TBMathM1co

c[[IPo 7toyflkf -Ao,afmtpq,xumu ! IP[IP’3fpG mkj o s5,rao xDlts;affu mjko [5ocl’ daopk;q’ mjko [5o,kd ]jP,cdh; mjko rq’lts;ao grQkltfk d;fcdh3fpG mkj o cv39 vkfzkl5d dt-;’lbdlkmydko lt4k[ao7Qo7;hk;ymtpklkfdkolbdlk ;P’9ao @W!W

7eoe xB,c[[IPo7toyflkf ,D! gs^A,oU wfhIP[IP’0Bogrnjvlvo8k,s^adl6f0v’ 7toyflkf-Ao,afmtpq,lbdlk8vo8Qolt[a[xa[x5’xu @WW_D .o0Ao,afmtpq,lbdlk xumu|bj’ X,D!? cdo0v’s^adl6fc,joG 9eo;ome,t-kf c]t 9eo;o4h;oF mt;u76o c]t v5xt76oF 9eo;orNo4ko8kj ’MF 7jkle[6oF leo;oF g,af c]t glAo-njF glAo-nj0t|ko c]t glAo-n8j A’lkdF dko;afcmd];’pk;F dkog7y’j 7nF ,5, c]t dko ;afcmd,5,F glAodk’lkdF glAoc[’j g7yj’,5,F ,kf8klj;oF d5j, c]t vq’xtdv[ 0v’d5j,F vto5d5j, c]t czo;kf0v’d5,j F dko8afdao0v’d5j, { dko3I,dao0v’d5j,F -vdskrqfmuj[+maoI6h0v’vaf8klj;orq;rao c]t dko7yfw]lj j;oIhvpD xB,c[[IPosq;oU wfhIa[dkoIP[IP’0Bo grnvj -;j p.sh76lvo8k,s^addko dkoIPo{dkolvoc[[gvqkoadIPogxao.9dk’ 3fp.shz6hIPo7Qorq[7;k,I6hfh;p 8qogv’ rkp.8hdko-Uoe0v’76D fa’j oAoF c8]j t[qf0v’xB,c[[IPosq;oU 9bj’xtdv[ fh;pdyf9tdeF .97;k, c]t [qfg/ydsafD dkola[-hvogoNv.o c]t [qfc,oj vu’.lj ltrk;t7;k,gxao9y’0v’dkoIPo{dkolvo.oxtgmfgIqkF s^addko0v’dkoIPo 7toyflkfle]a[-Ao ,D!F s^ad9yf8t;ymtpkdPj ;da[7;k,Ia[I6hF s^ad;y-k768k,95f \\kp c]t s^ad;y-k76muj9ecod7;k,c8d8jk’ ]ts;kj ’[5d7qo 3fplq,mq[da[;ymu dko96’.9oadIPoD 1kj ’.fd=8k, dkoIP[IP’xB,sq;oUc,oj .-hg;]klAo c]t d+7q’[+xklt9kd 0=h0kf8qd[qdrvj ’wfhF ltoAo 9bj’0=7;k,Ij;,,noemjkoz6hmujoe.-hxB,sq;oU 9qj’-j;pg[yj’ 95f[qdrjv’ s^n 95fzyfrkf c]h;lq’j 7e7yf7egsao0v’mkj owxpa’lt4k[ao7Qo7;hk ;ymtpklkfdkolbdlkD r;dgIqk9t4n;jkm5d7e7yfgsao0v’mkj ogxao0=h,6omu,j u75o7kj c]t gxaodko-j;pxa[x5’75ootrk[dkolbdlkr;dgIqk.shl6’0BoD lt4k[ao7Qo7;hk;ymtpklkfdkolbdlk d

lk]t[ko |hk 7eoe ສະຫງສ.ວວນ.ລິສຂະສິດ d lk]t[ko 0 [qfmu ! 9eo;ome,t-kf ! [qfmu @ 9eo;o4h;o !W [qfmu # dko[;d c]t dko]q[9eo;ome,t-kf !( [qfmu $ dko[;d c]t dko]q[9eo;o4h;o @& [qfmu & dko76o c]t dkosko9eo;ome,t-kf ## [qfmu * dko76o c]t dkosko9eo;o4h;o #) [qfmu ( mt;u76oIj;,|hvpl5f c]t v5xt76oIj;,s^kpl5f 0v’9eo;ome,t-kf $* [qfmu ) 9eo;orNo4ko8kj ’M &* [qfmu _ 7jkle[6o *& [qfmu !W leo;og]d7toyf (# [qfmu !! 7;k,leobddPj ;da[g,af c]t glAo-nj )! [qfmu !@ glAo-nj0t|ko c]t glAo-nj8A’lkd )( [qfmu !# dkoc8h,glAo-n0j t|ko c]t glAo-n8j A’lkd _$ [qfmu !$ dko[;d];’pk;mvj o-nj c]t dko76o9eo;.f|b’j !W! da[];’pk;0v’mjvo-nj !W) [qfmu !& dko;afcmd];’pk;0v’mvj o-nj !!* [qfmu !* dko;afcmd];’pk; !@_ [qfmu !( dkog7yj’7n !#* [qfmu !) ,5,F dko;afcmd,5, 0

[qfmu !_ I6[g7yj’7n0v’,5, c]t 0v’glAo-nj ສະຫງສ.ວວນ.ລິສຂະສິດ!$( [qfmu @W cdog7yj’7n0v’I6[F glAodk’lkdF glAoc[’j g7y’j ,5, !&* [qfmu @! ,kf8kl;j o !*( [qfmu @@ ,kf8kl;j o X8+? !(& [qfmu @# dj5, c]t vq’xtdv[0v’d5j, !)# [qfmu @$ g7njv’\\kp ∈ c]t ∉ c]t d5,j gmqkj dao !)) [qfmu @& vto5d5,j c]t czo;kf0v’d5j, !_# [qfmu @* dko8afdao0v’d5j,{dko3I,dao0v’d5j, !_) [qfmu @( vaf8kl;j o @W* [qfmu @) vaf8kl;j orq;rao @!! [qfmu @_ -vdskrqfmuj[+maoI6h0v’vaf8kl;j orq;rao @@# [qfmu #W dko7yfw]jl;j oIhvp @## [qfmu #! vaf8kl;j orq;raoxUo @#_ 7

rkdmu ! 9eo;ome,t-kf gg]t 9eo;o4h;o [qfmu ! 9eo;ome,t-kf !D dyf9tde dyf9tdemu ! dD 9q’j 8n,j 9eo;ooadIPo.oshv’0v’g9Qk.l[j jvo9Eg,afD 16j3I’IPo,afmtpq,csj’|bj’gryjowfhltcf’lt4y8y9eo;ooadIPoc8j]tshv’ fa’j 8k8t]k’]5j,oUD shv’ ,!D d ,!D 0 ,!D 7 ,!D ’ shv’g9Qkgv’ 9eo;oma’\\qf #& $W #) $! DDDD py’ !_ @@ @! @# DDDD 0D 9qj’oa[9eo;o\\kd[u16jc8j]tdt8jkc]h;8nj,.lj[jvo9Eg,afDສະຫງສ.ວວນ.ລິສຂະສິດ ,u DDDDDDD |;j p ,u DDDDDDD |;j p ,u DDDDDDD |;j p ,u DDDDDDD |j;p 7D 9eo;omuj.-h[aombd s^n .-h.odkooa[ g-aoG 9eo;ooadIPoc8]j tshv’ c]t 9eo;o \\kd[u16cj 8]j tdt8kj gryjogvUo;jkG 9eo;ome,t-kfD 9qj’pqdgsfdkomuj9egxao8hv’.-h9eo;ome,t-kfgrnvj [aombd,k9adlv’8q;1jk’D ’D gryjo[aombdzqodko;afcmdoE|ad c]t ];’l6’0v’oadIPo 3fpmqj;wx9eo;o muj.-hc,jo9eo;oco;.fL 1

ສະຫງສ.ວວນ.ລິສຂະສິດdyf9tdemu @ 9eo;o.f|b’j lk,kf0Pogxao8q;g]dS 8q;|a’ln X7evkj o?D dD dko0Po9eo;o.oI6[c[[dt9kpD 8q;1jk’G 41 375 = 41×1000 + 3×100 + 7×10 + 5 vjko;kj G ljuly[gvafraolk,Ihvpg9afly[shkD 9q’j vu’.l8j q;1kj ’c]h;0Po9eo;o]5j,oU.oI6[c[[dt9kp c]h;vjkorhv,D 507; 2532; 137 065; 45 700; 3 512 415; 25 718 272; 3 125 637 542. 0D dko0Po9eo;ogxao8q;g]dD 8q;1kj ’G lujraogdQkIhvpshkly[g9af = 4×1000 + 9×100 + 5×10 + 7 = 4957 9q’j vu’.lj8q;1kj ’c]h;0Po9eo;o]5,j oUgxao8q;g]dD { -k;gdQkraocxfly[ { lv’Ihvplulj y[g9afraosqdIhvplk,ly[gvaf { shk]hkog9afly[lk,rao|bj’Ihvpcxfly[sqd { ly[lv’8NlujIhvp-k;lv’]hkoshkIhvpgdQkly[g9afraocxfIhvpsqdly[sqd { @!# 8N (& ]hko #() rao &$WD dyf9tdemu # dkoxP[mP[9eo;o3fpoe.=-hg7nvj ’\\kp <F >F = 8q;1jk’G 4×1000 + 9×100 + 5×10 + 6 = 4956 2×1000 + 4×100 + 3×10 + 5 < 2×1000 + 4×100 + 3×10 + 7 9qj’vu’.lj8q;1jk’c]h;xP[mP[9eo;o]5,j oU 3fp0Pog7njv’\\kp <F >F = .l[j jvo 9Eg,afD dD 2×1000 + 4×100 + 3×10 +1 ....... 2430 0D 2×1000 + 5×100 + 7 ×10 + 5 ....... 2575 7D 4×1000 + 9×100 + 5×10 + 6 ....... 5×1000 + 9×100 + 5×10 + 6 ’D 6×1000 + 2×100 + 5×10 + 8 ....... 6×1000 + 2×100 + 4×10 + 8 2

dyf9tdemu $ dD 9q’j 0ufmvj o-nj ]5j,oU3fpg]u,j 9kd95fmudj eoqf.shfaj’I6[ wx8k,mk’ovog[Nv’0;kD [AB] pk; # cm A [MN] pk; & cm M [EF] pk; & cm E [PQ] pk; ) cm [RS] pk; !W cm P R 0D 9kdI6[c8h,F 9qj’xP[mP[];’pk;0v’mvj o-njgs^qjkoAoD 8q;1jk’G mvj o-nj [MN] pk;d;jkmjvo-nj [AB] fa’j oAo gryjog;Qk;kj G & s^kpd;jk #S 0PoG & > #S xUo7noG mvj o-nj [AB] lAod;jk [MN]. fa’j oAo gryjog;Qk;jkG # |hvpd;kj & ສະຫງສ.ວວນ.ລິສຂະສິດ 0PoG # < &D 9q’j xP[mP[];’pk;0v’c8j]tlv’mjvo-nj8+wxoUG [MN] c]t [EF]S [EF] c]t [PQ]S [PQ] c]t [RS]; [PQ] c]t [MN]S [AB] c]t [EF]S [AB] c]t [RS] @D .97;k,D !D 9eo;ome,t-kfD { 9eo;og-y’j gryoj .-hoa[ s^n [aombd9eo;ooadIPoc8j]tshv’ c]t 9eo;o\\kd[u16j c8]j tdt8jkg-ajoG #&S !_S $WS @@S #)S @!S $!S @#S @@S !)S @WS @$ gvUo;jkG 9eo;ome,t-kf { 9eo;ome,t-kfma’\\qf3I,g0Qkdao gxaod5,j |bj’,u-nj;jkdj5,9eo;ome,t-kf c]t laopt]adfh;p8q;vadlvo ` D gryoj 0Po ` = {1; 2; 3; 4; 5; 6; 7; 8; 9; 10;…} { gryjo.-hg7nvj ’\\kp ∈ cmo7e;kj <gxaovq’xtdv[0v’ < c]t ∉ cmo7e;kj < [+gxaovq’xtdv[0v’ < 3

sq;8N8q;1jk’G sq;3df! c,oj 9eo;ome,t-kfF gryoj g;Qk;jk ! gxaovq’xtdv[0v’d5,j ` . sq;]hko0Posp=h ! ∈ ` sq;clo#F& [+c,oj 9eo;ome,t-kfF gryoj g;Qk;kj #F& [+c,oj vq’xtdv[0v’d5,j ` . sq; j\\no0Posp=h #F& ∉ ` sq;rao @D dkovjko c]t dko0Po9eo;ome,t-kf sssqqq;;;lIyh j|[vສ;pະpຫງສ.ວວນ.ລິສຂະສິດ8k8t]k’dko0Po c]t dkovkj o9eo;ome,t-kf -Ao0v’sq;8N -Ao0v’sq;]hko -Ao0v’sq;rao -Ao0v’sq;|j;p sq; sq; sq; sq; sq; sq; sq; sq; sq; sq; sq; sq; Ihvp ly[ |j;p Ihvp ly[ |j;p Ihvp ly[ |;j p Ihvp ly[ |;j p &(@ $ (*& $# )_* @ &_* )WW # W ( @ WW) W$& 572 = 5×100 + 7×10 + 2 vjko;kj G shkIhvpg9afly[lv’ 4 765 = 4×1000 + 7×100 + 6×10 + 5 vkj o;kj G lujraog9afIhvpsqdly[shk 43 896 = 43×1000 + 8×100 + 9×10 + 6 vkj o;kj G lujly[lk,raocxfIhvpgdQkly[sqd s^n lu\\j noj lk,raocxfIhvpgdQkly[sqdD 2 596 800 = 2 +1000 000 + 596×1000 + 8×100 vjko;jkG lv’]hkoshkIhvpgdQkly[sqdraocxfIhvp 3 072 008 045 = 3×1000 000 000 + 72 ×1000 000 + 8×1000 + 4 ×10 + 5 vjko;kj G lk,8Ng9afly[lv’]hkocxfraolulj y[shk 4

#D dkoxP[mP[9eo;ome,t-kf { dko9af]Po9eo;ome,t-kfmu0j Pofh;p|bj’8q;g]dc8j|hvpsks^kpfa’j oUG 1<2<3<4<5<6<7<8<9 { dkoxP[mP[9eo;ome,t-kfmuj,u9eo;o8q;g]d8jk’dao D 9eo;o.f,u9eo;o8q;g]ds^kpd;kj c,jo9eo;ooAos^kpd;jkD 8q;1kj ’G !W c]t _ S !W ,ulv’8q;g]d c]t _ ,u|b’j 8q;g]d ltoAo !W > 9 !W WWW c]t _ ___S _ ___ ,ul8ju q;g]d c]t !W WWW ,ushk8q;g]d ltoAo _ ___ < !W WWWD { dkoxP[mP[9eo;ome,t-kfmuj,u9eo;o8q;g]dgmqkj daoD ສະຫງສ.ວວນ.ລິສຂະສິດ 3fpdkola’gdfc8-j hkpsk0;kF 9eo;o.f,u8q;g]dmu|j b’j s^kpd;jkc,jo 9eo;ooAos^kpd;jkD 8q;1kj ’G ()$& c]t &_)(F lv’9eo;ooU0Pofh;plu8j q;g]d c]t 8q;g]dmu|j bj’ c,jo ( c]t & g-yj’;jk ( > &F ltoAo ()$& > 5987D d=]tou8q;g]dmuj|bj’gmqjkdaoF gryjo9txP[mP[8q;g]dmulv’ s^n mulk, c]t 8q;g]d 8+MwxD 8q;1kj ’G *)!# c]t *&_)F gIqkgsao;kj 8q;g]dmuj|bj’gmqkj daoD faj’oAogIqkmP[8q;g]dmujlv’D gIqkgsao;jk ) > & ltoAo * )!# > * &_)D @ #)( *&W c]t @ #)_ !(&F gIqkgsao;kj 8q;g]dmu|bj’F mulv’F mulk,gmqjkdaoF gIqkmP[8q;g]dmuluj gsao;jk 7 < 9 ltoAo @ #)( *&W < @ #)_ !(&D { dko0v[0Ao9eo;ome,t-kf gryoj 0v[0Ao9eo;o.f|b’j fh;p7jk.dh7P’s^5f c]t 7kj .dh7P’]njo16js^ad8jk’M 0v’,aoD 8q;1kj ’G gryoj 0v[0Ao _ ()# 16sj ^ad8jk’M fa’j oUG 7jk.dh7P’s^5f 7kj .dh7P’]noj 16sj q;|j;p _ ()@ < _ ()# < _ ()$ 16jsq;ly[ _ ()W < _ ()# < _ (_W 16jsq;Ihvp _ (WW < _ ()# < _ )WW 5

$D g]d3ic,’ .og,nvj djvo -k;3ic,’0Po9eo;o3fp.-hg7njv’\\kpg9af8q; gvUo;kj g]d3ic,’ 8q;g]d3ic,a’ I V X L C D M 7evkj o |b’j shk ly[ shkly[ Ihvp shkIhvp rao s^addko0Po9eo;ofh;pg]d3ic,’,ulk,s^addkofa’j oUG { 8q;g]d I , X, C s^n M muj0Po]Po8yfdaos^kpgmnvj c,jo[;dg0QkdaoD g-jaoG II; III; XX; XXX; CC; CCC; MM; MMM. vkj o;kj G @S #S @WS #WS @WWS #WWS @WWWS #WWWD { m5d8q;g]dmuj,u7jk|hvp0Po16jg[Nv’0;k0v’8q;g]dmuj,u7jks^kpd;kj c,oj [;dg0Qk daoD g-ajoG VI; VII; VIII; XI; XII; XIII: XVII; LX; LVII; CCCXXI; MCCLXV vjko;jkG 6 ; 7; 8; 11; 12; 13; 17; 60; 57; 321; 1265 ສະຫງສ.ວວນ.ລິສຂະສິດ { m5d8q;g]dmu,j u7kj |hvp0Po8yf16jg[Nv’-hkp0v’8q;g]dmuj,u7jks^kpd;jk,aoc,oj ]q[ 7jk0v’8q;g]ds^kpd;jkoAovvdD g-jaoG IV; IX; XIV; XL; XC; CXL; CD; CM; MCDXLIX; MMCMLIV vjko;jkG $S _S !$S $WS _WS !$WS $WWS _WWS !$$_ S @_&$ #D [qfg/ydsaf !D dD 7kj 0v’9eo;ome,t-kf.f|bj’9txPocx’[+ 4hkgIqk8n,j W .ljg[Nv’-hkpL c]t 4hkgIqk8nj, W .ljg[Nv’0;kL 0D gryjo.sh9eo;o &#) D 9t8hv’gvqkg]d W .l[j vj o.fgrnjv.shwfh9eo;omuj,u7jks^kpl5fD DDDD&#)S &DDDD#)S &#DDDD)S &#)DDDD @D 9qj’0Po9eo;o8+wxoUgxao8q;g]d c]t 9afgxao\\;flk,8q;g]dD dD lk,]hkoluIj hvplv’rao|bj’Ihvp-k;gvafD 0D lv’Ihvp]hkocxfly[raoshkIhvpD 7D lk,8N|b’j Ihvp-k;]hko|b’j Ihvply[raog9afIhvp-k;shkD ’D lk,Ihvp8Nshkly[]hkog9afly[raogvafD 6

#D 9q’j 0Po9eo;o8+wxoUgxao8q;|a’ln.olv’I6[c[[8kj ’daoD 8q;1kj ’G @ W#& )WW (&!D { lv’8Nlk,ly[shk]hkocxfIhvpraog9afIhvpshkly[gvafD { lv’8Nlk,3dfshk]hkocxfclog9afIhvpshkly[gvafD dD !& @)# $((S 0D # !_W )(& $@WS 7D @( &)@ W&!S ’D !W &(* #_WD 9D !@ _(# !&( *@_D $D 9q’j .-hshk8q;g]d8+wxoUG #F !F WF WF @ 0Po9eo;omu,j ushk8q;g]d grnvj .shwfh9eo;omuj,u 7kj s^kpl5f c]t 9eo;omuj,u7jk|hvpl5fD &D 9q’j xtdv[8k8t]k’]5,j oU.sh4nd8hv’g\\ktlq,D 8k,4aoG dD 8q;g]dsq;|;j pc,oj # c]t 8q;g]dsq;Ihvpc,jo @D 0D 8q;g]dsq;|j;pc,oj ) c]t gxao9eo;o|hvpd;jk !WW 7D 8q;g]dsq;|j;pc,jo WD ’D 8q;g]dsq;raoc,oj @ c]t 8q;g]dsq;Ihvpgmqjklv’gmnjv0v’8q;g]dsq;rao ສະຫງສ.ວວນ.ລິສຂະສິດ 8k,c4;G A. 8q;g]dsq;raoc,oj # c]t 8q;g]dsq;ly[c,jo (D BD 8q;g]dsq;Ihvpc,oj W c]t 8q;g]dsq;ly[c,jo ) CD $ V !WW + & DD 8q;g]dsq;rao c]t 8q;g]dsq;|;j pc,oj 8q;g]dfP;daoD d07’ A B C D 7

ສະຫງສ.ວວນ.ລິສຂະສິດ*D 9qj’-vd|b’j 8q;g]d8k,.9 ,kxPj ocmo8q;g]d.f|bj’.o9eo;o @( $)@ grnvj .shwfhG dD 9eo;o,ushk8q;g]d,u7jks^kpl5fD 0D 9eo;o,ushk8q;g]d,u7jk|hvpl5fD (D 9qj’xP[mP[9eo;oc8j]t76]j 5,j oUfh;pg7njv’\\kp <; > s^n = D dD $ #)( c]t # __)S 0D &( @$) c]t &( $)@S 7D @ W!& WW! c]t !W WW# WWWS ’D ! WWW WW! c]t ___ ___ 9D (& W&# **( c]t (& W&@ **(S lD ! (WW c]t _ V !WW + ( V !W + & -D !& ]hko #$& rao (#@ c]t |bj’3dfg9af]hkogdQkIhvpraogvafD pD # V !W WWW + & V !WW + !W + ) c]t #W &!@D fD @ V !WWW + ( V !W + _ c]t @ V !W WWW + ( V !W + &D )D 9qj’la[xjPo[jvo0v’8q;g]dmuj.-h0Po9eo;o )!* _@# (_) rP’7A’fP;grnvj .shwfhG dD 9eo;owfh,kgxao9eo;o,u7jks^kpl5fmu0j Pofh;p8q;g]dgs^qkj oAoD 0D 9eo;owfh,kgxao9eo;o|hvpl5fmuj0Pofh;p8q;g]dgs^qkj oAoD _D 9qj’dkj p7no9eo;o8+wxoU3fpgvqk8q;g]d W mu1j 6js^ad[+,u7;k,\\kpvvdF c]h;9af]Po 9eo;ofaj’djk;c8j|hvpsks^kpD dD _WWS WW_S _W!S !_WS W_!S WW!_S !WW_S _W!WS _!WWS _WW!D 0D WW#W$!S WW#$!WS W#!WW$S WW$!#WS W$#!WWS WWW$#!S $WWW#!D !WD 9q’j 9af]Po9eo;o]5,j oUc8js^kpsk|hvpD dD ( *$@S * (@$S $ (*@S * ($@S @ $*(S $*@(S ( $*@S $ @*(S @ ($*S $ @(*D 0D @* W#&S *# &W@S @* &#WS @# *&WS @# &W*S @# &*WS &W *#@S &* W#(D 8

!!D 9q’j 0v[0Ao9eo;o]5j,oU fh;p7jk.dh7P’s^5f c]t 7kj .dh7P’]njo16js^ad8jk’MD dD DDDDDDDDDDDDDDDDD < ()$# < DDDDDDDDDDDDDDDDD 16jsq;|j;pD 0D DDDDDDDDDDDDDDDDD < !()&W < DDDDDDDDDDDDDDDDD 16jsq;IhvpD 7D DDDDDDDDDDDDDDDDD < _)!(# < DDDDDDDDDDDDDDDDD 16jsq;ly[D ’D DDDDDDDDDDDDDDDDD < ! @)( *WW < DDDDDDDDDDDDDDDDD 16sj q;raoD 9D DDDDDDDDDDDDDDDDD < ! @)( *WW < DDDDDDDDDDDDDDDDD 16jsq;\\noj D !@D w,h]e|b’j pk; &_#( mm. 9qj’0v[0Ao];’pk;0v’w,h]eoUfh;psq;|;j p m c]t cmD g]d39fG !#D I6h;kj ! xu ,u #*& ;aoS ! ;ao ,u @$ -q;j 3,’S ! -q;j 3,’ ,u # *WW ;yokmuD 4k,;jk 7qomu,j u vkp5 (& xu ,uvkp5pnog4y’8N;yokmu s^n [+ L !$D w]ptsjk’c8j|hk3]dsk8kg;ao,u !&W ]hkody3]c,af c]t 7;k,w;0v’cl’ c,oj #WW WWW km /s. 4k,;jk cl’8kg;ao.-hg;]k9ad;yokmugrnjv,kg4y’|hk3]d L ສະຫງສ.ວວນ.ລິສຂະສິດ !&D .shI6h ! 38ogmqkj !WWW kg c]t ! kg gmqjk !WWW gD [=]ylafzt]yf1kltzq,csj’ |b’j .-hw0gj s^nv’ #WW g 8+1kltzq, ! 38oD 9q’j 7yfw],j ;olko0v’w0jgs^nv’.o1k ltzq, @W kgD !*D ;af45vao|b’j |ad (& g S ;af45-toyffP;daooU]hkovao9t|adgmqkj .f mg L !(D w,h0ufw2vao|bj’pk; & cmD 4hkgryjogvqkw,h0ufw2-toyffP;daooU8+dao3dfvao9tpk; 9addy3]c,af L !)D g]d3ic,’D dD 9qj’0Po9eo;o]5j,oUfh;pg]d3ic,’D *_S (!S _(S #$(S *&(S )_$S # (*$S ! &@_S *W(S ! !W)D 0D 9q’j 0Po9eo;o]5j,oUfh;pg]dvkia[D CXLIX; CDLXXVIII; MCDLXIV; MMCCCXLIV; MMMCMXCIX; ICVIII; CVI; MI; MXIII; MVII. 9

[qfmu @ 9eo;o4h;o ! dyf9tde dyf9tdemu ! mhk; 7e g9Qk0v’Ihko0kpvkskol5dcs’j |b’j wfh[aombd]kpIa[ { ]kp9jkpxt9e ;ao0v’]k;.o|b’j vkmyffaj’oUD ,N 0kpwfhma’\\qf ]kp9jkpma’\\qf zqolts^5[xt9e;ao Xdu[? Xdu[? Xdu[? ;ao9ao )(& W&W *#W W(W + @$$ _)W ;aova’7ko &W_ &)& &)( W&W – (( $*& ;aor5f &@# &&W &@! W#W ;aortsaf *$& &(W *&@ (WW ສະຫງສ.ວວນ.ລິສຂະສິດ ;aol5d (#W &(& &$# (@& ;aoglqk ! W#& ()W )#* (_W ;aovkmyf ! WW( )&W )W_ &$W wfh7yfw]zj qolts^5[xt9e;ao fa’j oUG ;ao9aoG 0kpwfhs^kpd;kj ]kp9kj pF ]k;wfhdew]D zqolts^5[\\kp[;d 16j|hkzqo]q[ ]ts;kj ’9eo;og’yo0kpwfhma’\\qf .sh9eo;og’yo]kp9jkpma’\\qfD ;aova’7koG 0kpwfh|hvpd;jk]kp9jkpF ]k;0kfmboF zqolts^5[\\kp]q[ 16|j hkzqo]q[ ]ts;jk’]kp9jkpma’\\qf .sh9eo;og’yo0kpwfhma’\\qfD dD 9qj’djkp7no8k8t]k’gmy’oU c]h;8nj,.l.j sh4nd8hv’D 0D vu’.lj8k8t]k’mujwfh9kd0=h Xd? c]h;[vd;jk ,u;ao.fcfjmujmhk; 7e wfhdew]D 7D le]a[dko7hk0kpF wfhdew] s^n 0kfmbo c,jolv’gsfdkodq’dao0hk,dao.o lt4kotdkofP;daoD grnvj [aombdlv’gsfdkofh;p9eo;o gryoj deoqfgvqkg7njv’ \\kp[;d <+< -uh[vd;jkwfhdew] c]t g7njv’\\kp]q[ <–< -uh[vd;jk0kfmboD 10

ສະຫງສ.ວວນ.ລິສຂະສິດ9eo;omu,j ug7njv’\\kp <+< 168j +|hkc,oj 9eo;o[;d c]t 9eo;omu,j ug7nvj ’\\kp <–< 168j +|hk c,jo9eo;o]q[D 8q;1jk’G + !W vjko;jk [;dly[ c]t – !@ vjko;kj ]q[ly[lv’D 9qj’[vd9eo;o8=jwxoUF 9eo;o.fgxao9eo;o[;d c]t 9eo;o.fgxao9eo;o]q[D + &S + $S –13; + *; +!(S + 84; – 135; – 7; + 92; – 191; + 207; + 539; – 26; – 413D dyf9tdemu @ ok’ f;’fu pkj ’8k,glAo-njfh;p[kfpkj ’lt\\jeltg\\u3fpg]uj,8Qo9kdg,af o. dD 4hkgryjo\\kpIvp8uo0v’]k;.lgj lAo-noj Ao fh;p8q;vadlvofaj’oUG pjk’wx|hk \\kpfh;p A; B; C;……. ; 10 [kfdhk;D pjk’4vps^a’ \\kpfh;p A'; B'; C';……. ; 10 [kfdhk;D 9q’j ln[8+\\kpIvp8uo0v’ ok’ f;’fu 16gj lAo-njoAo.sh7q[4h;o.oc8]j tmyfmk’D O 0D grnjv-U[vdmyfmk’.odkopjk’ gryoj .-h9eo;o\\kp]efa[Ivp8uo faj’oUG \\kp W 16j95fg]uj,8QoF + ! 169j 5fIvp8uopjk’wx|hk dhk;mu|bj’F + @ 169j 5f[kf dhk;mulj v’ c]t 8+M wxD \\kp –! 16j95fIvp8uopkj ’4vps^a’dhk;mu|bj’F –@ 169j 5f Ivp8uopjk’4vps^a’dhk;mulv’ c]t 8+MwxD 9qj’ln[8+deoqfIvp8uo0v’ ok’ f;’fu fh;p9eo;oD 7D gIqkla’gdfgsao;kj g,af A dq’da[ + !D faj’oAogryjo;kj g,af A 4nddeoqffh;p + ! gryoj gvUo + ! ;jk< va[-yf < 0v’g,af A 0Posp=h A (+1). 9q’j [vdva[-yf0v’g,af8kj ’M mujgIqk\\kp16gj lAo-noj AoD # .97;k, !D 9eo;o4h;o Š 9eo;ome,t-kfmuj,ug7nvj ’\\kp < + < 168j +|hkgvUo;jk 9eo;o4h;o[;dF sn^ 4hk[+,u g7njv’\\kpd=gvUo;kj 9eo;o4h;o[;dF fa’j oAo 9eo;o4h;o[;dc,oj 1; 2; 3; 4; DDD 11

d5,j 9eo;o4h;o[;dlaopt]adfh;p ]+ D ]+ = { 1; 2; 3; 4; DDD } Š 9eo;ome,t-kfmu,j ug7nvj ’\\kp < – < 16j8+|hkgvUo;jk 9eo;o4h;o]q[F faj’oAo 9eo;o4h;o]q[c,jo –1; –2; –3; –4; DDD d5,j 9eo;o4h;o]q[laopt]adfh;p ]− D ]− = { –1; –2; –3; –4; DDD } Š 9eo;o W gxao9eo;o4h;o c8j[+gxao9eo;o4h;o[;d c]t [+gxao9eo;o4h;o]q[D Š 9eo;o4h;omu[j +gxao9eo;o4h;o]q[ c,jo 0; 1; 2; 3; 4; DDD Š 9eo;o4h;omu[j +gxao9eo;o4h;o[;d c,jo 0; –1; –2; –3; –4; DDD Š 9eo;o4h;oma’\\qf3I,g0Qkdaogxaod5,j |bj’,u-n;j jkd5j,9eo;o4h;o c]t laopt]ad fh;p ] D ] = {DDD – 4; –3; –2; –1; 0; 1; 2; 3; 4; DDD } Š m5dM9eo;ome,t-kf c,jo9eo;o4h;o c]t laopt]adfh;p ` ⊂ ] D   @D cdo9eo;o4h;o glAo-njmucj 8j]tg,af,uva[-yfgxao9eo;o4h;o gvUo;kj cdo9eo;o4h;o g-aoj glAo-nj]5j,oUG ສະຫງສ.ວວນ.ລິສຂະສິດ E  C  O  I  A B  D  −10 −  9  −8  −7  −6   −5  −4   −3  −2  −1  0   1  2   3   4   5   6   7   8   9   10   { g,afg7Qk O ,uva[-yf W 0Po OXW? { g,af I ,uva[-yf ! 0Po ; I (1) g,af A (+3) c]t C (– 3) 16jc8]j trkd0v’g,afg7Qk c]t ,uw]ptsjk’ltg\\umP[ .lgj ,afg7Qk O faj’oAo9eo;o +3 c]t 9eo;o – 3 c,oj lv’9eo;odq’dao0hk,daoD Š .sh a ∈] F −a c,jo9eo;odq’dao0hk,0v’ a D 8q;1kj ’G −(−3) = 3, − (+3) = −3 Š lv’9eo;o4h;odq’dao0hk,daoc,oj 9eo;ofP;daoc8j,ug7njv’\\kp8jk’daoD 12

ສະຫງສ.ວວນ.ລິສຂະສິດ#D dkoxP[mP[9eo;o4h;o Š I6h;kj 9eo;o4h;o[;d c]t 9eo;ome,t-kfc,oj 9eo;ovaofP;daoF dkoxP[ mP[9eo;o4h;o[;dd=7nda[dkoxP[mP[9eo;ome,t-kfD Š 9eo;o[;ds^kpd;jk9eo;o]q[g]NvpMD Š le]a[lv’9eo;o4h;o]q[F 9eo;o4h;o]q[.fmuj,uw]ptskj ’9kdg,afg7Qks^kpd;kj 9t,u7kj |hvpd;jkD 8q;1kj ’G – & ,uw]ptsjk’9kdg,afg7Qk & sq;|j;p c]t – # ,uw]ptsjk’9kdg,af g7Qk # sq;|j;pD gIqkg;Qkwfh;kj G – & ,u7jk|hvpd;jk – # 0Pofh;pxt3pdlaopt]adG – & < – # Š la’gdfwfh;jkG 16jcdo9eo;oF lv’9eo;o4h;o8k,.9F 9eo;o4h;omuj16gj [Nv’-hkp 9t,u7jk|hvpd;jk9eo;o4h;omuj16gj [Nv’0;kD 8q;1jk’G – * 16gj [Nv’-hkp0v’ – & fa’j oAo – * < – &D * 16gj [Nv’0;k0v’ & fa’j oAo * > &. zqogonjv’G { m5dM9eo;o4h;o]q[16jg[Nv’-hkp0v’ WF ltoAo m5dM9eo;o4h;o]q[9t,u7kj |hvpd;j k WD { m5dM9eo;o4h;o[;d16jg[Nv’0;k0v’ WF ltoAo m5dM9eo;o4h;o[;d9t,u7jk s^kpd;jk WD $ [qfg/ydsaf !D dD 9eo;o4h;o8+wxoUF 9eo;o4h;o.fgxao9eo;o4h;o[;d s^n 9eo;o4h;o]q[ L – 7; – 21; 19; – 15; – 23; 15; 9; 8; 12; 2; 0; 3; – 1; – 6 0D [aofk9eo;o4h;omuj.sh.o0=h Xd? 9eo;o4h;o.fgxaovq’xtdv[0v’ ]+ s^n ]– 8q;1jk’G + #∈]+ c]t − &∈]− 13

@D 9qj’8nj,g7nvj ’\\kp ∈ s^n ∉ .lj[jvo9Eg,af.sh4nd8hv’D + # DDDD ]+ S + (DDDD ]+ S –!W) DDDD ]+ S − $ DDDD ]–S + * DDDD ]+ S – 67 DDDD ]+ S − & DDDD ]– S + ) DDDD ]– S –13 DDDD ]+ S + @$ DDDD ]+ S + !@_& DDDD ]+ ; − #(DDDD ]–S + @(DDDD ]– S − &(DDDD ]– S − @W(DDDD ]– S +!W$ DDDD ]– #D 9qj’[vdva[-yf0v’g,af A; B; C; D; E; F c]t G D H  E  C  O  I  A  F  B  D  G   −4   −2   0  2  $D dD 9q’j c8h,cdo9eo;o4h;o 3fp;jk OI =1cm F O(0) c]t I(1)D 0D 9q’j ;k’g,af A; B; C; D; E; F c]t G mu,j uva[-yf + 5; –3; +7; +2; – 4; – 6; +8 8k,]efa[D 7D 9q’j ;k’g,af A',B', D' c]t F' muj,uva[-yfgxao9eo;odq’dao0hk,da[va[-yf0v’ g,af A; B; D; c]t F 8k,]efa[D ສະຫງສ.ວວນ.ລິສຂະສິດ &D 9q’j [vdva[-yf0v’[aofkg,af]5j,oUD dD C DB A −15   −10   P 0D N M F 108   103   7D M   D  C  E  A  3  −4   9qj’;k’ O(0) c]t I(1) 16j0=h 7D 14

*D .sh O(0)F I(1) c]t g,af9eo;o|b’j ]5j,oUG M  D  A  O  I  F  B  G   E  dD 9q’j [vdva[-yf0v’[aofkg,afgmy’oUD 0D 9q’j [vdlv’g,afmu,j uva[-yfdq’dao0hk,daoD 7D 9q’j [vdg,afg7y’j dk’0v’ [AB]; [DI]. ’D 9q’j ;k’g,afg7y’j dk’0v’mvj o-n8j +wxoUrhv,ma’deoqfva[-yf0v’,aoG [AD]; [IE]; [MI]. (D dD 9q’j xP[mP[9eo;o4h;o.oc8]j t768j +wxoUG −( c]t + @S @$& c]t @&!S #W( c]t − * !*_S − ( c]t + (S − _ c]t −!#S −( c]t +@S − _!$ c]t − _!$S @ $*) c]t @ &!& D ) c]t &S − _) &$@ c]t − _) #$@S # c]t *S )! #(& c]t )! !_&S ສະຫງສ.ວວນ.ລິສຂະສິດ 0D 9q’j 9af]Po9eo;o4h;o.o0=h d 8k,]efa[0Bo X c8j|hvpsks^kp ?D )D dD 9q’j 8n,j g7njv’\\kpG < ; >; = .lj[vj o9Eg,af.sh4nd8hv’D −!# DDDDD −!@S !! DDDDD !#S −!W DDDDD #S −(#@ DDDDD −(#&S −_#* DDDDD −*#_S !& DDDDD −!(S !W* DDDDD !*WS − $&#* DDDDD − $&#*S &(@! DDDDD− &(@!D −#& DDDDD − #&S −!$_ DDDDD − $!_S −)$@ DDDDD (@&S 0D 9q’j 9af]Po9eo;o.o0=h d 8k,]efa[cI, X c8js^kpsk|hvp ?D _D dD9qj’deoqf7kj 0v’9eo;o4h;o a; b; c; d; e 8k,]efa[ c]t 16j]ts;jk’ − # sk + @ 0D 9qj’deoqf7kj 0v’9eo;o4h;o x; y 3fp;kj −7 < x ≤ −6 c]t 25 > y > 23 7D 9q’j deoqf7jk0v’9eo;o4h;o m; n; p mugj xaova[-yf0v’g,af.ocdo9eo;o4h;o ]5,j oUG P   −16   −15   m   n  15

!WD dD 9q’j 0Po8nj,.lj8k8t]k’]5,j oUD c,jo9eo;odq’dao0hk, –@ W +( c,oj 9eo;odq’dao0hk, –!_ + & – ) +# 0D 9qj’la’gdfzqowfhIa[.oc4;mu ! c]t c4;mu # r;dohv’gsao;kj co;.fL gIqk4vowfh75o]adltot0v’9eo;odq’dao0hk,7nco;.fL !!D dD 9q’j [vd9eo;o4h;o x mu1j 6j]ts;jk’ – # c]t &D 0D 9qj’[vd9eo;o4h;o y 3fp;jk –!$ ≤ y < −12 D 7D 9q’j [vd9eo;o4h;o z 3fp;jk 8> z > 4. ’D 9q’j [vd9eo;o4h;o P 3fp;kj 0 < P ≤ 6 D ສະຫງສ.ວວນ.ລິສຂະສິດ 16

[qfmu # dko[;d c]t dko]q[9eo;ome,t-kf ! dyf9tde dyf9tdemu !  dD 9qj’0ufvhv,7e8v[mu4j nd8hv’D 0=h,6o s^n 7e4k, 7e8v[.shg]nvd zqo[;d 4825+ 25196 gmqkj ABC 29911 30021 29121 4 . 1+ 27 = 518 8q;g]dmuj0kf wx16[j jvo9Eg,af0v’rqfmu|bj’ 91 0 c,jo 2 . 6 − 79 = 137 8q;g]dmuj0kf 9 10 wx16j[jvo9Eg,af0v’rqfmu|bj’ ສະຫງສ.ວວນ.ລິສຂະສິດ c,jo 0D 9qj’fafcx’zqo0v’dko7yfw]jg]d]5j,oU.sh4nd8hv’D 2609 4725 2961 8632 + 352 + 3907 − 2609 − 4725 2591 7732 0362 4917 7D 9q’j la’gdfzqowfhIa[9kd0=h 0 c]h;xtdv[xt3spd]5j,oU.sh4nd8hv’D .shlv’9eo;ome,t-kf8k,.9 a c]t b; a > b F a −b c,jo9eo;omujgvqk,kDDDDDDDDX[;dF ]q[F 76oF sko? da[ b c]h;gmqjkda[ a D ltoAo le]a[9eo;ome,t-kf a,b c]t c g-yj’ a > b g,njv a −b = c c]h; a = DDDDDDDDDDDDDDDDD 17

dyf9tde @D mhk; mv’ |ad 47 kg F ok’ [5o,u |ad 45 kg c]t dtgxqJkg7njv’0v’mhk; mv’ |ad8 kg D dD 4hkmhk; mv’ gxhdtgxqJkg7njv’0v’]k;0Bo-a’j F c]h;oE|ad];,mujxtdqf16j|hk9=-y’ -aj’fa’j dkj ; c,oj gmqkj .fL 0D ok’ [5o,u gxhdtgxqJkg7njv’0v’]k; c]h;0Bo-a’j F oE|ad];,xtdqf169j =0v’-y’-a’j c,jo50 kg D 9q’j 7yfw]oj E|addtgxJqkg7nvj ’0v’ ok’ [5o,uD dyf9tde #D ສະຫງສ.ວວນ.ລິສຂະສິດ lk,vhkpohv’G mhk; 7erk ,u\\kd[uj 38 |;j pF mhk; 7erPo ,u\\kd[uj 35 |j;p c]t ok’ ,vo]u ,u\\kd[uj|hvpd;jkmhk; 7erPo 7 |j;pD dD 9q’j 7yfw]j9eo;o\\kd[uj0v’ok’,vo]uD 0D grnjv7yfw]9j eo;o\\kd[ujma’\\qf0v’lk,vhkpohv’gryjogIafco;.fL 7D 9q’j xtdv[xt3spd]5j,oU.sh4nd8hv’D • dko[;d c,oj dko7eo;ogrnjv7yfw]j DDDDDDDDDDDDDDDDDD 0v’lv’9eo;omuj,usq;|j;p vaofP;daog0qhkdaogxao9eo;ofP;D • dko]q[ c,oj dko7eo;ogrnjv7yfw]j DDDDDDDDDDDDDDDDDDD 0v’lv’9eo;omuj,usq;|;j p vaofP;dao3fp;jk 9eo;omu|b’j 8hv’s^kpd;kj s^n gmqkj da[9eo;omulv’D @D .97;k, dD dko[;d9eo;ome,t-kf dko[;d9eo;ome,t-kf c,oj dko7eo;omu7j yfw]jzqo[;d Xs^n 7yfw]jdko3I, gvqklv’ s^n s^kp9eo;o ? g0Qkgxao9eo;ofP;D laopt]adfh;pg7njv’\\kp < + < 8q;1jk’G • 2961gvUo;kj <zqo[;d0v’ 2609 da[352< 2609 + 352 = 2961 rqf0v’zqo[;d zqo[;d • 2609 c]t352gvUo;kj <rqf< 0v’zqo[;d 18

75o]adltot • zqo[;d0v’lv’ s^n s^kp9eo;ome,t-kfc,oj 9eo;ome,t-kfD \\kp7;k,;kj G .sh a∈` c]t b∈` gIqkwfh a + b ∈` s^n g;Qkvudco;|bj’;jkGdko[;d.odj5,9eo;ome,t-kf gxaodqfdko7eo;orkp.oD • zqo[;d0v’lv’9eo;ome,t-kf[+xjPocx’ 4hk;jkgIqkla[xPj o[vj orqf8jk’MD \\kp7;k,;kj G .sh a∈` c]t b∈` gIqkwfh a + b = b + a 8q;1jk’G 7 + 5 = 5+ 7 = 12 s^n g;Qkvudco;|b’j ;jkG dko[;d.odj5,9eo;ome,t-kf ,u75o]adltotla[xPj o [vj owfhD • zqo[;d0v’s^kp9eo;ome,t-kf[+xPj ocx’F 4hkgIqk3I,gvqk[k’rqf0v’r;d ,aog0Qkdaodjvo c]h;cmofh;pzqo[;d0v’r;d,aoD \\kp7;k,;kj G .sh a∈`, b∈` c]t c∈` gIqkwfh a + b + c = (a + b) + c = a + (b + c) s^n g;Qkvudco;|b’j ;kj G dko[;d.od5j,9eo;ome,t-kf ,u75o]adltot3I,\\6wj fhD ສະຫງສ.ວວນ.ລິສຂະສິດ 8q;1jk’G (7 + 5) + 8 7 + (5 + 8) 12 + 8 7 +13 20 20 faj’oAo 7 + 5 + 8 = (7 + 5) + 8 = 7 + ( 5 + 8 ) = 20 g8adoyddko[;d • djvovnoj oadIPo8hv’[;dlv’9eo;omu,j u|bj’8q;g]dwfh1jk’vaf8t3o,af c]t xklt9kdg7nvj ’7yfw]Dj 8q;1jk’G 7 + 5 = 12 ; 9 + 9 = 18 ; 6 + 8 = 14 . 19

• zqo[;d0v’lv’9eo;omu0j Pofh;ps^kp8q;g]dD 8q;1kj ’G dko7yfw]j 2+19+17+16 6 + 9 = 15 + 70 + 40 = 110 900 + 800 = 1700 5849 2000 + 5000 = 7 000 8825 2976 + 5849 = 8875 0D dko]q[9eo;ome,t-kf • dko]q[ g-yj’laopt]ad fh;pg7njv’\\kp< − < c,oj dko7eo;omuj7yf w]j7;k,c8d 8jk’ Xzqo]q[? 0v’lv’9eo;oD • lv’9eo;ome,t-kf8k,.9 a c]t b 3fp;jk a >b S a −b c,oj 9eo;omuj gvqk,k[;dda[ b c]h;gmqkj da[ a D 4hk a −b = c c]h; b + c = a D 8q;1kj ’G ສະຫງສ.ວວນ.ລິສຂະສິດ 8632 − 4725 = 3907 • 3907 gvUo;kj zqo]q[0v’8632da[ 4725 D • 9eo;o8632c]t 4725 gvUo;kj rqf 0v’ rqf0v’zqo]q[ zqo]q[ zqo]q[D • 8632 gvUo;jk 8q;8ah’]q[F 4725 gvUo;kj 8q; ]q[D 75o]adltot • zqo]q[lv’9eo;ome,t-kf3fpgvqk9eo;os^kpd;jk ]q[9eo;o|hvpd;jk c,jo9eo;ome,t-kfD \\kp7;k,;jkG .sh a∈` c]t b∈` , a > b gIqkwfh a −b ∈` • zqo]q[lv’9eo;ome,t-kfF ]efa[rqf,u7;k,le7aoF gIqk[+lk,kfla[xjPo [vj owfhD \\kp7;k,;kj G .sh a∈` ; b∈` F gIqkwfh a −b [+gmqjk b − a .  laopt]ad a −b ≠ b − a F   8q;1jk’G 9−5 ≠ 5−9 grkt;jk 9−5 = 4 S c8;j jk5−9 [=lj k,kf7yfw]jwfh.od5,j ` grkt;kj 5<9 D 20

g8adoyddko]q[G d;f7noG 8295 4588 − 4+15 8+18 +3707 3 70 7 8295 #D [qfg/ydsaf !D 9qj’7yfw]zj qo[;d c]t zqo]q[]j5,oU X[+.-hg7njv’7yfw]?j D dD 439 + 257 S 0D 1986 + 734 S 7D 984 + 3976 S lD 4075 + 318 + 1482 ’D 7849 + 251 + 9090 S 9D 15 + 2895 + 357 S fD 4583 − 956 S mD 107 − 59 D -D 357 −153S pD 4910 − 254 S 8D 13548 − 8937 S 4D 47201 − 28795 S @D 9qj’oe.-h75o]adltotdkola[xjPo[jvowfh c]t dko3I,\\6wj fh 0v’dko[;d grnjv,k 7yfw]jg]d]5j,oU.sh’jkp0bhoD ສະຫງສ.ວວນ.ລິສຂະສິດ dD 24 +13 +16 +17 S 0D 25 + 39 + 75 + 85 + 61 S 7D 438 + 247 +192 + 62 +153 S ’D 131+ 547 + 53 + 70 D #D 9q’j gvqk8q;g]d8nj,.l[j jvo9Eg,af grnjv.shdko7yfw]jgxaowxwfh c]t 4nd8hv’D 438 .27 ..8 . 659 16 0. 3872 . + . 5. +1 0 3 +54. + 8. . + 7.5 + 90 . 4 +2.9 9 + ... 70 7.9 7. . 973 2.63 .4 41 64 6 3 9 .53 . 49 6. . . 6 91 30. 2 − .. −15 . −. 87 −1. 2 . −. 386 222 3. 6 16 4 66.6 7. 7. 21

$D 9q’j gvqk8q;g]d8n,j .lj[jvo9Eg,afgrnjv.shdko7yfw]jgxaowxwfh c]t 4nd8hv’D dD .386 + 45 .+ 6.97 = .16.2 S 0D 38.+ 6.3 +.85 = 1547 7D .362 + 45.0 + 80. = 6.59 S ’D 997.+ 3.96 +.537 = .04.6 9D ..79 + 90.2 + 63. = .4428 S lD .987 −..3 = 633. -D .38.− 2.6 = 5.08 S pD .367 −..9 = 388.S fD .34.0 − 68.98 = 2.61 S 8D .38.− 42.4 = 2.91 S &D 9q’j -vdsklv’9eo;ome,t-kf]Po4afdaoF I6h;jkzqo[;d0v’r;d,aogmqjk 1275 D *D 9qj’-vdsklk,9eo;ome,t-kf]Po4afdao x, y c]t z F I6h;kj x < y <z c]t x + y +z = 435 D (D xB,sq;|b’j ,u 146 |hkD 4k,;kj grnjv0PooEg[uh|hk0v’xB,Ivf|hkmu 146 gryoj 8hv’ 0Po8q;g]d9adgmjnvL XI6h;jk0Po|bj’gmnvj wfh|b’j 8q;g]d?D )D gryjo0PooEg[u|hk0v’xB,sq;|bj’3fpg]uj,9kd|hk1,2,3,4,..., gryoj wfh0Po8q;g]d 8kj ’M 516 gmnvj D 4k,;kj xB,sq;faj’djk;,u9ad|hkL XI6h;jk0Po|b’j gmnjvwfh|b’j 8q;g]d?D ສະຫງສ.ວວນ.ລິສຂະສິດ _D lt|k,dy]kcs’j |b’j [ao95wfh 10000 7qoF 4hk,uz6hg0Qk-q,dy]k 78757qoD 4k,;jk ,u9ad[vj ooaj’mugj xqjks;jk’L !WD 3I’IPo,afmtpq,csj’|bj’,uoadIPoma’\\qf 1522 7qoD .ooU 0Ao ,D! ,u 439 7qoF 0Ao ,D@ ,u 413 7qo c]t 0Ao ,D$ ,u 295 7qoD 4k,;kj oadIPo0Ao ,D# ,u9ad7qoL !!D xB,c[[IPo7toyflkf |b’j sq; c]t xB,c[[IPorklk]k;lv’sq; ]k7k];, dao17000 du[D xB,c[[IPo7toyflkf lv’sq; c]t xB,c[[IPorklk]k;lk, sq; ]k7k];,dao 29000 du[D dD 4k,;kj xB,c[[IPo7toyflkf @ sq; c]t xB,rklk]k; 4 sq; ,u]k7k];,dao gmqkj .fL 0D 4k,;kj xB,c8]j t-toyf,u]k7ksq;]tgmqjk.fL 22

ສະຫງສ.ວວນ.ລິສຂະສິດ!@D mhk; 7es^hk ,ug’yo 3550 du[F r+0v’]k;gvqk8jn,.shvud 4850 du[D mhk; 7es^hk [vd;kj .og;]koU]k;,ug’uos^kpd;kj ok’ pvf2hk gvNvp0v’]k; 1700 du[D 4k,;kj ok’ pvf2hk ,ug’yogmqjk.fL !#D ok’ mPj ’me wfhltlq,ltc8,ma’\\qf 1253vaoD .ooU,u 357 vao c,oj ltc8,gxao I6[laf8kj ’MD 4k,;kj ltc8,0v’ ok’ mjP’me mu[j +c,oj I6[laf ,u9advaoL !$D 4a’|j;p|b’j [ao95oEwfhs^kpd;kj 75 3]yf c]t |hvpd;jkws 15]yfD .sh I6h;kj ws[ao95oEwfh 25 ]yfD 4k,;kj 4a’|;j poAo[ao95oEwfh9ad]yfL c]t 75[ao95wfh9ad]yfL !&D 3I’IPo,afmtpq,cs’j |b’j F 0Ao ,D! ,ulv’shv’7nG shv’,!Dd c]t shv’,!D0F shv’ ,!Dd ,uoadIPopy’ 237qo c]t oadIPo-kp 20 7qoD shv’,!D0 ,uoadIPoma’\\qf 39 7qoD .olv’shv’0Ao ,D! ,uoadIPopy’ma’\\qf 41 7qoD dD 4k,;kj 9eo;ooadIPoma’\\qf0v’0Ao ,D! ,u9ad7qoL 0D 4k,;jk 9eo;ooadIPopy’.oshv’,!D0 ,u9ad7qoL !*D mhk; 7volts;ao wfh9af’ko;aogduf0v’]k;7q[Iv[12xu .o,N;aomu 1585]kF r+0v’ ]k;wfhg;Qkda[]k;;jk xuoUg9Qk,uvkp5|hvpd;jkvhkpdqd0v’g9Qk 25 xur=fuD 4k,;kj xu oUvhkpdqd0v’ mhk; 7volts;ao ,uvkp59adxuL !(D ok’ ;aofu wx8ts^kfF ]k;,ug’yo16j.odtgxqJk .[ 50000 du[ |bj’.[ c]t .[5000du[ lk,.[D g,nvj da[,kIvfgInvo g’yo0v’]k;pa’gs^nv16j.odtgxqJk ,u.[10000 du[ |bj’ .[F .[ 2000 du[ ,uluj.[ c]t .[1000 du[ ,ulk,.[D 4k,;jk ]k;9kj pg’yo16j8ts^kf ma’\\qfgmqkj .fL !)D ]qfg,7ao|bj’ ,uz6h3fplko 40 7qo vvd9kd8ts^kfg-QkD g,nvj wxIvfxhkp9vf[jvo|bj’ z6h3fplkowfh]q’]qf9eo;o|bj’ c]t 0Bovud 20 7qoD dvj ovvd9kdxhkp gryoj oa[z6h3fp lko.o]qf,u 367qoD 4k,;jk z6h3fplko]q’]qf9ad7qoL 23

!_D gryjo.-h8q;g]d1,3 c]t 5 grnjv0Po9eo;omu,j ulk,8q;g]d8kj ’daoD ສະຫງສ.ວວນ.ລິສຂະສິດ 3 5 gIqkwfh 135 1 35 5 dD 9q’j xtdv[Ijk’’jkw,hgmy’oU.sh7q[4h;oD 0D 4k,;jk gryjolk,kf0Powfh9ad9eo;o L 7D 9qj’9af]Po]efa[9eo;owfh,kc8j|hvpsks^kpD ’D 9qj’7yfw]jzqo[;d9eo;omuwfh,kma’\\qfD 9D 9q’j 7yfw]j7;k,s^5f]njo]ts;kj ’9eo;os^kpl5f c]t 9eo;o|hvpl5fD @WD dD 9qj’xtdv[ c]t 7yfw]jdko7eo;o]5,j oUD S = 1+ 2 + 3+ 4 + 5 + 6 + 7 + 8+ 9 +10 S = 10 + 9 + 8 +............................+1 S +S = 11+11+................................ +11 2S = 11×.................. S = ......................... 0D oe.-hdko7yfw]j.o0=h Xd? grnvj 7yfw]jzqo[;d]5,j oUG a. S = 1+ 2 + 3+ 4 + 5 +...+ 99 b. S = 1+ 2 + 3+ 4 + 5 +...+ 200 @!D 9q’j xtdv[8k8t]k’]5j,oU.sh7q[4h;o c]t 4nd8hv’ 3fp.shzqo[;d8k,4aoF c4; c]t 8k,glAogoa’j 9v,gmqkj daoD dD 0D 27 6 18 13 15 17 38 24

[qfmu $ dko[;d c]t dko]q[9eo;o4h;o !D dyf9tde dyf9tdemu ! mhk; 7e pjk’8k,cdo9eo;o4h;o d fh;p[kfdhk;lt\\jeltg\\umuj,u7;k,d;hk’ OA g-yj’ O(0), A(1) c]t g]u,j 8Qo9kdg,af O. d C' B' A' O A B C dD 4hkmhk; 7e pkj ’wx|hk & [kfdhk;9kd O c]h;wx|hk8nj,vud # [kfdhk;F gsao;kj mhk; 7e wxg4y’95f HX)?D ltcf’;kj w]pt mk’wxwfh ) sq;|;j p s^n (+5) + (+3) = +8 D 4hk]k; 4vps^a’9kd O luj [kfdhk; c]t 4vps^a’vud shk [kfdhk; c]h;9twx1j6 J'(−8) D ltcf’;kj (−4) + (−5) = −9 D ສະຫງສ.ວວນ.ລິສຂະສິດ 9qj’xtdv[8k8t]k’]5,j oUD dkog7njvomuj9kd O mu8j A’l5fmhkpmuwj xg4y’mP[.lj ltcf’dkog7njvo g,af O mujfh;p9eo;o wx|hk & [kfdhk; 8k,fh;p wx|hk8n,j vud # [kfdhk;D wxg4y’g,af HF ltcf’;jkwx (+5) + (+3) = +8 4vps^a’ $ [kfdhk; c]t |hk ) sq;|;j psjk’9kd O D 4vps^a’vud & [kfdhk;D (−4) + (−5) = −9 wxg4y’g,af J′F ltcf’;jk4vp wx|hk @ [kfdhk; 8k,fh;p s^a’ _ sq;|;j pskj ’9kd O D wx|hk8n,j vud * [kfdhk;D 4vps^a’ # [kfdhk; c]t 4vps^a’vud ( [kfdhk;D 25

wx|hk _ [kfdhk; c]t 4vps^a’vud $ [kfdhk;D wx|hk # [kfdhk; c]t 4vps^a’vud & [kfdhk;D 4vps^a’ # [kfdhk; 8k, fh;pwx|hk _ [kfdhk;D 4vps^a’ ) [kfdhk; c]t 8k,fh;pwx|hk $ [kfdhk;D 7D 3fpla’gdf9kdzqowfhIa[.o4aomulk,0v’8k8t]k’gmy’oUF 9qj’4vogvqks^addko 0v’dko[;d9eo;o4h;o .od=]tou8jk’MD dyf9tdemu @ g-ajofP;daoda[16j.o ` F .o ] 4hk a− b = c c]h; c c,oj 9eo;o4h;omu[j ;dda[ b gmqkj da[ a \\kp7;k,;jk b+ c = a. 8q;1jk’G dD (+3) + (+1) = +4 ltcf’;jk (+4) − (+3) = +1 s^n (+4) − (+1) = +3 0D (−3) + (+5) = +2 ltcf’;jk (+2) − (−3) = +5 s^n (+2) − (+5) = −3 9q’j xtdv[8k8t]k’]5,j oU8k,8q;1kj ’gmy’oUD ສະຫງສ.ວວນ.ລິສຂະສິດ ab a−b= a + X−b? = +4 + 3 (+4) − (+3) = +1 (+4) + (−3) = +1 +4 +1 +2 +5 +2 −3 +7 −4 −5 +8 −9 +6 −4 −4 −3 −7 26

−8 −4 −12 − 9 9q’j xP[mP[zqowfhIa[.o4aomulk, c]t luj c]h;4vogvqks^addko0v’dko]q[9eo;o 4h;oD @D .97;k, !D w]ptskj ’9kd 0 D .sh A; B; C; D gmy’cdo9eo;o4h;o O(0), I(1) : B C OI AD −6 −3 +4 +6 g,af D(+6) F fa’j oAo ];’pk;0v’mvj o-nj [OD] c,jo OD = 6 gryjogvUow]ptsjk’ 0v’9e o;o4h;o +6 9kd W c,oj *D g,af C(−3)F faj’oAo w]ptsjk’0v’9eo;o 4h;o –3 9kd W c,jo];’pk;0v’ mvj o-nj [OC] c,jo OC = 3D ສະຫງສ.ວວນ.ລິສຂະສິດ ‚ 4hk g,af M ,uva[-yf x, w]ptsjk’0v’9eo;o4h;o x 9kd W c,oj ];’pk;0v’ mjvo-nj [OM] D ‚ w]ptsjk’9kd W 0v’9eo;o4h;o8kj ’Mgxao9eo;o[;dg]NvpMD @D dko[;d9eo;o4h;oD ‚ grnvj 7yfw]jzqo[;d0v’lv’9eo;o4h;omuj,ug7njv’\\kp7ndaoF gryjoxt8y[affaj’oUG [;dw]ptsjk’9kd W 0v’r;d,ao c]h;gvqkg7nvj ’\\kp7ndao0v’r;d,ao.lj8+|hk 0v’zqo[;dD 8q;1kj ’G dD (+3) + (+7) = +10S grkt;kj +3 ,uw]ptsjk’9kd W gmqkj # c]t +7 ,uw]ptskj ’ 9kd W gmqkj ( c]t 3 + 7 =10 D 0D (−8) + (−12) = −20S grkt;jk − 8 ,uw]ptskj ’9kd W gmqkj ) c]t −12 ,uw]pt sjk’9kd W gmqjk !@ c]t 8 +12 = 20 D 27

‚ grnvj 7yfw]zj qo[;d0v’lv’9eo;o4h;omu,j ug7njv’\\kp8jk’daoF gryjoxt8y[affaj’oUG gvqkw]ptskj ’0v’9eo;omusj jk’9kd W s^kpd;kj ]q[w]ptsjk’0v’9eo;omujskj ’ 9kd W |hvpd;jk c]h;gvqkg7nvj ’\\kp0v’9eo;omusj kj ’9kd W s^kpd;jk.lDj 8q;1kj ’G dD (−6) + (+8) = +2 S grkt;jk 8 − 6 = 2 c]t g7njv’\\kpc,jog7nvj ’\\kp0v’ + ) grkt + ) 16sj jk’9kd W s^kpd;jk − *D 0D (−7) + (+4) = −3S grkt;jk 7 − 4 = 3 c]t g7njv’\\kpc,jog7njv’\\kp0v’ −7 grkt −7 16sj jk’9kd W s^kpd;kj + 4 D 75o]adltotD ‚ dko[;d.odj5,9eo;o4h;o gxaodqfdko7eo;orkp.oD ‚ dko[;d.od5j,9eo;o4h;o ,u75o]adltotla[xPj o[jvowfh c]t 3I,\\6wj fhD 8q;1kj ’G dD (−7) + (+4) = (+4) + (−7) = −3 Xla[xPj o[jvo? 0D (+8) + (−6) + (−5) = [(+8) + (−6)] + (−5) = +8 + [(−6) + (−5)] = −3 ສະຫງສ.ວວນ.ລິສຂະສິດ X3I,\\j6? ‚ dko[;ds^kp9eo;o4h;o7yfw]wj fhfh;plv’;ymufa’j oUG ;ymumu !G gryoj 7yfw]zj qo[;dgmnjv]tlv’rqfc8j-hkpsk0;k9qog4y’rqfl5fmhkpD 8q;1kj ’G A = (+7) + (+4) + (−3) + (−4) + (5) A = +11 + (−3) + (−4) + (+5) A = +8 + (−4) + (+5) A = +4 + (+5) A=9 ;ymumu @G [;d9eo;omuj,ug7njv’\\kp7ndaodvj o 9kdoAo9b’j [;dzqo[;dmujwfh,k.ldj aoD 8q;1kj ’G A = (+7) + (+4) + (–3) + (–4) + (+5) + (–2) A = (+ 7) + (+4) +(+5) + (–3) + (–4) + (–2) A = (+ 16) + (–9) A=7 28

‚ zqo[;dlv’9eo;odq’dao0hk,daogmqkj l6og]NvpMD ‚ xUo7noG4hkzqo[;dlv’9eo;ogmqjkl6oc,jolv’9eo;ooAogxao9eo;odq’dao0hk,daoD 8q;1kj ’G dD X–5) + (+5) = 0 0D 7 + x = 0 gIqkwfh x c,jo9eo;odq’dao0hk,0v’ 7 F fa’j oAo x = –7. #D dko]q[9eo;o4h;oD gIqkI6hc]h;;kj G 9eo;odq’dao0hk,0v’ x c,oj −x ‚ grnvj 7yfw]zj qo]q[0v’lv’9eo;o4h;oF gryjo7yfw]zj qo[;d8q;8A’]q[da[9eo;odq’dao 0hk,0v’8q;]q[D .shlv’9eo;o4h;o8k,.9 a c]t b ; a − b = a + (−b). 8q;1kj ’G 7D −3 − (−9) = −3 + 9 = 6 S dD 7 − 5 =7 + (−5) = 2 S ’D −12 − (−5) = −12 + 5 = −7 0D 3 − (−5) =3 + 5 = 8 S 9D −13− 9 = −13 + (−9) = −22. ສະຫງສ.ວວນ.ລິສຂະສິດ $D dko7yfw]j\\;f7eo;o0v’9eo;o4h;oD dD \\;f7eo;omu[j +,u;q’g]a[D \\;f7eo;omuj[+,u;q’g]a[lk,kf7yfw]jwfhfh;plv’;ymufa’j oUD 8q;1jk’G A = −14 − (−8) + 15 − (−7) + (−8) −9. ;ymumu!G 0Po\\;f7eo;o A gxaozqo[;d c]h;7yfw]j7nda[dko[;ds^kp9eo;o4h;o A = −14 − (−8) + (+15) − (−7) + (−8) −(+9). A = −14 + 8 + 15 +7 + (−8) +(−9). A = −14 + 15 +7 +(−9). A = −14 +(−9) + 15+7 A = −23 +22 A = −1 ;ymumu @G 0Po\\;f7eo;o A gxaoI6[0Posp=hc]h;3I,gvqkrqfmuj,ug7njv’\\kp16|j hk7n dao,kw;hoedaoF 7yfw]7j nda[dko[;ds^kp9eo;o4h;oD 8q;1jk’G A = −14 − (−8) + 15 − (−7) + (−8) −9 A = −14+ 8+ 15+7 + (−8) +(−9). A = 30+(−31) A = −1 29

0D \\;f7eo;omu,j u;q’g]a[D \\;f7eo;omuj,u;q’g]a[F gIqk7yfw]j.o;q’g]a[djvoD 8q;1kj ’G A = (− X!+)−!@? +X−_ + !!? A = (− X−#? +X+@? A = (+#+@ A = !@ #D [qfg/ydsaf B = (−11) + (+12); ສະຫງສ.ວວນ.ລິສຂະສິດ D = (+4) + (+15); !D 9qj’7yfw]jg]d]5j,oUG F = (−18) + (+7); H = (−18) + (+15); A = (−6) + (+11); J = (−6) + (+6); C = (+12) + (−5); L = −23 + 28; E = (−6) + (−11); N = −17 + (−21); G = (−15) + (−12); S = (−601) + (−11) I = (−19) + (−18); K = (+18) + (−24); M = 53 + 76; P = (−127) + (+161); 2D 9qj’7yfw]gj ]d]5,j oUG B = X+12) − (−5); A = X−16) − (+11); D = X−14) − (+21); C = X−6) − (−15); F = X+13) − (+15); E = X−12) − (−12); H = X+25) − (+11); G = X+7) − (−18); J = X+96) − (−101); I = X+16) − (−71); L = X−9) −(+21); K = X−167) − (+311); 3D 9q’j 8nj,9eo;omuj4nd8hv’.l[j vj o9Eg,afD dD DDD+ (+7) = −11 0D DDD + (−12) = 15 7D (−3) + DDD = −6 ’D (−18)+ DDD = 2 9D DDD − (+7) = −12 lD DDD − (−3) = −6 -D DDD − (−15) = +27 pD DDD − (+16) = 18 fD 7 + DDD = −3 8D (−13) − DDD = −5 4D 18 − DDD = 11 mD 3 − DDD = 8 30

$D 9qj’7yfw]\\j ;f7eo;o]5j,oUG A = (−18) +(−15)+ (+12)+ (−7)+ (+15); B = (−26) +(−4)+ (+17)+ (−3)+ (−15) C = (+16) +(+11)+ (−5)+ (−13)+ (+2) D = (−3) −(+12)+ (+18)− (−7)+ (+17) E = 12 − (−15)+ (−6)− (−9)− (+12) F = (−5) −(−6)+ 16− (+12) −15 G = (−18) −[(−15)+ (+12)+ (−7)]+ (+15) H = (−18) +[(−15)− (+12)]−[−7+ 15] I = [(−18) +(−15)]−[(+12)+ (−7)]+ 35 J = (−18) −[(−15)+ (+12)]+ [(−47)+ (−15)] &D 9qj’-vd7kj 0v’ x ]5,j oUG 0D (−8)− x =2; ສະຫງສ.ວວນ.ລິສຂະສິດ dD x + (−7) =15; ’D x+ (+14) = −5); 7D x −(−9) = −4; lD (+13) −x= −1 9D (−15) −x= 8; pD x+(−9) =18; -D x− (+3) = −15; 8D 3+(x−6) = 3−(−4); fD (19−3) + x=(25−18); *D 9qj’8nj,8k8t]k’]5j,oU.sh4nd8hv’D a) x –6 5 4 y –7 –3 –2 x+y 8 12 –11 –9 7 –16 b) x 12 −12 17 y –9 8 5 x−y −16 5 –7 3 –5 –1 c) x 13 57 y –3 –4 4 9 z –1 20 0 –5 x+y+z 5 4 –10 1 0 –1 31

(D 9q’j 8nj,8k8t]k’]5,j oUfh;p !* 9eo;o4h;oc8j −( 6 –7 4 sk ) 3fp;jk9eo;o|bj’.-h wfh7A’fP; c]t zqo[;d – 4 –2 7 0v’9eo;o16j8k,c8j]t4aoF c8]j tc4; c]t c8]j t 53 glAogoaj’9v, gmqjkdaoD –3 )D 9qj’xtdv[I6[fk;]5,j oU3fp.shzqo[;d8k,c8j]tc4;gmqkj da[ @D 6 −4 −7 47 −3 ສະຫງສ.ວວນ.ລິສຂະສິດ 0 _D 9kdI6[]5,j oUF gryoj g7nvj omjuwx8k,mk’]6dlvo3fpdko[;d.sh9eo;omujrq[D +8 −3 −6  −1 + 2 =......... +12  −9  −1 dD ,u9admk’g]nvdg,nvj g7njvomu9j kd −1 wxsk +2D 0D 9qj’deoqfglAomk’ g-y’j zqo[;d9eo;omjurq[ma’\\qfgmqkj da[l6oD 7D glAomk’.fmu.j shzqo[;d9eo;omurj q[s^kpl5fD ’D glAomk’.fmuj.shzqo[;d9eo;omujrq[|hvpl5fD 9D ,uglAomk’mujzqo[;d9eo;omurj q[gmqjklk, s^n[+L 4hk,u 9qj’[vdglAomk’oAoD 32

ສະຫງສ.ວວນ.ລິສຂະສິດ[qfmu & dko76o c]t dkosko9eo;ome,t-kf !D dyf9tde dyf9tdemu ! !D mhk; lk]u -NxB,0Posqdsq;M]t !&WW du[D g9Qk0v’Ihko7yfw]j]k7kxB,0Posqdsq; fh;pdko[;d !&WW -Edaosqdgmnvj faj’oUG !&WW + !&WW + !&WW + !&WW + !&WW + !&WW = _WWW dD 4k,;jkovd9kd;ymuoUgryoj lk,kf7yfw]j]k7kxB,sqdsq;wfhfh;p;ymu.fvudL 0D !&WW + !&WW + !&WW + !&WW + !&WW + !&WW = _WWW                gmnjv0v’ 1500 = 9000 ×1500 = 9000 D 7D 9qj’4vogvqkdkorq;rao]ts;jk’dko[;d c]t dko76o3fpdkoxtdv[xt3spd ]5j,oUG ‚ dko76o !&WW da[ * c,jodko[;d DDDDDDDDDDDDDDD -Edao DDDDDDDDD gmnjvD @D xB,7toyflkf ,D! lk,sq;]k7k];,dao @!WWW du[D 9q’j 7yfw]]j k7kxB,7toyflkf fa’j dkj ; 9eo;oly[shksq;F ly[sqdsq; c]t -k;sq;fh;ps^kp;ymu8kj ’daoD dyf9tdemu @ gryoj wfhc8h,I6[ltcf’dkocpd @$ vvdgxaozqo76o0v’lv’9eo;ome,t-kffa’j oUG ) )+  3×8= 24  )+  3+3+ 3 +3 + 3 +3+3+ 3   8×3 dD 9kdzqodko7yfw]wj fh,k ltcf’.shgIqkgsao;kj dko76o0v’lv’9eo;ome,t-kf ,u75o]adltotco;.fL 33

0D 3fpoe.-h;ymufP;da[0=h dD 9q’j 8nj,.lj[vj o9Eg,af.sh4nd8hv’D 2 x 4  …..x…..= 24  + 2 x 4  + 2 x 4  (3 x 2) + (3 x 2)  + (3 x 2) + (3 x 2)  ….x…..= 24  dyf9tde # mhk; ]ugIu c]t ok’ [5o rhv,da[\\6j];,ma’\\qf _ 7qo c[j’xao\\kd[uj !W) |;j p g-y’j m5d7qo8hv’wfhgmqjkdaoD mhk; ]ugIu wfhgvqk\\kd[ujma’\\qf,k1kpgxaogdQkdv’3fpgvqk \\kd[uj8nj,.lcj 8]j tdv’gmnvj ]t|;j pF ok’ [5o g;Qk;kj c[’j c[[oU,aopkd c]t glpg; ]kS r;dgIqkrhv,dao7yfw]dj jvo c]h;9bj’c[j’dao8k,zqodko7yfw]j faj’oUG ...× 9 = 108 108 9 ສະຫງສ.ວວນ.ລິສຂະສິດ ... = 108 ÷ 9   ... 9eo;omu4j nd8hv’16[j vj o9Eg,af c,oj zqo0v’dkoc[j’xao s^n zqosko 108 .sh 9 D 9qj’1A’1nodko7yfw]7j no;jk4nd8hv’fh;pdko]q[ !W) vvdgmnjv]t _ 8yf8+dao9qowfh zqoIa[gmqkj l6oF g-aoj G !W) − _ = 99S __ − _ = _WS _W − _ = )!S DDD 4k,;kj G gIqk]q[8yf8+dao9adgmnjvF zqo]q[9bj’gmqkj l6oL gIqk4vowfhdkorq;rao]ts;jk’dko]q[ c]t dkoskoco;.fL @D .97;k, !D dko76o9eo;ome,t-kfD ‚ .sh9eo;ome,t-kf a c]t n S dko76o a da[ n laopt]adG n× a c,oj dko7yfw]j0v’dko[;d a -Edao n gmnvj D \\kp7;k,;kj G a + a + a + a + … + a = n× a. n gmnvj 34

laopt]adG n ×a vjko;jk n 76oda[ a F zqowfhIa[gvUo;jk zqo76o n da[ a. l;j o;jk n c]t a mujgvqk,k76odaogvUo;jk lj;o76oD ‚ dko76oc,jodko7eo;omu.j -hcmodko[;d9eo;o.f|bj’muj-EdaoD 8q;1jk’G gIqkwfhG !#) × @$ = ##!@ 138 × 24 552 276 3312 s^n 138+138+ 138 +...+1 38 = 3312 l;j o76o zqo76o 24 gmnvj @D 75o]adltot ‚ dko76ola[xjPo[jvowfh \\kp7;k,;jk le]a[m5dM9eo;ome,t-kf a c]t b gIqkwfh a × b = b × a ສະຫງສ.ວວນ.ລິສຂະສິດ ‚ dko76o3I,\\6jwfh \\kp7;k,;jk le]a[m5dM9eo;ome,t-kf a; b c]t c gIqkwfh Xa×b?×c = a×(b×c) = a×b×c. ‚ .sh9eo;ome,t-kf a 8k,.9S gIqkwfh a × ! = a. 8q;1jk’G dD )×( = (×) = 56 0D (2× (?×# = !$ ×# = $@ ( 2 × 7) ×3 = 2 ×(7×3) 2×X(×#? = @×@! = $@ = 2 × 7 × 3 = 42 7D 7 ×1 = 7 #D dkosko9eo;ome,t-kf .sh9eo;ome,t-kf a c]t b 8k,.9 3fp;jk a > bF zqosko a .sh b laopt]ad a ÷ b c,oj dko7yfw]jcmodko]q[ b vvd9kd a 8yf8+dao9qozqo]q[gmqjk r 3fp;jk r < b. 35

a − b − b − … − b = rF g-yj’ r < b. ltcf’;kj a b q gmnjv r q gryoj gvUo q ;jk zqosko c]t r ;kj 8q;glf 0v’ a sko.sh b D 8q;1jk’G 23 − 7= 16; 16 − 7= 9; 9 − 7= 2 c,jo @# ( @# s^n 23 − 7− 7 − 7 = 2 < 7 D # c,oj zqpsko c]t @ c,oj 38glfD 3gmnvj ‚ dkoskoc,jodko7eo;omu.j -hgrnjv-vdskzqosko c]t 8q;glfD # [qfg/ydsafD !D 9q’j 8A’[A’g]dc]h;7yfw]zj qo76o]5j,oUD dD !(& × !&S 9D *() × *_S fD # (W! × !WWS 8D *W& × !W × !WWS 0D @$)# × _&S lD )(_ × )(S 4D $ WW$ × !WW × ! WWW × !WS mD $!( × ! WWW × !WW × !WD 7D _* @@( × !&*S -D _*_ × _(*S ສະຫງສ.ວວນ.ລິສຂະສິດ ’D (&# × *!)S pD *#& × ! WWWS @D 9q’j 7yfw]jzqo76o]5,j oU X[+.-hg7njv’7yfw]j?D dD X!@ × &? × (S 7D @& × !@ × )S 9D X) × $ × &? × *S lD X!& × *? × X@& × $ × )?D 0D *W × X!W × (?S ’D (× _ × & × !WS #D 3fpoe.-hdkorq;rao]ts;jk’dko76o c]t dkosko 9eo;ome,t-kf aF b c]t c, a = b × c gIqkwfh a ÷ b = c s^n a ÷ c = b. 9q’j -vdsk7jk0v’ x ]5j,oUD dD * × x = $@S lD *W ÷ x = &S 0D x × @! = @!WS -D x ÷ @& = $S 7D !@ × x = $)S pD x ÷ !! = !$S ’D #& × x = !W&S fD !@* ÷ x = !)S 9D & × x = (&S 4D !#* ÷ x = !(D 36

$D 9qj’cmo[jvo9Eg,affh;p8q;g]dF grnjv.shdko7yfw]jzqo76o c]t zqosko]5,j oUgxao9y’D dD 0D 7D ’D 6 91 • • •1 •••• •357 × × × •3 × 7•8 34 • 56 ••• ••• 7611 •••4 •••5 • • • •8 • • • •8 ••••9 •••3 19743 • • •2 ••••5 •••••• ••••••• 3• 4 • •7 • •••••• 9D lD -D 653 ‚ 2‚ ‚‚‚‚ ‚2 7548 2‚ .‚ .  416 48 173 2.2  54 5.8.   0 &D !WF !#& 9kd9eo;ooAoD149.k  .. ,  ;kj 9eo;ooAoc,oj 76o9eo;o|bj’da[ zqoIa[gruj,0Bo gmqjk.fL *D 9q’j -uhc9’;jk @ $&( × $W( = 2849 × #&! X 3fp[+.sh7yfw]jzqo76o0v’r;d,ao ?D ສະຫງສ.ວວນ.ລິສຂະສິດ (D mhk; [q;7e ,uglNvlk,zno c]t 3lh’lu3j 8D 4k,;kj ]k;,u9admk’g]nvdgrnvj o5j’g7njv’ faj’djk;3fp[+.sh-EdaoL )D l;o8vo|bj’gxaoI6[lucj 9lkdF ,ugoNvmju &&#& m2 c]t 0hk’d;hk’cmdwfh $& mD 4k,;jk];’pk;0v’l;o8vooAo,u9adc,afL _D ok’ [5o,u 9af\\kddhP’ #*W |;j p.lj !@ cdafF 3fp.shc8j]tcdaf[ao959eo;o \\kddhP’gmqkj daoD 4k,;kj c8j]tcdaf,u\\kddhP’9ad|j;pL !WD dq[8q;|bj’g8Aolt\\ej ltg\\u !@ gmnjv 8k,mk’-njF 9kd95f Ο g4y’95f Α muj16sj jk’ )$W cm 9kd Ο D 4k,;kj dq[g8Aogmnjv|bj’wxwfhwd9ad-a’8uc,afL !!D xB,0Po _ sq; ]k7k !#&WW du[ c]t xB,0Po-toyffP;dao * sq; ]k7k _ WWW du[D 9qj’7yfw]]j k7kxB,0Po-toyffP;daooU !& sq;D !@D g0Qk[kg] !WW dy3]dik, zt]yfcxh’wfh (& dy3]dik,D cxh’g0Qk[kg] @& dy3]dik, zt]yfg0Qk9ujwfh #W dy3]dik,D 4k,;jkgrnvj zt]yfg0Qk9uj $&W dy3]dik,9t8hv’.-hg0Qk [kg]9addy3]dik,L !#D ]qf9ad7ao|b’j ]k7k @@ (&W WWW du[F xtdkf0kpzvj ofa’j oUG 9kj p,N,kIa[]qf $ WWW WWW du[F lj;opa’gs^nv9jkpzvj oc8j]tgfnvogmqkj dao rkp.o|b’j xuD 4k,;jk c8]j tgfnvo9kj p8n,j gmqjk.fL 37

[qfmu * dko76o c]t dkosko9eo;o4h;o !D dyf9tde dyf9tdemu ! 9eo;o4h;o[;d c]t 9eo;ome,t-kfc,jo9eo;ofP;daoF gryjo7yfw]zj qo76olv’ 9eo;o4h;o[;d7nda[7yfw]jzqo76olv’9eo;ome,t-kffaj’oUG 5×3 =5 + 5 + 5 =15 D dD 9q’j xtdv[[A’l6f[A’ # ]5,j oUD 3× 5 =15 –#  3× 4 = 12 –#  3×3 = 9 3×2 = ສະຫງສ.ວວນ.ລິສຂະສິດ 3×1 = 3×0= 3×(−1) = 3× (−2) = 3× (−3) = 0D gIqk7yfw]wj fhvudco;|b’j fa’j oUG X−#? × # = X–#? + X–#? + X–#? = – _    9q’j 7yfw]zj qo76o]5j,oUG c. 6 × X–$? = DDDDDDDDDDD S a. X−#? × $ = DDDDDDDDDD S d. ( × X–5? = DDDDDDDDDDDD b. X−&? × $ = DDDDDDDDDDDD S 7D gIqkla’gdfgsaozqo76o0v’lv’9eo;o4h;omuj,ug7nvj ’\\kp8jk’daogxao7nco;.fL 38

dyf9tdemu @ dD 9qj’xtdv[[A’l6f[A’ X−#? ]5,j oUD (−3) × 4 = −12 +# (−3) × 3 = − 9 +#  (−3) × 2 = − 6 (−3) ×1 = (−3) × 0 = (−3) × (−1) = (−3) × (−2) = 0D gIqkla’gdfgsaozqo76o0v’lv’9eo;o4h;o]q[gxao7nco;.fL 7D 9kdzqowfhIa[.odyf9tde ! c]t @ 9qj’xtdv[xt3spd]5,j oUD ‚ zqo76o0v’lv’9eo;o4h;omuj,ug7njv’\\kpfP;daogxao DDDDDDDDDDDDDDDDD ‚ zqo76o0v’lv’9eo;o4h;omuj,ug7nvj ’\\kp8kj ’daogxao DDDDDDDDDDDDDDDDD dyf9tdemu # ສະຫງສ.ວວນ.ລິສຂະສິດ !D gIqkI6hc]h;;jkG !& = & × # 4vowfh !& ÷ & = # s^n !& ÷ # = & 9q’j 8nj,9eo;omuj4nd8hv’.l[j vj o9Eg,afD dD !@ = # × DDDDDDDD 4vowfh !@ ÷ # = DDDDDDDDDDDD s^n !@ ÷ DDDDDDDD = DDDDDDDDDD 0D @& = & × DDDDDDDD 4vowfh @& ÷ & = DDDDDDDDDDDD 7D !@ = X−#? × DDDDDDDD 4vowfh !@ ÷ X− #? = DDDDDDDDDDDD s^n !@ ÷ DDDDDDDD = DDDDDDDDDD ’D @W = X−&? × DDDDDDDD 4vowfh @W ÷ X− &? = DDDDDDDDDDDD s^n @W ÷ DDDDDDDD = DDDDDDDDDD 9D X−!@? = $ × DDDDDDDD 4vowfh X−!@? ÷ $ = DDDDDDDDDDDD s^n X−!@? ÷ DDDDDDDD = DDDDDDDDDD lD X−!)? = _ × DDDDDDDD 4vowfh X−!)? ÷ _ = DDDDDDDDDDDD s^n DDDDDDDDDD ÷ DDDDDDDD = DDDDDDDDDD -D X−#*? = X−$? × DDDDDDDD 4vowfh DDDDDDDDD ÷ X− $? = DDDDDDDDD s^n DDDDDDDDDDD ÷ DDDDDDDD = DDDDDDDD pD X−@$? = X−)? × DDDDDDDD 4vowfh DDDDDDDDDD ÷ X− )? = DDDDDDDDDDDD s^n DDDDDDDDDD ÷ DDDDDDDD = DDDDDDDDDD fD X−@(? = −# × DDDDDDDD 4vowfh DDDDDDDDDD ÷ DDDDDDDDDDD = DDDDDDDDDD s^n DDDDDDDDDD ÷ DDDDDDD = DDDDDDDDDD @D 9qj’xtdv[xt3spd ]5j,oUD ‚ zqoskolv’9eo;o4h;omuj,ug7njv’\\kpfP;daogxao DDDDDDDDDDDD ‚ zqoskolv’9eo;o4h;omuj,ug7nvj ’\\kp8kj ’daogxao DDDDDDDDDDDD 39

@D .97;k, !D dko76o9eo;o4h;o s^addkoG grnjv7yfw]zj qo76o0v’lv’9eo;o4h;oF gryjo76ow]ptskj ’9kd W 0v’r;d ,aoc]h;deoqfg7njv’\\kp0v’zqo76ofaj’oUG ‚ zqo76olv’9eo;o4h;omu,j ug7njv’\\kpfP;daogxao9eo;o[;dD ‚ zqo76olv’9eo;o4h;omuj,ug7nvj ’\\kp8kj ’daogxao9eo;o]q[D 8q;1kj ’G a. 7×5 = 35; b. (−8)× (−3) = 24 c. 9× (−3) = − 27; d. (−4)× 6 = − 24 zqogonvj ’D ‚ zqo76o9eo;o4h;o|b’j .sh8q;,aogv’gvUo;jk < de]a’lv’< 0v’9eo;ooAoD .sh9eo;o4h;o a, zqo76o a × a 0Posp=h a × a = a2 vjko;jk < a de]a’lv’ < 8q;1jk’G dD )@= ) × ) = *$S 0D X−&?@ = X−&? × X−&? = @&D ‚ m5dM 9eo;o4h;o a gIqkwfh a2 ≥ 0 D ສະຫງສ.ວວນ.ລິສຂະສິດ @D 75o]adltot ‚ dko76o9eo;o4h;o,u]adltotla[xjPo[jvowfh c]t 3I,\\6wj fhD .sh9eo;o4h;o8jk’l6o8k,.9 a; b; c gIqkwfhG a × b = b × a. a × b × c = a × ( b × c) = (a × b ) × c 8q;1jk’G dD !& × @ = @ × !& = #WS 0D X− )? × $ = $ × X− )?= − #@D 7D & × X− #? × * = [& × X− #? ] × * = & × [X− #? × *] = X− !&? × * = & × X− !)? = − _WD ’D & × $ × ( = X& × $? × ( = & × X$ × (? = @W × ( = & × @) = !$W 9D X−!W? × X− #? × X−$? = [X−!W? × X− #? ] × X−$? = X−!W? × [X− #? × X−$?] = #W × X−$? = X−!W? × !@ = − !@WD 40

#D dko76o0v’s^kp9eo;o4h;o ‚ zqo76o0v’s^kp9eo;o4h;o8jk’l6ogxao9eo;o[;dF 4hkskd;kj 9eo;ol;j o76o]q[ gxao9eo;o76Dj ‚ zqo76o0v’s^kp9eo;o4h;o8kj ’l6ogxao9eo;o]q[F 4hkskd;jk9eo;olj;o76o]q[ gxao9eo;o7udD ‚ grnvj 7yfw]jzqo76o0v’s^kp9eo;o4h;o8jk’l6oF 7;odeoqfg7nvj ’\\kpdvj oc]h;7yfw]j zqo76o w]ptskj ’9kd W 0v’m5dl;j o76oD 8q;1jk’G A = (−20)× 3× (−2)× (−4)× 5 S ,ulk,l;j o76o]q[7nG − @WS − @S − $F = − (20× 3× 2× 4× 5) g-yj’ 9eo;ol;j o76o]q[gxao9eo;o7ud fa’j oAo = − 2 400 zqo76ogxao9eo;o]q[D B =3× (−2)× (−5)× (−4)× (−1)× 6 S ,ulujlj;o76o]q[7nG − @S − &S −$S −! g-yj’ =3× 2× 5× 4×1× 6 9eo;ol;j o76o]q[gxao9eo;o76j faj’oAo ສະຫງສ.ວວນ.ລິສຂະສິດ =720 zqo76ogxao9eo;o[;dD $D dkosko9eo;o4h;o s^addkoG grnvj 7yfw]jzqosko0v’lv’9eo;o4h;oF gryjoskow]ptsjk’9kd W 0v’ r;d,ao c]h;deoqfg7nvj ’\\kp0v’zqoskofaj’oUG ‚ zqosko0v’lv’9eo;o4h;omu,j ug7njv’\\kpvaofP;dao gxao9eo;o[;dD ‚ zqosko0v’lv’9eo;o4h;omuj,ug7nvj ’\\kp8kj ’dao gxao9eo;o]q[D \\kp7;k,;jkG m5dM9eo;o4h;o8k,.9 a c]t b 3fp;jk b ≠ 0 S 4hk a c]t b ,ug7nvj ’\\kpfP;dao (a ÷ b) ≥ 0 4hk a c]t b ,ug7njv’\\kp8jk’dao (a ÷ b) ≤ 0 . 8q;1jk’G dD @) ÷ ( = $S 7D X− @$? ÷ # = − )S 0D X− @(? ÷ X− _? = #S ’D !@ ÷ X− $? = − #D 41

&D dkoskogvu d]uf (Euclide) dkosko gvu d]ufc,oj dkoskolv’9eo;ome,t-kf g-yj’zqosko c]t 8q;glf gxao9eo;ome,t-kf s^n l6oD .sh9eo;ome,t-kf a c]t b, a > b gIqkwfh a = b×q + r 3fp;kj r < b. 8q;1kj ’G 342 16 gIqkwfh 342 = 16 × 21+ 6 22  21  6  *D dkosko0kfD 4hk8q;glf.odkosko gvu d]uf a .sh b skdgmqjkl6oD gryoj g;Qk;jk a sko0kf.sh b c]t gIqkwfh a = b×qD 8q;1kj ’G #* $ gryjog;Qk;kj 36 sko0kf.sh 4 c]t gIqkwfh 36 = 4×9D 0  9 \\kpgsfG 4hk a sko0kf.sh b gryjog;Qk;kj < a c,jomt;u76o0v’ b < c]t < b c,jov5xt76o0v’ a < ສະຫງສ.ວວນ.ລິສຂະສິດ faj’oAogIqklts^5[wfhdkorq;raofaj’oUG a c,jomt;u76o0v’ b mP[gmqkj b c,jov5xt76o0v’ a ,ao,u9eo;ome,t-kf8jk’l6o k 3fp;jk a = b × kD ‚ 9eo;ome,t-kf|b’j sko0kf.sh @ wfhd=8+g,nvj 8q;g]dsq;|;j p0v’,aoc,oj G WS @S $S *S )D ‚ 9eo;ome,t-kf|b’j sko0kf.sh & wfhd=8+g,njv8q;g]dsq;|j;p0v’,aoc,oj G W c]t &D ‚ 9eo;ome,t-kf|bj’sko0kf.sh # wfhd=8+g,nvj zqo[;d8q;]dma’\\qfmu.j -h0Po9e o;ooAosko0kf.sh #D ‚ 9eo;ome,t-kf|bj’sko0kf.sh _ wfhd=8+g,nvj zqo[;d8q;g]dma’\\qfmu.j -h0Po9e o;ooAosko0kf.sh _D ‚ 9eo;ome,t-kf|b’j sko0kf.sh $ wfhd=8+g,njv9eo;omuxj tdv[fh;plv’8q;g]d l5fmhkpsko0kf.sh $D 42

8q;1jk’G dD !#!* sko0kf.sh $ grkt !* sko0kf.sh $D 0D !#!$ sko.sh $ [+0kf grkt !$ sko.sh $ [+0kfD s^adgdoG ‚ 9eo;ome,t-kf|bj’sko0kf.sh a c]t b 8hv’sko0kf.shzqo76o0v’r;d,aoD 8q;1jk’G ()* sko0kf.sh @ grkt;jk8q;g]dsq;|;j pc,jo *D ()* sko0kf.sh # grkt;jk ( + ) + * = @! c]t @! sko0kf.sh # phvo;jk @ + ! = #D ltoAo ()* sko0kf.sh * grkt @ × # = *D #D [qfg/ydsafD !D 9qj’7yfw]jzqo76o]5j,oUG dD & × X−*?S 9D X−!@? × _S fD X−*? × X−@&?S 0D X−(? × X−_?S lD ( × !!S 8D (× X−&? × X−)? × $ × X−!W?S -D X−!)? × X−*?S 4D X−$?@ × X−#?S 7D X−#? × #S ສະຫງສ.ວວນ.ລິສຂະສິດ ’D W × X−(?S pD ) × X−@&?S mD X−@? × X−*? × @W × X−&? × X−_?S @D 9q’j xtdv[8k8t]k’dko76o]5,j oU.sh4nd8hv’D × −* −$ −! @ & −# !) −@ # $ #D 9q’j xtdv[8k8t]k’]5j,oU.sh4nd8hv’D X Y Z A=X×Y B=Y×Z C=A×Z D=X×B & −@ −!# −!W −_ * −!$ −$ −# −& #$) 43

$D .sh 2  −3  vu’8k,dko7yfw]0j hk’oUF 9qj’xtdv[I6[]5,j oU.sh4nd8hv’D −6  0D 7D ‐ 3 5 ‐2 ‐1 dD ‐2 5 ‐6 1 ‐1 2 &D 9qj’7yfw]zj qo76o]5j,oU [ ]D = (−4)2 × (−3)× 22 × 6 [ ]E = 52 ×8× (−1)2 × (−2) × (−4)2 A = (−8) × (−2) × (−5) × 4 F =(−2)2 × 32 ×[4× (−11)] B =[(−25) × 4× (−2)]× 5 C =5× (−3) ×[(−2) × (−10) × (−4)] *D 9q’j xtdv[ltg\\uzqo]5j,oU.sh4nd8hv’D 9D X−$? V DDDDDDDD = WS dD # V DDDDDDDDD = − *S lD $ V DDDDDD V X−)? = *$S 0D X−@? V DDDDDDDDDD = −#WS -D X−$? V X−#? V DDDDDDD = (@S 7D DDDDDDDDDD V X−#? = @(S pD X−!#? V #& V DDDDDD V !! V X−(? = WS ’D X−&? V DDDDDDDDDDDD = − @&S ສະຫງສ.ວວນ.ລິສຂະສິດ (D .sh a × b vu’8k,dko7yfw]j0hk’oUF 9qj’xtdv[I6[]5,j oU.sh4nd8hv’D ab −80  50  8  15  −2  4  −3  )D 9q’j 7yfw]zj qosko]5,j oUD lD !@ ÷ X−!?S dD X−!W? ÷ @S -D #* ÷ !@S 0D @$ ÷ X−)?S pD X−$&? ÷ X−&?S 7D X−!*? ÷ X−)?S fD W ÷ X−(?S ’D # ÷ X−#?S 8D W ÷ _D 9D X−@(? ÷ X−_?S 44

_D 9qj’xtdv[ltg\\uzqo]5j,oU.sh4nd8hv’D dD DDDDDDDD ÷ # = −_S 9D DDDDDDD ÷ X−$? = −#S fD $_ ÷ DDDDDDDD = −(S 8D DDDDDDDD ÷ $ = −!@S 0D X−@@? ÷ DDDDDDDD = !!S lD DDDDDDDDDD ÷ X−#?= &S oD X−&*? ÷ DDDDDDD = (S [D $@ ÷ DDDDDDDD = −*D 7D X−)? ÷ DDDDDDD= $S -D DDDDDDDD ÷ $ = −&S ’D X−!@? ÷ DDDDDDD =−@S pD −#* ÷ DDDDDD = _S !WD 9q’j xtdv[8k8t]k’]5j,oU.sh4nd8hv’D x − &* −!* $& *W #@ y −( $ −!! −@ −!@ x÷y −_ # _ −) !!D 9qj’-vd7kj 0v’ x le]a[ltg\\uzqo]5j,oUD 9D x ÷ (−6) = 6; dD X−@*? ÷ x = !#S lD @W ÷ x = −5; 0D x ÷ 8 = −3; -D x ÷ (−8) = −3; 7D !$ ÷ x = @S pD −#* ÷ x = 3 ’D X−$_? ÷ X−(? = x S ສະຫງສ.ວວນ.ລິສຂະສິດ !@D 9q’j 7yfw]zj qosko]5,j oUD 9D #W$WW ÷ X−!WW?S dD !&WW ÷ !WS lD X−#@WW? ÷ X−!W?S 0D # $W@ WWW ÷ ! WWWS -D X−&WWW? ÷ X−!W?@S 7D !W WWW ÷ ! WWWS pD X−#$ WWW? ÷ !WW ’D @WW WWW ÷ !W@ S !#D 9qj’8A’[A’g]dc]h;7yfw]zj qoskoc[[gvud]yfD dD $@)$ ÷ *S ’D _* @@( ÷ (_S -D * *#W ÷ )&S 0D !@ &(@ ÷ !@S 9D *# W$# ÷ !_S pD !@ (*& ÷ **S 7D #&($ ÷ #&S lD (! W&W ÷ _@S fD $@ &)$ ÷ !*D !$D 3fp[+8hv’7yfw]dj koskoF 9qj’[vd;jk9eo;o]5,j oU,u9eo;o.fsko.sh @S #S $S &S *S ) s^n _ 0kfD !$$S *WS !_#@S @@ &@*S $$#S &@&S !!(*S $&_S !) W!&S !@S @#S @$&WS !@ &$*S @$&S #$#D 45

[qfmu ( mt;u76oI;j ,|hvpl5f c]t v5xt76oIj;,s^kpl5f 0v’9eo;ome,t-kf !D dyf9tde …  dyf9tdemu ! dD 9q’j 8n,j .lj8k8t]k’]5j,oU.sh4nd8hv’D ! @ # $ & * ( ) _ !W * #W 9eo;o.oc4;mu @ wfh,kco;.fL 9eo;ofa’j dkj ;gvUo;kj co;.fLສະຫງສ.ວວນ.ລິສຂະສິດ 0D 9q’j 0Po9eo;ome,t-kfc8j ! g4y’ $@ c]h;0ufvhv,9eo;omugj xaomt;u76o0v’ $D 7D 9qj’0Po9eo;omugj xaomt;u76o0v’ $ sqd9eo;o8k,.9D ’D 9qj’7yfw]jzqo]q[lv’mt;u76o0v’ $ 4afdao8k,]efa[ 8k,.9,klu7j 6Fj la’gdfzqowfhIa[gxao7nco;.f L dyf9tdemu @ dD 9qj’0ufvhv,9eo;o]5j,oUmujgxaomt;u76o0v’ (D @!S @&S @)S @_S #&S #(S $@S $&S $)S *#S _!S _#S !$WS !$#D 0D 9q’j 0Po9eo;omujgxaomt;u76o0v’ ( sqd9eo;o8k,.9 D 7D 4hk9eo;ome,t-kf a c,jomt;u76o0v’ (F gIqk0Po a gxaozqo76o0v’ ( faj’oAoG a = 7V DDDDD 4hk9eo;ome,t-kf a [+c,oj mt;u76o0v’ (F gIqk0Po a gxaozqo76o0v’ ( fa’j oAoG a = 7 V DDDDDDD + DDDDDDDDD 8q;glfgxao9eo;ome,t-kf|hvpd;kj (D d=]toumqj;wxG .sh9eo;ome,t-kf a c]t bF gIqkwfhG a c,oj mt;u76o0v’ b 4hk;kj ,u9eo;ome,t-kf q g-yj’;jk a = bq. 4vowfh a ÷ b = q c]t 8q;glfgmqjk WD gryjog;Qk;jk a sko0kf.sh b 46