "DusS' FXnsbpw Ayamb FImcw kzcw]camIptm hnIevt]\ tem]npw.

DZm: t]ms+ Ah= t]mh, t]msbh
hcmsX+ Ccpp= hcmXncpp, hcmsXbncpp
imanbpsS+ ANvO= imanbpSNvO, imanbpsSbNvO
hmXneneqsS+ Cdpp= hmXneneqSndpp, hmXneneqsSbndpp

ChbpsS {lkzamb FImchpw ap]dt]mse _esSpt kvYefn ZoLamn DcnpIbpw {]m[m\yanmsXhcp kvYefn Xosc DcnmXncnpIbpw sNmhpXmWv. AXn\m Ch tem]npXn\p bpnbpw ap]dXpXs BIpp.

(1) Hpw+G= Ht, At			(2) BXp+ G= BtX
(3) DSb= DsS			(4) DuSv = (Dv)+ G= DutS

CXn (3) HgnsI Fmnepw "G' \n]mXamIpp. AXn\m CXpIv Aytem]w hcptm "G' \n]mXw tNm ^ew hcptX Dp. "DSb' Npcpn "DsS' BbXn\pta FImchpwIqSn tem]nnptm aqei_vZn\p hfsc sshcq]yw hpt]mIpp DtS Fp ZoLw sNpXn\pw \ymbw t]mcm.

2. BKakn

hPvPnq kzckwtbmKw
b h tNp bYmhte;
]qw XmehyamsWn
bImcaXntebvWw;
]qtamjvTykzcw hm
hImcw tNpsImpI.

kzcv kzX{amb DmcWw DXn\m cp kzc tNpcnpXn kuIcypdhpv. AXn\m DmcWn kzckwtbmKw hPvPyamIpp. cp kzc ASpSpp hm a[yn bImctam hImctam bpwt]mse tNv Dcnp sImWw. ap]nes kzcw XmehyamsWn a[yn bImcw tNpXv bpw. ap]netXv HmjvTykzcambn hcpnSv a[yn hImcw bpw. Xmehyfmb A, B, C, Cu, F, G, sF Chbvv kzcw ]camIptm cnsbpw a[yn BKaambn bImchpw, HmjvTyfmb A, B, D, Du, H, Hm, Hu Chbvv kzcw ]camIptm a[yn BKaambn hImchpw hptNcpsaw.

DZm: Xmehyv			HmjvTyv
Ic+ D= Icbp			XS+ Dp= XShpp
t]mcm+ CXv= t]mcmbnXv			Nm+ Dp= Nmhpp
hgn+ BIpw= hgnbmIpw			Xncp+ HmWw= XncpthmWw
Xo+ Bv= Xobmv			]q+ Av= ]qhv
Xs+ Ah= Xsbh			t]m+ Dp= t]mhpp
tNt+ Dp= tNtbpp
ssI+ Dv= ssIbpv

{lkzamb HImcnepw HuImcnepw Ahkm\np ]Ztfm {]IrXnt]meptam aebmf`mjbn AXnZp`amWv. kuIcyw sImpam{Xw B cp kzcsfbpw IqSn ChnsS DsSpn HmjvTykzc Fp hn[nbn ]dncnppsht Dp.

bmhnpmhpw hmhnp
Imhpw IrXnbn hnSmw
{ipXnam[pcymambpw
mKaw NnseSfn

IrXnIfn Xmehykzcn\p kzcw ]camIptm bImcw t]mse Imchpw BKaambn hcpXmWv. A{]ImcwXs HmjvTykzcn\p kzcw ]cambm hImcwt]mse IImchpw BKaambn hcpw.

DZm:
Imn+ G= Imnt\, Imntb
hnekn+ B= hnekn\m, hneknbm
t]m+ Dp= t]mIpp, t]mhpp
XS+ Dp= XSIpp, XShpp.

CXn Xmehyp ]d ImcmKaw BJymXfnset]mse t]scfnepw hcpw.

DZm:
hnekn+ A= hnekn\, hneknb
IcpXn+ A= IcpXn\, IcpXnb

A\p\mknI{]nbmcmb aebmfnI Nne kvYefn IkpJn\pthnbpw Imcw BKaambn tNmdpv.

DZm:
Icn+ ]pen= Icnpen
]pfn+ Ipcp= ]pfnpcp

Cu DZmlcWfn "Icnpen' FXn aImchpw "]pfnpcp' FXn MImchpw BWv ImWpsXnepw AXp cpw Imcn\v BtZiambn hcpXmsWp taen kv]jvSamIpw.

hImccmKata tNcp
NpsgppI aqnepw;
Adnhm\dnths\
hImcw `mhnNnlv\amw

Xmehykzcp kzcw ]cambm a[yn bImcamWv BKaw Fp hn[np tZmjw inv B tZmjw hcpXsv ]cnlcnbmWv Cu kq{Xns ]qmwsImp sNbvXncnpXv. A, C, F F hnthNIk\map "Npsgpv' Fp Xang t] sNbvXncnpp. CXp aqpw XmehykzcamsWnepw CXn\p kzcw ]camIptm hImcmKamWv, bImcmKaasv A]hmZw.

DZm: A+ A= Ah, C+ A= Ch, F+ A= Fh

Cu k\masfs t`ZIfmn D]tbmKnptmgpw hImcmKaw Xs.

DZm: A+ CSw= AhnSw, C+ CSw= ChnSw, F+ CSw= FhnSw

t]sc{]Xybamb AImchpw NpsgppXs BIbm AXnepw hImcmKawXs thWw.

DZm: sNbvX+ A= sNbvXh
sNmm+ Atbmy= sNmmhtbmy

Fm B[p\nIaebmfn "sNmmbtbmy' Fv D hn[n{]ImcwXs {]tbmKw hncnpp. Ctm ]Zayknbnepw, "AhnSw' "ChnSw' apXemb kamk{]mbfmb ]Z{]tbmKfnepw am{Xta hImcmKaw D]tbmKnmdpp.

C\n thsd Hcp hyXykvXn\p kam[m\wIqSn ImWnpp:

Adn+ B F {]IrXn{]XybfpsSbpw, AXpt]mse Adn+ G F {]IrXn{]XybfpsSbpw kybn ayn hImcmKaw, hv "Adnhm' Fpw "Adnth" Fpw BbnmWp ImWpXv. AmsX Xmehykzcn\p ]d bImcmKaw hv "Adnbm' Fpw "Adntb" Fpw Bbn. AXn\m taevmWn bImcmKaw Xmehykzcfn \nXya Fp i. B ]Zfn ImWp hImcw kzckn{]Imcw BKaambn hcpX, `mhn{]XybNnlv\amb hImcamWv. AXpsImp taev]d hn[n \nXywXsbmWv Fp ]cnlmcw. Adnhm FXp ]nhn\sbamWv; AXp `mhnImes Ipdnbpw sNpp. Adnbpw+ B= Adnbphm Fp `mhncq]n\npXs AXns\ Dmbpw BImw. "D' F `mhn{]Xybw tNmtmgpw `mhnImekqN\bvmWv hImcmKaw. `mhywiw Fmw hnSp]w Adn+ B= Adnbm F Dcq]hpw BImw. \mtSmSn_v`mjbn CXpXsbmWp \Spw.

"Adnth' FXns\ "Adntb FmpI Hcnepw ]Xnhn; AXn\pImcWw AXp {K`mjbn am{Xw D]tbmKnp cq]ambntmbXpw, "Adntb' FXp \ntj[cq]{`aw DmpXpw BIpp.

bhmKaw \n]mXn
kzcnekpcw.

"\n]mX' F C\ns ]ZfpsS Ahm\n hcp kzcp kzcw ]cambmepw a[yn bImctam hImctam BKaambn tNpcnpXp \mbncnbn. kwkvIrXsshbmIcWm "{]Krlyw' Fp t]cnnp i_vZ DcnpXpt]mse bmsXmcp knImcyhpw IqssSsXXs DcnpXmWp \mbncnpI.

DZm: Ftm + Ah= Ftm Ah; GtXm + Hv = GtXm Hv; AXm+ Hcph = AXm Hcph.

Cs\ AmsX, Ftmhh, GtXmshmv, AXmsbmcph Fsmw DcnpXp `mjbpsS kz`mhn\p tbmPnpX.

ZoLnhkm\n
_mlptey\ bhmKaw.

B, Cu, Du, sF, Hm Fp ZoLkzc Ahkm\nep i_vZv AXmXns bpwt]mse B ZoLkzcn\ptijw bImctam hImctam AmKaambn hcpw. Fm Cu bhmKa Nne i_vZfnte DmIbpp. C C i_vZfnemWv DmIpsXp hymIcWimkv{Xm \nbanmhp Hcp hyhkvYbpw CXm\pw. AXpsImmWv "_mlptey\' Fp ]dncnpXv.

DZm: Im= Imbv; ]m= ]mbv, \m= \mbv, \mhv; cm= cmbv; cmhv
Xo= Xobv; \o= \obv, ]q= ]qhv, tKm= tKmhv; ssI= ssIbv.

_lpeamIbm Xo, ]q, ssI, Fv BKawIqSmsXbpw Kpcphv, ]cphv CXymZn {lkzfn BKaw tNpw {]tbmKw ImWpw.

{]XybmZnImcn
ap]pw XmehybmKaw.

Hcp {]Xybns BZnbnencnp Ccn IImcn\p ap]nep Xmehykzcp tijhpw bImcw BKaambn hcpw.

DZm: Xe+ v= Xebvv; he+ pp= hebvpp
Xe+ = Xebv; Nncn+ pp= Nncnbvpp;
lcn+ pp= lcnbvpp

{]XybmZnbsn,
NmSn+ ISpp= NmSnSpp.
XSn+ IjWw= XSnjWw
Ac+ I= Ac (Ins ]IpXn)

Chbnse IImcw {]XybmZnb. app \nevpXp Xmehyasn,

Idp+ pp= Idppp
shdp+ pp= shdppp

ChnsS apkzcw HmjvTyamIbm {]XybmZnImcamsWnepw bImcmKaw C.

3. ZznXzkn

JcmXnJcaqjvamhpw
arZptLmjfpw ZrVw;
]aw ayaw lmhpw
inYnem`n[ambv hcpw.
hntijWhntijy
]qtmc]Zfmbv
kamknmencnp
ZrVw ]c]ZmZnKw.

kzc kzXt{mmcWafmIsImpw AXpIfpsS [z\n hnhrXamIsImpw AXpIsf Dcnptm IWvTmZnkvYm\fn \mv (kv]inv) Xn \npt]mImXpsImpw Ah hyRvP\sftmse IqSntv Iqcw DmIpn. kzctbmKw Dmbmepw hyRvP\tbmKwt]mse hya. hyRvP\fpsS ZznXzkvYm\p kzcp ZoLamWv; IqckvYm\p kychpw. AXn\m ZznXzhn[nbpsS hnjbsamw hyRvP\w am{XamIpp. AXnepw aebmfns GmSn XovW[z\nbp hp ZznXzw [mcmfw ImWpw; tImaf[z\nbphbvp Ipdbpw. AXn\m ZznXzns Bhiyntepthn hyRvP\sf cpXcambn ]ncnpp Jc 6; AXnJc 5; arZp 5; tLmj 5; Dujvam 3; Cs\ 24 hyRvP\ ZrV, ]a 6; aya 6; l; Cs\ 13 hyRvP\ inYne. C\n knImcyw hnhcnmw.

hntijhmNIambnpXp ]q]Zhpw hntijyhmNIw ]c]Zhpambn kamknptm ]c]Zns BZynep ZrVhyRvP\ Ccnpw.

DZm:
Xe+ sIv= Xesv			]ip+ Zm\w= ]ipm\w
Xmac+ Ipfw= Xmacpfw			]q+ Xw= ]qw
Dh+ [ndpXn= DhndpXn			amXr+ Iv= amXrv
tXm+ IqSv= tXmqSv			ssI+ sXmgn= ssIsmgn
amm+ ]pc= ammpc			abn+ Ipn= abnpn
aSn+ ioe= aSnoe			Ihn+ XSw= IhnSw
]Wn+ ]pc= ]Wnpc			Xn+ InSmhv= XnnSmhv
Xo+ I\= Xo\			ae+ s]mSn= aesmSn
]pgp+ tISv= ]pgptSv			Xfn+ Iqw= Xfnqw

Ch XXv]pcpjkamkn hcpXn\v DZmlcW. I[mcbn:

Acw+ ]Wn= AcWn			X+ Nn= Xn
]mXn+ Ipf= ]mXnpf			t]+ ]n= t]n
]pXp+ Iew= ]pXpew			Xr+ ssI= Xrss

_lp{holnbn:

Xmac, ssan, tN\eb, s\ddnm.

Zzzkamkw hntijWhntijy tNp kamknpXmbvIbm Cu ZznXzw AXn hcpIbn.

DZm: ssI+ Im= ssIIm; B\+ IpXncI= B\IpXncI;
Cun+ Ip= CunIp; cma+ IrjvWm= cmaIrjvWm.

tLmjpw AXnJcpw ZznXzw hcptm ]qn\v apdbvv arZphpw Jchpw BtZiwIqSn hcpXmWv.

DZm: B\+ {`mv= B\_v{`mv; ]mXn+ ^ew= ]mXn]v^ew.

cpw hntijyamIbm "Aeic' FnSv ZznXzan.

Aep]vXmJykamkn
[mXp]qhnepw hcmw.

]q]Znep enwKhN\{]Xybp tem]anm kamkn\v Aep]vXkamksap t]. AXnepw, tIhe[mXpXs ]q]Zambp kamknepw Csm Dc]ZmZnbnse ZrVn\p ZznXzw hcpIbn.

DZm:

A + tIm= AtIm (A hN\{]Xybw tem]nnn)
aeaI + NcWw= aeaINcWw (A enwK{]Xybw tem]nnn)
Fcn + Xo= FcnXo (Fcn tIhe[mXp)
IS + tIm= IStIm (IS tIhe[mXp)

Aep]vXkamkw A]qamWv; IhnXbnte ImWpIbpp.

hn\sb aqsw
ap X ]mnIkwnXw,
B[mcnIm`mkamw G
BenIse hn`nI
Chbvp ]n ZrVnp
ZznXzw hmIynepw hcpw.

aphn\sbw, Xhn\sbw, ]mnIhn\sbw ChbpsSbpw, B[nImcnImhn`ym`mk{]Xybamb G FXnsbpw, B, C, I, F F hn`n {]XybfpsSbpw ]n]p hcp ZrV kamkanm kvYefnepw Ccnpw. DZm:

aphn\sbw		NmSn+ ]pdsp= NmSnpdsp
Sn		t]mbn+ ]dp= t]mbndp
Xhn\sbw		Ccns+ Ip= Ccnsp
Sn		IqsS+ t]mp= IqsStmp
Sn		apdpsI+ ]nSnp= apdpsInSnp
]mnIhn\sbw		ImWpIn+ sNmmw= ImWpInsmmw
Sn		Im+ Xmw= Immw
B[nImcnIm`mkw		Pente+ t]mfI= PentetmfI
Sn		s\ddnbnte+ Iv= s\ddnbntev
B (hn`n)		a\m+ sImSpp= a\msmSpp
Sn		hSnbm+ XSpp= hSnbmSpp
C (hn`n)		AXn+ Xs= AXns
Sn		In+ sImp= Insmp
I (hn`n)		Xebv+ ]nSnp= XebvnSnp
Sn		]Snbv+ sNp= ]Snbvsp
F (hn`n)		\ns+ Ip= \nsp
Sn		_mes\+ ]nSnp= _mes\nSnp

\ncpiXzw\nanw IhnI Nnetm ZznXzwIqSmsXbpw {]tbmKnpImWppv. DZm:

""Imanmhsf _te\ sXmpIqSm'' amSnbm
""\mept] IqSn Pth\ sNm'' \nbm

Cs\ ZznXzw sNmXncnpXv arZpp am{Xta Dq; AXn\p bpnbpw Dv. ZrVfn arZpp addphsb At]nv XovWX Ipdbpw. AXpsImv Ahsb tImaf[z\nIfmb inYnefpsS Iqn Xnfbphm \ymbw Dv.

Iqcancn
]Za[ynemhpIn

Hcp ]Zns CSbn Ccnp Iqcn BZytXv Ccnpw.

DZm: evan= evan, `kvkvaw

FgpXptm evan, `kvaw Fsmta FgpXpI ]Xnhpp. Fnepw Dcnptm CcntmSpIqSns thWw. ]Zaynesn Cu ZznXzw th.

DZm: ]d+ {]Imcw= ]d{]Imcw
`mcy+ {]khnp= `mcy {]khnp
ta\n+ oWnp= ta\n oWnp
Kpcp+ {`anp= Kpcp {`anp

""\Xp asbtmp hknn''
""\mcoaWnamsp {]knw''
""]q]m]wsImp hym[nbpw Zmcn{Zyhpw''

kwkvIrXn "A\Nn N' F kq{XwsImv kzcn\np ]camb kwtbmKns\msbpw ZznXzw hcpw; AXn\m `mjbnepw appw]npw D ]Z kwkvIrXambncpm ZznXzw sNWw.

DZm:
""PmXnivt{ijvTXbp P\s''
""DXv]v{]Imcfmw\b\tbmcKv{Kls.''

Fm,

""aqmw amcoNs\Iv{Iqc''
""sasz]vt\ hcppcnIn''

CXymZnIfn `mjbnepw ZznXzw ImWpXv 15mw kq{Xm knnXmIpp. Fnepw ]Zyfnepw addp kwtbmKn\v ZznXzw hcpmsX,

""IgenW]Wnbp {]mWnI
gsImSp t]Sn tW\ `amn
KncnkpXsbmSp t{]aamphmgpw
lc\psS {ioNcW IpnSpt''

Fpw addpw {]tbmKnm BbXv kwkvIrX]nXmp ImcppZambntmpw. Fm "]Wnbp]v{]mWnI', "t]SnIvtW\' CXymZn ZznXzw sNpXv Hcp aebmfnbpw klnpXpa. Cu kvYnXnv Cs\bp ZpLS{]tbmKv CSta sImSpmsX,

""IgenW]Wnbpw P\ns\m
agsImSp t]Sn Pth\ `amn
KncnkpXsbmSp cmKamp hmgpw
lc\psS XrcW IpnSpt''

CXymZn coXnbn kkXamb an {]tbmKnpsImm aXn Fv D]tZinmsaphm AXv imkv{XImc\v Kuchpdhnp hIbmIpp. AXn\m FmwIqSn t\mptm kwkvIrXimkv{XnamcpsS Bt]sf hIshmsX `mjmcoXnb\pkcnp ZznXzw IqSmsX BZyw ImWn{]Imcw tmI sNpXn\p bmsXmcp ktmNhpw hnNmcnm\nsp Fs A`n{]mbn almP\fpw tbmPnpsap hnizknpp. A{XXsbpa, aljnamcpw ]ucmWnImcpw [mcmfambpw,

""ZznXobw tla{]mImcw IphZv`ncnh hm\sscx''
""QSnXn {]hni tKlw am _lnkvXnjvT _mte!''

CXymZnt]mse almIhnI A]qambpw, kwkvIrXns Cu ZznXzs Dt]nmdpspw, almcmjv{Sn, iuctk\n apXemb {]mIrX CXns\ KWnnnspw, "AaNn N' F kq{Xs `tmPn{]`rXn \ho\m "hm' i_vZs A\phnv hnIev]npspw, AtNm clm`ymw tZz', "A\Nn N', "\mZn\ymt{Imti]p{Xky', "itcmNn', "{Xn{]`rXnjpimISmb\ky', k{X imIeyky', ZoLmZmNmcymWmw' Cu Ggp kq{Xsfsmv ]mWn\nXs ZznXzhn[nbn aXt`ZmZnIfm F{Xtbm Ipgfpmnoppspw Ah \hw BtemNnm\pXmIpp.

Npsgpn\p ]n hm
satXpw ZznXzamnSpw;
]cw inYneamsWn
entX ZoLnnepw aXn.

k\mafmb A, C, F, Chbvv "Npsgpv' Fpt]. NqnmWnp]dbp hmpIfmb Ah, Ch, Fh FnhbpsS {]IrXnv Npsgps t] A\zamWtm. B hI NpsgppIv ]cambn hcp Fm hyRvP\fpw Ccnpw. DZm:

A+ Imew= Amew		F+ \cIw= FcIw
A+ Zzmcw= Azmcw		F+ amXncn= FmXncn
C+ I= C		C+ \o= Co
C+ Znn= Cnn		C+ hw= Cw
F+ Imcy= Fmcyw		A+ Xcw= Acw
F+ tZh= Fth		A+]Sn= ASn

]cambn hcpXv inYnemcamsWn BhI NpsgppIXs ZoLamnbmepw aXn.

DZm: A+ amXncn= AmXncn; BamXncn		C+ B= C+ bm=
A+ hn[w= Bhn[w		Cm; Cubm
C+ hw= Cw; Cuhw		Cbm (hyXykvXw)

aebmf`mjbn Hcp am{X am{Xambnp ]Z hcpX. AXmWv NpsgppI ZoLnpItbm CcnpItbm sNpsapXns Xzw. CXpXs ImcnIbn hyampp.

GIam{X{]IrXyt
\mtkmw bfefpw
CcnnpXm \nevpw;
]Zansmddam{Xbn.

GIam{XIamb i_vZns Ahkm\n hcp A\p\mknIfpw b f e Fnhbpw Ccnt \nevpIbpp. Hcpam{X am{Xambnp ]Z aebmf`mjbn C.

DZm:
XMv+ C= Xn		\av+ F= \s
s]mMv+ C= s]mn		Iav+ Dp= Ipp
s\Rv+ Dv= s\v		sNbv+ Dp= sNpp
s]+ Cs= s]ns		s\bv+ Bdv= s\mdv
F+ Bbncw= Fmbncw		Xev+ Dp= Xpp
F\v+ F= Fs		I+ Dv= Iv, I
s]m+ D= s]mp		apfv+ Dv= apv, ap
		Ffv+ Dv= Fv, F

]Zmt hyRvP\w hm
kwhrXw tNp sNmpI;
Nnpam{Xw kwhrXn
kmlywIqSmsXbpw hcpw.

Hcp ]Zw hyRvP\n Ahkm\npXmsWn B hyRvP\n\ptijw kwhrXkzcw tNWw. Fm NnpIfn am{Xw kwhrXw tNWsap \n_an.

hbkv= hb, Imiv= Im, hmIv= hm, DeIv= De

NnmsWn

I= I, I		]m= ]m, ]m
tX= tX, tX		tXm= tXm, tXm
\o= \o, \o		Io= Io, Io

4. BtZikn

""XhnSpp
ShtW kh\w,
NhtmSpaw
htmSpas\.''

Xhw ShtmSp IqSntcptm Shambnpw, NhtmSp tNcptm Nhambnpw htmSp tNcptm hambnpw amdpw. DZm:

hn+ Xew= hnew		\n+ Xp= \n (\np)
X+ Xm= Xm		heRv+ Xp= hep (hep)
F+ \qdv= Fqdv		s]mcnRv+ Xp= s]mcnp (s]mcnp)
F+ Xp= F(Fp)

F, \n CuhI i_vZ "Fp' "\np' F cq]n BnbXv ]noSmWv. AXpt]mseXs hep, s]mcnp CXymZnI hep, s]mcnp FsmambXpw ]nevmepmb A\p\mknImXn{]kcw hgnbmIpp.

eImcw hyambvocpw
Xhw ]camhpIn;
fImcsan aq\yw;
hn\ma{InbbmWnXv.

eImcn\v Xhmc ]camIptm hyw F hw BtZiambv hcpw; AXpt]mse fImcn\v Xhmc ]cfmIptm aq\yamb Shhpw BtZiambv hcpw. Cu amn\v "hn\maw' Fp t]cnSmw. "kvYm\w XmgpI' Fv AXns Aw.

DZm:


AI+Xpp	=	AI+Xpp	=	(AIddpp)
hn+Xp	=	hn+Xp	=	(hnddp)
th+Xp	=	thSv+Xp	=	(thp)
tI+Xp	=	tISv+Xp	=	(tIp)
\+\q	=	\+\q	=	(\q)
sXm+\qdv	=	sXm+\qdv	=	(sXmqdv)

"AIXpp' FncnpXv ]ns "AIddpp' FmpXv apImcnIbn hnhcn \nba{]ImcamsWdnpsImI.

hn\maw efItmXmw
hytaXns ap]nepw;
Fnepw aebmfn
{]XybmZnXImctam,
\mtkmtam ]cw thWw
hn\ma{InbsNbvXnSm.

e X \ Zyw, f hyw; c S W aq\yw takvYm\p hw IogvkvYm\nse hambn amdpIbmWv hn\maw. Atm hn\mawsImv eImcw Ifnsemmbnpw, fImcw SWIfnsemmbnpw amdpw. Ifnepw SWIfnepw CsXp XosSptXp s]mcpwsImpthWw. ]camb hyw Jcsan SI; A\p\mknIsan WI. eImcfImcp hymcfn GXp ]cambmepw apdbvp Ifpw SWIfpw BtZiw thWsamWv {ZmhnUhymIcWnse GmSv. AXn{]Imcw Xangn,

hnpI= hnpI		hnm= hnm
Xmew= Xmew		tIdnndm= tISvndm

Fp hn\maw hcppapv. Fm aebmfn BIs, {]XybXImctam A\p\mknItam ]cw hmte hn\maw sNmdpp.

DZm:
hnev+Xp= hnXp= hnddp		AIev+Xpp= AIddpp
s\ev+aWn= s\Wn		Dfv+a= Da
Iev+aZw= IZw		\ev+\q= \q

Jcnap]pemtZiw
ap]ns]msmhn\v.

ap, ]n, s]m Cu ]Zfnse Ayamb Imcw Jcw ]camIptm eImcambn amdpw.

DZm: Xncpap+ImgvN= XncpapevmgvN ]n+]mSv= ]nev]mSv
s]m+IpSw= s]mevpSw.

CXv hn\man\p hn]coXamb Dmawt]mse tXmpp; Fm Xangn ChnsSbpw hn\mawXsbmWv: ]nmSp, s]mpSw FmWv Xangnse cq]w.

A\p\mknIamtZiw
]n{]XybJcn\v
ISmhpIn,nXt{X
AXn{]kcsaXv.

Hcp {]Xybns BZnbn hcp Jcn\v A\p\mknIw BtZiambn hcpw. B {]Xybw "I' FpXmsWnepw, {]XybmZnbmb Jcw SImcamsWnepw Cu A\p\mknImtZian. Cu \nba{]Imcw hcp A\p\mknImtZiamWv "A\p\mknImXn{]kcw' Fp t]cn \bw.

DZm:
]dp= ]dp		Xnp= Xnp
hep= hep		Ipp= Ipp

"I' {]XybamsWn:

X+I= XI(X A)
cma+I= cma (cmaemhpIbn);

{]Xybw SImcmZnbmsWn:

I+Sp = Ip (Iph); D+Sp = Dp(Dph).

25 (a) aImcwXm\\pkzmcw,
kzcw tNm sXfnnSpw.

(1) hy ]cambv hm
AXmXn ]aw hcmw

(2) kapb\n]mXn
tNcptcw hImcamw;
`mhn{]Xybamhnpw
hImcw{]Xytb ]tc;

(3) {]Xybkzctaevptm
fncn XImcamw;

(4) mtZis hnIev]nm
tamsSp hn`nbn.

A\pkzmcw Fp ]dbpXv ]Zmhkm\n hcp aImcw XsbmWv. AXnt\mSv GsXnepw kzcw tNpcnpt\mnbm acmcamsWv kv]jvSamIpw. acw+A, acw+C FpsXmw aca, acan FsmsbmpXv A\p`hknamWsm.

(1) Cu aImccq]amb A\pkzmcn\v hmc ]cfmIptm ]cw GXp hn tNtXm B hnse Amacw BtZiambn hcpw.

DZm: hcpw+Imew= hcpmew Nw+ tN= Nt
t]mIpw+ tXmdpw= t]mIptmdpw s]cpw+ ]d= s]cpd

CXv \n_ambnsgpXnmWnpI ]Xnhn.

(2) Cu A\pkzmcw kapb\n]mXamb "Dw' FXv ]camIptm hImcambn amdpw. AXpt]mse "Dw' F `mhn{]Xybnse A\pkzmchpw asddmcp{]Xybw ]camIptm hImcamIpw.

DZm: [\w+ Dw= [\hpw		Ipew+ Dw= Ipehpw
Fmw+ Dw= Fmhpw		hmcw+ Dw= hmchpw

CXv aebmfn\p Hcp hntijamWv. Xangn X\apw Ipeapw, Fmapw FpXs cq]w.

`mhn{]Xybn\v DZmlcWw:

hcpw+ A= hcph		hcpw+ F= hcpsh
hcpw+ A= hcph		hcpw+ C= hcphn
hcpw+ A= hcph		hcpw + B = hcphm
		hcpw+ Hm= hcpthm

(3) {]Xybkzcw GsXnepw ]camIptm Cu A\pkzmcw Ccn XImcambn amdpw; Fpshm "v' Fmbn amdpw.

(4) HmSv F hn`n{]Xybw ]camIptm Cu "v' F BtZiw hnIev]ambnt hcnIbpp.

DZm: Xpemw+ Cs= Xpemns		[\w+F= [\s
[\w + B = [\m		[\w + C = [\n
[\w+ HmSv= [\tmSv([\tamSv)

Cu mtZiw thsd NnteSpw hcpXmWv.

DZm: I]nIpew+ AcN= I]nIpecN
Bbncw+ Bv= Bbncmv

(b) I{]Xybnpap]p
ap kh\w

"I' F _lphN\{]Xybw ]camIptm ap]p \nevp aImc Imcv kh\w sImv MImcw hcpw.

DZm: acw+ I= ac(ac); \n+ I= \n(\n)

Ipfw+Ic= Ipfc F t]mse thtd NnseSpw Cu kh\w ImWpw. kh\w Fm ASpncnp hfn Hn\v kvYm\{]bXv\fpsS kmaywsImv atddXn\p s]mcpap hw BtZiamnsbvI. AXn ]qn\v ]cns khw hcnIbmsWn ]ckhw; adnv ]cn\v ]qns khw hcnIbmsWn AXv ]qkhw. {]IrXn a v ]camb IImcns khamtZiamIbm CXv ]ckhw.