emeronti vismikar. edahrsayvismikaragerkam³ k> ( y ) ( y) lmhat; + > ( ) K> y+ < y X> ( ) > g> ( ) ( + ) + + y y < c> ( ) ( ). edahrsayrbbn ½vismIkarageRkam³ k> K> y+ ( y+ ) ( y+ 7 ) <. edahrsayvismikaragerkam³ > X> + k> ( ) + < 6 > X> 6 7 + 7> 9 8 6( + ) > 6 + < K> + + >. kmnt;témø c énvismikar + 7+ 9> 8+ c mancmemøiy cmebahrkb; témø.. ek[smnmubir Anig B ducagerkam³ = { 6 7 > }, B= { 8< } A currkcmnynkt;kñúgcmenamfatuén A B. 6. curbgðajvismpabagerkamcmebah a, b, c,, y k> a+ b+ c+ 8 b c a >( ab+ bc+ ca) abca ( + b+ c) K> y y X> > > > > >. + y ( y ) 7. ekmanvismikar( a ) ( a ) +. kmnt;témøa edim,i[ vismikarman rws. - - - -
. edahrsayvismikaragerkam³ k> ( y ) ( + y) y+ 8+ y 7 8 y y y y y cmeliy y y [, + ) > (+ ) + ( ) + 8+ + 6+ K> y 8 6 = [, + ) + < y y+ < y 6 y< 8 y> 8 y (8, + ) X> ( ) > ( ) > > 8+ > > 8 8 > = 7 (7, + ) g> < ( ) < < 6 > 6 > (, + ) c> ( y ) ( y) 6( y ) ( y). edahrsayrbbn ½vismIkar³ k> 6y + y 6y 8 8 y = 6 smmulnwg y (,] ( ) ( ) - - - -
> K> X> eyig)an vismikarkµancmeliy + 7> 9 > 7 9 8 smmulnwg + 8 6 > 6 < 6 < 6 eyig)an y+ ( y+ ) ( y+ 7 ) < y y y< y< smmulnwg [, ) y+ y+ y+ 7 < eyig)anrbbn ½vismIkarKµancMelIy y ( ) 6( + ) > 6 smmulnwg 6+ 6> ( 6 ) 6 > 6. edahrsayvismikar³ ducenh > [, + ) + k> ( ) 6 ( ) + < 6( ) + 6 ( ) < > > + < eyig)an > ducenh 6 + < 7 -smikar > smmulnwg 6 + < 6 (, ] 6 6 + 8< 6+ 8= man 8 + 6 > = ' = 9 8= 9< eday a= > enah 6 + 8> ducenh vismikarkµancmeliy K> + > + 7 > + = man -smikar 7 - - - -
= 9 = -sbaøarbs; 7+ 7 = =, = = + 7 ducenh (,) + + X> -smikar + + + + = man = 69 = 9 -sbaøarbs; + 7 7 = =, = = + + -edim,i[vismikarmancmeliycmebahrkb;tmél luhrtaet 9 c + man < = (9 c) < smmul a> a= > c <, c< ducenh c< c. rkcmnynkt;kñúgcmenamfatuén A B eyigman, (, ) c < a = > = { 6 7 > }, B= { 8< } A A :6 7 -taragsbaøa > smikar = 9+ = 69 6 7 = man 7+ 7 = =, = = ducenh [, ]. kmnt;témø c énvismikar + 7+ 9> 8+ c mancmemøiycmebahrkb; témø eyigman + 7 + 9> 8+ c c + 9 > eyig)an,, + ( ) B : 8 < smikar ' = + 8= 9 8= man = + =, = = - 6 - - 7 -
- 8 - -taragsbaøa eyig)an (,)( ) -tam( )&( ) enah A B eyig)an³ eday A BCacMnYnKt; enah {,} 6. bgðajvismpabcmebah a>, b>, c>, >, y> k> a+ b+ c+ 8 b c a eyigman a a, a + b b b a a+ ( ) b b duckñaedreyig)an b b+ ( ) c c c c+ ( ) a a -yk( ) ( ) ( ) eyig)an a b c a+ b+ c+ b c a b c a a b c 8 = 8 b c a ducenh a b c + + + 8 b c a >( ab+ bc+ ca) abca ( + b+ c) eyigman ( ab+ bc+ ca) abca ( + b+ c) = ( ) ( ) ( ) = ab + bc + ca + abca ( + b+ c) abca ( + b+ c) ( ) ( ) ( ) = ab + bc + ca abca ( + b+ c) ( ) ( ) ( ) ab bc ca abca ( b c) + + + + = ( ) ( ) ( ) ab + bc + ca abca ( + b+ c) = ( ) ( ) ( ) ( ) ( ) ab abc+ ac + bc abc+ ab + ac abc + ( bc) = ( ab ac) + ( bc ab) + ( ac bc) = eyig)an ( ab+ bc+ ca) abca ( + b+ c) ducenh ( ab+ bc+ ca) abca ( + b+ c) K> y y y+ y BinitüemIl smikar y y + = man = ( y) y = 9y 6y = 7y <, y> ehiy a= > ducenhvismpab epþógpþat; y+ y y, > ducenh y y - 9 -
X> + y ( y ) y y + y y + + + y y + + + + ( ) + ( + ), ducenh y y + y ( y ) 7. kmnt;témøa edim,i[vismikarman rws eyigman ( a ) ( a ) + edim,i[ vismikarmanrws luhrtaet ( a ) ( a ) + man ' ' = ( a ) ( a ) smmulnwg a a ( a ) [( a ) ] ( a )( a ) a a -BinitüsBaØarbs;( a )( a ) ducenh a (,) [, + ). KNna lmhat;cmbuk k>( 7)( 7 7) ( + i) ( i) K> X> + i + i + + + i + i + i i + >. currktméléncmnynbit a nig b énsmikaragerkam³ k>( a b) abi i + + = + > + ( a+ b) i= a b+ i K>( + i)( a+ bi) = i X>( + ia ) ( ) ib+ (+ 6) i =. edahrsaysmikaragerkam³ k> ( ) + ( ) + = >( )( )( ) = K> X> ( ) ( ) = ( ) + ( )( ) ( ) =. edahrsayrbbn ½smIkar k> y z= y z= y+ z= y= > y+ y = 6. edahrsayvismikaragerkam³ k> 8+ > > + > K>( )( + ) > 8 X> 8+ 6< 6 =. 6. k> edahrsaysmikar - - - -
>ekmankensam f ( ) = 9 + A+ B Edl Anig B CacMnYn efr. kensamenhecknwg + rweckdac;nwg mansmnl; duckña. > rktmél Anig B. > edahrsaysmikar f ( ) =. > edahrsaysmikar f( ) =. 7. ekman a, b,, y CacMnYnviC man nig a b + =. bgðajfa a+ by a + b y. 8. ekmansmikar ( k ) + k= EdlmanrwsBIr α nigβ. k> rktmél k kñúgkrniedlrwstamgbirvic man. > rktmél k kñúgkrniedlrwstamgbirgvic man. 9. ek[abc CaRtIekaNEdlmanmMuACB CamuMEkg ducrub k> erbirtwsþibtbitakrsresrsmikaredl mangbaøat. > edahrsaysmikar =. K> rképþrklarbs;rtiekan ABC.. kmµkrbirnak;cysculpþhmyy. kmµkrti efvi)anbak;knþal ryc[kmµkrti begðiykic kar EdlenAsl;. kic karenh)ancb;kñúgryhebl hmn EtebIkmµkrTaMgBIreFVIkarrYmKña enahekgacbba b; kic karenhkñúgryhebl 6h. etikmµkrnimyy RtUvcMNayeBlb:unµanema:g edim,icysculpþhenahehiy.. Rkumh unmyyplitvitüúlk;tmél $ kñúgmyyerkog. m as;rkumh un )a:n;sµan fa ebieklk;tmél$ kñúgmyyerkogenahep óvnwgtijrbehl 8 vitüúkñúg E². k> rktmélcmnynefr A nig B EdlR)ak;cMeNjrbs;Rkumh un[eday kensam + A+ B. > kmnt;tmélvitüú ebieklk;)ancmenj$. K> etir)ak;cmenjgaceligdl;$ rwet?. synpáamyyragctuekanekgmanttwg m nigbenþay m. ekcg; efvi pøúvbirkat;kña kñúgsynpáaenah ducrub EdlmanTTwgpøÚvesµI²Kña nigépþrklapøúv esµinwg m. KNnaTTwgpøÚvenaH. - - - -
. KNna cmeliylmhat;cmbuk k>( 7)( 7+ 7) ( )( ) = ( i 7)( 7+ 7 i ) = i 6 + i = ( i 6) ( + i) = 6( i 6)( + i) ( i i i) ( i) = 6 + 8 = 6 = 6 6i ducenh ( 7)( 7+ 7) = 6 6i > ( + i) ( i) ( ) ( ) ( i) ( i) i ( + i) ( i) = + i i = + = ducenh K> ( + i) ( i) = i + + i i = = = = i+ + i + + i i ( + i)( i) i + ducenh + + i i X> + i + i + + i + i = ( + i) ( ) + i + i + i i + = + + i + i + i i + i ( ) ( + i)( i ) + ( i )( + i) ( i)( i ) + i i = + = + i i + i 9i + 9i i = i 8 = = ducenh i + + i + = + i + i. rktméléncmnynbit a nig b énsmikar³ k>( a+ b) + abi= + i edaypþwmepñkbit nigepñknimitþeyig)an³ a+ b= ab= -yk( ) CMnYs ( ) a= b( ) ab= ( ) eyig)an ( bb ) =,b b =, b b+ = = 9 = + b=, b= + -cmebah b= CMnYskñúg ( ) )an + a=, a= -cmebah b= CMnYskñúg ( ) )an + a=, a= ducenh + a=, b= + rw a=, b= - - - -
> + ( a+ b) i= a b+ i edaypþwmepñkbit nigepñknimitþeyig)an³ -yk( ) CMnYs ( ) a b= a+ b = eyig)an b+ + b = b + b = a= + b ( ) a+ b = ( ), b =, b = ( a+ b+ c= ) -cmebah b= CMnYskñúg ( ) )an a= + = -cmebah b= CMnYskñúg ( ) )an a= = ducenh ( a=, b= ) rw( a=, b= ) K>( + i)( a+ bi) = i ( + i)( a+ bi) = i a bi+ ai b= i a b+ ( a bi ) = i edaypþwmepñkbit nigepñknimitþeyig)an³ -yk( ) buk ( ) a b= a b= a+ b= ( ) a b= ( ) )an a=, a= CMnYskñúg ( ) )an + b=, b= ducenh a=, b= X>( + ia ) ( ) ib+ (+ 6) i = ( + ia ) ( ) ib+ (+ 6) i = a+ ai b+ bi+ + 6i= a b+ + ( a+ b+ 6) i= = + i edaypþwmepñkbit nigepñknimitþeyig)an³ a b+ = a+ b+ 6= -yk( ) buk ( ). edahrsaysmikar³ k> 6a b+ = ( ) a+ b+ 6= ( ) )an 7a+ =, a= CMnYskñúg ( ) )an + b+ 6=, b= ducenh a=, b= ( ) + ( ) + = tag t = eyig)an t + t+ = manrws t t -cmebah t= =, = -cmebah t= =, = ducenhsmikar manrws =, = >( )( )( ) = ( + )( ) = + 9 + 6= 6 + = + + 6 = ( ) ( ) + 6( ) = ( )( + 6) = =, = - 6 - - 7 -
. =, =. + 6= = = = i + i i =, = ± i =, = ducenh smikarmanrws K> ( ) + ( )( ) ( ) = ( ) + ( )( ) ( ) = + + ( + ) ( + 9) = + 6 = ' = 9 = 6= 6i 6i + 6i =, = i i + =, = ducenhsmikarmanrws 6 6 X> ( ) ( ) = tag t = eyig)an t t = ' = + = 6 t= + = 6, t= = t= 6 =, 6= man -cmebah = + = + = =, = = t= =, + = man -cmebah = 8= 7= 7i + 7i 7i =, = i i + =, =, =, = ducenhsmikarmanrws 7 7. edahrsayrbbn ½smIkar k> y z= y z= y+ z= -yk( ) ( ) + eyig)an smmulnwg 6y=, y= ( a) -yk( ) ( ) -yk( a) ( b) + eyig)an y= ( b) tam ( )&( ) + eyig)an a b eyig)an y= a y= ( b) y z= ( ) y z= ( ) y + z = ( ) ( ) 7y=, y= CMnYs ( ) b eyig)an =, = yk =, y= CMnYskñúg ( ) )an z=, z= ducenhrbbn ½smIkarmanrws ( =, y=, z= ) - 8 - - 9 -
y= > y+ y = 6 -yk( ) CMnYskñúg ( ) smmulnwg eyig)an³ (y+ ) (y+ ) y+ y = 6 y + y+ 9 8y y+ y = 6 = = =± y, y, y = y+ ( ) y+ y = 6( ) -cmebah y= CMnYskñúg ( ) )an = + = -cmebah y= CMnYskñúg ( ) )an = + = ducenhrbbn ½smIkarmanrws (, y ). edahrsayvismikar³ k> 8 + > 6 + > ( ) 6( ) > ( )( 6) > eyig)antaragsbaøa = = rw ( =, y= ) eyig)antaragsbaøa + > + > ( ) ( ) > ( )( ) > ducenh K>( )( + ) > 8 eyig)antaragsbaøa (, ) (, + ) + 8 8> + 7 > + > ( ) + ( ) > ( )(+ ) > ducenh (,) (6, + ) > + > X> ducenh 8+ 6< 6 eyig)an (, ) (, + ) - - - -
+ < 8 6 6 + > 8 6 6 -tam( ) eyig)an ( 8) < (,8) -tam( ) eyig)an -tam( )&( ) smmul 8< a ( ) 8+ > 8 + > 6 + > ( 6) ( 6) > ( 6)( ) > 8< ( ) 8+ > ( ) (,) ( 6, + ) ( b) a b eyig)an 6. k>edahrsaysmikar ducenh (,) ( 6,8) = = + 6+ = ( ) + ( ) + ( ) = ( )( + + ) =. =, =. + + = + + + = ( + ) + ( + ) = ( + )(+ ) = =, = ducenhsmikarmanrws =, =, = > > rktmél Anig B -tambmrab; f ( ) = 9 + A+ B Ecknwg + rweck dac;nwg mansmnl; duckña eyig)an f( ) = f() = f( ) = ( ) 9( ) + A( ) + B= A+ B= ( ) f() = ( ) 9( ) + A( ) + B= A+ B= ( ) -yk( ) ( ) eyig)an³ A= 9, A= CMnYskñúg ( ) + B=, B= 7 ducenh A=, B= 7 )an - - - -
> edahrsaysmikar f ( ) = eyig)an f ( ) 9 7 f = + + enah ( ) = 9 + + 7 f = + + 9 ( ) =, 9 + + = 9 + + = + + + = ( + ) ( + ) + ( + ) = ( + )( + ) = ( + )( 7+ ) = ( + ) [ ( ) 7( ) ] = ( + )( )( 7) = 7 =, =, = =, =, = ducenhsmikarmanrws 7 > edahrsaysmikar f( ) = eyigman f ( ) = 9 + + enah f( ) = ( ) 9( ) + ( ) + f( ) =, ( ) 9( ) + ( ) + = tag t = enahsmikaretaca t t + t+ = tamsmnyr k> eyig)ansmikarmanrws 7 t, t, t -cmebah t= =, = -cmebah t= =, = 9 = = = 7 7 -cmebah t= =, = ducenhsmikarmanrws 7,, = = = 7. eyigman a, b,, y CacMnYnviC man nig a+ b= bgðajfa a+ by a + b y eyig)an ( a+ by) ( a + b y) a+ by a+ ab y+ by a+ ab y+ by a+ by ab ab y a by + + y+ b a b b a y by + y b b a a a b ( ) + y( ) y b b a a a b y + y b a eday a+ b= enaha b, b a = = eyig)an b a y + b a + y y y+ y ( y) eyig)an ( y) ducenh a+ by a + b y 8. k> rktmél k kñúgkrniedlrwstamgbirvic man y Bit - - - -
- 6 - eyigman ( k ) + k= manrwsbir α nigβ -edim,i[ smikarmanrwstamgbirvic mankalna ' > S > P> smmul k k+ > 9 k > k> -taragsbaøa( ) -eyig)an ' = ( k ) k > b S = = ( k ) > a c P= = k> a ( k )( k 9) > ( ) k> ( ) ducenh k> 9, k ( 9, + ) > rktmél k kñúgkrniedlrwstamgbirgvic man -edim,i[ smikarmanrwstamgbirgvic mankalna ' > S < P> smmul ' = ( k ) k > b S = = ( k ) < a c P= = k> a -eyig)an k k+ > 9 k < k> ducenh k (,) ( k )( k 9) > ( ) k < ( ) k > ( ) 9. k> erbirtwsþibtbitakrsresrsmikaredlmangbaøat tamrtwsþibitakreyig)an AB = AC + BC (+ ) = ( ) + ( ) + + 9= + + + + + = = > edahrsaysmikar ( >, > ) = = man ' = + = = +, = smikarmanrws ( = +, = ) K> rképþrklarbs;rtiekan ABC S ABC = AC BC SABC= ( )( ) - 7 -
eday tmél S ABC = + > enah = ( + )[ ( + ) ] = ( 8 + )( 9 + ) S = 8+ 6 ABC ducenhrklaépþ 8 6 ABC S = + ÉkataépÞ. k> rktmélcmnynefr Anig B eyigmanr)ak;cmenj tageday + A+ B -kñúge Rkumh unlk;)an ( 8 ) erkog edayéfø myyerkogman tmél enah R)ak;lk;srubKW ( 8 ) -vitüúmyyerkogéfø $ lk;)an ( 8 ) erkog enahr)ak;cmnay srubkw ( 8 ) eyig)an R)ak;cMenjsrubKW = R)ak;lk;srub- R)ak;cMNaysrub R)ak;cMenjsrub = ( ) 8 (8 ) = + 9 8 EtkenSamR)ak;cMenjKW A B + A+ B= + edaypþwmggátamgbireyig)an A= 9, B= 8 + + eyi)an 9 8 > kmnt;tmélvitüú ebieklk;)ancmenj$ eyigman kensamr)ak;cmenjkw + 9 8 eyig)an + = 9 8 + = 9 8 =, = 6 ducenhtmélvitüúkw$ rw6$ K> R)ak;cMeNjGaceLIgdl;$ rwet? ]bmafa R)ak;cMeNjeLIgdl; $ eyig)an + = 9 8 + = 9 8 = = < ' 8 9 smikarkµanrwset ducenh R)ak;cMeNjminGaceLIgdl;$ )anet. KNnaRbEvgTTwgpøÚv -RkLaépÞpøÚv m tag RbEvgpøÚv < < -S P RkLaépÞpøÚv -S H RkLaépÞpøÚvkat;tamTTwgsYn -S W RkLaépÞpøÚvkat;tambeNþaysYn eyig)an SP= SH+ SW SEFGH ( ) ( ) ( ) S = AD + AB P P = + = 7 S = = 7-8 - - 9 -
7 + = =, = 6> ducenhrbevgttwgpøúvkw= m sumrg;camgannuv PaK EdlnwgecjpSaynaeBlqab;²enH>>> sumgrkun>>>>>> eyig k¾manttylbergónknitvitüa dl;ekhdæanelakgñkpgedr BIfñk;TI 7 dl; TMnak;TMng³ 9 9-6 -