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
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 }
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 }
9eo;o W gxao9eo;o4h;o c8j[+gxao9eo;o4h;o[;d c]t [+gxao9eo;o4h;o]q[D
9eo;o4h;omu[j +gxao9eo;o4h;o]q[ c,jo 0; 1; 2; 3; 4; DDD
9eo;o4h;omu[j +gxao9eo;o4h;o[;d c,jo 0; –1; –2; –3; –4; DDD
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 }
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
.sh a ∈] F −a c,jo9eo;odq’dao0hk,0v’ a D
8q;1kj ’G −(−3) = 3, − (+3) = −3
lv’9eo;o4h;odq’dao0hk,daoc,oj 9eo;ofP;daoc8j,ug7njv’\kp8jk’daoD
12
ສະຫງສ.ວວນ.ລິສຂະສິດ#D dkoxP[mP[9eo;o4h;o
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
9eo;o[;ds^kpd;jk9eo;o]q[g]NvpMD
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 – & < – #
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 1 38 + 13 8 + 13 8 + ... +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
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