ມ.2_ຄະນິດສາດ_Math

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(D 9qj’9afg]dlj;o8+wxoU.l8j k8t]k’ 21 ; 17 ; 19 ; 1291 ; 19 ; 15 ; 17 23 17 17 19 19 19 g]dl;j o|hvpd;j k ! g]dl;j ogmqjkda[ ! g]dlj;os^kpdj;k ! )D mkh; le]u,u\\kd[u @!W |;j pS ]k;gIafglp 3 D 4k,;k\\j kd[u]k;glp9ad|;j pL 7 c]t pa’gs^nv9ad|;j pL _D -k;l;ozh6|b’j gda[\\kdoafwfh 240 kg S ]k;0kpc]h; 3 D 5 4k,;jk\\kdoaf]k;pa’gs^nvgmqkj .fL !WD g0Qk|q,vq,4q’|b’j ,u _* dhvoS dD 4hkdyowxc]h; 5 F 4k,;kj g0Qk|q,vq,pa’gs^nv9addhvoL 12 0D 4hkdyowxc]h; 3 F 4k,;jkg0Qk|q,vq,pa’gs^nv9addhvoL ສະຫງສ.ວວນ.ລິສຂະສິດ 8 !!D ws|j;p|bj’[ao95oEwfh !@ ]ufD dD 4hk[ao95c8j 2 9t,uoE9ad]ufL 3 0D 4hk[ao95c8j 3 9t,uoE9ad]ufL 4 !@D .o’ko;k’ltcf’]qfgdqjkcsj’|bj’ ,u]qfma’\\qf (@W 7aoD ]qflu2hk,u 1 S ]qflucf’,u 3 1 c]t ]qflu0k;,u 5 D 9qj’-vdsk9eo;o]qfc8]j txtgrf Xlu?D 4 12 !#D mkh; d=]tdaoF mkh;la’mv’ c]tok’ verk Ij;,dao]q’mbogIafm5]tdyf,u,6o7kj !! W!* WWWdu[D c8]j t7qo,umbo 5 S 1 c]t 2 8k,]efa[D 24 8 3 4k,;jkmboc8j]t7qo,ugmqkj .fL !$D 9vdgxqjk|;j p|b’j ,u[=]y,kf[ao95 20 dl F gryjo4vdoE\\kdw,hlqf8kj ’M.l9j vdfa’j dkj ;fa’j oUG oE\\kddhP’ 1 S oE \\kdoaf 3 S oE\\kd,j;’ 2 S oEvhvp 1 D 4 10 5 20 4k,;jk [=]y,kf0v’oE\\kdw,hc8]j t-toyf,ugmqjk.fL 4

[qfmu @ mt;u76o c]t v5xt76o0v’g]dlj;o !D dyf9tde dyf9tdemu ! .shlv’g]dlj;o 2 c]t 4 D 5 15 dD 9qj’[vdmt;u76oIj;,|hvpl5f0v’ @ c]t $D 0D 9q’j [vdv5xt76oI;j ,s^kpl5f0v’ & c]t !&D @ c]t $ mt;u76oI;j ,|hvpl5f0v’ & c]t !& 7D 9qj’8n,j .lj[vj o,9Eg,afG  v5xt76oI;j ,s^kpl5f0v’ = … … 4 gvUo;kj mt;u76oI;j ,|hvpl5f0v’ 2 c]t 4 D 5 5 15 dyf9tdemu @ .shlv’g]dl;j o 2 c]t 4 D 3 7 dD 9qj’[vdv5xt76oIj;,s^kpl5f0v’ @ c]t $D ສະຫງສ.ວວນ.ລິສຂະສິດ 0D 9qj’[vdmt;u76oIj;,|hvpl5f0v’ # c]t (D v5xt76oI;j ,s^kpl5f0v’ @ c]t $ … 7D 9qj’8nj,.lj[vj o,9Eg,afG mt;u76oI;j ,|hvpl5f0v’ # c]t (= … 2 gvUo;kj v5xt76oI;j ,s^kpl5f0v’ 2 c]t 4 D 21 3 7 @D .97;k, !D mt;u76oI;j ,|hvpl5f0v’g]dl;j o mt;u76oI;j ,|hvpl5f0v’lv’ s^n s^kpg]dlj;o gmqjkda[zqosko]ts;kj ’mt;u76oI;j , |hvpl5f0v’9eo;or6f c]t v5xt76oI;j ,s^kpl5f0v’r6fD mt;u76oIj;,|hvpl5f0v’g]dlj;o = mt;u76oIj;,|hvpl5f0v’9eo;or6f v5xt76oIj;,s^kpl5f0v’r6f 8q;1kj ’ !D -vdmt;u76oIj;,|hvpl5f0v’ 2 c]t 5 D 9 12 5

gIqk,u mt;u76oI;j ,|hvpl5f0v’ 2 c]t 5 c,oj 10 v5xt76oI;j ,s^kpl5f0v’ 9 c]t 12 c,oj 3 faj’oAo 10 c,oj mt;u76oIj;,|hvpl5f0v’ 2 c]t 5 D 3 9 12 @D v5xt76oI;j ,s^kpl5f0v’g]dlj;o v5xt76oIj;,s^kpl5f0v’lv’ s^n s^kpg]dl;j o gmqkj da[gmqjkda[zqosko]ts;jk’v5xt 76oIj;,s^kpl5f0v’9eo;or6f c]t mt;u76oIj;,|hvpl5f0v’r6fD v5xt76oIj;,s^kpl5f0v’g]dl;j o = v5x76oI;j ,s^kpl5f0v’9eo;or6f mt;u76oI;j ,|hvpl5f0v’r6f 8q;1kj ’ @D -vdv5xt76oI;j ,s^kpl5f0v’ 12 c]t 2 5 25 gIqk,u v5xt76oI;j ,s^kpl5f0v’ 12 c]t 2 c,oj 2 D mt;u76oIj;,|hvpl5f0v’ 5 c]t 25 c,jo 25 9bj’wfh 2 c,jov5xt76oI;j ,s^kpl5f0v’ 12 c]t 2 25 5 ສະຫງສ.ວວນ.ລິສຂະສິດ25 #D [qfg/ydsaf !D .shlk,g]dlj;o 2 , 3 c]t 4 D 5 10 15 dD 9q’j [vdmt;u76oI;j ,|hvpl5f0v’ @F # c]t $D 0D 9q’j [vdv5xt76oIj;,s^kpl5f0v’ &F !W c]t !&D 7D 9q’j 4vogvqkmt;u76oIj;,|hvpl5f0v’ 2 , 3 c]t 4 D 5 10 15 @D 9q’j -vdv5xt76oI;j ,s^kpl5f c]t mt;u76oI;j ,|hvpl5f0v’ 2 c]t 8 D 7 21 #D 9q’j -vdmt;u76oIj;,|hvpl5f0v’ 2 , 7 c]t 32 D 3 24 45 $D 9q’j -vdv5xt76oIj;,s^kpl5f0v’ 25 , 15 c]t 10 D 3 7 9 &D 9qj’-vdr6fIj;,0v’g]dl;j o 5 , 21 c]t 11 D 48 120 90 *D 9q’j 7af9hvog]dlj;o 126 3fp.-hv5xt76oIj;,s^kpl5f0v’ 126 c]t 189 D 189 6

[qfmu # dkosp=h c]t dko0Bor6fIj;,g]dlj;o !D dyf9tde dyf9tdemu ! dD 9q’j 8n,j 9eo;omugj \\ktlq,.lj[jvo9Eg,afD 210 = 42 × = 42 = 14 × = 14 165 33 × 33 11× 11 0D 9qj’7af9hvog]dl;j o]5j,oUG 30 ; 24 ; 42 ; 27 ; 75 36 32 18 81 45 dyf9tdemu @ 8k,I6[c8h,]5,j oUF 9q’j 8n,j 9eo;omujg\\ktlq,.l[j vj o9Eg,afD dD 2 =… = … = … = ສະຫງສ.ວວນ.ລິສຂະສິດ… 3 9 … … 6 0D 2 = 2×… = 2×… = … × 4 = 2×… 3 3× 2 3×… … × 4 …×5 dyf9tdemu # dD 9qj’8nj,9eo;omugj \\ktlq,.l[j vj o9Eg,afD 5 = 5 ×… = … ; 7 = 7 ×… = … 6 6 ×… 12 4 4×… 12 0D 9qj’0Bor6fI;j ,g]dlj;o]5,j oUG 3 ; 7 ; 3 ; 5 4 8 24 2 @D .97;k, !D dko7af9hvog]dl;j oD dko7af9hvog]dlj;o c,oj dko7aflj;o76oI;j ,0v’9eo;or6f c]t 0v’r6fvvdD 7

8q;1kj ’ !D 30 = 6/ × 5 = 5 @D 36 6/ × 6 6 138 = 2/ × 3/ × 23 = 23 24 2/ × 3/ × 4 4 @D dko0Bor6fI;j ,0v’g]dl;j oD dko0Bor6fIj;,0v’g]dl;j o c,jodkogIaf.shg]dl;j o,ur6fgxao9eo;ofP;daoD 8q;1jk’ !D 0Bor6fIj;,g]dlj;o 7 c]t 5 @D 18 12 gIqk,u 7 = 7×2 = 14 18 18 × 2 36 c]t 5 = 5×3 = 15 12 12 × 3 36 9b’j wfh 7 c]t 5 ,ur6fIj;,fP;daoc,jo #*D 18 12 0Bor6fI;j ,g]dl;j o 8 c]t 5 3 24 gIqk,u 8 = 8× 8 = 64 ສະຫງສ.ວວນ.ລິສຂະສິດ 3 3× 8 24 9bj’wfh 8 c]t 5 ,ur6fI;j ,fP;daoc,jo @$D 3 24 #D g]dl;j ogmqjkdaoD 4hk a gxaog]dl;j o|bj’S k gxao9eo;ome,t-kf8k,.98kj ’ WS b gIqkwfh ka = a kb b 8q;1jk’ !D 48 c]t 16 15 5 gIqk,u 48 = 3/ ×16 = 16 15 3/ × 5 5 ltcf’;jk 48 16 15 = 5 @D 5 c]t 15 4 12 gIqk,u 5 = 5×3 = 15 4 4×3 12 8

ltcf’;kj 5 = 15 4 12 #D [qfg/ydsaf !D 9q’j 8nj,9eo;omugj \\ktlq,.l[j jvo9Eg,afD dD 76 = … = 19 ; 7 = 28 = … 12 6 … 5 … 100 0D 1,5 = … = 0,3 ; 16 = … = 1,6 2,5 25 … 100 25 … @D 9q’j 8nj,g7njv’\\kp < = s^n ≠ < .l[j vj o9Eg,afD dD 6 …… 42 ; 72 …… 8 ; 100 ……100 5 35 63 9 1000 0D 12 …… 1 ; 400 …… 1 ; 48 …… 8 32 3 800 2 18 3 #D 9q’j 7af9hvo]dl;j o]5j,oUG dD 8 ; 18 ; 32 0D −2 ; − 120 ; 81 ສະຫງສ.ວວນ.ລິສຂະສິດ 12 33 96 4 − 360 −9 7D 33 ; 15 ; 27 ’D 24 ; 14 ; 36 44 20 24 30 16 24 $D 9qj’0Bor6fI;j ,g]dl;j o]j5,oUG dD 5 c]t 10 ; 7 c]t 1 ; 4 3 5 4 0D 7 c]t 11 ; 5 c]t 2 ; 8 16 18 27 7D 9 c]t 1 ; −3 c]t 5 21 3 4 6 &D .og]dl;j o]5j,oU ,ug]dlj;o0=h.fgmqkj daoL dD 28 c]t 32 0D 8 c]t 10 35 40 12 15 7D 12 c]t 24 ’D 33 c]t 11 16 18 21 3 9D 55 c]t 10 lD 7 c]t 14 33 6 16 32 *D dD 9q’j 0Pom5dMg]dlj;omujgmqkj da[ 2 c]t ,ur6f|hvpd;jk @!D 3 9

0D 9qj’0Pom5dMg]dl;j omugj mqjkda[ 5 c]t ,ur6f|hvpd;jk $WD 4 7D 9q’j 0Pog]dl;j omujgmqjkda[ 16 c]t ,u9eo;or6fgmqkj @D 24 ’D 9q’j 0Pog]dl;j omugj mqkj da[ 18 c]t ,ur6fgmqjk &D 30 (D gryjo.shg]dlj;o 7 ; 6 ; 70 ; 9 ; 63 ; 21 3 4 30 5 27 9 .o[aofkg]dl;j ogmy’oAo ,ug]dl;j o.fmujgmqjkda[ 35 ? 15 )D .o[aofkg]dl;j o8+wxoUG 6 ; 6 ; 6 ; 0,1 ; 4 ; 15 ,ug]dl;j o.fL 24 8 12 0,2 16 20 dD gmqkj da[ 1 S 0D gmqjkda[ 1 S 7D gmqkj da[ 3 D 4 2 4 _D 9q’j -vdg]dlj;omujgmqkj da[ 1 c]t ,ur6fc,oj G 2 dD $S 0D !WS 7D !)S ’D #WD !WD 9qj’-vdg]dl;j omujgmqjkda[ 3 c]t ,ur6fc,joG ສະຫງສ.ວວນ.ລິສຂະສິດ 4 dD )S 0D !@S 7D @)S ’D #*D 10

[qfmu $ dkoxP[mP[g]dlj;o !D dyf9tde dyf9tde ! dD 9qj’8n,j 9eo;omugj \\ktlq,.l[j jvo9Eg,af grnjv.shwfhg]dl;j ogmqkj daoD 3 = ; 7 = 91 5 75 3 0D 9qj’7af9hvog]dl;j o]5,j oUG 32 ; 35 ; 42 ; 30 ; 90 36 60 108 20 75 dyf9tdemu @ 9q’j 8n,j g7nvj ’\\kp < > S < sn = < .lj[jvo9Eg,afD dD 1 5 ; 7 5 ; 68 13 3 3 12 12 5 55 0D 6 6 ; 12 1; 33 ສະຫງສ.ວວນ.ລິສຂະສິດ 10 5 12 82 dyf9tdemu # dD 9qj’0Bor6fI;j ,g]dl;j o]5j,oUG 9;2; 7 ;5 4 5 20 2 0D 9qj’9af]Pog]dlj;o0hk’gmy’oAoc8|j hvpsks^kpD @D .97;k, dkoxP[mP[g]dlj;o dD g]dl;j omuj,ur6f7ndao { 4hkg]dl;j o.f,u9eo;or6fs^kpd;kj c,oj g]dlj;ooAos^kpd;jkD 8q;1kj ’ !D xP[mP[ 7 c]t 5 3 3 gIqk,u 7 >5F fa’j oAo 7 > 5 3 3 11

@D xP[mP[ 4 c]t 11 15 15 gIqk,u 4 < 11F faj’oAo 4 < 11 15 15 0D g]dlj;omu,j u9eo;or6f7ndao { 4hkg]dlj;o.f,ur6fs^kpd;jk c,oj g]dl;j ooAo|hvpd;kj D 8q;1kj ’ !D xP[mP[ 9 c]t 9 8 5 gIqk,u 8 > 5 F faj’oAo 99 8<5 @D xP[mP[ 3 c]t 3 17 14 gIqk,u 17 > 14 F faj’oAo 3 < 3 17 14 7D g]dl;j omuj,ur6f8jk’dao 1kdxP[mP[g]dl;j omu,j ur6f8jk’dao gIqk8hv’0Bor6fIj;,dvj oc]h;9bj’xt8y[af ສະຫງສ.ວວນ.ລິສຂະສິດ dkoxP[mP[7nda[g]dl;j omu,j ur6f7ndaoD 8q;1kj ’ !D xP[mP[ 4 c]t 3 5 7 .og,nvj 4 = 4×7 = 28 c]t 3 = 3× 5 = 15 5 5×7 35 7 7× 5 35 gsao;jk 15 < 28 fa’j oAo 3 < 4 35 35 7 5 @D xP[mP[ 7 c]t 5 18 12 .og,nvj 7 = 7×2 = 14 c]t 5 = 5×3 = 15 18 18× 2 36 12 12 × 3 36 gsao;jk 14 < 15 F fa’j oAoF gIqkwfh 7 < 5 36 36 18 12 \\kpgsfG gIqklk,kfxP[mP[g]dl;j o fh;pdko76ow0;jD 8q;1kj ’G xP[mP[ 4 c]t 3 gIqk,u 4× 7 = 28 c]t 5×3 = 15 gsao;kj 28 > 15 5 7 fa’j oAo 4 > 3 5 7 12

#D [qfg/ydsaf !D 9qj’9af]Pog]dlj;o]5j,oUc8|j vhpsks^kpG dD 5 ; 7 ; 1 ; 3 ; 12 0D 3 ; 3 ; 3 ; 3 ; 3 8888 8 8 12 9 6 2 7D 2,5 ; 1 ; 20 ; 1,5 ; 4 ; 1,7 7 7 70 7 7 7 ’D 2 ; 12 ; 1 ; 3 ; 23 ; 1,5 4,3 43 4,3 4,3 43 4,3 @D 9qj’xP[mP[g]dl;j o]5j,oUG dD 1 c]t 7; 1 c]t 5; 1 c]t 7 6 36 3 18 8 48 0D 1 c]t 1,9 ; 1 c]t 5; 3 c]t 7 1,5 3 0,8 3,2 5,1 10,2 7D 1 c]t 0,1; 1 c]t 10 ; 1 c]t 2 10 0,1 0,5 #D dD gryoj [;d !W .lj9eo;or6f c]t r6f0v’g]dlj;o 7 D 9qj’0Pog]dl;j o.\\mj ujwfhIa[D 10 ສະຫງສ.ວວນ.ລິສຂະສິດ 0D 9q’j xP[mP[ 7 c]t g]dlj;o.\\oj Ao 10 $D 9qj’0Bor6fI;j ,g]dlj;o]j5,oU c]h;9af]Poc8j|hvpsks^kpG dD 3; 5; 4; 9; 5 4 8 16 2 3 0D 2 ; 7 ; 9 ; 3 ; 1 5 10 20 4 2 7D 11 ; 2; 5; 3; 7 12 3 6 4 12 &D 9q’j 8nj,9eo;ofP;dao.l[j vj o,9Eg,afD dD … < 6 < … 0D 4 < … < 7 8 … 5 … 5 … *D oadIPoshv’ ,@ shv’|bj’,adg8t[ko 2 ; ,ad[ko[h;’ 3 c]t ,ad[kolq’j 7 3 4 12 4k,;jkoadIPoshv’oU,addy]kxtgrf.fs^kpd;kj L (D lv’I6[lucj 9lkd R1 c]t R2 ,u];’d;hk’gmqkj dao c]t gmqkj da[ 3,5 cm c]t ,u 13

goNvmujgmqjkda[ 19,6 cm2 c]t 19,25 cm2 8k,]efa[D 4k,;kj I6[lucj 9lkd.f,u];’pk; pk;d;jkL )D oadIPoshv’ ,@}! .ljc;jo8k !$ 7qo.o9eo;o #W 7qoF lj;oshv’ ,@}@ .lj !) 7qo.o9eo;o #) 7qoD 4k,;kj oadIPoshv’.f.ljc;jo8ks^kpd;jkdaoL _D .odko3po\\kd[u|;j p|b’j ]q’05, • mhk; cl’3po]q’ & gmnvj .odko3po _ gmnjv • mhk; 7e,u3po]q’ & gmnvj .odko3po ) gmnvj • mhk; ]5o3po]q’ $ gmnjv.odko3po _ gmnjv • mhk; gof3po]q’ * gmnvj .odko3po ) gmnjv 4k,;jk.o9eo;olu7j qooU c,jo.z3pogdaj’d;jkdaoL !WD .oI6[lk,c9lkdF .sh A c,oj goNvmuFj b c,oj rNo c]t h c,oj ];’l6’murj q; raodaofh;pl6f h = 2A b ສະຫງສ.ວວນ.ລິສຂະສິດ b (cm) # ( !! h h (cm) h h’ h” 9kd8k8t]k’0hk’gmy’F gryoj .shlk,I6[lk,c9muj A ,ugoNvmujgmqjkdao c]t gmqjkda[ 13cm2 b dD 9qj’7yfw]j];’l6’0v’c8]j tI6[lk,c9.oI6[Ikj ’g]dlj;oD 0D 9q’j 9af]Polk,g]dlj;ooAoc8j|hvpsks^kpD 7D 4k,;jkI6[lk,c9.f,u];’l6’ l6’d;jkdaoL 14

  [qfmu & dko[;d c]t dko]q[g]dlj;o !D dyf9tde dyf9tdemu ! .oI6[]5j,oUF 9qj’8n,j g]dlj;omugj \\ktlq,.lj[vj o9Eg,afD dD rkdl;j omuj0uf]kpdq’da[g]dl;j o DDDDDDDDDD 0D rkdl;j omumj kludq’da[g]dlj;o DDDDDDDDDD 7D rkdl;j omu0j uf]kp c]t rkdl;j omumj kludq’da[g]dlj;o DDDDDDDDD s^n DDDDDDDDDD \\kp7;k,;kj 1 + 5 = \" = \" 10 10 10 \" ’D rkdl;j omuj[+mklu c]t [+0uf]kpdq’da[g]dlj;o DDDDDDDDD s^n DDDDDDDDDD \\kp7;k,;jk 10 − 6 \" = \" ສະຫງສ.ວວນ.ລິສຂະສິດ 10 10 = 10 \" dyf9tdemu @ ok’ 8kfegIafoExaoj grnvj .shohv’MdyoS mevyf]k;xajo\\kdoaf 4 ]ufF 8+,k]k; 7 xaoj \\kd]t,5f8n,j 6 ]ufD 7 dD 4k,;jk ok’ 8kfegIafoExajowfhma’\\qf9ad]ufL 0D ohv’lk;0v’]k;dyowxc]h; 2 ]ufF 7 4k,;kj oExajomuoj k’8kfegIafpa’gs^nv9ad]ufL dyf9tdemu # 9qj’7eo;og]dl;j o]j5,oUG dD 15 + 18 ; 11 − 5 ; 23 + 18 ; 10 − 7 7 7 12 12 2 2 23 23 0D 13 + 8 ; a + b ; 8 − 6 ; a − b b b 9 9 b b 3 3   15

  @D .97;k,G !D dko[;dg]dlj;omuj,ur6f7ndao dko[;dg]dlj;omuj,ur6f7ndao c,oj gryoj [;d9eo;or6foedaoc]h;Iadlkr6foAow;hD 4hk;kj a c]t c c,jog]dl;j omu,j ur6fc,jo b g-yj’ b ≠ 0 gIqkwfhG b b a + c = a + c b b b 8q;1jk’ !D 12 + 8 = 12 + 8 = 20 11 11 11 11 @D 2 + 13 = 2 + 13 = 15 = 5 9 9 9 9 3 @D dko]q[g]dl;j omuj,ur6f7ndao dko]q[g]dlj;omu,j ur6f7ndaoc,jogryjo]q[9eo;or6foedaoc]h;Iadlkr6foAow;hD 8q;1kj ’ !D 8 − 1 = 8 −1 = 7 ສະຫງສ.ວວນ.ລິສຂະສິດ @D 3 3 3 3 12 − 17 = 12 −17 = −5 = − 1 25 25 25 25 5 #D [qfg/ydsaf !D 9qj’7eo;og]dlj;o]j5,oUG a = 4 + 3 ; b = 7 + 5 ; c = 5 + 3 5 5 4 4 16 16 d = 4 − 3 ; e = 7 − 5 ; f = 5 − 3 5 5 4 4 16 16 @D 9qj’[;dc]h;7af9hvog]dlj;o]5,j oUG a = 6, 4 + 1, 5 ; b = 6, 2 + 2, 8 ; c= 7 + 5 24 24 21 21 1,2 1,2 d = 8, 5 − 0, 5 ; e = 7,8 − 1,8 ; f = 28 − 3 12 12 15 15 0,5 0,5 #D 9qj’8n,j 9eo;omugj \\ktlq,.lj[vj o9Eg,afD   16

  a) 15 = 11 + \" = 18 − \" b) 19,5 = 11 + \" = 30 − \" 7 7 7 7 7 13 13 13 13 13 c) 1 = 8 + \" = 20 − \" d) 2 = 17 + \" = 35 − \" 11 11 11 11 8 8 8 8 $ D 9qj’7eo;og]dlj;o]5,j oUG a = 13 + 5 + 1 ; b = 2 + 8 + 7 4 4 4 13 13 13 c = 20 − 2 − 17 ; d = 7 + 2 − 4 3 3 3 15 15 15 e = 12 − 8 + 2 ; f = 2 − 1 + 11 7 7 7 9 9 9 &D 9qj’7eo;og]dlj;o]5j,oUG a = ⎛⎜ 2 + 4 ⎞⎟ + 13 ; b = 5 + ⎜⎛ 4 + 9 ⎞⎟ ⎝ 3 3 ⎠ 3 8 ⎝ 8 8 ⎠ c = 2 + ⎛ 5 + 1 ⎞ ; d = ⎛ 8 + 15 ⎞ +1 ⎜⎝ 4 4 ⎠⎟ ⎝⎜ 3 3 ⎠⎟ e = ⎛⎜ 7 + 1⎟⎞ + 5 ; f = ⎜⎛5 + 7 ⎞⎟ + 4 ສະຫງສ.ວວນ.ລິສຂະສິດ ⎝ 12 ⎠ 12 ⎝ 9 ⎠ 9 *D 9q’j 7eo;og]dl;j o]5j,oUG a = 15 + ⎛ 13 − 1 ⎞ ; b = 16 − ⎛ 4 − 1 ⎞ 2 ⎝⎜ 2 2 ⎟⎠ 3 ⎜⎝ 3 3 ⎠⎟ c = ⎛ 29 + 1 ⎞ − 2 ; d = ⎛ 25 − 9 ⎞ + 1 ⎝⎜ 12 12 ⎠⎟ ⎝⎜ 4 4 ⎠⎟ 4 (D 9q’j 7eo;og]dlj;o]5j,oUG a = 17 + 2 × 2 ; b = 5 × ⎜⎛ 11 + 1 ⎟⎞ 15 15 ⎝ 5 5 ⎠ c = 7 × ⎛ 9 − 5 ⎞ ; d = 15 × ⎛ 7 + 9 ⎞ ⎜⎝ 17 17 ⎟⎠ ⎜⎝ 15 15 ⎠⎟ )D 9qj’7eo;og]dlj;o]5j,oUG a = ⎛ 1 + 4 ⎞ + ⎛ 7 + 5 ⎞ ; b = ⎛ 5 + 7 ⎞ + ⎛ 8 − 1 ⎞ ⎜⎝ 3 3 ⎟⎠ ⎜⎝ 3 3 ⎟⎠ ⎜⎝ 6 6 ⎟⎠ ⎜⎝ 6 6 ⎟⎠ c = ⎛ 9 − 1 ⎞ + ⎛ 13 + 3 ⎞ ; d = ⎛ 7 + 9 ⎞ − ⎛ 1 + 9 ⎞ ⎝⎜ 20 20 ⎟⎠ ⎜⎝ 20 20 ⎟⎠ ⎜⎝ 19 19 ⎠⎟ ⎝⎜ 19 19 ⎠⎟   17

  e = ⎛ 19 − 7 ⎞ + ⎛ 9 − 6 ⎞ ; f = ⎛ 15 − 1 ⎞ − ⎛ 8 − 2 ⎞ ⎝⎜ 2 2 ⎠⎟ ⎜⎝ 2 2 ⎟⎠ ⎜⎝ 9 9 ⎟⎠ ⎜⎝ 9 9 ⎟⎠ _D 9q’j 0uf8+.l7j e8v[mu4j nd8hv’D 7 + 13 • • 3 12 12 5 19 − 5 • • 1 28 28 2 7 + 23 • • 5 20 20 3 9,1 − 0,1 • • 2 15 15 3 7,3 + 4,7 • • 3 18 18 2 !WD 9qj’7yfw]j8k,c4;8A’ c]t c4;ovog]dlj;o.o8k8t]k’]5j,oUG ສະຫງສ.ວວນ.ລິສຂະສິດ dD 0D 7 13 13 5 30 + 30 = 24 − 24 = + −+ −+− 11 1 = 30 − 30 1+ =7 24 24 = == = ==   += −=   18

  [qfmu * dko[;d c]t dko]q[g]dl;j o X8+? !D dyf9tde dyf9tdemu ! .oI6[]5,j oUF 9qj’8n,j g]dlj;omujg\\ktlq,.lj[vj o9Eg,afD dD rkdlj;omu0j uf]kpdq’da[g]dl;j o DDDDDDDDDD 0D rkdl;j omumj kludq’da[g]dl;j o DDDDDDDDDD s^n DDDDDDDDDDD 7D rkdl;j omuj0uf]kp c]t rkdl;j omumj kludq’da[g]dl;j o DDDDDDDDDD \\kp7;k,;jk 1 + 3 = 1 + \" = \" 20 10 20 20 20 ’D rkdlj;omuj[+mklu c]t [+0uf]kpdq’da[g]dl;j o DDDDDDDDDD \\kp7;k,;kj 10 − 7 = \" − 7 = \" ສະຫງສ.ວວນ.ລິສຂະສິດ 10 20 20 20 \" dyf9tdemu @ c,j8h6 ve,k]k ,ug0Qk|q,vq,|b’j 4q’grnvj c[’j .shs^kolk; 2 c]t s^ko-kp 3 5 25 0v’g0Qk|q,ma’\\qfD dD 4k,;kj c,8j h6c[j’g0Qk|q,.shs^koma’lv’gmqkj .fL 0D g0Qk|q,c,j86hpa’gs^nvgmqkj .fL dyf9tde # 9qj’7yfw]Gj dD 4 + 7 ; 13 − 7 ; 5 + 1 ; 2 − 5 15 30 6 18 12 3 6 0D 3 + 14 ; 7 − 7 ; 5 + 1 ; 7 − 2 25 75 2 8 4 3 5 3  19

  @D .97;k,G @D dko[;dg]dl;j omuj,ur6f8jk’dao dko[;dg]dlj;omuj,ur6f8jk’dao c,oj gryoj 0Bor6fIj;,dvj oc]h;9b’j xt8y[afdko [;d7nda[dko[;dg]dl;j omu,j ur6f7ndaoD 8q;1jk’ !D 5 + 3 = 5× 2 + 3×3 = 10 + 9 = 10 + 9 = 19 @D 6 4 6× 2 4×3 12 12 12 12 4 + 5 = 8 + 35 = 43 7 2 14 14 14 $D dko]q[g]dlj;omuj,ur6f8jk’dao dko]q[g]dl;j omuj,ur6f8jk’dao c,oj gryoj 0Bor6fIj;,djvoc]h;9b’j xt8y[afdko ]q[g]dlj;omu,j ur6f7ndaoD 8q;1jk’ !D 11 − 7 = 44 − 21 = 23 @D 6 8 24 24 24 7 − 2 = 35 − 6 = 29 3 5 15 15 15 ສະຫງສ.ວວນ.ລິສຂະສິດ #D [qfg/ydsaf !D 9qj’7yfw]j c]h;7af9hvog]dlj;o]5j,oUG a) 4 + 7 ; b) 55 5 10 4 + 12 c) 2 + 13 ; d) 1 + 7 7 28 6 30 @D 9q’j 7af9hvo]dl;j o]5j,oUF c]h;7yfw]Gj a = 15 − 2 ; b = 49 + 9 ; c = 24 − 36 + 11 20 8 35 15 8 16 44 d = 26 + 11 ; e = 14 − 25 ; f = 12 − 15 + 9 48 24 12 30 12 30 6 #D ltg|uzqo]5,j oU 4nd s^n zyfL a) 0 = 0; b) 13 = 13 9 1 c) 1 − 9 = 0 ; d) 8 + 1 = 9 3 27 5 5 10  20

  e) 7 − 1 = 6 ; f) 3 + 3 = 4 ; g) 3 − 9 = 0 12 6 6 3 3 $D 9q’j 7yfw]lj eo;o]j5,oUG a = 4 + 20 ; b = 3 + 4 ; c = 7 − 2 9 27 5 15 3 d =1+ 1 ; e = 5 − 1 ; f = 3 − 2 4 5 5 g = 6 − 3 ; h = 3 − 2 ; k = 15 − 2 20 3 4 &D 9q’j [;dg]dlj;o]5,j oUG a = 13 + 4 + 1 ; b = 8 + 2 + 3 ; c = 20 + 2 + 12 6 5 2 5 3 4 3 7 21 d = 13 − 4 − 1; e = 8 − 2 − 3 ; f = 20 − 2 − 12 6 5 2 5 3 4 3 7 21 * D 9q’j -vdsk7jk0v’ x grnjv.shltg\\uzqo]5j,oUgxao9y’D a) x + 2 = 7 ; b) x + 2 = 10 ສະຫງສ.ວວນ.ລິສຂະສິດ 5 5 3 c) x − 1 = 3 ; d) 2 − x = 1 8 4 7 14 (D ltg\\uzqo 3y − 4x = 2 4nd s^n zyfL 4hk;kj G dD x= 1 c]t y =1 4 0D x= 3 c]t y= 5 2 2 7D x = 5 c]t y= 3 8 2 ’D x = 1 c]t y = 67 100 100 9D x=0 c]t y= 2 3 )D ok’ ,k]y,ug0Qk|q,g7addhvo|b’j F ]k;c[’j dyo8vog-Qk 1 , dyo8vol;kp 7 . 4 12 4k,;jkg0Qk|q,muj ok’ ,k]ygs^nvw;hdyo8voc]’,ugmqjk.fL _D 3I’IPo,afmtpq,cs’j |b’j wfhc[’j g’yo[6]tot3I’IPofaj’oUG -Ao ,! wfh 1 ; 6  21

  -Ao ,@ 5 ; -Ao ,# 1 S -Ao ,$ 1 c]t l;j omujpa’gs^nv,v[.sh-Ao,=xkpD 24 4 12 dD -Ao,=8Qowfhg’yoma’\\qfgmqkj .fL 0D -Ao,=xkpwfhg’yogmqjk.fL !WD gryoj lk,kfg;Qkwfhco;.fdPj ;da[g]dlj;o]5j,oUG dD 7 ; 28 ; 49 c]t 21 5 20 35 15 0D 2 ; 10 ; 6 c]t 8 25 125 75 100 !!D 9qj’7eo;og]dlj;o.o8k8t]k’]5,j oU a b a + b 3×a + 2×b 3× a − 2×b 11 28 11 3 12 33 5 20 ສະຫງສ.ວວນ.ລິສຂະສິດ  22

[qfmu ( dko76og]dl;j o !D dyf9tde dyf9tdemu !    9qj’7yfw]j c]t 8n,j 9eo;omujg\\ktlq,.l[j jvo9Eg,afD a = 3 × 40 ; b = 28 × 5 10 7 c = 15 = 18 × \" ; d = 21 = 35 × \" 6 5 dyf9tdemu @ 1 m      .shI6[9t85]af|b’j muj,u0hk’gmqkj 1mD 1j6.oI6[        C  9t85]af gryjolhk’I6[lujc9lkd ABCD B       3fp.sh0hk’ AB = 2 m c]t 0hk’ AD = 4 m . 2   1 m  3 5 3   9qj’8nj,9eo;omugj \\ktlq,.lj[jvo9Eg,afD       m   ສະຫງສ.ວວນ.ລິສຂະສິດ dD goNvmuj0v’I6[lucj 9lkd ABCD gmqjkda[ AB × AD = \" × \" m2 A 4 D \" \" 5       m 0D goNvmu0j v’I6[lujc9lkdohvpmujmklufegmqjkda[ 1 m2 \" 7D goNvmuj0v’I6[lujc9lkd ABCD gmqkj da[ DDDDDDD gmnvj 0v’goNvmu0j v’I6[lucj 9 lkdohvpmujmklufe g-y’j gmqkj da[ \" m2 , ltcf’;kj 2 × 4 = \" \" 3 5 \" dyf9tdemu # 9q’j 7yfw]jzqo76o]5j,oU dD a = 3 × 2 ; b = 4 × 3 ; c = 4 × 3 ; 5 7 9 5 9 5 d = 7 × 3 ; e = 7 × 5 3 7 8 3 0D 9q’j 8n,j leo;o8+wxoU a × c = \" b d \"   23

@D .97;k, !D dko76og]dlj;o dko76og]dlj;o c,oj gryoj 76o9eo;or6fda[9eo;or6f c]t 76or6fda[r6f c]h;7af9hvo X4hk7af9hvowfh?D .shlv’g]dlj;o a c]t c g-y’j b ≠ 0 c]t d ≠0 b d gIqkwfh a × c = a×c b d b×d 8q;1jk’ !D 4 × 5 = 4×5 = 20 3 7 3×7 21 @D 7 × 2 = 7 ×2 = 2 3 7 3× 7 3 #D 4 × ⎛⎜ − 7 ⎞⎟ = 4 × (−7) = − 28 5 ⎝ 3 ⎠ 5×3 15 $D 8 × 3 = 8× 3 = 8 = 2 ×4 = 4 ສະຫງສ.ວວນ.ລິສຂະສິດ 3 14 3 ×14 14 2 ×7 7 @D 9eo;oxUo0v’9eo;omu8j jk’l6o dD oypk,G lv’9eo;o9txUodaod+8+g,nvj zqo76o0v’r;d,aogmqjk !D 4hk a gxao9eo;o8kj ’l6oS 1 c,jo9eo;oxUo0v’ a. a 8q;1kj ’ !D 9eo;oxUo0v’ @ c,oj 1 2 @D 9eo;oxUo0v’ WF( c,oj 1 10 0,7 = 7 0D 9eo;oxUo0v’g]dl;j o .sh a c,oj g]dlj;og-y’j a ≠0;b≠0. b 9eo;oxUo0v’ a c,jo b grkt;jk a × b = ab = 1 b a b a ba ¾ \\kpgsfG W [+,u9eo;oxUo   24

8q;1jk’D 9eo;o 2 −6 − 1 − 4 0 9eo;oxUo 3 7 5 3 2 − 1 −7 − 5 L 6 4 #D [qfg/ydsaf !D 9qj’7yfw]j c]h;7af9hvo X4hk7af9hvowfh? a = 2 × 6 b = 1 × 2 c = 7 × 16 3 5 2 3 8 14 d = 4 × 1 e = 5 × 2 f = 4 × 0 3 6 2 5 3 @D 9q’j 7yfw]j c]h;7af9hvo X4hk7af9hvowfh?D a = 3 × 5 b = 1 × 1000 c = 1 × 24 7 11 100 55 2 7 d = 3 × 44 e = 100 × 3 f = 5795 2 44 5 7 100 2 × 5795 #D 9qj’8n,j 9eo;omugj \\ktlq,.lj[jvo9Eg,afD ສະຫງສ.ວວນ.ລິສຂະສິດ dD 2 × \" = 8 0D 2,5 × \" = 10 9 \" 27 7 \" 77 7D 17 × \" = 1 ’D 3 × 7 = 7 19 \" 5 \" 10 $D 9q’j 7yfw]zj qo76o]j5,oUG A = 5 ×16 B = 7 × 6 C = 1 × 2 × 3 8 12 35 2 3 5 D = 3 × 21 × 4 E = 1 × 15 × 4 × 6 7 8 9 18 14 5 &D 9q’j 7eo;og]d]5,j oUG dD 7 + 4 × 1 0D ⎜⎛ 7 + 4 ⎞⎟ × 1 3 3 4 ⎝ 3 3 ⎠ 4 7 ⎛⎜ 7 − 4 ⎞⎟ × 1 ’D 7 − 4 × 1 ⎝ 3 3 ⎠ 4 3 3 4   25

*D 9q’j 7eo;og]d]5j,oUG dD 9 + 1 × 13 − 1 0D 40 + 5 × 25 − 5 8 + 1 16 − 1 35 + 5 80 − 5 7D 3 × 4 + 7 × 12 ’D 3 × ⎛⎜ 4 + 7 ⎞⎟ × 12 2 15 5 9 2 ⎝ 15 5 ⎠ 9 7D 9qj’-vd9eo;oxUo0v’9eo;o]5j,oU: a = 40 b = −0,8 c = −3,2 d = 0,016 e = 3 f = − 1 4 7 )D 9qj’8n,j 9eo;omugj \\ktlq,.lj8k8t]k’]5j,oUG dD 0D 4× = 8 ×32 5 9 27 43 6 5 ×× × 3 × 9 = 3 2 4 4 ສະຫງສ.ວວນ.ລິສຂະສິດ == = 9 ×= 77 88 5 _D gryjo.sh a = 1 , b = 1 , c = 1 8 4 2 dD 9qj’7yfw]j ab, ac, ab + ac, a + bc, a (b + c). 0D la’gdfgsaoco;.fL !WD gryoj .shlv’leo;o A = 1 × x − 1 × 4 c]t B = ⎛⎜ 1 × x − 1 ⎟⎞ × 4 4 8 3 ⎝ 4 8 ⎠ 3 9q’j 7yfw]jleo;o A c]t B le]a[ x = 7 ; x = 5 ; x = 4 2 4 3   26

[qfmu ) dkoskog]dl;j o !D dyf9tde dyf9tdemu ! dD I6h;jk 7 ÷5 = 7 5 c]t 7× 1 = 7 ×1 = 7 5 5 5 gsao;jkzqosko0v’ ( .sh & gmqkj da[zqo76o0v’ ( da[9eo;oxUo0v’ &D { 9q’j xjPodkosko]j5,oUgxaodko76oD 4 ÷ 3; 3, 5 ÷ 7, 5; 4,8 ÷ 5; 4 ÷1, 3 0D 9qj’-vdglAozkj o.9dk’0v’;q’,qomu,j u];’Iv[ @ m, I6h;jk 1 ≈ 0,31 π 7D .sh a gxao9eo;o|bj’8kj ’l6oD 9q’j sko a .sh 0,5; 0, 25; 0,1; 0, 2 dyf9tdemu @ ສະຫງສ.ວວນ.ລິສຂະສິດ 8k,I6[c8h,F 9q’j 8n,j xt3spd7eg;Qk]5j,oUG dD rkdlj;omklu0v’I6[ X!? dq’da[g]dlj;o DDDDDDDD s^n DDDDDDDF 0D rkdlj;omklu0v’I6[ X@? dq’da[g]dlj;o DDDDDDDD s^n DDDDDDDF ltcf’;kj 3 = 3 ÷2 = 3 × \" 8 4 4 \"      7D 9qj’8n,j 9eo;omujg\\ktlq,.lj[jvo9Eg,afD (1 ) 4 ÷ 2 = 4 × \" ; 5 5 \"      9 ÷ 3 = 9 × \"   2 3 \"      4 ÷ 3 = 4 × \" = \" ; 5 \" \" (2 ) 8÷ 2 = 8 × \" = \"      3 \" \"   7 ÷ 2 = 7 × \" = \" ; 2 ÷ 8 = 2 × \" = \" 3 5 3 \" \" 3 5 3 \" \" 27 

@D .97;k, oypk,G dD .shlv’9eo;o a c]t b g-yj’ b ≠ 0. dkosko a .sh b c,jodko76o a da[9eo;oxUo0v’ b a ÷ b = a × 1 ; b≠0 b 8q;1jk’ !D 81 ÷ 4 = 81× 1 = 81 4 4 @D 48 ÷1,2 = 48 × 1 = 48 ×1 = 48 1,2 1,2 1,2 #D 4,5 ÷ 2,4 = 4,5 × 1 = 4,5 ×1 = 4,5 2,4 2,4 2,4 0D .shlv’g]dl;j o a c]t c g-yj’ a, b, c c]t d 8kj ’l6oF b d gIqkwfhG a ÷ c = a × d c]h;7af9hvozqowfhIa[ X4hk7af9hvowfh?D b d b c ສະຫງສ.ວວນ.ລິສຂະສິດ 8q;1kj ’ !D 5 ÷ 2 = 5 × 7 = 5× 7 = 35 3 7 3 2 3× 2 6 @D 2 ÷ 11 = 2 × 1 = 2×1 = 2 9 9 11 9 ×11 99 #D 5 ÷ 4 = 5× 13 = 5 ×13 = 65 13 4 4 4 $D 0 ÷ 6 = 0× 7 = 0×6 = 0 = 0 7 6 6 6 &D − 6 ÷ 3 = − 6 × 5 = − 2 × 3/ × 5 = − 10 17 5 17 3 17 × 3/ 17 #D [qfg/ydsaf !D 9qj’7yfw]jG a = 4 ÷ 3 ; b = 1 ÷ 4 ; c = 2 ÷2S d = −7 ÷ 42 ; e = 3 ÷ 3 ; f = 8 ÷ 4 7 5 3 3 5 4 4 21 7 @D 9q’j 7yfw]j c]h;7af9hvoG a = 2 ÷ −7 ; b = −36 ÷ 30 ; c = − 12 ÷ − 28 ; d = 34 ÷ 38 3 5 7 9 27 51 57 28 

#D 9q’j -vd9eo;oxUo0v’9eo;o]5j,oUG 0, 25; 0,1; 0, 01; 0, 02; 1 ; 5; 8 ; 6 7 3 19 $D 9q’j 7yfw].j o.9 a = 7 ÷ 0, 5; b = 12 ÷ 0, 25; c = 100 ÷ 0,1 d = 0, 6 ÷ 0,1; e = 0, 25 ÷ 0, 01; f = 3 ÷ 0, 02 5 &D 9q’j 7yfw]j c]h;7af9hvoD dD 7 ÷ 12 ; 9 ÷ 7 ; 13 ÷ 7 10 100 100 10 100 100 0D 13 ÷ 9 ; 47 ÷ 9 ; 64 ÷ 32 10 100 100 10 100 100 *D 9q’j 0Po − 27 rkp.8hI6[c[[8kj ’M]5,j oUG 8 dD − 27 gxaozqo76o0v’lv’g]dlj;oD 8 0D − 27 gxaozqosko0v’lv’g]dlj;oD ສະຫງສ.ວວນ.ລິສຂະສິດ 8 7D − 27 gxaozqo76o0v’lk,g]dl;j ogmqjkdaoD 8 ’D − 27 gxao9eo;omqfltoypq,D 8 (D 9qj’7af9hvog]dl;j o]5j,oUG 19 ; 16 ; 36 ; 11 ; 72 ; 13 95 65 64 110 27 325 )D 9q’j 8nj,9eo;omugj \\ktlq,.l8j k8t]k’]5j,oU −1 × 4 = 1÷ = -8 3 4 ÷× ÷ ×× ÷ −8 ÷ −1 = ÷1 = 2 === == = −8 × = ×=   29 

_D 9qj’8n,j g7njv’\\kp0v’dko7eo;o.ljcmog7nvj ’\\kp <∗<G dD 2 ∗ 6 = 5 ; 5 ∗ 5 = 3; 3 ∗ 5 = 6 3 5 9 4 12 7 14 5 0D 3∗ 3 ∗ 1 = 14 ; 3 ∗ 3 ∗ 1 = 17 2 6 3 2 6 6 7D 3 ∗3 ∗ 1 = 1 ; 3∗3 ∗ 1 = 3 2 6 2 2 6 4 !WD 4a’oE,ao4a’|b’j [ao95oE,aog8a,S mevyfgryoj gvqkvvd 1 0v’4a’S gmnvj mulv’gvqkvvd 6 1 0v’4a’ c]t oE,aopa’gs^nv &(W ]ufD 5 9qj’-vd[=]y,kf[ao950v’4a’oE,ao|j;poAoD !!D 4a’oE4a’|b’j ,uoE1j6 1 0v’4a’S gryoj 8nj,oE.lj @) ]uf g;]koAooE.o4a’,u 3 0v’4a’D 6 4 4k,;kj [=]y,kf[ao950v’4a’,ugmqjk.fL !@D 9eo;o0k0v’c,’,6,gmqjkda[ 4 0v’9eo;o0k0v’[Q’dnD 4k,;kj [Q’dn,u9ad0k 21 I6h;kj c,’,6,,u ) 0kL ສະຫງສ.ວວນ.ລິສຂະສິດ !#D dtc-afgra’d=h|bj’lk,kfvafwfh !@W okmuD |hk A 4ndvafwx 7 ; |hk B 4ndvafwx 2 . 12 5 4k,;kj c8]j t|hkpa’gs^nvg;]kgmqkj .fL !$D 9q’j c8h,I6[]5j,oU7no.\\j c]h;mklurkdlj;omulj tcf’g4y’ dD 2 ÷ 3 0D 4 ÷ 5 5 7                                                      a b a b a ×b        c !&D .shleo  ;o A= + S B = − C = c c 9qj’7yfw]lj eo;o A, B c]t C 3fp.sh a = 2 ; b = 3 ; c = 2 3 7 7 30 

[qfmu _ 9eo;oxqddt8y !D dyf9tde dyf9tdemu ! 9q’j skog]d]5j,oUG 14 ; 5 ; 23 ; 31 ; 109 . 35 3 99 90 50 dyf9tdemu @ mhk; cdh; -N\\kd[u _ |;j pgxaog’yo @)&W du[D 4k,;jk\\kd[u|;j p|b’j ]k7kgmqkj .fL dyf9tdemu # mhk; 7e,u c]t \\6j @ 7qorkdaowxdyog0Qk16Ij hkovkskocs’j |b’j 7yfgxaog’yo &*WWW du[D dD g0qkg9Qk8hv’9jkp7kj vkskozh6]tgmqkj .fL 0D .o8q;9y’8hv’9kj pz6h]t9addu[L ສະຫງສ.ວວນ.ລິສຂະສິດ @D .97;k, !D 9eo;oxqddt8y oypk, 9eo;oxqddt8y c,jo9eo;omuj0Po.oI6[Ijk’ a g-y’j aF b gxao9eo;o4h;o c]t b b 8jk’l6o (b ≠ 0)D 8q;1jk’ !D 2 ; − 8 ; 17 ; 1 = 12 ; −1 = − 5 @D 5 13 4 12 5 6 = 6 ; − 1 ; 17 ; 3 = 15 ; 4 1 3 8 5 9 @D dkoxjPog]dl;j o gxao9eo;omqfltoypq, m5dM9eo;omuj0Po.oI6[Ijk’g]dl;j o a g-y’j a, b gxao9eo;o4h;o c]tb 8kj ’l6o b (b ≠ 0)D lk,kf0Pogxao9eo;omqfltoypq,wfh3fp.-hdkosko a .sh b . 8q;1kj ’ !D 9 = 2,250…; 5 = 1,666… 4 3   31

@D − 7 = −0,6363 ; 23 = 0,2555… 11 90 9eo;omqfltoypq,7n0hk’gmy’ gvUo;kj 9eo;omqfltoypq,Iv[;Po g-y’j gryoj oypq,0Po fa’j oUG !D 11 = 2,75000… s^n 0,750 Iv[;Poc,jo 0 4 @D 5 = 0,555… s^n 0,5 Iv[;Poc,jo 5 9 #D 7 = 0,4666… s^n 0,46 Iv[;Poc,oj 6 15 $D − 50 = −1,515151… s^n − 1, 51 Iv[;Poc,oj 51 33 &D 11 = 0,297297297… s^n 0, 297 Iv[;Poc,oj 297 37 *D 83 = 0,30181818… s^n 0,3018 Iv[;Poc,jo 18 275 gryjo9af9eo;omqfltoypq,Iv[;Pogxaolv’d5j,7nG dD 9eo;omqfltoypq,Iv[;PolYol5f XIv[;Pol6o? gvUo;kj 9eo;omqfltoypq,D ສະຫງສ.ວວນ.ລິສຂະສິດ .od5,j oUgryjo[+oypq,0PoIv[;Pol6o0v’,aoD 8q;1kj ’D 0,30 0Pogxao 0,3 2,650 0Pogxao 2,65 − 1,83520 0Pogxao −1,8352 0D 9eo;omqfltoypq,Iv[;Po[+lYol5f XIv[;Po8kj ’l6o? gvUo;jk 9eo;oxqddt8yD 8q;1jk’ !D 4,7 ; − 3,05 ; − 0,137 @D 1,18 ; − 2,063 ; − 2,1327 .od5j,oUgryoj [+lk,kf0Po7kj mujcojovo0v’,ao c8gj ryjolk,kf0v[0Ao7kj xt,ko 0v’,aowfhD 8q;1kj ’D 707 = 7,141414 s^n 7,14 99 fh;p7;k,-afg9oIvf ! fh;p7;k,-afg9oIvf WF! gIqkwfh 7 < 707 < 8 99 7,1 < 707 < 7,2       99 7,14 < 707 < 7,15 fh;p7;k,-afg9oIvf WFW! 99   32

#D [qfg/ydsaf !D 9qj’0Pog]dl;j o8+wxoUgxao9eo;omqfltoypq,D 1) 2 2) − 15 3 8 3) 35 4) − 6 24 11 5) − 5 6) 15 132 37 7) 21 8) − 75 15 6 @D .o[aofk9eo;o]j5,oU 9eo;o.fgxao9eo;oxqddt8yL 1) 11 2) 3,2 5 1,3 3) 2 + 5 4) − 3,56 − 4,81 3 7 5) − 4,6 × 0,7 6) 3 × 5 ສະຫງສ.ວວນ.ລິສຂະສິດ 7 2 7) − 5,6 ÷1,1 8) − 3,5 ÷ 6 #D 9q’j sko $ .sh # c]h;0v[0Ao 4 fh;p7;k,-afg9oIvf WFWW! 3 $D 9qj’sko @& .sh ( c]h;0v[0Ao 25 fh;p7;k,-afg9oIvf WFW! 7 &D 9q’j sko #&& .sh !!# c]h;0v[0Ao 355 fh;p7;k,-afg9oIvf WFWWWWW! 113 *D c,-j N\\kdok; # |j;p8+ !WWW du[D xhk-Nwfh !WW |;j p8+ ##WWW du[D 4k,;kj c,jo.z-N \\kdok;wfh4ndd;jkdaoL (D c,jc]tok’ 4k;vo wxIhko7hkcs’j |bj’grnvj -Ncx’46c0h;Ia[.-h7v[7q;D cx’46c0h; vao|bj’]k7k !$&W du[F |bj’3s^]k7k !&)WW du[D c,oj c]t 4k;vo9t8aflyo.9-N cx’46c0h;c[[.fL phvosap’L )D cdh;F dhvp c]t d5h’ vvdg’yodao-N0v’0;ao;aogduf.sh\\6j ];,gxaog’yo #@&WW du[D 4k,;jkG dD c8]j t7qo9t9jkpg’yo7kj 0v’0;aogmqkj .fL 0D .o8q;9y’c8j]t7qo9t9jkpg’yolqfwfh8k,9eo;o.o0=h dD [+L phvosap’L   33

7D c8j]t7qo8hv’9jkpxt,kogmqkj .fL _D 9q’j 0Pog]dl;j o s^n 9eo;omqfltoypq,mujdq’da[muj8A’0v’g,af A; B; C c]t D A B  C  D  0  3 1  6 2  4  4  !WD dD 9qj’[vdva[-yf0v’g,af A; B c]t C. A B  C  0  1 1  2  1 ; 2 c]t 7 8k,]efa[D 0D 9qj’;k’g,af D; E c]t F muj,uva[-yf 6 3 6 !!D dD 9q’j [vdva[-yf0v’g,af P; Q c]t R. PQ R 0 LL 1L 0D 9qj’;k’g,af M; N; K; c]t L mu,j uva[-yf 1 ; 3 ; 1 + 3 ສະຫງສ.ວວນ.ລິສຂະສິດc]t11 5 10 10 10 8k,]efa[D   34

  [qfmu !W 9eo;oxqddt8y X8+? !D dyf9tde dyf9tdemu ! 9q’j xjPo9eo;omqfltoypq,]j5,oUgxaog]dlj;omuj[+lk,kf7af9hvowfhD 4,6; 1,04; 0,168; 0,054; 1,275; 0,0196 dyf9tde mu @ gryoj .sh9eo;o x = 0,777… dD 9q’j 7yfw]j 10 x; 10 x − x 0D 9q’j 0Po9eo;o x rkp.8hI6[c[[g]dlj;omuj[+lk,kf7af9hvowfhD dyf9tdemu # ສະຫງສ.ວວນ.ລິສຂະສິດ gryjo.sh9eo;o x = 0,545454… dD 9q’j 7yfw]j 100 x; 100 x − x 0D 9qj’0Po9eo;o x rkp.8hI6[c[[g]dl;j omuj[+lk,kf7af9hvowfhD dyf9tdemu $ gryjo.sh9eo;o x = 0,2333… dD 9q’j 7yfw]j 100 x; 10 x; 100 x −10 x 0D 9qj’0Po9eo;o x rkp.8hI6[c[[g]dl;j omuj[+lk,kf7af9hvowfh @D .97;k, ¾ dkoxPj o9eo;omqfltoypq, gxaog]dl;j o { m5dM9eo;omqfltoypq,Iv[;Pol6o lk,kf0Pogxaog]dl;j owfh 8q;1jk’ 1. 0,11 = 11 100 2. − 0,137 = − 137 1000 3. 2,19 = 219 100 { le]a[9eo;omqfltoypq,Iv[;Po8jk’l6o lk,kf0Pogxaog]dlj;owfh 35

  3fpdkoxt8y[affaj’oUG 8q;1jk’ !D 0Po 0,6 gxaog]dlj;o gryjo;k’ N = 0,6 = 0,666… 76oma’lv’rkdda[ !W gIqkwfh 10 N = 10 × 0,666 = 6,666… faj’oAo 10 N − N = (6,666 ) − (0,666…) 9N =6 6, 666... − 0, 666... 9bj’wfh N = 6 9 6, 000... ltcf’;kj 0, 6 = 6 s^n 2 9 3 8q;1jk’ @D 0Po 0,725 gxaog]dl;j o ສະຫງສ.ວວນ.ລິສຂະສິດ gryoj ;k’ N = 0,725 = 0,7252525… ............(1) 76oma’lv’rkd0v’ X!? da[ !WWW gIqkwfh 1000 N = 1000 × 0,7252525… = 725,252525… ............(2) 76oma’lv’rkd0v’ X!? da[ !W wfh 10N = 10 × 0,7252525 = 7,252525… ............(3) gvqk X@? ]q[vvd X#? wfh 1000 N −10N = (725,252525 ) − (7,252525…) 990 N = 718 725, 252525... − 7, 252525... fa’j oAo N = 718 718, 000000.... 990 ltcf’;jk 0, 725 = 718 = 359 990 495 lts^6[G gIqklk,kf0Pog]dlj;o.shgxao9eo;omqfltoypq,Iv[;Powfh c]t 0Po9e o;omqfltoypq,Iv[;Po.shgxaog]dlj;owfh faj’oAogIqklk,kfg;Qkwfh;kj G 9eo;oxqddt8y c,oj 9eo;omuj0Pocmowfhfh;p9eo;omqfltoypq,Iv[;Po s^n g]dlj;o a g-y’j a, b gxao9eo;o4h;o c]t b 8kj ’l6o (b ≠ 0)D b 36

  9kdzqowfhIa[0hk’gmy’gIqkgsao;jk9eo;oxqdt8y c,jo9eo;o-toyf8jk’Mfaj’ czo;kf]j5,oUG 9eo;oxqdt8y 9eo;o4h;o g]dlj;omu[j +c,oj 9eo;o4h;o 9eo;o4h;o]q[ l6o 9eo;o4h;o[;d #D [qfg/ydsaf !D 9qj’0Po9eo;omqfltoypq,8+wxoUgxaog]dlj;omu[j +lk,kf7af9hvowfh dD 3,6; 1, 08; 0, 231 ສະຫງສ.ວວນ.ລິສຂະສິດ 0D 0, 076; 1, 025; 0, 2318 @D .o[aofk9eo;oxqdt8y8+wxoU9eo;o.fgxao9eo;omqfltoypq,L dD 14 ; 90 ; 60 35 48 140 0D 6600 ; 250 ; 99 2400 28 225 #D 9qj’[vdIv[;Po0v’c8j]t9eo;o]5j,oUG dD 0,222…; 5,141414…; 3,8; 41,29 0D 6,21555…; 0,452; 28,373737; 8, 209 $D 9q’j xPj o9eo;omqfltoypq,Iv[;Po8+wxoUgxaog]dlj;omu[j +lk,kf7af9hvowfh dD 0,777…; 0,52; 0,48; 0,236236236… 0D 0,215; 0,453; 0,24838383…; 0,290 &D la’gdfgsao;jk9eo;o x = 2,999… lk,kf0Powfh.oI6[c[[ x = 2,999… = 2 + 0,999… 9qj’0Po9eo;o x rkp.8hI6[c[[g]dlj;omu[j +lk,kf7af9hvowfh *D 9q’j xjPo9eo;omqfltoypq,Iv[;Po8+wxoUgxaog]dl;j omu[j +lk,kf7af9hvowfh 37

  dD 1,888…; 5,12; 2,34; 4,2393939… 0D 8,215; 7,152; 9, 2; 23,40 (D 9qj’ltcf’.shgsao;jk 1,999… = 2 )D 9qj’-vdIv[;Po0v’9eo;oxqddt8y]j5,oUG 23 ; 77 ; 221 ; 317 ; 21 91 40 90 407 37 _D w0wj dcj 8t|bj’,u #W |j;p ]k7k @_WWW du[D 4k,;kj G dD w0|j j;p|bj’]k7kgmqjk.fL 0D 4hk0kppvj pgryjo8hv’0kp|;j p]t9addu[L !WD 9q’j 7yfw]jzqo[;d]5j,oUG dD 9 + 9 + 9 + 9 + 10 100 1000 10000 0D 23 + 23 + 23 + 100 10000 1000000 7D 2 + 8 + 8 + 8 + ສະຫງສ.ວວນ.ລິສຂະສິດ 10 100 1000 10000 ’D 3 + 54 + 54 + 54 + 10 1000 100000 1000000 9D 2 + 3 + 3 + 3 + 3 + 10 100 1000 10000 38

[qfmu !! 9eo;ovtxqddt8y !D dyf9tde 9qj’[vdIv[;Po0v’c8]j t9eo;o]j5,oUG dD 0,151515…; 0,63917256…; −1,8555…; 3,14168306701… 0D 4,089201537…; − 2,543626262…; 0,666…; 0,758130165… @D .97;k, oypk, 9eo;omuj[+lk,kf0Pogxao9eo;omqfltoypq,Iv[;Po s^n g]dlj;o a g-y’j a,b b gxao9eo;o4h;o c]t b 8jk’l6o (b ≠ 0) gvUo;jk 9eo;ovtxqddt8yD 8q;1jk’D !F@#$&*()_!!!@!#!$DDD #F$#@#@@#@@@DDD ສະຫງສ.ວວນ.ລິສຂະສິດ !*F(_((_(((_((((_DDD {$F#__##__###__DDD 9kd8q;1jk’0hk’gmy’ la’gdfgsao;jk gIqk[+lk,kf9af8q;g]ds^a’95f.shgxaoIv[;PoF vud8q;1jk’|bj’0v’9eo;ovtxqddt8y7n π g-y’j ,u7kj gmqkj da[ #F!$!&_@*&#&)_(_#@#)$*@DDD π c,oj vaf8kl;j o]ts;kj ’];’Iv[0v’;q’,qo c]t glahozkj o.9dk’0v’,aoD g,nvj gIqk-vdskgonhvmu0j v’czoj ,qo3fp.-hl6f π r2 s^n -vdsk];’Iv[0v’;q’,qo 3fp.-hl6f 2π r g-y’j r c,jo]aflt\\u0v’;q’,qo gryoj ,ad.-h7jkxt,ko0v’ π gxao 22 s^n #F!$!* s^n #F!$@ g-y’j gxao9eo;oxqddt8yma’\\qfD 7 9eo;omujgxao9eo;oxqddt8y sn^ 9eo;ovtxqddt8y gvUo;jk 9eo;o9y’ 8q;1jk’G 7; − 4; 5 ; − 7 ; 6, 3; 0; −15, 02; 8,999...; −1, 40555; 2 8 2,121221222...; −0, 2534970128...; 28,346 39 

czo;kfltcf’9eo;o-t|yf8jk’M 9eo;o9y’ 9eo;oxqddt8y 9eo;ovtxqddt8y 9eo;o4h;o g]dl;j omuj[+c,jo9eo;o4h;o 9eo;o4h;o]q[ l6o 9eo;o4h;o[;d ;ymultcf’9eo;ovtxqddt8y.lcj do9eo;o zkj o,kgIqklk,kfltcf’9eo;oxqddt8yc8j]t9eo;o.lcj do9eo;owfhD ສະຫງສ.ວວນ.ລິສຂະສິດ le]a[9eo;ovtxqddt8y gIqkd=,u;ymultcf’c8j]t9eo;o.ljcdo9eo;owfh fa’j ]5,j oUG 8q;1jk’G le]a[ π ];'Iv[0v';'q ,oq gonjv’9kd π = glAozkj o.9dk' faj’oAo 4hkgIqkc8h,;q’,qo|bj’mu,j uglAozkj o.9dk’ ! sq;|j;pF ];’Iv[0v’;q’ ,qod=9tgmqjkda[ π , ltcf’;jk π c,jo];’Iv[0v’;q’,qomu,j uglAozkj o.9dk’ ! sq;|j;pD faj’oAo gIqklk,kfltcf’9eo;o π .ljcdo9eo;o3fpdY’;q’,qo muj,uglAozjko.9dk’ ! sq;|j;p9kdg,af W wxg[Nv’0;k,n.shwfh|b’j Iv[D 40 

#D [qfg/ydsaf !D 9eo;o8+wxoU 9eo;o.fgxao9eo;oxqddt8y c]t 9eo;o.fgxao9eo;ovtxqddt8yL 1) 9 2) − 7 7 3 3) 0 4) − 2 + 0,2 5) 0,842 6) 1,366666 7) 2,4313113111 8) − 5,9326483264832648 9) 2,137137137 10) − 0,1666676869 @D dD 9qj’cmd0t|kf0v’g9hp A4 . 0D 9qj’-vdvaf8klj;o]ts;j k’];’pk; c]t ];’dh;k’0v’,aoD #D fh;pg7njv’7yfw]jg]dF 9qj’-vdsk 2 ; 3 c]h;.sh0=hla’gdfD $D 9q’j ltcf’9eo;ovtxqddt8y π c]t − π .ljcdo9eo;ofP;daoD &D ,u9eo;o9y’.f[+ mujgxaoma’9eo;oxqddt8y c]t 9eo;ovtxqddt8yL 7jk0v’ π ສະຫງສ.ວວນ.ລິສຂະສິດ dko7eo;o 7kj 0v’ π .ovtfufgryjo.-h;ymumujc8d8jk’dao g-aoj G { lhk’I6[s^kpc9ltg\\uco[.o c]t 8yfovd;q’,qo { 7yfw]j];’Iv[0v’I6[s^kpc9ma’lv’I6[ { 7yfw]7j jkxt,ko0v’];’Iv[0v’lv’I6[;q’,qo { -vdvaf8kl;j o]ts;jk’];’Iv[0v’;q’,qo c]t glAozkj o.9dk’0v’;q’,qo 7jkmuwj fh9tgxao7jkxt,ko0v’ π dko7eo;o 7kj 0v’ π fh;p;ymuoU0Boda[9eo;o0hk’0v’I6[s^kpc9ltg\\upyj’9e o;o0hk’0v’I6[s^kpc9ltg\\upy’j s^kp0Bogmqkj .f7jkmujwfhpy’j .dh7P’da[7jk0v’ π gmqjkoAoD 41 

vkd-uc,f XArchimèdesG xt,ko @)({@!@ xu djvo7iy8tladdtikf? oad7toyflkf c]t ;ymtpklkf -k;gdiadltcf’.shgsao;jk π ,u7jk]ts;jk’ 223 c]t 22 71 7 vkd-uc,f  • -695’9u XTsu Ch’ung chih: xt,ko 7DlD $#W{&W!) ;ylt;tdvo-k; 9uowfh7kj xt,ko0v’ π c,jo 355 s^n xt,ko #F!$!&_@_ 113 • m=g],u XPtolemy: xt,ko 7DlD @WW) wfh7kj xt,ko0v’ π c,oj #D!$!* •.o7iyf8tlt8t;afmuj !) ,udko.-h8q;vadlvogdiad7n π cmovaf8kl;j ofaj’dkj ; c]t ,udko7eo;osk7jk0v’ π fh;ps^kp;ymudkomujc8d8jk’daogIaf.shwfh7jk0v’ π c8d8kj ’daogIaf.shwfh7jk0v’ π mujgxao9eo;omqfltoypq,s^kp8ec\\j’ g-aoj G { .o 7D lD !)(# c-’ (Shanks) oad7toyflkf-k;va’dyf wfh7eo;o7kj 0v’ π ສະຫງສ.ວວນ.ລິສຂະສິດ gxao9eo;omqfltoypq,g4y’ (W( 8ec|j’3fp.-hg;]k7eo;os^kpd;j k !& xuD { .o 7D lD !_$_ g7nvj ’7v,ru;g8ulk,kf7eo;o7kj 0v’ π gxao9eo;omqfltoypq, wfhg4y’ @WWW 8ec|’j { .o 7D lD !_*! π gxaomqfltoypq,wfhg4y’ !WW@*& 8ec|’j { .o 7D lD !_*( π gxaomqfltoypq,wfhg4y’ &WWWWW 8ec|j’ { .o 7D lD !___ g7njv’-5[gxu7v,ru;g8umuj,tsk;ymtpkw]38dP;7eo;owfhg4y’ @W* !&) $#W WWW 8ec|’j c]t .oxt95[aowfh9eo;omqfltoypq,s^kpd;kj lv’rao]hko 8ec|’j D 42 

rkdmu @ dkog7yj’7n [qfmu !@ dkog7yj’7nmP[.ljg,af|b’j !D dyf9tde dyf9tdemu ! .shglAo-nj Δ F g,af A, B, C C c]t D. A { 9qj’lhk’g,afg7yj’7n A' , B', C' c]t D' 0v’ A, B, C c]t D D mP[.lgj lAo-nj Δ. la’gdfgsao co;.fL B { 9qj’8n,j xt3spd8+wxoUfh;pG glAo-nj Δ F A, A' , glAodk’lkdF Δ ສະຫງສ.ວວນ.ລິສຂະສິດ mvj o-nj [AA'] D i g,af DDDDDDDDDDDDDDD c]t DDDDDDDDDDDDDDDg7y’j 7ndao mP[.lj DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD i DDDDDDDDDDDDDDDDDDDDDDDc,oj DDDDDDDDDDDDDDDDDDDDDDDDDDD0v’ DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD dyf9tdemu @ .shg,af O, A, B c]t C. A { 9q’j lhk’g,af A', B' c]t C' g7yj’7n da[g,af A, B c]t C mP[.ljg,af O. B { g,af A' gvUo;kj g,afg7yj’7n0v’ A mP[.lgj ,af O. O { 9q’j lhk’g,afg7yj’7n O' 0v’ O mP[ C .l8j q;,aogv’ c]h;.sh0=hla’gdfD A' { 9qj’8n,j xt3spd8+wxoUfh;pG g,afg7y’j dk’FA, O, A' c]t [AA']. i g,af DDDDDDDDDDDDDDDDDD c]t g,af DDDDDDDDDDDDDDDDDDDDg7yj’7ndaomP[.ljg,af DDDDDDDDDDDDDDDDDDDD i g,af DDDDDDDDDDDDDDDDDDc,jo DDDDDDDDDDDDDDDDDDDDDDDDDDDDD 0v’mvj o-nj DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD { 9q’j xP[mP[w]pt OA', OB' c]t OC' da[ OA, OB c]t OC 8k,]efa[D 43

{ I6[lk,c9 A'B'C' c,oj I6[g7yj’7n0v’I6[lk,c9 ABC mP[.ljg,af O. 9qj’xP[mP[lv’I6[lk,c9faj’dkj ;D { g,af O gvUol6odk’dkog7yj’7nD @D .97;k, !D dkog7yj’7nmP[.ljglAo-n|j bj’D A { g,af A’ c,oj g,afg7y’j 7n0v’g,af A mP[.lj glAo-nj Δ. Δ { glAo-nj Δ c,joglAodk’lkd0v’mjvo-nj [AA’]. { g,afmu1j 6gj lAo-nj Δ ,ug,afg7yj’7nc,oj 8q;,aogv’D A' { glAo-nj Δ gvUo;kj G cdog7y’j 7nD @D dkog7yj’7nmP[.ljg,af|b’j D dD P’, Q’, R’ c,oj g,afg7y’j 7n0v’ P, Q, R R 8k,]efa[mP[.ljg,af O. ສະຫງສ.ວວນ.ລິສຂະສິດ { O c,oj g,afg7yj’dk’0v’mjvo-nj [PP’], Q [QQ’] c]t [RR’]. 180 P O { g,afg7yj’7n0v’ O c,oj 8q;,aogv’D P' { g,af O c,jo l6odk’g7yj’7nD 0D lv’I6[g7yj’7ndaomP[.lgj ,af|bj’ Q' 9tg8a’daor=fu4hkgIqkxyjoI6[|bj’vhv, l6odk’g7yj’7nwfh 180 D R' POP' = QOQ' = ROR' = 180 { lv’I6[g7yj’7nmP[.lgj ,af|bj’ gmqjkdaoD ΔPQR = ΔP'Q'R' #D dkolhk’g,afg7y’j 7nmP[.ljg,af|b’j D M lhk’g,af M’ g7y’j 7nda[g,af M mP[.lgj ,af O. O d=]toumu ! • g,af M g8a’da[g,af O, g,afg7y’j 0v’ M c,oj 8q;,aogv’D M' D ma’lk,g,af O, M c]t M' g8a’dao16gj ,affP;D 44