வாக்கிய கணிதம்- இலக்னம் காண

[hjf fzpj Kiwfs;;

    md;G> = mfj;jpau; N[hjplg;gapw;;rp ika mUik khzth;fNs ! N[hjplg; gapw;rpapy; kpf Kf;fpakhd gFjp [hjf fzpjk; nra;tjhFk;. xUtUf;F Vd; [hjfk; vOjg;gLfpwJ ? - vd;gjw;Fg; gytpj tpsf;fq;fs; nfhLf;fg;gLfpd;wd. vdpDk;> gok;ngUk;; ghly;fspy; [hjfk; vOJtijg;gw;wp vd;d Fw;g;gplg;gpLfpd;wd vd;gijg;gw;wp ghh;g;Nghk;.

cj;jpufhykph;;jk; nrhy;tnjd;d ?;

        “[ddP [d;k nrsf;fpahdhk;>th;j;jdPk; Fy rk;gjhk;

        gjtPk; G+h;t Gz;ahdhk; ypf;aNj n[d;k gj;hpfh.”

    

    [hjfk; vOJk;NghJ Nkw;nrhd;d RNyhfNk  md;W Kjy; ,d;Wtiu Kjyhtjhf vOjg;gLfpwJ. mNjNghd;W ‘IahRthkpak;’ -; vDk; jkpo;r; nra;ASk; ,Nj fUj;ij typAWj;Jtjhf mike;Js;sJ.

         “rPuhUk; nrd;kRfk; nra;FtJ ck; Fyj;jpd;

         VuhUk; nry;t tsk; Vw;WtJk; -- Njuh

         Kdk; nra;j ey;tpidapd; Kd;D topAkhk;

         rdd gj;jpuk; vOJjy;.”

    ,g;ghly;fspd; %ykhf ehk; RUf;fkhf mwpeJnfhs;Sk; fUj;Jf;fs; ahnjdpy;>xUth; jdJ Kw;gpwg;gpy; nra;j ghtGz;zpaq;fSf;F>,g gpwtpapy; mDgtpf;fg;NghFk; ey;tpid>jPatpidg; gyd;fisAk;>mjdhy; [hjfUf;F Vw;gLk; Rfnrsf;fpaq;fs;>nry;t tsq;fs; ngUFjy;>gjtpfs; miljy; Nghd;witfis mwpe;J nfhs;tjw;Nf mtUf;F [hjfk; vOjg;gLfpwJ.;

    Foe;ij gpwe;jJ Kjy; jpUkzkhfp tho;f;if elj;Jk; fhyj;jpy;>Kf;fpakhd 16 rk;];fhuhf;fs; -- vDk; flikfspy; ‘[hjf fh;kh’ – vDk [hjfk; vOJjy; kpf Kf;fpakhd xd;whFk;.

    ,t;thW “vOjg;gLk; [hjfkhdJ> [hjfUf;F mth; gzk; rk;ghjpf;Fk; fhyj;jpy; xU ez;gidg; NghyTk;>gazq;fs; Nkw;nfhs;Sk; NghJ ke;jphpiag; NghyTk>Mgj;Jf; fhyq;fspy; flypy; jj;jspf;Fk; NghJ fpilf;Fk; kpjit NghyTk; cjtp nra;fpwJ”;--vd rhuhtspia vOjpa fy;ahzth;kh Fwpg;gpLfpwhu;.;

    xU [hjfj;ijf; fzpf;f ehk;> gQ;rhq;fj;ijAk;>’vgpnkhP];’ – vDk; fpuf epiyfisf; nfhz;l ml;ltiziaAk; gad;gLj;JfpNwhk;. ‘thf;fpa Kiw’ kw;Wk; ‘jpUf;fzpj Kiw’ -- vd ,UNtW tpjkhd [hjf fzpj Kiwfs; gad;gLj;jg;gLfpd;wd. ,jpy; jpUf;fzpjg; gQ;rhq;fNk ;;Jy;ypakhd fpuf epiyfisj; jUtjhFk;.

   

thf;fpa Kiwg;gb [hjfk; fzpf;fj; Njitahd tpguq;fs; ahit ?

1.   [hjfh; gpwe;j Neuk;.

2.   gpwe;j ehs; :- jkpo; Mz;L>khjk;> Njjp; kw;Wk; fpoikAld; mjw;F ,izahd Mq;fpy tUlk;>khjk; kw;Wk; Njjp Mfpait;

3.   gpwe;j ,lk; :- gpwe;j Chpd; mlrhk;rk;;>jPh;f;;fhk;rk; kw;Wk; ,uhrpkhdq;fs.;

4.   gpwe;j ehspd;> me;j CUf;fhd #hpa cjak;.

5.   md;iwa ,yf;d ,Ug;G.

6.   fpufepiyfs; mwpa ghjhrhu tpguq;fs;.

7.   gpwe;j ehspd; el;rj;jpuk;> jpjp>Nahfk;>fuzk; gw;wpa tpguq;fs;. NjitfF Vw;g Kjy;ehs; my;yJ kWehSf;fhd el;rj;jpu ,Ug;G. gb 2 Kjy; 7 tiuf;fhd tpguq;fs; mwpa me;j Mz;Lf;fhd thf;fpa gQ;rhq;fk; MfpaitahFk;.

 ;

    ,dp>[hjfk; vOJtjw;F Kd;dh; ehk; ,yf;dk; vd;why; vd;d ? mij vt;tpjk; mikg;gJ ? ,yf;dj;jpd; Kf;fpaj;Jtnkd;d ? – vd;gij mwpjy; mtrpakhfpwJ. fzpjq;fisr; nra;Ak;NghJ xU Ritahd ehtiyg; gbg;gJ Nghd;w czh;it khzth;fs; ngwNtz;LNkad;wp> rypg;ngh>rq;flNkhapd;wp Gjpa fiyiaf; fw;Wf;nfhs;Sk; Mh;tk; kl;LNk neQ;rpy; epyt Ntz;Lk;.

    xUth; gpwe;j Neuj;ijf; nfhz;L [hjfj;jpy; ve;j ,lj;jpy; ,yf;fpdk; mikAk; vd;gijf; fz;Lgpbf;f Ntz;Lk;. ,yf;dk; mikg;gJ fbdkhd Ntiy ,y;iy vd;whYk;> khzth;fs; fzpjq;fisf; ftdkhfj; njhpe;J nfhs;sNtz;ba Kf;fpag; gFjpahFk;.

   

    G+kpahdJ 24 kzp Neuj;jpw;F xUKiw jd;idj;jhNd Rw;wpf;nfhs;fpwJ vd;gij ehk; mwpNthk;. mt;tpjk;> G+kp jd;idj;jhNd Rw;Wfpd;w 24 kzpNeuj;jpy;> ve;j Ntisapy; xU [ddk; epfo;fpd;wNjh me;j ,lj;ijj;jhd; ‘,yf;fpdk;’ vd;fpNwhk;. rhjuzkhfg; Ngr;Rthf;fpy; khiy 4 kzpf;Ff; Foe;ij gpwe;jJ vdf; Fwpg;gpLtij> ,uhrpf;fl;lj;jpy; FwpapLtNj ,yf;fpdk; MFk;. 12 ,uhrpfSk; fhiy Kjy;> kWehs; fhiy tiuAk; te;J Nghfpd;wd. xU [Ptd; cjpj;j fhyj;jpy; ve;j ,uhrpapd; fhyk; epfo;e;jJ vd;W nrhy;YtNj ,yf;fpdkhFk;. gQ;rhq;fj;jpYs;s gy;NtW tpguq;fisj; jpul;b [hjfh; mDgtpf;ff; $ba ey;y kw;Wk; jPagyd;fisj; jPh;khdpf;Fk;> [hjfj;jpd; Jtf;fg; Gs;spNa ,yf;dkhFk;.

    xU ehspy; #hpad; xt;nthU ,uhrpf;Fk; xsp jUk; fhy msitj; njhpe;J nfhs;sj; jf;fthW ekJ Kd;Ndhh; Nk\ ,uhrp Kjy;>kPd ,uhrptiuahd xt;nthU ,uhrpf;fhd ,uhrp ghpkhz msit>mjhtJ jpdrhp ,uhrpfs; epfOk; fhy msit ehopifapy; $wpAs;sdu;. mijf; fhz;Nghkh ? ,it nghJthd ,uhrpkhdk; Mjyhy;> khzth;fs;> gQ;rhq;fq;fspy;>me;je;j CUf;fhd ml;rhk;rj;jpw;Ff; nfhLf;fg;gl;Ls;s ,uhrpkhdg;gb fzpjk; nra;tNj Jy;ypakhf ,Uf;Fk; vd;gij epidtpy;; nfhs;s Ntz;Lk;.

kPdk;; 4.15	Nk\k; 4.15	up\gk; ;4.45	kpJdk;; 5.15		1.42		1.42		1.54		2.06
Fk;gk;; 4.45	,uhrpkhdk; ehopif-tpdhb                                ;		flfk; ;5.30		1.54	,uhrpkhdk; kzp – epkplj;jpy;				2.12
kfuk;; 5.15			rpk;kk; ;5.15		2.06					2.06
5.30	5.15	5.00	5.00		2.12	2.06		2.00		2.00

,yf;d fzpjk; nra;Ak; Kiwia tpsf;Ff:

1.   [hjfh; gpwe;j Neuj;jpypUe;J>gpwe;j CUf;fhd #hpa cjaj;ijf; fopf;f ntz;Lk;

2.   fopj;J te;j kzp – epkplq;fis>1 kzp Neuj;jpw;F 2½ ehopif vd;w fzf;fpd;gb> ehopif – tpdhbfshf khw;w Ntz;Lk;. ,ij cjahjp ehopif my;yJ [dd ehopif vdf; Fwpg;gpLfpNwhk;.

3.   ( 1 epkplk; = 2½ tpdhb ;  60 tpdhb = 1 ehopif ;  24 epkplk; = 1 ehopif ;     1 kzp = 2½  ehopif;  24 kzp = 60 ehopif. epidtpw; nfhs;f.)

4.   ,e;j [dd ehopifapypUe;J md;iwa ,uhrp ,Ug;igf; fopf;f Ntz;Lk;.

5.   fope;J te;j ,uhrpkhdj;jpypUe;J>xt;nthU ,uhrpf;fhd ,uhrpkhdq;fisf; fopj;Jtu>ve;j ,uhrpapy; fopf;f Kbahj epiy tUfpwNjh> me;j ,uhrpNa ,yf;dkhFk;.;

6.   [d;k yf;dkhf mike;j ,uhrpapd; nkhj;j ,uhrpkhd ehopifia> 9 My; tFf;f tUk; <thdJ xU ghj;jpd; msthFk;. ,ij itj;J xh; ,uhrpapYs;s 9 ghjq;fspy; ve;jg; ghjj;jpy; ,yf;dk; mikfpwJ vdf; fzf;fpl Ntz;Lk;;. ,e;jpa N[hjplj;jpy; ,yf;dk; vd;gJ ,d;wpaikahj xd;whFk;.

cjhuz thf;fpa fzpjk;:

1.   ,yf;d fzpjk; :-

     [hjfh; [dd ehSf;Fhpa gQ;rhq;fj;ijf; nfhz;L chpa tpguq;fisr; Nrfhpf;f Ntz;Lk;. [hjfh; gpwe;j Neuj;jpypUe;J>[dd ehSf;Fupa #hpa cjaj;ijf; fopj;J tUk; tpilia 2½ klq;fhf Mf;f Ntz;Lk; my;yJ ,uz;liuahy; ngUf;ff; fpilg;gNj #hpa cjahjp ehopifahFk;.

   

     kJiuapy; gpwe;j> [hjfh; gpwe;jNjjp> tp[a tUlk; Mb khjk; 12 k; ehs; Qhapw;Wf;fpoik (28-7-2013) md;W mtu; gpwe;j Neuk; fhiy 10 – 43. kJiuapy;; md;iwa jpd #hpa cjak; fhiy 6 – 13 kzpahFk; (gf;fk; 85);. flf ,uhrpapUg;G 3—22 MFk;. ,e;j tpguq;fs; tp[a tUl =rPdpth]d; gQ;rhq;fj;jpypUe;J ngwg;gl;lJ. gf;fk; 83 y; cs;sgb kJiu ml;rhk;rj;Jf;fhd rpk;k ,uhrpkhdk;; 5.02 MFk;.;

   

mLj;J>cjahjp ehopif fhz;Nghk; :

            

                                       kzp- epkp

             [hjfh; gpwe;j Neuk; =     fhiy 10 – 43

            #hpacja Neuk; =;        fhiy  6 -- 13

4 - 30

 #hpacjaj;jpw;Fg;gpwF =                     4  -30
             [hjfh; gpwe;j Neuk;.

 ,e;j 4 kzp 30 epkplj;ij ehopifahf;f 2½ My;; ngUf;fNtz;Lk;.

    4 – 30 × 5/2 =  11 ehopif 15 tpdhb.  my;yJ ,e;j 4-30 I ,uz;liuklq;fhf;fpAk; #hpa cjahjpehopifiaf; fhzyhk;.

1 klq;F-2klq;F- miuklq;F

 

 

1.30     + 4.30  + 2.15 = 11 eh. 15 tp. ,JNt cjahjp ehopif.

khzth;fNs ! cjahjp ehopif fzf;fpLtJ Rygk;jhNd. Ghpe;jjh ? mLj;J mLj;J tUk; ,uhrpkhdq;fisf; fopj;J ,yf;d epiyia mwpNthkh ?

[hjfh; kJiuapy; gpwe;jtuhjyhy;>kJiuapd; ml;rhk;rkhd 10° f;F chpa ,uhrpkhdj;ijNa gad;gLj;j Ntz;Lk.;;

  

   cjahjp ehopif     11 – 15  vdf; fz;Nlhk; ,jpypUe;J md;iwa flf ,uhrpkhd ,Ug;igf; fopf;f          3 --  22.

7  -- 53

  rpk;k ,uhrpkhdk;         5  -- 02         

2  -- 51

 ,e;j 2 – 51 fd;dp ,uhrpapy; nrd;w ehopif>fd;dp ,uhrpapd; nkhj;j ,uhrpkhdkhd 4-58 ,y; ,Ue;J fopf;f KbahJ. vdNt fd;dp ,uhrapNyNa ,yf;dk; mikfpwJ vdf; nfhs;f.