Embed Size (px)
Citation preview
1
˘RÙPdLdLpÆ B£¨Vl ThPVd LpÆDIPLOMA IN ELEMENTARY EDUCATION D.El.Ed.
T⁄Ym ˛ 1
RÙs ˛105L¶Rm ˛ Lt\¤m Lt©jR¤m
˙LW[ AW—˘TÙ’d LpÆj’˚\
UÙ®Xd LpÆVÙWÙnf£ Utflm T´t£ ®flY]m (SCERT), ˙LW[m2019
2
12345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678
State Council of Educational Research and Training (SCERT)Poojappura, Thiruvananthapuram 695012, Kerala
Website : www.scertkerala.gov.in, e-mail : [email protected] : 0471 - 2341883, Fax : 0471 - 2341869
Typesetting and Layout : SCERT
©Department of EducationGovernment of Kerala
3
ÿu‡˚W
2018˛19 LpÆ Y⁄Pm ÿRp ∫W˚UjR •.Gp.Gh TÙP HtTÙh•p SÙm TVQm Nn°˙\Ùm
ApXYÙ. L¶Rd Lt\p L•]UÙ] J⁄ TÙPm Gu\ GiQm Cuflm £X CPeL∞p
®˚X®t°\’. B]Ùp ˘Nu\ C⁄T’ Y⁄PeL[ÙL SÙm ˙LW[jßp
S˚Pÿ˚\T”jßV ˘NVpTÙh” A•lT˚P´X˚UkR TÙP HtTÙ” L¶Rm
Lt©jR¤dœ ◊ßV T¨UÙQj˚RŸm, ®˚XL˚[Ÿm HtT”jߟs[’. UÙt\lThP
L¶R A‘œÿ˚\dœ UÙ\ÙL, TVuTÙ”m, ˘NVpTÙ”m NÙokR ÿ˚\´p
L⁄j’L˚[l ◊¨k’˘LÙiP ©u]˙W "Ap˙LÙ¨R ÿ˚\' ´p ÷˚ZV ˙Yi”m
GuT˙R ◊ßV Li˙QÙhPUÙœm. Ap˙LÙ¨R ÿ˚\´p œ±l©hP J›eœÿ˚\´p
Uh”˙U ˘RÙPok’ Y⁄m ®˚XL∞p ˘NpX˙Yi”m G] G∞RÙLd ·\XÙm.
B£¨VlT´t£ UÙQYoLs ◊ßV L¶R Lt©jRp ÿ˚\L˚[Ÿm,
A‘œÿ˚\˚VŸm N¨VÙLl ◊¨k’˘LÙs[ ˙Yi”m. L¶R Es[PdLj˚R Ltfld
˘LÙsY˚R ÆP ÿd°VUÙ]’ L¶R A‘œÿ˚\L˚[Ÿm, Lt©jRp ÿ˚\L˚[Ÿm
Ltfld ˘LÙsYRÙœm.
•.Gp.Gh L¶R HtTÙ” SÙuœ T⁄YeL[ÙLl ©¨dLlTh”s[’. ÿRp T⁄Yjßp
L¶Rjßu Ru˚UŸm, A˚Ul◊m, L¶Rd Lt©jRp A‘œÿ˚\Ÿm ÿ˚\Lfim,
L¶R A±ÆV≠u Y[of£Ÿm ˙UmTÙ”m Gu‡m AXœLs EhT”jRlTh”s[].
˘RÙPok’ YWd·•V AXœLs L¶Rjßu Es[PdLm NÙokR LXk’˚WVÙPpLfidœ
˘Np°\’. CWiPÙm T⁄Yjßp ÿ˚\˙V ˘RÙPdL®˚X, EVo ˘RÙPdL®˚X L¶Rl
TÙPeL˚[ ÿRu˚UlT”jß Ts∞d·P A‡TY®Lr‹Lfim, Lt©jRp T´t£Ÿm
TXuR⁄m ÿ˚\´p SPj’YRtœj ’˚Q◊¨Ÿm AXœLs EhT”jRlTh”s[].
A’Uh”UpXÙ’ Ts∞d·P A[Æp Ußl¿” ˘NnRp, Lt©jRp ÿ˚\´XÙ]
TœlTÙn‹ (˘TP˙LÙ¥dLp A]Ù≠≥v) B°VYt˚\Ÿm Lt©jRp T⁄YeL∞p
LXk’˚WVÙPp SPjR ˙Yi”m.
£\kR L¶R B£¨VoL[ÙLj ßLrYRtLÙ] EhLÙh£Ÿm, T´t£Ÿm ◊ßV TÙP
HtTÙh•p, GpXÙ UÙQYoLfidœm °˚Pdœ˘U] GßoTÙod°˙\Ùm. ARtLÙL
UÙQYoLfidœj ’˚Q◊¨Ÿm L⁄ÆLs Uh”UpXÙ’ Ae∏L¨dLlThP ©\
ËpL˚[Ÿm TÙo˚Y´P ˙Yi”m GuT˚RŸm ®˚]‹T”j’°˙\Ùm. EeLfi˚PV
B£¨V¨u ERÆŸm, ˘NVpTÙh” A‡TYÿm CRtœ EeLfidœ ERÆVÙ´⁄dœm.
PÙdPo. _. ©WNÙjCVdœSo
Gv.£.C.Bo.• ˙LW[m
12345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678123456789012345678901234567812345678901234567890123456781234567890123456789012345678
4
Es[PdLm
T⁄Ym ˛ 1
RÙs˛105 L¶Rm ˛ Lt\¤m Lt©jR¤m
AXœ 1 - ˛ L¶RÆV≠u CVp◊m A˚Ul◊m
AXœ 2 ˛ L¶RÆV≠u Y[of£Ÿm ÿu˙]t\ÿm
AXœ 3 ˛ L¶RdLpÆ A‘œÿ˚\Ÿm Lt\p Lt©jRp ÿ˚\Lfim EjßLfim
AXœ 4 ˛ ©Wf£˚]j æo‹ L¶RdLpÆ´p
AXœ 5 ˛ RLYp RÙØp÷hTm L¶Rd LpÆ´p
AXœ 6 ˛ ˘RÙPdL ®˚X Yœl◊L∞p L¶Rm ˛ Es[PdLl TœlTÙn‹.
5
AXœ ˛ 1
L¶RÆV≠u CVp◊m A˚Ul◊m
ÿu‡˚W
Jq˘YÙ⁄ TÙPl©¨Ætœm AR]Ru £\l©Vp◊Ls Ei”. ˘UÙØlTÙPeLfim, A±ÆVp
TÙPeLfim AYt±u Ru˚U´p ®˚\V ˙YflTÙ”L˚[d ˘LÙi”s[]. A±ÆVpL∞u
AW£VÙ] L¶RÆVp ©\ TÙPl ©¨‹LL∞≠⁄k’ £X £\l©Vp◊L[Ùp R≤j’Ym
™dLRÙL Æ[eœ°\’. ÷i L⁄j’dLfidœm, ©Wf£˚]l TœlTÙn‹dœm, TœjR±‹f
£kR˚]dœm A∞dœm ÿd°Vj’Yÿm, L¶RÆV˚X R≤j’Ym E˚PVRÙL UÙtfl°\’.
J⁄ ˘RÙPdLlTs∞ B£¨Vo Gu\ ÿ˚\´p L¶RÆV≠u £\l©Vp◊L˚[Ÿm,
A˚Ul˚TŸm Tt±j ˘R¨k’ ˘LÙsY’ ÿd°VUÙœm. Gu\Ùp Uh”˙U, N¨VÙ] TÙP
A‘œÿ˚\˚Vl ◊¨k’ ˘LÙsYRtœm. ˙Y˚X´p ◊Lr˘T\‹m CV¤m.
Lt\p A˚P‹Ls
$ L¶RÆVp Gu\Ùp Gu]? GRtœ?
GuT˚R R]’ ˘UÙØ´p Æ[dœRp.
$ L¶Rm J⁄ ˘UÙØ. L¶Rm YÙ´XÙLd
L⁄j’l T¨UÙflYRtLÙ] YÙnl◊Ls.
$ L¶Rm Au\ÙP YÙrÆp J⁄
L⁄ÆVÙœm.
$ L¶Rf £kR˚]˚V Y[olTßp A±‹
ÈoYUÙ] £kR˚]´u ÿd°Vj’Yj˚R
A±Rp.
$ L¶Rjßu CVp˚T A•lT˚PVÙLd
˘LÙi” £kR˚]´u L¶R
UVUÙdœR˚Xj ˘R¨k’ ˘LÙsfiRp.
ÿd°V L⁄j’dLs
$ GiL∞u A±ÆV˙X L¶Rm.
$ GpXÙ A±ÆVpLfidœm L¶Rm
A•lT˚PVÙœm.
$ A˚Ul◊L˚[Ÿm ˘RÙPo◊L˚[Ÿm
œ±jR LpÆ˙V L¶Rm.
$ L¶Rm J⁄ ˘UÙØ.
$ L¶Rm YÙ´XÙLd L⁄j’l T¨UÙt\m.
(Y˚WTPm, AhPY˚Q, Æ[dLlTPm)
$ L⁄j’L˚[ ÷hTUÙL‹m, —⁄dLUÙL‹m
˘R¨ÆlTRtLÙ] J⁄ YØÿ˚\
$ Au\ÙP YÙrÆp Tp˙Yfl
„r®˚XL∞p ©¨f£˚]j æoÆtœ
L¶Rm TVuT”°\’.
$ L¶Rm A±‹ ÈoYUÙ] £kR˚]˚V
A•lT˚PVÙLd ˘LÙiP’.
$ GkR J⁄ L¶Rd L⁄j’m LÙWQ˛LÙ¨Vj
˘RÙPo˚T Li” ©•j’ A±‹
ÈoYUÙLf £kR˚] Nn’ ®fl‹YRtLÙ]
ß\˚] A•lT˚PVÙLd ˘LÙi”s[’.
6
L¶RÆV≠u CVp◊
""˙LÙh˚P´u Es˙[RÙu ®ß TÙ’LÙdLlT”°\’. ˙LÙh˚P´u SÙuœ YÙNpL∞¤m
LÙYpLÙWoLs ®fljRlTh”s[]o'' AYoL∞Pm AWNo ·±]Ùo. ÿR≠p ®tTYo Jq˘YÙ⁄
Íufl U¶ ˙SWm ÿ•Ÿm˙TÙ’ Tß˚]k’ ®™PeLs Kn˘Y”dLXÙm. CWiPÙYRÙL
®tTYo Jq˘YÙ⁄ SÙuœ U¶ ˙SWm ÿ•Ÿm˙TÙ’ Tß˚]k’ ®™PeLs Kn˘Y”dLXÙm.
Íu\ÙYRÙL ®tTYo Jq˘YÙ⁄ Ik’ U¶ ˙SWm ÿ•Ÿm˙TÙ’m SÙuLÙYRÙL ®tTYo
Jq˘YÙ⁄ BflU¶˙SWm ÿ•Ÿm˙TÙ’m Tß˚]k’ ®™PeLs Kn˘Y”dLXÙm.
AlT•VÙ]Ùp J⁄Yo Kn˘Y”dœm˙TÙ’ Ut\YoLs LÙY¤dœ ®tTÙoLs G] AWNo
$ L¶Rm Lt\≠p LÙh£l T”j’R≠u
YÙnl◊dL˚[ A±k’ ˘LÙs°u\]o.
$ ’p≠Vm, ÷hTm CYt±tœ L¶Rm
Lt\≠p Es[ ÿd°Vj’Ym
©¨jR±°u\]o.
$ L¶Rm Lt\≠p ÷iL⁄j’L˚[
T ⁄ l ˘ T Ù ⁄ h L s / A ‡ T Y e L ∞ u
ERÆŸPu Æ[dœ°u\]o.
$ ©\ TÙPeL˙[Ù” L¶Rjßtœs[
˘RÙPo˚T LiP±k’ Æ[dœ°u\]o.
$ L¶Rm Lt\≠u TœßVÙL UÙQYoL∞p
E⁄YÙL ˙Yi•V StTi◊L˚[Ÿm,
U]lTÙu˚U˚VŸm LiP±°u\]o.
$ ÷i L⁄j’dL˚[ (G∞RÙLl
◊¨k’˘LÙs[) LÙh£l T”jRp
TVuT”°\’.
$ LÙh£lT”j’Rp YÙ´XÙL L¶Rd
L⁄j’⁄YÙdLÿm, ©Wf£˚]j æo‹m
G∞RÙL A˚UŸm.
$ ©Wf£˚] æo‹dœ, ’p≠Vÿm, ÷hTÿm
L˚P©•dL ˙Yi”m.
$ ’p≠Vm, ÷hTm, CYt±u A[‹˙LÙp
NkRoTeLfidœ HtT UÙflT”m (Context
sensitive)
$ ÷iL⁄j’/ T⁄dL⁄j’
$ ÷i˚UVÙ] L¶Rd L⁄j’dL˚[
T ⁄ l ˘ T Ù ⁄ h L s / A ‡ T Y e L ∞ u
ERÆŸPu ˘Y∞´P CV¤m.
$ GpXÙ TÙPeLfim L¶Rj˙RÙ”
˙SWÙL˙YÙ/U˚\ÿLUÙL˙YÙ ˘RÙPo◊
˘LÙi”s[’.
$ L¶Rd Lt\≠p Tp˙Yfl StTi◊Ls
UÙQYoLs, Eh˘LÙs°u\]o.
$ S˚Pÿ˚\ A±‹
$ U]˚R J⁄ ÿLlT”jRp
$ AZœQof£
$ A±ÆVp Li˙QÙhPm.
7
LQd°hPÙo. B]Ùp, ◊jßNÙ≠VÙ] J⁄ LsYu SÙuœ˙T⁄m J˙W ˙SWjßp Kn˘Y”dœm
˙SWm Y⁄m GuT˚Rl ◊¨k’˘LÙi” ®ß˚Vd L[YÙ•f ˘Nu\Ùu. OÙ´tfld°Z˚U UßVm
Tu≤˘Wi” U¶dœd LÙYp ˙SWm ˘RÙPe°V’ G≤p GkR ˙SWjßp ®ß
L[YÙPlTh•⁄dœm?
Æ˚P Li”©•dL‹m.
Æ˚P Li”©•lTRtœ øeLs B˙XÙ£jR YØÿ˚\ G’?
L¶Rm NÙo £kR˚]L∞p GYt±u YØVÙL øeLs ˘Nu»oLs?
£u] Yœl◊ ÿRp CjR˚LV CWN˚] ™dL’m £kR˚]˘VÙ∞ R⁄°u\’UÙ] L¶Rd
L⁄j’dL˚[ SÙm Eh°W°lT’Pu Tp˙Yfl „ZpL∞p S˚Pÿ˚\lT”jR‹m
˘Nn°u˙\Ùm. B£¨Vl T´t£d LpÆ´u J⁄ Lt\p ˘TÙ⁄s Gu\ ®˚X´p L¶Rm
œ±j’m ARu £\l©Vp◊Ls œ±j’m C≤Ÿm SÙm A±k’˘LÙs[ ˙Yi•V] Es[].
L¶RÆVp GuT’ Gu]? GRtLÙL?
GiLs YÙ´XÙL EX˚Ll TœlTÙn‹ ˘NnY’Pu Æ[dœRp GuT’m L¶Rjßu
A•lT˚PVÙ] CVpTÙœm. TX˙Y˚[L∞¤m L⁄jRÙdLeL˚[ GiL˚[l TVuT”jß
ÆY¨dœm ˙TÙ˙R ˘R∞YÙLl ◊¨k’˘LÙs[ CVp°u\’. CVt˚LNÙo At◊R ®Lr‹L∞p
EhLXk’s[ GiNÙo RÙPo◊L˚[ CVtL¶Rf NUuTÙ”L∞p Y∞´”m˙TÙ˙R ÷hTUÙ]
A±‹Lfim ™LfN¨VÙ] ·tfldLfim °˚Pd°u\]. CqYÙfl TÙodœm˙TÙ’ Gi
ÿd°Vj’Ym YÙnkR J⁄ ˘UÙØ˙V L¶Rm G]d ·\XÙm. S˚Pÿ˚\ NÙokR
T¨UÙt\eLfidœm ˙UXÙLl T˚Pl◊jß\u NÙokR J⁄ Ti◊ ˘UÙØdœ Es[’ ˙TÙu˙\
L¶Rjßtœm S˚Pÿ˚\ NÙokR LQd∏”s Uh”UpXÙUp A±YÙokR —RkßWf
£kR˚]˙VÙhPeLfim Es[].
L⁄jRÙdLeL˚[ GiL˚[l TVuT”jß Eh°W°lT’Pu Æ[dœR¤dœm L¶RÆVp
LpÆ ™L ÿd°Vj’Ym YZeœ°\’. YÙrd˚L´u A˚]j’j ’˚\L∞¤m L¶RÆVp
LpÆ ™œkR ˘NpYÙdœf ˘N¤j’°u\’. L¶R YÙnl◊L˚[l TVuT”jRÙR J⁄ SÙs
·P Cp˚X G]XÙm. L¶RÆVp LpÆ £kR˚]˚Vj ˘R∞‹T”j’Y’Pu L⁄j’dL˚[
A±ÆVp NÙok’ TœjRÙWÙV‹m ’˚Q ◊¨°u\’. ©\ TÙPeL˚[d LtTߤm L¶RÆVp
©¨dL ÿ•VÙR LÙW¶VÙL A˚Uk’s[’. EeLfidœd L¶Rm NÙokR TX Y˚WV˚\Ls
A±ÿLm B°´⁄dLXÙm. ¥.Gf.aÙo• L¶Rm Tt±d ·±V˚Rd TÙWdL‹m.
“A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more
permanent than theirs, it is because they are made with ideas”:
G.H. Hardy, 1940
L¶R A±Oo J⁄ L˚XO⁄m LÆO⁄m BYÙo Gu\ Au]Ù¨u L⁄j’ EeLfidœ
Ht◊˚PVRÙ? Tp˙Yfl LÙXLhPeL∞p YÙrkRYoL∞u L¶RÆVp œ±j’s[ TÙo˚Y
™Lf —˚YVÙ]’. AYtfls £XYt˚\l TÙol˙TÙm.
$ VRÙ£LÙUÎWÙQÙm
SÙL]ÙmUQ˙VÙ RRÙ
8
Rj Yj ˙YRÙeL NÙvjWÙQÙm
L¶Rm ÍojR≤ vßRm - ˙YRÙeL ˙NÙßPm °.ÿ. 1200
(U´≠u R˚X Ef£ ˙TÙX‹m TÙm©u UÙ¶dLdLp ˙TÙX‹m ˙YReL∞p L¶R J∞
≈—°u\’.)
$ L¶Rm A˚Ul◊L˚[Ÿm (Pattern) AYt±u ˘RÙPo◊L˚[Ÿm œ±j’s[ LpÆVÙœm.
$ L¶Rm L⁄j’lT¨UÙt\jßtœ E¨V L⁄ÆVÙœm.
$ A[‹Ls, GiLs GuT]Yt±u ˘RÙPo◊Ls EhT”°u\ A±ÆV˙X L¶Rm.
$ L¶Rm J⁄ L˚XŸm J⁄˘UÙØŸm A±ÆV¤m Bœm. (Mathematics is an art, lan-
guage and science)
$ L¶Rm GiL∞u, Y•YeL∞u, A[‹L∞u A±ÆVXÙœm. (Mathematics is a science
of numbers, shapes and measurements).
L¶Rm Tt±f £Xo ·flm L⁄j’dL˚[l TÙodL‹m.........
$ The science of quanttiy – Aristotle
$ The abstract science which investigates deductively the conclusions implicit in the
elementary conceptions of spatial and numerical relations, and which includes as its main
divisions geomtery, arithemetic, and algebra – Oxford English Dictionary.
$ The study of measurements, properties, and relationships of quantities and sets, using numbers
and symbols – American Heritage Dictionary, 2000.
$ The science of structure, order and relation that has evolved from elemental practices of
counting, measuring and describing the shapes of objects – Encyclopedia Britannica,
2006.
L¶RÆV˚Xd LÆj’Yj’Pu LÙQd·•V Y˚WV˚\Lfim Es[].
$ Mathematics is the music of reason – James Joseph Sylvester.
$ Go down deep enough into anything and you will find mathematics. - Dean Schlicter.
$ "Just because we can't find a solution, it doesn't mean that there isn't one." -
Andrew wiles.
$ "Mathematics is the gate and key to science." - Roger Bacon.
$ Mathematics is not about numbers, equations, computations, or algorithms: it is about
understanding." - William paul Thurston.
$ Without mathematics, there's nothing you can do. Everything around you is mathematics.
Everything around you is numbers." - Shakuntala Devi
L¶R ËpL∞≠⁄k’m C˚QVR[jß≠⁄k’m ˙U¤m £X Y˚WV˚\L˚[d
Li”©•dL‹m. Jq˘YÙ⁄ Y˚WV˚\´‡˚PV‹m A±YÙokR TÙo˚Y´˚]Ÿm
YÙnl©˚]Ÿm ˙NÙßj’lTÙodL‹m. CYt±u A•lT˚P´p —VUÙL Y˚WV˚\L˚[
E⁄YÙd° —YolTjߨ˚L´p ˘Y∞´P‹m. ˙YflThP]Yt˚\ EhT”jß R≤STo
T˚Pl◊Ls RVÙWÙd° ˘Y∞´P‹m ˙Yi”m.
9
Ußl¿”
$ R≤STo RVÙWÙd°V Y˚WV˚\L∞u —YolTjߨ˚L
L¶Rm Tt±V ˘Yq˙Yfl Li˙QÙhPeLs
∏rdLÙ‘m TPeL˚[l TÙodL‹m.
Tjߨ˚L, CRrL∞p Yq˙Yfl UiPXeLfiPu RÙPo◊ LÙi” Yk’s[ A±d˚LL˙[
C˚Y. CYt±p L¶Rm NÙokR L⁄j’dLfim œ±¬”Lfim GqYÙfl Es[]? A˚Y VÙ˚Y?
Tp˙Yfl UiPXeLfiPu RÙPo◊s[ RLYpL˚[j R∞YÙL‹m ƨYÙL‹m Suœ Æ[dL
L¶Rd œ±¬”Ls GjR˚LV TVuTÙ” E˚PV]?
˘NVpTÙ”
øeLs ßWh”°u\ CjR˚LV A±d˚LL∞u ERÆŸPu L⁄j’lT¨UÙt\jßtœd L¶Rm
GqYÙfl ’˚Q◊¨°u\’ G]d LXk’˚WVÙ•d Li”©•dL‹m. œ±l◊Ls RVÙWÙd°
Yœl©p ˘Y∞´P‹m.
$ L¶Rm J⁄˘UÙØ
˘UÙØ L⁄j’lT¨UÙt\jßtœ E¨V’. L¶Rjßu —⁄dL G›j’dLs, AhPY˚QLs,
Y˚WTPeLs, TPeLs, Y•YeLs, A˚PVÙ[eLs GuT]Yt˚\ EtfllTÙolTRu
YÙ´XÙL HWÙ[UÙ] ÆYWeL˚[f —⁄dL Y•Æp Æ˚WYÙL‹m G∞RÙL‹m NÙRÙWQ
U≤RoLfidœd ·Pl ◊¨k’˘LÙs[ ÿ•Ÿm. AR]Ùp L¶Rj˚RŸm J⁄ ˘UÙØVÙLd
L⁄RXÙm.
˘T¨V’, £±V’, NUm, ·hPp G‡m L⁄j’dLfidœ <, >, =, + G‡m œ±¬”Ls
TVuT”jRlT”°u\].
G.”. 4 ‡˚PV‹m 5 ‡˚PV‹m ˘RÙ˚L 9 Gu\ L⁄j˚R 4+5=9 G]d L¶R ˘UÙØ´p
—⁄d° G›RXÙm.
10
L⁄j’l T¨UÙt\d L⁄Æ.
˛ L¶Rm J⁄ £\kR L⁄j’lT¨UÙt\ YØÿ˚\ Bœm.
˛ Y˚WTPeLs, AhPY˚QLs, TPeLs ˙TÙu\Yt±u YÙ´XÙLd L⁄j’lT¨UÙt\m
™L Æ˚WYÙL‹m G∞RÙL‹m S˚P˘T\ÿ•Ÿm.
˛ ßWh•V RLYpL˚[ Y˚WTPeLs, AhPY˚QLs G‡m œ±l◊Ls TVuT”jßl
Tß‹ ˘NnYRtœ E¨V ß\˚] Y[ojRp GuT’ L¶RdLpÆ YÙ´XÙL CVp°\’.
TÙPl◊jRLjßp Es[ ˘TÙ⁄jRUÙ] YÙnl◊L˚[d Li”©•j’, LXk’˚WVÙ• CkRf
£kR˚]˚V ˙UmT”jRXÙm.
Ußl¿”
$ ˘UÙØ YÙd°VeLfim ARtœf NUUÙ] L¶R YÙd°VeLfim G›ßV œ±l◊
$ L¶R A˚PVÙ[eL˚[l TVuT”jßV Lh”˚WL∞u ˘RÙœl◊. AYt±u
L¶RYÙnl◊L˚[d Li”©•jR œ±l◊
L¶Rm Au\ÙP YÙrd˚L´u J⁄ L⁄Æ
Au\ÙP YÙrd˚LŸPu ˘RÙPo◊˚PV ©Wf£˚]L˚[ ˙S¨”YRtœm ©Wf£˚]L˚[
A±‹lÈoYUÙLl TœjRÙWÙnk’m ©Wf£˚]j æo‹ YØÿ˚\L˚[d Li”©•j’m
AYt±u YÙ´XÙL U≤R ÿu˙]t\j˚R A˚PY’˙U L¶Rd LpÆ´u ÿd°V CXdœ.
CRtœd œZk˚R˚Vj RœßŸ˚PVY]ÙL BdœY˙R B£¨V¨u LP˚UVÙœm.
S˚Pÿ˚\ YÙrd˚L´p L¶Rm Ge˘LpXÙm TVuT”°\’? £kßj’l TÙodL‹m.
£X G”j’dLÙh”Ls C˙RÙ..................................
$ œ”mT YW‹ ˘NX‹jßhPm RVÙWÙdœRp
YW‹m NX‹m Jufl˙NWÙR TX „ZpLs Y⁄m˙TÙ’ YÙrd˚L Cu]p ™dLRÙL A˚U°\’.
CRtœ A±‹lÈoYUÙLj RVÙWÙdœm YW‹˘NX‹jßhPm AY£VUÙœm.
$ ˘TÙ⁄hL∞u ˘LÙ”dLp YÙeLp
$ TQlT¨UÙt\j’Pu ˘RÙPo◊ ˘LÙi”
11
$ ˘RÙؤPu ˘RÙPo◊ ˘LÙi” (˙Y[Ùi˚Uj ˘RÙØp, Y¶Lm N˚UVp)
CqYÙfl TX „ZpL∞p L¶Rm J⁄ ÿ˚\´p ApX’ Ut˘\Ù⁄ ÿ˚\´p TVuT”Y˚Rl
TÙodLXÙm.
Ußl¿”
Au\ÙP YÙrd˚L´p L¶Rm TVuT”m Tp˙Yfl „ZpLs APe°V LXk’˚WVÙPp
œ±l◊
Sm LQd∏”Ls RY\ÙL A˚UŸm˙TÙ’ YÙrd˚L´p ˙RÙpÆŸm LQd∏”Ls N¨VÙL
A˚UŸm˙TÙ’ YÙrd˚L´p ˘Yt±Ÿm ®Lr°u\] Gufl ˘NÙpYÙoLs Cp˚XVÙ?
4˛Bm Yœl◊ Ltfl ÿ•kR œZk˚R SÙpY˚Lf ˘NVpLs EhThP ©Wf£˚]Lfidœj
æo‹LÙ‘m ß\u ˘Tt±⁄dL ˙Yi”m. ˙TÏkßp TVQm ˘NnŸm˙TÙ’m, L˚P´≠⁄k’
˘TÙ⁄hLs YÙeœm˙TÙ’m, N˚UVp ˘NnŸm˙TÙ’m, L¶Rl ©Wf£˚]L˚[f Nkßd°˙\Ùm.
CjR˚LV ©Wf£˚]Lfidœj æo‹LÙ‘m ß\u ˘RÙPdL®˚Xl Ts∞ Yœl◊L∞≠⁄k˙R
Jq˘YÙ⁄ œZk˚RŸm ˘TflRp˙Yi”m.
Tp˙Yfl A[‹Ls (ø[m, G˚P, LÙs[[‹, SWm, SÙQVeLs)
Au\ÙP YÙrd˚L´u TX ©Wf£˚]Lfim A[‹LfiPu ˘RÙPo◊˚PV]YÙœm. AR]Ùp
AYt±u ™Lf £fl A[‹Ls œ±j’m A˚Y EhThP ©Wf£˚]Lfidœj æo‹LÙQ‹m
˘R¨kß⁄dL˙Yi”m. ˘TÙ⁄hL∞u ˘LÙ”dLp ˛ YÙeL¤Pu ˘RÙPo◊s[ ˙YflThP
A[‹Ls Y⁄°u\]. Aj’Pu SÙQVeLfim ÏTÙn ˙SÙh”Lfim Tt±
A±kß⁄dL˙Yi”m. A[‹L˚[ F°lTRtœm N¨VÙL A[lTRtœm E¨V ß\˚]
CkRdLhPjßp ˘Tt±⁄dL˙Yi”m.
L¶Rjßu A±YÙokR £kR˚] (Logic)
L¶Rjßp A±YÙokR A˚Ul◊ Es[’. G”j’dLÙhPÙL, GiQp GiLs
Y•Y˚UdLlTh”s[’ ‘Jufl’ Gu\ Gi¶u ¡i”m ¡i”m Y⁄m ·hPp YÙ´XÙL
Bœm. (AhPY˚Q´p RWlTh”s[ ·hPp ÿ˚\˚Vl TÙodL‹m). GqYÙfl A±YÙokR
ÿ˚\´p GiQp GiLs Y•Y˚UdLlTh”s[] Gufl LXk’˚WVÙP‹m. 9 Y˚W Es[
GiQp GiLfidœ A”jRRÙL, Tj’ (10) Y•Ym ˘TflYRu Ejßÿ˚\ G’? 0, 1, 2, ˛ ˛ ˛ ˛9
Y˚W Es[ Tj’ CXdLeL˚[l TVuT”j’m˙TÙ’ 9dœ A”jRRÙL YW˙Yi•V C⁄CXdL
Gi 10 Bœm. CjR˚LV A±YÙokR A˚Ul©˚]d L¶Rm ÿ›Yߤm LÙQ CV¤m.
1 = 1
1+1 = 2
2+1 = 3
3+1 = 4
4+1 = 5
12
L¶RdLpÆ J⁄Y¨u ◊jßNÙokR A˚]j’j ß\uLfim Y[of£ ˘TflYRtœj ’˚Q
◊¨°\’. A±‹lÈoYUÙLf £kßdœm ß\u GkR J⁄ U≤R¨u YÙrÆu ˘Yt±dœm ™L
AY£VUÙœm. L¶Rl©Wf£˚]L˚[l TœjRÙWÙnRp YÙ´XÙL‹m æo‹YÙ´XÙL‹m
Cjß\u ˙U¤m Y[of£ ˘Tfl°\’. Etfl˙SÙdœm ß\u, U]lTß‹jß\u,
◊’dL⁄j’dL˚[d Li”©•dœm ß\u, E⁄YÙdœm ß\u ˙TÙu\ ß\uL˚[Ÿm L¶Rd
LpÆ YÙ´XÙLl ˘T\CV¤m.
A±YÙokR £kR˚] Y[of£dœj ’˚Q◊¨Ÿm Lt\p ˘NVpTÙ”L˚[l
TÙPl◊jRLjß≠⁄k’ Li”©•dL‹m.
G.”. J⁄ ˘NqYLm Es[’. CkRf ˘NqYLjßu ø[m 10% ·P‹m ALXm 10% œ˚\V‹m
˘NnRÙp TWlT[Æp UÙt\m EiPÙœUÙ? Ei˘P≤p GjR˚] NR≈Rm? CjR˚LV GkR
J⁄ S˚Pÿ˚\l ©Wf£˚]Ÿm A±YÙokR £kR˚] Y[of£dœj ’˚Q◊¨°\’.
£kR˚]˚Vd L¶RUVUÙdœRp (Mathematisation of thought process)
˙R£V TÙP HtTÙ” Y˚XfNhPLjßu TœßVÙ] (NCF 2005) L¶Rm œ±j’s[
L⁄j’lTßÆp L¶Rjßu ÿd°V ˙SÙdLUÙLf ˘NÙpXlTh”s[’ L¶R UVUÙRp Gu\
œQjß˚] TiT”j’YRtœ Bœm. ˘R∞kR £kR˚], A•lT˚Pd˙LÙhTÙ”L∞≠⁄k’
A±‹lÈoYUÙ] ÿ•‹L˚[ ˙SÙd°f ˘Np¤m ß\u, ANÙRWQ L⁄j’dL˚[d ˚LVÙfim
ß\u, ©Wf£˚]L˚[ J›eLÙLl TœlTÙn‹ ˘NnY’Pu æo‹LÙ‘m ˘TÙfll◊Qo‹
GuT]Yt˚\ TiT”jR≠u TœßL[ÙL Gu.£.C.Bo.•. CkRl TßÆp œ±l©h”s[’.
C˚Y A˚]j’m ©\ TÙPeL∞u Lt\≠p Es[] G≤‡m ™L‹m ˘Y∞lT”Y]
L¶RjßXÙœm.
Ußl¿”
A±YÙokR £kR˚] Y[of£dœj ’˚Q◊¨Ÿm ◊ßoL∞u / Lt\p ˘NVpTÙ”L∞u
˘Y∞¬” / ˙NL¨l◊.
L¶Rj˚Rd LÙh£UVUÙdœRp
L¶Rf „ZpL˚[d LÙh£UVUÙdœY’Pu LÙh£UVUÙdœY]Yt˚\ Æ[dœYRtœm E¨V
ß\˙] œZk˚R´u ©Wf£˚]jæo‹jß\˚]j æoUÙ≤d°u\’. L¶Rd L⁄j’Lfidœm
L¶Rf £kR˚]Lfidœm Y•ÆVp Li˙QÙhPm A∞j’ ˘Y∞´”YRu YÙ´XÙL CkRd
LÙh£UVUÙdL≠u A±ÆVp ÿ˚\˚Vd œZk˚RLfim ◊¨k’˘LÙsYo. GkR J⁄ L¶Rd
L⁄jߤm ÿd°VUÙ]Yt˚\d LÙh£UVUÙdœR≠p TVuTÙ” GuT’ L⁄j’⁄YÙdLj˚R
G∞RÙdœYRÙœm. ©u]o CkRd LÙh£jR[jß≠⁄k’ L¶Rj˚R ARu
ÿd°Vj’Yj’P˙]˙V Htfld˘LÙs[‹m AYoL[Ùp CVp°\’. Lt\p ©Wf£˚]˚V
13
1+3+5+7+9 = 25 = 52
1+2+3+4+5+4+3+2+1 = 25 = 52
Ht˘\”lTß≠⁄k’ ©Wf£˚]j æoÆ˚] ÿ›˚UlT”jß HWÙ[UÙ] ˘R∞‹L∞u
˙RPpL˚[ ˙SÙd° AYoL∞p £kR˚]˚Vf ˘N¤j’YRtœ CjR˚LV LÙh£UVUÙdœRp
™L‹m Cu±V˚UVÙR’ Bœm.
J⁄ Lt\p A‡TYjßu GkR ®˚X´¤m LÙh£ A‡TYj˚Rd œZk˚Rdœ A∞dLXÙm.
L¶R A±‹⁄YÙdLjßtœ ™Lf ˙NokR J⁄ YØÿ˚\˙V TPUÙdœRp. RWlTh”s[
ÆYWeL˚[ AhPY˚QlT”j’Rp, AYtflPu ÆYWeL˚[l TœjRÙWÙnY’Pu
˘RÙPo◊s[ Y˚WTPm Y˚WRp GuT] YÙ´XÙL Æ[dœR¤m œZk˚Rdœ G∞RÙ°u\’.
TX £dLXÙ] L¶Rl ©Wf£˚]L˚[d œZk˚R Eh˘LÙs[ AYt±u TPY•YeLs YÙnl◊
A∞d°\’.
G.”: (1)
G.”. (2)
N’W GiL˚[ CqYÙfl LÙh£UVUÙdœRp YÙ´XÙL GjR˚LV ÿ•‹Lfidœ Yk’ ˙NWXÙm?
G.”. (3)
2 ××××× 3 = 3 ××××× 2
0 0 0 0 0 0
0 0 0 0 0 0
3 Cu 2 œ›dLs 2 Cu 3 œ›dLs
2 ××××× 3 3 ××××× 2
14
G.”. (4)
1/2 C˚]d LÙh£UVUÙdœRp.
G.”. (5)
3/4 C˚]d LÙh£UVUÙdœRp.
Cmÿ˚\´p ˘Yq˙Yfl L¶RdL⁄j’dL∞u LÙh£VÙdLm RVÙWÙdL‹m.
˘NVpTÙ”
JlT˚Pl◊:
1, 2, 3, 4 Yœl◊L∞u TÙPl◊jRLeL∞p L¶Rd L⁄j’dL∞u LÙh£VÙdLm
Li” ©•dL‹m.
L¶RdLpÆ´u ’p≠Vÿm ÷hTÿm
$ F£´p Ëp ˙LÙolTRtœ Es[ ˙TÙh• SPd°u\’. J⁄Y⁄dœ 20 Æ]Ù•Ls ≈Rm
Es[ 5 YÙnl◊Ls Es[]. CWUÙ 5 Cp 4 YÙnl◊L∞¤m N¨VÙL Ëp ˙LÙojRÙs.
£VÙUÙ‹dœ J⁄ RP˚Y Uh”˙U Ëp ˙LÙodL ÿ•kR’. Ëp ˙LÙolTßp ™L
÷hTj’Pu ˘NVXÙt±VYo GYo?
$ L•LÙWjßp ˙SWm 10 U¶ LØk’ 29 ®™PeLfim 30 Æ]Ù•Lfim B´tfl. CkR ˙SWj˚R
J⁄Yo TjR˚W B°ÆhP’ Guflm J⁄Yo Tj’U¶ 29 ®™PeLs Guflm Ut˘\Ù⁄Yo
Tj’ U¶ 29 ®™PeLs 15 Æ]Ù•Ls Guflm ·±]Ùp GYo ·±V’ ™Lf N¨VÙL C⁄dœm?
$ L¶RÆVp L⁄j’dL˚[ ÷hTUÙLl TœlTÙn‹ ˘Nn°\’. L¶RÆVp L⁄j’dL˚[j
’p≠VUÙL ˘Y∞´”°u\’. SUdœd °˚Pj’s[ A[‹, N¨VÙ] A[‹Pu GkR
A[Æu A⁄LÙ˚U´p ®t°u\’ GuTRtœ HtT ARu ’p≠Vm ·”°u\’.
J⁄Yo 100 ¡hPo K”YRtœ G”jR ˙SWj˚R 10 ®™PeLs Gufl ˘NÙpY˚R 10.25 Æ]Ù•Ls
Gufl ˘NÙp¤m˙TÙ˙R ™LfN¨VÙL Es[’.
˘NnŸm ˘NVpTÙ” ™Lf N¨VÙL C⁄jRp ˙Yi”m G] ™Ld LY]m ˘N¤j’Rp ApX’
·o˚UVÙLl TÙojRp GuT˙R ÷hTl TÙo˚Y. ÷hTlTÙo˚Y Ti◊ NÙokR’. ’p≠VUÙ]’
GuT’ AqYÙfl C⁄dL ˙Yi”m Gu±p˚X. ∏˙Z RWlTh”s[ TPm TÙodL‹m.
15
Ußl¿”
÷hTm, ’p≠Vm GuT]Yt˚\l ◊¨k’ ˘LÙs[d·•V „ZpLs G›ßV œ±l◊
LÙQd·•V (T⁄l˘TÙ⁄s) / LÙQ CVXÙR (÷i˘TÙ⁄s) CVp◊
$ L¶RdL⁄j’dL∞p ˘T⁄YÙ¨VÙ]˚Y ÷hTjRu˚U Es[˚YVÙœm. Im◊XuL[Ùp
◊¨k’ ˘LÙsYRtœd L•]UÙ] CjR˚LV L⁄j’dL˚[d LÙQd·•V]YÙL UÙt±
YœlT˚\´p ˘Y∞´PXÙm.
$ G.”. 5 Gu\ Gi˚Q A±ÿLlT”jR 5 LÙQd·•V ˘TÙ⁄hLfiPu (˙T]Ù, ◊jRLm,
ÿj’dLs ˙TÙu\˚Y) AR˚]j ˘RÙPo◊lT”jR ˙Yi”m. 5 ˘TÙ⁄hLfim ˘YqYfl
ÿ˚\L∞p œ›dL[ÙL BdLlT”°u\]. A˚Y Æ[dLlT”°u\].
$ 3 + 4 = 7 Gu\ ·hPp ˙LÙhTÙh˚Pl ◊¨k’˘LÙs[ Íufl UXoLs, SÙuœ UXoLs
GuT]Yt˚\f ˙Noj’ H› UXoLs G]d LÙQd·•V ÿ˚\´p Æ[dLlT”°\’.
$ 7 Gu\ Gi˚Q Gk˘RkR ÿ˚\L∞p Æ[dLXÙm?
$ ˘NqYLjßu TWlT[‹ = ø[m x ALXm G‡m L¶Rd ˙LÙhTÙh”f ˘NVpTÙ”
YÙ´XÙL‹m LÙh£lT”j’Rp YÙ´XÙL‹m (˘UÙjR AXœ N’WeL∞u Gi¶d˚L)
EQojRlT”°u\’.
C˚Rl˙TÙufl Tp˙Yfl ˘NqYLeL˚[ AXœ N’WeL[ÙL Y˚LlT”jß ˘NqYLjßu
TWlT[‹ = ø[m x ALXm Gu\ ˙LÙhTÙh•˚] A˚P°˙\Ùm.
Ußl¿”
LÙQCVXÙR L⁄j’dL˚[d LÙQd·•V]YÙL UÙt± ˘Y∞´”m ˘NVpTÙ”Ls
APe°V A±d˚L RVÙ¨dL‹m.
©\ TÙPl˘TÙ⁄hLfiPu L¶Rjßu ˘RÙPo◊
GkR J⁄ TÙPl˘TÙ⁄˚[d LtTߤm L¶Rm ÿd°Vl Teœ Y°d°\’. CVt©Vp,
˙Yß´Vp, ˘TÙ⁄∞Vp, ◊Æ´Vp, ˙NÙßPÆVp G‡m TÙPlTœßL˚[d LtTߤm ÿd°V
CPm L¶Rjßtœ Ei”.
Tp˙Yfl ˘UÙØLs, L˚XdLpÆ, EPtT´t£d LpÆ, ˘NVp A‡TYdLpÆ GuT]
LtTߤm L¶Rj˚Rl TVuT”j’°u\]o.
5 ˘N.¡
4 ˘N.¡
16
G.”. Ko Ek’ Tk’ Æ˚[VÙhPWeL Volyball court E⁄YÙdLm (©\ G”j’dLÙh”L˚[Ÿm
˙Noj’d˘LÙi” LXk’˚WVÙP‹m)
1. L¶RÆV¤m YWXÙflm
YWXÙfl GuT’ LPkRLÙX ®Lr‹L˚[ Y¨˚NlT”jß Ko J›eœÿ˚\´p LtTRÙœm. Cßp
®Lr‹L∞u Æ[dLeL˚[f N¨VÙL G”j’˚WdL L¶RÆVp A±‹ ™L AY£VUÙœm.
°.©.326˛Cp UÙ≈Wo A˘XdNÙiPo CkßVÙ˚Y Bd°W™jRÙo GuT’ YWXÙfl. GjR˚]
Bi”Lfidœ ÿu]o AYo CkßVÙ˚Y Bd°W™jRÙo G] A±V˙Yi”˘U≤p L¶R
A±ÆV≠u ERÆ AY£Vm ApXYÙ. Ce˙L YWXÙfldœm L¶Rjßtœm C˚P˙V Es[
˘RÙPo◊ LÙi©dLlT”°\’. YWXÙfl NÙokR GkR ÿd°Vl TßYÙ´‡m ˙SWm, LÙXm Tt±V
N¨VÙ] œ±l◊Pu ˘NÙpXlTPÙÆ•p AkRlTß‹ ˘TÙ⁄[t\RÙ°Æ”m. ÿu]o
RWlTh”s[ G”j’dLÙh•p A˘XdNÙiPo Gl˙TÙ’ CkßVÙ˚Y Bd°W™jRÙo Gufl
Uh”˙U YWXÙfl SUdœf ˘NÙp≠jR⁄°u\’. ARu ©u]o Cl˙TÙ’ GjR˚] Bi”Ls
LØk’ÆhP] Gufl Li”©•dL SÙm L¶RÆV˚X A‘L˙Yi•V’ AY£VUÙœm.
C˚Rl˙TÙufl ˙SWd˙LÙ”, Jq˘YÙ⁄ AWNlTWmT˚W´u Bh£dLÙXm ˙TÙu\ YWXÙtflf
˘NnßL˚[l Tt±V A±˚Y ÿ›˚UVÙLl ˘T\ UÙQYoLs L¶RÆVp
A±˚Yl˘Tt±⁄dL˙Yi”m. Ce˙L YWXÙfldœm L¶Rjßtœm C˚P˙V Es[ ˘S⁄e°V
˘RÙPo˙T LÙhPlT”°\’.
2. L¶RÆV¤m ◊Æ´V¤m
◊Æ´Vp GuT’ ◊Æ˚Vl Tt±Ÿm ©WTgNj˚Rl Tt±Ÿm Es[ A±Y´p NÙokR’m,
L¶RÆVp NÙokR’UÙ] Æ[dLm Uh”UÙœm. ◊Æ´Vp LpÆdœd L¶R A±ÆVp A±‹
™L AY£VUÙœm. CRtœl TX G”j’dLÙh”L˚[d LÙhP CV¤m. È™´u A[‹, T⁄Uu
TLp ˛ CW‹ Y⁄˚L, „¨V°WLQm, NkßW°WLQm, LPpø˙WÙhPeLs, LÙX®˚X UÙt\m,
U˚ZdLÙXm ˙TÙu\ ◊Æ´V≠u TX ®Lr‹L˚[Ÿm œ±j’s[ N¨VÙ] A±˚Yl ˘T\d
L¶RÆVp ’˚Q◊¨°\’. ◊ÆUÙt\eLs Tt±V ™Lf N¨VÙ] L¶l◊ UÙQY⁄dœ
HtTP˙Yi”˘U≤p AYo L¶RÆVp A±‹ ˘Tt±⁄dL˙Yi”m. C˚Y A˚]j’m
◊Æ´V¤dœm L¶RÆV¤dœUÙ] ˘S⁄e°V ˘RÙPo˚T˙V LÙh”°u\].
3. L¶RÆV¤m E[ÆV¤m.
E[ÆVp EQo‹NÙo A±ÆV¤Pu ˘RÙPo◊˚PV’ G≤‡m A[‹LfiP‡m A’
Cl˙TÙ’ ™Lj ˘RÙPo◊˚PVRÙL A˚Uk’s[’. U≤R¨u ◊jßd·o˚U˚V A[k’
LQd°P E[ÆVp YØÿ˚\ Li”©•j’s[’. CRtœ E¨V J⁄ L¶R YÙd°Vm
L¶RÆVp NÙok’ E⁄YÙdLlTh”s[’.
IQ =M.A 100
C.A
×
GuT˙R AkRd L¶R YÙd°Vm. C˚Rj ˘RÙPok’
U≤R¨u A˚]j’j ß\uL˚[Ÿm A[k’ LQd°P E[ÆVp RVÙWÙ]’. C’
E[ÆV˚Xd L¶RÆV¤Pu ™L C˚QjR’. ˙PÙl˙TÙ[Ù¥dLp ˚NLÙXÙ¥ E[ÆV≠u
Cu˚\V J⁄ S≈]d °˚[j’˚\VÙœm. C’ L¶RÆVp °˚[j’˚\VÙ]
˙PÙl˙TÙ[¥ŸPu ˘RÙPo◊ ˘LÙi”s[’. C’ GR˚]d œ±l©”°u\’? E[ÆV¤dœm
L¶RÆV¤dœm Es[ ˘RÙPo˚T ApXYÙ.
17
4. L¶RÆV¤m CVt©V¤m
CVt©V¤m L¶RÆV¤m Ju˙\Ù˘PÙufl C˚QkR˚YVÙœm. CVt©V≠u ˘RÙPdL®˚X
A[‹Ls L¶RÆV≠u J⁄ °˚[j’˚\VÙL˙Y Es[’. L¶RÆVp A±Æu±
CVt©Vp LpÆ LtL CVXÙ’ Gu˙\ ·\XÙm. CVt©Vp LpÆ´u A•lT˚PVÙ]
TÙPl˘TÙ⁄˙[ L¶RÆVp. CVt©V≠p TXS˚Pÿ˚\l ©Wf£˚]Lfidœj æo‹Ls LÙQ
Es[’. CYt±u æo‹Lfidœd L¶RÆVp A±‹ ™L AY£Vm. CVt©V≠p Es[
A‘dL⁄ CVt©Vp CVt©V˚Xd L¶RÆV¤Pu ™L‹m C˚Qj’s[’. A‘dL⁄
CVt©Vp Rt˙TÙ’ L¶RÆV˚X CVt©V≠u J⁄ ˘NVpTÙh” UiPXÙUÙL UÙt±Ÿs[’.
C˚Rl˙TÙufl ˙Yß´¤dœm L¶RÆV¤Pu ˘RÙPo◊ Es[’. ˙Yß´V≠u
A‘˙Yß´Vp L¨U˙Yß´Vp ˙TÙu\ °˚[j’˚\L∞p L¶RÆVp ˙LÙhTÙ”L∞u
˘NVpTÙ”Ls AY£VUÙœm. Cmÿ˚\Ls A˚]jߤm CVt©V¤Pu L¶RÆVp
™Lj˘RÙPo◊˚PVRÙLd LÙQÿ•°u\’.
5. L¶RÆV¤m „Z≠V¤m.
L¶RÆV¤Pu GkRj ˘RÙPo◊Ut\RÙL˙Y „Z≠Vp TÙodLlTh•⁄kR’. B]Ùp,
E´odL¶Rm (Bio mathematics) G‡m Ko A±ÆVp ’˚\ Y[of£V˚PkR ©u]o
Cf£kR˚]dœ UÙt\m Yk’s[’. L¶RÆVp YÙnlTÙ”Ls YÙ´XÙL‹m NUuTÙ”Ls
YÙ´XÙL‹m E´W˚Ul◊L˚[l ˘TÙfljRY˚W ÷hTUÙLd LtL CV¤m Gufl Cl˙TÙ’
˘R∞YÙdLlTh”s[’. UW◊d·flLs Y⁄eLÙXjR˚Xÿ˚\´]¨Pm LÙQlT”YRu
˘RÙPoTÙL ‘˘UuPp’ SPjßV Bn‹ ®fl‹Y’ „Z≠V¤dœm L¶RÆV¤dœm Es[
˘RÙPo©˚]VÙœm. CqYÙfl S≈]dLÙXjßp L¶RÆV¤m „Z≠V¤m ˘LÙi”s[
˘RÙPo◊ ™L Y¤YÙ]RÙL E⁄l˘Ttfl Y⁄°\’.
6. L¶RÆV¤m˘TÙ⁄[ÙRWÆV¤m
˘TÙ⁄[ÙRÙWd LpÆ´¤m UÙQYoLs L¶RÆVp A±Æ˚]Ÿm ARu˘UÙØ˚VŸm
TVuT”j’°u\]o. TÙ⁄[ÙRÙW´V≠p EhT”Y] YW‹ NX‹Ls, Y¶Lm TÙu\Yt±p
APe°Ÿs[ Tp˙Yfl ·flL∞p Es[ ˘RÙPo◊L[Ùœm. Tp˙Yfl ÷Lo‹l˘TÙ⁄hL∞u
EtTjß, ÆtT˚], YÙeœRp, T¨UÙt\m ˙TÙu\˚Y L¶RÆV≠u ERÆŸP˙]
J›eœT”jRlT”°u\]. Tp˙Yfl SÙ”Lfidœ C˚P˙V Es[ SÙQVl T¨UÙt\d
LhP˚Ul©˚] æoUÙ≤dœm LÙW¶ L¶RÆVp Bœm. Ye°Ls, LÙl¿h” ®flY]eLs
˙TÙu\ A˚]j’ ®ß®flY]eLfidœm L¶R Y[m °˚PdLl˘Tfl°\’. CR]Ùp
˘TÙ⁄[ÙRÙWÆV¤dœm L¶RÆV¤Pu ˘S⁄e°V ˘RÙPo◊ Es[’ G]l ◊¨k’˘LÙs[XÙm.
7. ˘UÙØ, KÆVm, C˚N ˙TÙu\]
˘UÙØ, KÆVm, C˚N, SP]m, EPtT´t£d LpÆ, ˘NVpTÙh” A‡TYm ˙TÙu\
A˚]jߤm L¶RÆVp ˘RÙPo◊ ˘LÙi”s[’. ˘UÙØdLpÆ´u TœßVÙ] CXdLQm
Lt\≠p L¶RÆV≠u ˘NpYÙd°˚] SmUÙp LÙQCV¤m. ˘UÙØ˚Vd ˚LVÙfim ˙TÙ’
Ge˙L LÙt◊s∞, ÿtfll◊s∞ ˙TÙu\] AY£Vm G]d LQd°P L¶RÆVp ER‹°\’.
L⁄j’ ˘Y∞¬h•tœj ˘R∞‹ RW‹m L¶RÆVp LpÆ ER‹°\’ G] ÷hTlTÙo˚Y
18
YÙ´XÙL‹m ◊¨k’˘LÙs[XÙm. LÆ˚Rdœ K˚N °˚PlT’ G›j’dL˚[l LQd°h”l
TVuT”j’m˙TÙRÙœm. ARÙY’ VÙl◊ N¨VÙL A˚UŸm˙TÙRÙœm. VÙl◊ GuT’
Ei˚U´p L¶RÆVp TVuTÙ˙P Bœm. C˚Rl˙TÙufl ©\’˚\LfiPu L¶Rjßtœ
Es[ ˘RÙPo˚Td Li”©•dL‹m.
Ußl¿”
Jq˘YÙ⁄ ’˚\dœm L¶Rj’Pu Es[ ˘RÙPo˚T Æ[dœm L⁄j’ Y˚WTPm
L¶RÆVp LpÆ YÙ´XÙLl T\lT”m Ußl◊Ls
L¶Rd LpÆ´u YÙ´XÙL UÙQY¨Pm Tp˙Yfl Ußl◊Lfim U]lTÙu˚ULfim
E⁄YÙdLm ˘Tfl°u\]. AYt±p £XYt˚\ A±k’˘LÙs˙YÙm.
$ S˚Pÿ˚\NÙo Ußl◊
$ ◊jßd·o˚U
$ U]dLh”lTÙ”
$ AZ°Vp EQo‹
$ ˙R£Vm ˛ EXLj˙R£Vm
$ ˘RÙØp NÙo Ußl◊
$ TiTÙ”
$ NÍLm
S˚Pÿ˚\NÙo Ußl◊
Au\ÙP YÙrd˚Ll ©Wf£˚]L˚[d ˚LVÙsYRtœd œZk˚RL˚[j Rœß E˚PVYWÙL
E⁄YÙdL ˙Yi”m GuT˙R L¶RdLpÆ´u ÿd°V CXdœ. YœlT˚\L∞≠⁄k’ ˘Tt\
A‡TYeL˚[j ß]N¨ YÙrd˚L´p S˚Pÿ˚\lT”j’YRtœ E¨V Ru]m©d˚L˚Vd
L¶RdLpÆ YÙ´XÙL˙Y ˘T\˙Yi”m. S˚Pÿ˚\NÙo Ußl◊L˚[l ˘TflYRtœj
˘RÙPo◊˚PV £X „ZpLs ∏˙Z ˘LÙ”dLlTh”s[].
$ Ye°f ˘NVpTÙ”L∞p D”T”Rp
$ L˚P´≠⁄k’ ˘TÙ⁄hLs YÙeœRp
$ LhPPm Lh”Rp
$ A[‹Ls Tt±V LQd∏”
$ ˙U¤m TX G”j’dLÙh”L˚[d Li”©•dL‹m.
◊jßd·o˚U NÙo Ußl◊
œZk˚R´u £kR˚]j ß\‡dœ FhPm A∞dœm J⁄ TÙPl˘TÙ⁄˙[ L¶Rm. EPpNÙo
Y[of£dœ ER‹m Y˚L´p ∏˙Z ˘NÙpXlTh”s[ £X £kR˚]jß\uL˚[d
œZk˚RL∞Pm Y[Wf˘NnVXÙm.
19
$ A±‹f £kR˚]˚V ƨY˚PVf ˘NnRp
$ ÆU¨N]m NÙo £kR˚]
$ Eh°W°dL‹m LÙWQLÙ¨Vj ˘RÙPo˚Td LÙiTRtœm E¨V ß\u
CkRÿ˚\´p ◊jßd·o˚U NÙo Ußl◊ L¶Rm Lt\p YÙ´XÙL Y[⁄m.
U]dLh”lTÙ”NÙo Ußl◊
L¶RÆVp Ltœm˙TÙ’ UÙQYoL∞Pm £X R≤lThP £kR˚]ÿ˚\Lfim
U]lTÙu˚ULfim TZdLYZdLeLfim Y[o°u\]. CYt±u TX]ÙL UÙQY¨Pm
U]dLh”lTÙ” EiPÙ°\’. CkR U]dLh”lTÙ” YÙrd˚L´u A˚]j’j ’˚\L∞¤m
Es[ œZk˚R´u ˘NVpTÙh”dœ ERÆ◊¨°u\’. A±YÙokR £kR˚]jß\u, A±ÆVp
U]lTÙu˚U ˙TÙu\] Y[Wf˘Nn’ ◊jßd·o˚U˚V ƨ‹T”jR L¶RÆVp LpÆ
’˚Q◊¨°u\’. Yh•ŸPu ˘RÙPo◊s[ TÙPm, Au\ÙP YÙrd˚L´p ÆWVm ˘NnVÙ’
YÙ›m TZdLj˚Rd L˚P©•jRp, ˘T⁄mYh•dœl TQm LPu YÙeLÙß⁄jRp,
LPuYÙe°]Ùp N¨VÙLl TQj˚Rj ß⁄l©f ˘N¤j’Rp G‡m YZdLeL˚[l
©uTtflYRu YÙ´XÙL U]dLh”lTÙ” NÙo Ußl˚Td œZk˚RVÙp ˘T\ CVp°u\’.
L¶RÆVp LpÆ TX SpX TZdLYZdLeL˚[ E⁄YÙdœYRtœj ’˚Q◊¨°u\’.
A±ÆVp NÙo £kR˚], ÷hTl TÙo˚Y, —⁄eLd·flm ß\u, Lh”lTÙ” ˙TÙu\] AYtfls
£X. U]dLh”lTÙ” NÙo Ußl◊ L¶Rjßp Es[PdLjßp ApXÙUp, LtTRu ÿ›˚U˚Vf
NÙokß⁄d°u\’ GuT˚R Eh˘LÙi˙P L¶R B£¨Vo ˘NVpTP˙Yi”m.
AZœQo‹ NÙo Ußl◊
GiT¨UÙt\jßp Es[ K˚NSVm, Y•ÆVp Y•YeL∞u AZœ, Y•˘YÙl◊˚ULs
GuT]Yt±tœ J⁄ R≤lThP AZœ Ei”. œZk˚RL∞u Es[eL∞p J˙W ˙SWjßp
ÆVl˚TŸm B]kRj˚RŸm ˙RÙtflÆdL CR]Ùp CV¤m.
C˚Rl˙TÙufls[ Gi Y¨˚NLs YÙ´XÙL‹m Y•ÆVp Y•YeL[Ùp ˘Nn°u\ AZLÙ]
Y•YUÙߨLs YÙ´XÙL‹m œZk˚RL∞u AZ°Vp EQo‹j ß\˚]l ˘T⁄dL CV¤m.
œZk˚RL∞u T˚PlTÙfi˚U˚V Y[od°u\ ÿ˚\´XÙ] Ëp A˚Ul◊Ls,
Æ[dLlTPeLs, Gi A˚Ul◊Ls, TP A˚Ul◊Ls ˙TÙu\] E⁄YÙdœYRtœ E¨V
YÙnl◊L˚[ A∞dL˙Yi”m.
˙R£Vm ˛ EXL ˙R£Vm NÙo Ußl◊
˙R£V EQo‹m EXL ˙R£Vm NÙo £kR˚]l ˙TÙdœm Y[oYRtœd L¶RÆVp LpÆ
TVuT”m. CkßVÙÆu ÿtLÙXdL¶R A±OoL˚[d œ±j’m AYoLs L¶RÆV¤dœ
A∞j’s[ Su˘LÙ˚PL˚[l Tt±Ÿm œZk˚RLs A±kß⁄dL˙Yi”m. TjR•UÙ]
Gi‘⁄ TZdLm, ÈwVm G‡m L⁄j’ ÿRXÙL HWÙ[UÙ] Li”©•l◊Ls CkßV¨u
Su˘LÙ˚PL[Ùœm. CkßVo A∞j’s[ ©\ Su˘LÙ˚PLs Tt±V ˙RP˚X SPjRXÙm.
BoVThPo, YWÙL™°Wo, TÙvLWÙfNÙ¨VÙo, CWÙUÙ‡_o B°˙VÙo A±ÆV≠u
UÙ˙U˚RL[ÙLd L⁄RlT”°u\]o. CdL⁄j’dL˚[l ◊¨k’˘LÙsfim J⁄ UÙQYo Sm
SÙh˚P Gi¶l ˘T⁄™Rm ˘LÙs[XÙm. ˙R£Vm NÙokR CkR EQo‹Pu EXL ˙R£Vm
20
NÙo £kR˚]l˙TÙdœ Y[oYRtœd L¶RÆVp LpÆ ’˚Q◊¨°u\’. ˚TR˙LÙWv,
BodL™ºv, Îd∞h G‡m L¶R A±OoLs A∞j’s[ Su˘LÙ˚PL∞u ULjRÙ]
TVuTÙh˚P SÙÿm A‡TÆd°˙\Ùm Gu\ Ko ELXj˙R£V EQoÆ˚]d œZk˚RL∞Pm
Y[Wf ˘NnV CV¤m. AkR UÙ˙U˚RL˚[l Tt± A±k’ ˘LÙi” AYoL˚[ Ußj’m
YWXÙtfld Li˙QÙhPj˚R EhL˘LÙi”m AYt±u YÙ´XÙL B£¨VlT´t£
UÙQYoL˚[ YWXÙt±u ÿd°Vj’Yj˚R ˙SÙd° A˚Zj’f ˘NpX‹m ˙Yi”m.
˘RÙØpNÙo Ußl◊
GßoLÙXjßp GkRj ˘RÙØ˚XŸm Htfld˘LÙsfim U]lTÙu˚U œZk˚RL∞Pm
Y[oYRtœd L¶RdLpÆ ER‹°\’. GkR J⁄ ˘RÙØ≠¤m D”T”m J⁄ STo ™Lf N¨VÙLl
T¶ ˘NnRp, LÙX EQo‹Pu Æ˚WYÙL‹m ™LfN¨VÙL‹m T¶˘NnRp GuT˙R
˘RÙØpNÙo Ußl◊. C’ L¶Rm Lt\p YÙ´XÙL˙Y ˘T\lT”°\’. ’p≠Vm, ÷hTm, Æ˚W‹,
ßhP™h”f ˘NVXÙtflRp G‡m Ti◊L˚[ J⁄ L¶R B£¨Vo L¶R YœlT˚\´p
˘NVpT”j’Y’Pu J⁄ SpX ÿuUÙߨ STWÙL B£¨Vo UÙflYRu YÙ´XÙLd
œZk˚RL∞Pm RÙØpNÙo Ußl◊ Y[o°\’. GßoLÙXjßp GkRj RÙØ˚X Htfld˘LÙsfim
˙TÙ’m ÿuUÙߨVÙ] B£¨Vo Tt±V ®˚]‹ œZk˚RL∞Pm AØVÙß⁄lT’Pu A’
œZk˚R Teœ˘Tflm ˘RÙØp ’˚\´p ™L Ht\m ˘TflYRtœm E¨V RœßŸ˚PVY]ÙL
Bdœ°\’.
TiTÙ” NÙo Ußl◊
L¶RÆVp LtTRÙp œZk˚RL∞u TiTÙh”jRWm EVo°\’ GuT˙R ARu TiTÙ” NÙo
UßlTÙLd L⁄RlT”°\’. AZ°V˚X CW£dœm ß\˚] Y[ojRp, Kn‹˙SWl
˘TÙ›’˙TÙdœ G]d L¶RÆVp ˘RÙPoTÙ] ˘NVpTÙ”L˚[ Htfld˘LÙsfiRp,
TÙWÙhPjRdL TZdLYZdLeL˚[ E⁄YÙdœRp G‡m ÿ˚\Ls YÙ´XÙL˙Y TQTÙh”
Y[of£ EiPÙ°\’. B£¨V¨u N¨VÙ] A‘œÿ˚\ YÙ´XÙLd œZk˚RL∞Pm
L¶RÆVp LtTßp ™œkR BoYm ˙RÙu\f ˘NnVXÙm. C’ ®˚\˙Y±]Ùp Ko Kn‹˙SWl
˘TÙ›’˙TÙdœ G]d L¶Rm ˘RÙPoTÙ] ˘NVpTÙ”L∞p D”TP BoYm LÙh”Yo. UkßWf
˘NqYLeL∞u E⁄YÙdLm, L¶RÆVp EhT”m Æ˚[VÙPpLs, BoYm Fh”m
◊ßoLfim Æ˚[VÙh”Lfim A∞jRp CYt±u YÙ´XÙL‹m, L¶RÆVp LpÆ˚V
SpX˙RÙo Jn‹LÙXl ˘TÙ›’˙TÙdLÙL UÙt\XÙm. Y•ÆVp Y•Y UÙߨLs E⁄YÙd°
®\m æh”Rp, SpX UÙߨLs A∞j’ YNßdœ HtT AYt˚\ E⁄YÙdLd ·flRp, L¶RÆVp
YWXÙt±u ÿd°V A±OoL∞u YÙrd˚L YWXÙflL˚[Ÿm, —˚YÎh”m ®Lr‹L˚[Ÿm
UÙQYoLfidœ A±ÿLm ˘NnRp, A˚Y EhT”m ËpLs Tt±V œ±l©˚] A∞j’
YÙ£lTRtœ E¨V YÙnl◊ A∞jRp GuT] TiTÙh” Y[of£dœj ’˚Q◊¨Ÿm.
L¶RÆVp YWXÙt±u ©u]¶˚Vl ◊¨k’˘LÙi” L¶Rm Ltœm J⁄ UÙQY¨Pm
TiTÙ”NÙo Ußl◊ Y[⁄m GuTßp IV™p˚X.
NÍLm NÙo Ußl◊
J⁄ NÍL E´¨ Gu\ ®˚X´p J⁄Y¨Pm LÙQlT”m U]lTÙu˚UL∞p £X˙R
·h”\‹Qo‹, _]SÙVL EQo‹, R˚X˚Ul Ti◊Qo‹, AfN™u± ˘NVXÙtflm EQo‹,
21
˙So˚U, ˙SWm RY\Ù˚U, L⁄j’lT¨UÙt\j ß\u GuT]. CYt˚\d L¶Rm LtTRu
YÙ´XÙL‹m B£¨V¨u TVu ™dL D”TÙ”Ls YÙ´XÙL‹m J⁄ œZk˚R ˘TflY’Pu
GßoLÙXjßp, NÍLj’dœl TVuT”m ÿ˚\´p œZk˚Rdœ CYt˚\l TVuT”jR‹m
CV¤m. CRu YÙ´XÙL NÍLjßu Ko AeLjß]WÙL NÍLj’Pu C˚Qk’ SpX˙RÙo
Bfi˚Udœ E˚P˚UVÙ[WÙL‹m Y[W J⁄ UÙQYWÙp CV¤m.
Jq˘YÙu±tœm ˘TÙ⁄jRUÙ] G”j’dLÙh”Ls Li”©•j’ L⁄jRWe°p ˘Y∞´P‹m.
Ußl¿”
L¶RdLt\p Ußl◊Lfim U]lTÙu˚ULfim L⁄jRRWeLm.
(˘Y∞¬”, A±d˚L.)
22
AXœ ˛ 2
L¶RÆV≠u Y[of£Ÿm ÿu˙]t\ÿm
ÿu‡˚W
˘RÙPdL ®˚X Yœl◊L∞p L¶Rm Lt©dœm B£¨VoLfidœ AkRl TÙPjßp
ÿ›˚UVÙ] ◊¨Rp ˙Yi”m. TÙPjßp Es[ A±‹ Lt\p ˛ Lt©jRp ˘NVpÿ˚\˚V
N¨VÙL‹m TV‡s[YRÙL‹m ˘NnYRtœ ™L‹m ER‹°\’. L¶RÆV≠u Y[of£˚VŸm,
˙UmTÙh˚PŸm œ±j’l ◊¨k’˘LÙsYRtœ ERÆ◊¨Ÿm L⁄j’L˙[ CkR AX°p
LXk’˚WVÙPl Th”s[].
Lt\p A˚P‹Ls ÿd°VdL⁄j’dLs
a) L¶R YWXÙtfld LpÆ´u AY£Vÿm
ÿd°Vj’Yÿm
$ Tp˙Yfl LÙXLhPeL∞u L¶R b) L¶R YWXÙfl
Y[of£Ÿm ÿu˙]t\ÿm A±k’ 1) GiL∞u ˙RÙt\m
˘LÙsfiRp $ ˘Yq˙Yfl TjR•UÙ] Gi‘⁄
TZdLeLs
$ ˙WÙUu, CkßV ˛ A˙W©V
$ L¶R A±OoL∞u YÙrd˚L Nominal, Ordinal
YWXÙflLfim AYoL∞u 2) GiL¶R YWXÙfl
Su˘LÙ˚PL˚[Ÿm A±k’ 3) CVtL¶R YWXÙfl
˘LÙi” ˘Y∞´”Rp 4) Y•Y´p YWXÙfl
c) CkßVd L¶R YWXÙfl
d) L¶R A±OoLs
$ ˙LW[ A±OoLs (Ne°WÙU UÙRYu,
øXLiP ˙NÙUVÙ¥, ˙NÙUÙߨ,
TW˙UvYWo)
$ ©\ CkßV A±OoLs (B¨VThPo,
TÙvLWÙfNÙ¨VÙo CWÙUÙ‡_o, LÙ©˙WdLo.
$ AVpSÙh” A±OoLs (Îd≠h,
˚TR˙LÙWv, ˙LÙv, ©l˙]Ùf£
RdLÙoj˘R
23
L¶R YWXÙtfl Lt\≠u Cu±V˚UVÙ˚U
Tp˙Yfl LÙXLhPeL∞p L¶RÆV≠u Y[of£˚V ÿ›˚UVÙL A±YRtœm Jq˘YÙ⁄
’˚\´¤m Es[ Li”©•l◊L˚[l ◊¨k’˘LÙs[‹m YWXÙfl Lt\p SUdœj
’˚Q◊¨°u\’. L¶RÆVp Li”©•l◊L˚[ Y¨˚NVÙLl ◊¨k’˘LÙs[‹m Cj’˚\´p
Jq˘YÙ⁄ A±OoLfim A∞j’s[ TeœL˚[lTt± ˘R¨k’˘LÙs[‹m YWXÙfl
A±kß⁄dL˙Yi”m. Jq˘YÙ⁄ SÙh•¤m L¶RÆVp Bn‹Lfim Li”©•l◊Lfim
®Lrk’s[˚U Tt± YWXÙtfl ËpLs SUdœj ˘R¨Æd°u\].
˙LW[ÙƤm CkßVÙƤm CkßVÙÆtœ ˘Y∞˙VŸm Es[ TX L¶R A±OoL∞u
˘NVpTÙ”L∞u TV]ÙL Cufl SÙm ˘Tt\ L¶RÆVp œ±j’m AYoL∞u LÙXLhPeLs,
˘NVpTÙ”Ls, YÙrd˚Ll ©u]¶Ls CYt˚\d œ±j’m ◊ßV R˚Xÿ˚\´]o
A±kß⁄dL˙Yi”m. Jq˘YÙ⁄ LÙXLhPjߤm L¶Rjßp A˚Pk’s[ Y[of£ ARtœ
ÿu]o Li”©•jR L⁄j’dL∞p ®u\Ùœm. ®ÎhPu ·±V ˘NÙtLs Ceœd LY]jßtœ
E¨V]. " "SÙu H˙R‡m EeLfidLÙLf ˘Nnß⁄d°˙\u G≤p ARtœd
LÙWQUÙ]YoLs Gu ÿu˙]ÙoLs Bœm. AYoL∞u ˙RÙ∞p H± SÙu ®t°˙\u''
˛ ®ÎhP≤u CkRl ©u]¶ ◊’ R˚X´]⁄dœ FdLm A∞lT’Pu AYoL∞u ◊ßV
Li”©•l◊Lfidœm L¶RÆV≠u Y[of£dœm ER‹°u\].
L¶Rjßu Y[of£ Jq˘YÙ⁄ LÙXLhPjßu ˙R˚Ydœ HtT ÿu˙]±Ÿs[’. G°lßu
˘Ys[l˘T⁄dœm AR]Ùp HtT”m ©Wf£˚]Lfim Y•ÆVp Y[of£dœ ERÆ].
A˚Rl˙TÙufl A±ÆVp, ◊Æ´Vp GuT]Yt±u Li”©•l◊L∞¤m L¶Rjßtœ Teœ
Es[’. A±ÆV≠u Y[of£ L¶Rjßu Y[of£dœm ’˚Q◊¨k’s[’.
L¶RÆV≠u YWXÙfl
L¶RÆV¤dœ UÙ≤PlTiTÙh”dœ E¨V LÙXlTZdLm Es[’. CkßVÙ, ∫]Ù, G°l’,
DWÙu, °˙WdLm B°V SÙ”L∞p RÙu ÿR≠p L¶RÆVp Y•Ym ˘Tt\’.
CkßVÙÆp L¶R ËpLs ˘NnŸs Y•Æp G›RlTh•⁄kR]. U]lTÙPm ˘NnV G∞RÙL
C⁄kR˙R ˘NnŸs Y•Yjßu TVuTÙ”. B]Ùp, ™Lf ˘N±YÙL ™Ld œ˚\kR
G›j’dLfim NÙtLfim TVuT”jß˙V NnŸhLs T˚PdLlTh•⁄kR]. øi”˙TÙR˚Xj
RÆodLXÙm Gu\ ˙Uu˚U C⁄l©‡m L⁄j’dL˚[ ÿ›˚UVÙLl ◊¨k’˘LÙs[ CVXÙR
J⁄ ˘R∞Æu˚U˚V CRu J⁄ œ˚\TÙPÙLf —h•dLÙhPXÙm. CqYÙfl
—⁄dLlTh•⁄kR˚UVÙp L¶RÆßL∞u NÙuflLs TX ˙Y˚[L∞¤m
◊\dL¶dLlTh”s[]. NÙu˙\ÙoLs ◊¨k’ ˘LÙs[h”m Gu\ GiQ˙U A±OoL∞Pm
LÙQlThP’. B]Ùp, ©tLÙXjßp Yk’s[ Æ[dL E˚W ËpL∞p ˘R∞‹Ls
LÙQÿ•°u\].
Ti˚PV TÙ©˙XÙ≤Vo Tj˚R A•lT˚PVÙLd ˘LÙiP GiL˚[Ÿm 60 I
A•lT˚PVÙLd ˘LÙiP GiL˚[Ÿm TVuT”jß´⁄kR]o. °.ÿ.2300dœm 1600dœm
C˚P˙V T˚PdLlThP C⁄ ËpL∞p 60 Y˚WŸs[ YodL GiLs LÙQlT”°u\]. 60I
A•lT˚PVÙLd ˘LÙiP ©u] GiL˚[Ÿm AYoLs TVuT”jߟs[]o. AYoLs Jo
Bi”dœ 360 SÙhLs LQd°h•⁄kR]o. CR]Ùp J⁄ YhPjßtœ 360 •°¨ G]d
24
LQd°PlTh•⁄dLXÙm. ÈwVm Tt±V A±‹ C⁄kRRÙ G] EflßVÙLd ·\ CVXÙ’.
B]Ùp, CPUßl◊ Tt± AYoLs A±kß⁄kR]o. °±v’Ætœ 200 Bi”Lfidœ ÿu]o
T˚PdLlThP T˚Pl◊L∞p GiL∞p CP Ußl˚T LÙhP Gi Cp˚X˘V≤p Ko
A˚PVÙ[m ET˙VÙ°jß⁄kR]o G] A±Vÿ•°\’. B]Ùp, CR˚]d LQd∏”L∞p
TVuT”jRÆp˚X.
˘T⁄dLp, YœjRp, YodLUÙdœRp, YodLÍXm LÙ‘Rp, ÿd˙LÙQm, ˘Ne˙LÙQ
ÿd˙LÙQm, N’Wm GuT]Yt±u TWlT[‹ LÙQ ÿtLÙXm ÿR˙X AYoLs A±kß⁄kR]o
π dœ AYoLs 3 Gu\ Ußl◊ A∞jß⁄kR]o. L¶RÆV≠u ©\l©Pm G°l’ G]
A¨vPÙh•p œ±l©h”s[Ùo. (Y•ÆVp LÙQ‹m) °.ÿ.1700˛Cp AyUv T˚PjRl
‘TlT˚Wv’ ÿRp L¶RÆVp Ëp Gufl ·\XÙm. C⁄NUTdL ÿd˙LÙQm, ˘Ne˙LÙQ
ÿd˙LÙQm CYt±u TWlT[‹Ls, ©u] GiLs ÿR≠V HWÙ[m L¶Rd L⁄j’dLs
Cßp œ±l©PlTh”s[]. AyU≥u LÙXm G°lßp L¶Rjßu YNkRLÙXm G]d
·\XÙm. ©u]o B´Wm Bi”Ls Y˚W œ±l©h”f ˘NÙp¤m A[Æp AYoL[Ùp
ÿu˙]\ CVXÆp˚X.
°.ÿ.7˛Bm Ët\Ùi” ÿRp G°l’dœm °˙WdLjßtœm C˚P˙V Y¶L E\‹Ls C⁄kR].
A±ÆVp T¨UÙt\eLfim S˚P˘Tt\]. °˙WdLjßu ÿd°VUÙ] A˚]j’ ˙U˚RLfim
L¶R A±OoLfim ©W™”L∞u SÙPÙ] G°l’dœf ˘Nu±⁄kR]o. ®Z≠u ø[j˚R
A[k’ ©W™•u ø[j˚Rj "˘R´pv' LQd°hP’ A˚]Y⁄m A±kR˙R. L¶RÆV≠u
Rk˚R G]l TX⁄m ˘R´p˚^ A˚ZjR]o. ˘R´pv J⁄ ◊Lr˘Tt\ ˙NÙßPÆVp
A±O⁄m BYÙo. °˙WdL SÙh•p Y•ÆVp BnÆ˚] ˘Y∞´hPYo ˘R´pv BYÙo.
„¨V°WLQm Tt± ÿR≠p æodLR¨N]m ·±VYo ˘R´pv BYo. ˚TR˙LÙW≥u LÙXm
°˙WdLd L¶RÆV≠u ˘TÙtLÙXm G]d ·\XÙm. —VUÙLd L¶Rjßp TX Li”©•l◊Ls
®LrjßV’ Uh”UpXÙUp L¶R BnÆtLÙL J⁄ NÍLj˚Rl T˚PjR∞jR˚U Au]Ù¨u
Su˘LÙ˚PVÙœm. EX°u A˚]j’ At◊R ®Lr‹L˚[Ÿm L¶Rj’Pu C˚QjR˙R
˚TR˙LÙW≥u ™Ll ˘T¨V TeL∞l◊.
◊Lr˘Tt\ Rj’Yf£kR˚]VÙ]o ©˙[h˙PÙ ®flÆV ‘ALÙP™’ °˙WdL SÙh•u
L¶RÆVp Y[of£dœl ˘T⁄m Teœ A∞j’s[’. ""L¶Rm ˘R¨VÙRYoLfidœ
÷˚ZY‡Uß Cp˚X'' Gu\ A±Æl◊l TX˚LŸm AdLÙP™´p ˘TÙ⁄jRlTh•⁄kRRÙL
YWXÙt\Ù£¨VoLs ·fl°\ÙoLs. Au]Ù¨u ∫PWÙ] ˙PÙX™ °.ÿ 338 ˛Cp AXdNÙi•¨VÙ
TpL˚XdLZLm ®flÆ]Ùo. C’ ™LÆ˚WÆp ◊Lr˘Tt\ LpÆ˚UVUÙL E⁄l˘Tt\’. Cßp
L¶Rl©¨˚Y SPj’m ˘TÙfll◊ Îd≠h A±O¨Pm JlT˚PdLlTh•⁄kR’. Y•ÆV≠u
Rk˚R G] A±VlT”m Îd≠h CkRj’˚\´u Y[of£dœ ™L ÿd°Vl Teœ Y°jRÙo.
BodL™ºv, Al˙TÙ˙[Ù¶Vv, P˙VÙTÙWuWv, A]Ùd≥UÙu, ˘a˙WÙu, ß˙VÙu
(˚NvUoQÙ) ß˙VÙu (AXdNÙi•¨VÙ) B°V TX L¶R A±OoLfim °˙WdL SÙh•u
Su˘LÙ˚PLs BYo.
£\kR SÙs LÙh• ˙WÙUÙ≤Vo L¶RÆV¤dœ A∞jR ÿd°V Su˘LÙ˚PVÙœm.
GiL¶Rm RÆW ©\ L¶RÆVp ’˚\L∞p œ±l©PjRdL Li”©•l◊Ls Jufl˙U
˙WÙUÙ≤Vo ®LrjRÆp˚X. ˙WÙUu Giÿ˚\ EXL˘Ueœm Ae∏L¨dLlThP’ G≤‡m
25
A’ AYoL∞u Su˘LÙ˚PVÙ Gu\ IVm Cu‡m Es[’.
CkßVÙƤm °˙WdLSÙh•¤m ˙RÙu±V ™Lf £\kR L¶RÆVp ËpLs A˚]j˚RŸm
A˙W©VoLs Gtfld ˘LÙi” TÙ’LÙj’ ˙UmT”jß I˙WÙl©V SÙ”L∞p TWYf ˘NnR]o.
TÙWRjߤm, °˙Wd°jߤm ˙RÙu±V ÿd°V L¶RÆVp T˚Pl◊dLs A˚]j˚RŸm
A˙W©V ˘UÙØ´p ˘UÙØ˘TVojR’Pu CkßVd L¶R A±OoL˚[ A˙W©VÙÆtœ
A˚Zj’m ˘Nu\]o. CVtL¶Rjßtœl TVuT”j’m "Bp¥lWÙ' Gu\ ˘NÙp¤dœ
Ap_T⁄Pu ˘RÙPo◊ Es[’. Ap˘LÙWÙv™ ˛ Al’p˘YT, EUoLnVÙm, ApLÙo°
Gu˙TÙo L¶RÆV˚X Y[Wf ˘NnR A˙W©VoL∞p ™L ÿd°VUÙ]YoLs BYo.
CkßVÙÆu ™L ÿd°VUÙ] L¶RÆVp Su˘LÙ˚P ÈwVm Gu\ CXdLÿm
TjR•UÙ] Giÿ˚\Ÿm Bœm. L¶R YWXÙt±u ™Ll˘T¨V Li”©•l˙T ÈwVm
Gu\ CXdLm. L¶RÆV¤dœ Uh”UpX A˚]j’ A±ÆVp ’˚\L∞u Y[of£dœm
ÈwVm G‡m Li”©•l◊ TVuTh”s[’. ÈwVÿm Juflm TVuT”jߟs[ CWiP•UÙ]
ÿ˚\˚V TVuT”j’°u\ L¶≤ S≈] EX°p ˘N¤j’m ˘NpYÙdœ ™Ll ˘T¨RÙœm.
£k’SßdL˚W SÙL√Lm ˙RÙu±V LÙXjß≠⁄k˙R CkßVÙÆp L¶Rm Y[of£
A˚Pkß⁄kR’ GuTRtœ HWÙ[UÙ] NÙuflLs Es[]. A[lTRtœm LQd°”YRtœm
ß\u™dLYoL[ÙL C⁄kR]o. ˘UÙLgNRÙ˙WÙÆ≠⁄k’m aWlTÙÆ≠⁄k’m °˚Pj’s[
G˚PdLtLs, ÷hTj’P‡m ’p≠Vj’P‡m G˚P Li•⁄kR]o GuT˚Rj
˘R∞‹T”j’°u\]. G˚P ˙SÙdL Au≥VŸm ø[m A[dL CuPv v˙L´¤m
TVuT”jß]o.
Ti˚PV TÙWRjßu L¶RÆVp ËpL∞p ™Ll TZ˚UVÙ]’ —pT „jßWeL[Ùœm G]
SmTlTh•⁄kR’. CYt±u T˚Pl◊dLÙXm °±v’Ætœ ÿu 800dœm 500dLm C˚P˙V
G] Sm©]o. ˘Yq˙Yfl LÙX LhPeL∞XÙL TXWÙp G›RlThP˚Y˙V —pT„jßWeLs.
ÿd˙LÙQm, C˚QLWm, N¨YLm B°VYt±u £\l©Vp◊L˚[d —pT„jßWeL∞p
LÙQXÙm. 2 ,
3
G] Es[ Æ°Rÿ\Ù GiL˚[l Tt±Ÿm LXk’˚WVÙ•Ÿs[]o.
°±vRY Bi” 476 Cp ©\kR BoVThP˙W £\kR CkßVd L¶R A±Oo.
B¨VThºVm Gu\ ◊Lr˘Tt\ Ë≠u T˚PlTÙ∞VÙ] B¨VThPo ˙LW[ÙÆp ©\kRÙo
G] J⁄ L⁄j’ Es[’. B¨VThPo BWm©j’ ˚YjR L¶RÆVp Bn˚Y ©WmUœlRo
˙U¤m S≈] YØÿ˚\dœ A˚Zj’f ˘Nu\Ùo. ©WmUv◊P £jRÙkR˙U Au]Ù¨u ™Ll
˘TVo ˘Tt\ Ëp. B¨VThPo ÆYÙßjR A˚]j’d L¶Rl TœßL˚[Ÿm ©WmUœlRo
ÆYÙßj’s[Ùo. ˙U¤m B¨VThP⁄dœ YkR ©˚ZLfim ß⁄jRlThP] G]j ˘R¨°\’.
Gp˚X (Limit) Gu\ L⁄j˚Rl Tt±j ˘R∞Yt\ £X FLeLs (A‡UÙ]eLs)
©WmUœlR¨Pÿm LÙQlThP]. °±vRY Bi” GhPÙm Ët\Ùi•p YÙrkß⁄kR cRWu,
TjRÙm Ët\Ùi•p YÙrkß⁄kR CWiPÙm B¨VThPo Gu˙TÙ⁄m CkßVd L¶Rjßtœl
˘T⁄m Su˘LÙ˚PLs A∞j’s[]o. ƒXÙYß Gu\ Ëp YÙ´XÙLl ◊Lr˘Tt\
TÙvLWÙfNÙ¨VÙo J⁄ L˚XO¨u TÙo˚YŸPu L¶Rj˚R A‘°Ÿs[Ùo. C’
Au]Ù⁄dœl ˘T⁄m◊˚Zj ˙R•d˘LÙ”jR’. ‘£jRÙkR £˙WÙU¶’ Gu\ Ë≠u J⁄ Tœß
Uh”˙U ‘ƒXÙYß’ Gu\ Ëp.
14, 15 Bm Ët\Ùi”L∞p ˙LW[Ù CkßVd L¶Rjßtœ ®LWt\ Su˘LÙ˚PLs
A∞j’s[’. ◊’U˚] ˙NÙUÙߨL∞u LWQTjRß, ˙L[pÌo øXLiP ˙NÙUVÙ¥´u
26
RkßWNe°WLm. GuT] £\kR L¶R ËpL[Ùœm. CYt±p ÆY¨j’s[ ˘T⁄YÙ¨VÙ]
L⁄j’dLfim ©u]o 17, 18˛Bm Ët\Ùi”L∞p YÙrkß⁄kR AVpSÙh” A±OoL∞u
Su˘LÙ˚PL[ÙL A±VlThP]. ◊’U˚] ˙NÙUÙߨŸm, øXLiP ˙NÙUVÙ¥Ÿm
Y•Y˚UjR "LX]NÙjßWm’ Gu\ L¶Rj’˚\ Ët\Ùi”Lfidœl ©u]o ˘XT≤—m
®ÎhP‡m ‘LÙpœXv’ (÷iL¶Rm) Gu\ ˘TV¨p ˘Y∞´hP]o.
®ÎhP‡m ˘XT≤—m J⁄Y⁄d˘LÙ⁄Yo A±VÙU˙X LÙpœXv Li”©•jR]o.
RkßWNe°WLjߤm LWQTjRß´¤m ÆYÙßjR λ Cu Ußl◊ Li”©•lTRtœ E¨V Y¨˚N
Ët\Ùi”Lfidœl ©u]o ©\kR ˙UtLjßV L¶R˙U˚R °¨L¨´u ˘TV¨p
A±VlThP’.
Tj˘RÙuTRÙm Ët\Ùi•u L˚P£lTœß´¤m C⁄TRÙm Ët\Ùi•u ˘RÙPdLjߤm
CkßVÙÆp YÙrkß⁄kR ™Ll ◊Lr˘Tt\ L¶R˙U˚R ∫≤YÙNCWÙUÙ‡_o Bœm. 32
YV’Y˚W Uh”˙U YÙrkR CWÙUÙ‡_o EX°p AdLÙXjßp YÙrkß⁄kR ™L ÿd°Vd
L¶R˙U˚RL∞p J⁄Yo. Ramanujam Institute of Mathematics, Tata Institute of Fundamental
Research Cu L¶Rl ©¨‹ GuT] CkßVÙÆp Cu˚\V ◊Lr˘Tt\ L¶RÆVp BWÙnf£
®flY]eL[Ùœm.
GiL∞u YWXÙfl
GiL˚[l Tt± G’‹m ˘R¨VÙR J⁄LÙXLhPm C⁄kß⁄kR’. Auflm Sm ÍRÙ˚RVWÙ]
BßU≤RoLs AYoLfid˙L E¨V ÿ˚\L∞p LQd°h•⁄kR]o. LÙ”L∞¤m TÙ˚\
A”dœL∞¤m YÙrkß⁄kR Bß U≤RoLs TX ÆXeœL˚[Ÿm CQd° Y[odLj
˘RÙPe°]o. AYoLs ˙Y[Ùi˚U ˘Nn’m ˙Yh˚PVÙ•Ÿm LuflLÙ≠L˚[ Y[ojߟm
YÙrkR]o. B”L˚[ UndLl˙TÙœm TÙ’ Jq˘YÙ⁄ C˚PV‡dœm ·hPUÙL B”L˚[
˙Undœm ˘TÙfll◊ C⁄kR’. LÙ˚X´p ˘LÙhP˚LL∞≠⁄k’ B”L˚[ Jq˘YÙu\ÙL
˘Y∞˙V Æ”m˙TÙ’ Jq˘YÙ⁄ Bh•‡˚PV‹m Gi¶d˚L˚Vd œ±dL Jq˘YÙ⁄
Lp˚Xl ˚T´p ˙TÙ”YÙu. B”Ls GpXÙm ˘Y∞˙V YkR ©u]o ˙TÙ”YÙu. ˚T´u
Es˙[ C”°u\ LtL∞u ·hP˙U B”L∞u Gi¶d˚L. A’ GqY[‹ Gufl
˘NÙpYRtœm ˘R¨VÆp˚X. UÙ˚X´p B”L˚[j ß⁄l©d ˘LÙi”Yk’, Jq˘YÙu\ÙLd
˘LÙhP˚L´p A˚Pdœm˙TÙ’ ˚T´p Es[ LtL˚[ Jq˘YÙu\ÙL ˘Y∞˙V ˙TÙ”YÙu.
˚T´p LtLs ¡ßVÙL YkRÙp AkR Gi¶d˚L´uT• B”Ls SxPUÙ´] GuT˙R
˘TÙ⁄[Ùœm. A˚Y Y]ÆXeœLfidœ EQYÙL˙YÙ, ·hPm RY±l ˙TÙL˙YÙ
˘Nnß⁄dLXÙm G]d L⁄’Yo. CqYÙfl Ju˙\Ù˘PÙufl ˘TÙ⁄k’m ÿ˚\´˚] Bß
U≤Ru Gi¶d˚L˚Vd LQd°”YRtœl ©u Tt±]Ùu.
H\dœ˚\V 2000 Bi”Lfidœ ÿu]˙W U≤Ru GiL˚[l TVuT”jß´⁄kRÙu Gufl
YWXÙfl œ±l©”°\’. °.©. Íu\Ùm Ët\Ùi•p A˙NÙLl ˙TWWN¨u Ah£ LÙXjßp
∏˙Z ˘LÙ”dLlTh”s[ Giœ±¬”Ls TVuT”jß C⁄kRRÙLd LÙQXÙm.
AD CWiPÙm Ët\Ùi•p ""SÙ£d''œ˚LL∞≠⁄k’ Li”©•jR Lp˘Yh”L∞p ∏˙Z
RWlTh”s[ Gi œ±¬”Ls TVuT”jß´⁄kR˚U˚Vd LÙQXÙm.
27
˘RÙPok’ AD 8˛Bm Ët\Ùi•p TVuT”jßV ˙RYSÙL¨ GiLs S˚Pÿ˚\´p Es[
Giœ±¬”LfiPu ™L Jl◊˚U E˚PV] G]d LÙQXÙm. A˚Y ∏˙Z ˘LÙ”dLl-
Th”s[].
CkR Gi œ±¬”L∞≠⁄k’ SÙm TVuT”j’°u\ Giÿ˚\dœ Yk’ ˙Nok˙RÙm. AD
9˛Bm Ët\Ùi•p ˙UÙvXm (Moslem) Bh£j R˚XYoLs ReL∞u SÙh˚P ˙UtLjßV
SÙPÙ] v˘T´u, °ZdLjßV SÙPÙ] CuPv Y˚W ƨY˚PVf ˘NnR˙TÙ’ ‘CkßV
A˙W©V GiLs’ Gu\ ˘TVo TWYj˘RÙPe°V’. Cufl SÙm TVuT”j’Y’ CkR Gi
ÿ˚\˚V˙V. Sm Gi ÿ˚\´u £\l◊L∞p ÿRu˚UVÙ]’ 0, 1, 2......................9 Y˚W Es[
CXdLeL˚[ (10 Gi¶d˚L) Uh”m TVuT”jß˙V Giÿ˚\ E⁄YÙdLlTh”s[’
GuTRÙœm. AR]Ùp TjR•UÙ] Gi‘⁄ Guflm CR˚]f ˘NÙpY’i”.
CWiPÙm £\l©Vp◊ GuT’ CPUßl◊ (place value) Gu\ L⁄j’ TVuT”jRlTh”s[’
GuTRÙœm. GjR˚] T¨V Gi˚QŸm ARu CPUßl©tœ HtT Tß‹˘NnV GkR £WUÿm
Cp˚X.
Íu\Ùm £\l©Vp◊ ‘ÈwVm’ Ko Gi‘⁄YÙL TVuT”jRlTh”s[’ GuTRÙœm.
˘Yfl˚U˚Vf —h”m (Juflm CpXÙR Ru˚U) GiQÙLl ÈwVm TVuT”jRlT”Y’ Sm
Giÿ˚\´u G”j’˚WdL ˙Yi•V £\l◊j Ru˚UVÙœm.
GjR˚]? GjR˚]VÙY’?
Gi‘YRtœ C⁄ UÙflThP ˙LÙQeLs Es[].
J⁄œ›Æp GjR˚] Gi¶d˚L Es[] Guflm
Jq˘YÙuflm GjR˚]VÙY’ Guflm A±k’˘LÙs[
˙Yi• Es[’.
G.”. 1. Tjßu ∏r GjR˚] CWh˚P GiLs Es[]?
2. CkRl Tg£p GjR˚] œZk˚RLs Es[]?
3. GjR˚]VÙY’ CWh˚P Gi6
4.U˙]Ùw GjR˚]VÙY’ œZk˚R?
Cßp ÿRp C⁄ Æ]ÙdLfim ‘GjR˚]’ Gu\ Æ˚P R⁄Y]. GiL∞u CkR CVp◊
‘cardinality’Gufl ˘NÙpXlT”°\’. 3, 4 Æ]ÙdLs GjR˚]VÙY’ Gu\ Æ˚P R⁄Y]. £X
£\l◊dL˚[ A•˘VÙt± (A[‹, YV’, CPm....) Y¨˚NlT”jß CP Ußl˚Tf ˘NÙpX‹m
Gi TVuT”jRlT”°\’. CkR CVp◊ ‘ordinality’Gufl ˘NÙpXlT”°\’.
˙WÙUu Giÿ˚\
Ti˚PV ˙WÙ™p TVuT”jßV Giÿ˚\˙V ˙WÙUu Giÿ˚\. ÿRp Tj’ ˙WÙUu
GiLs ∏˙Z RWlTh”s[].
I, II, III, IV, V, VI, VII, VIII, IX and X
GjR˚] ˙To ?
˙TÙh•´p SÙu CWiPÙY’
CPjßp Es˙[u. B]Ùp
L˚P£VÙL ®tTYu Íu\ÙYRÙL
YkR˙TÙ’ SÙu KhPj˚R
®fljß˙]u. Gu ©u]Ùp ˙Yfl
Íufl˙To Es[]o.
28
˙WÙUu Gi‘dœ ˙Up ˙LÙ”˙TÙhPÙp AkR Gi¶u B´Wm UPe˚Lf —h”°\’.
˙WÙUu Giÿ˚\ Tt±d ·”Rp ÆYWeLs ßWh•d L⁄jRWe°p ˘Y∞´P‹m
CkßV ˛ A˙W©V Gi ÿ˚\
CkßV A˙W©V ÿ˚\´p Cufl TWYXÙLl TVuT”j’°u\ A˚PVÙ[eLs 0, 1, 2, 3, 4,
5, 6, 7, 8, 9 GuT] Bœm. SÙm Cufl A˚]j’ CPeL∞¤m TVuT”j’°u\ RNUÿ˚\
ApX’ Tj’ A•UÙ] ÿ˚\´p GkR Gi˚Q G›R‹m CkRl Tj’ CXdLeL[Ùp CV¤m.
CkßV A˙W©V Giÿ˚\ Tt±d ·”Rp ÆYWeL˚[j ßWh• YœlT˚\´p Y∞´P‹m.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
29
Tp˙Yfl Gi‘⁄ ÿ˚\Ls
L¶R YWXÙt±u ÿd°V ˚UpLtL˙[ Gi‘⁄ ÿ˚\Ls (Number System). 0, 1 G‡m
GiL˚[ A•lT˚PVÙLd ˘LÙiP’ CWiP•UÙ]ÿ˚\ (Binary System).
0, 1, 2, 3, 4, 5, 6, 7, 8, 9 G‡m GiL˚[ A•lT˚PVÙLd ˘LÙi”s[’ TjR•UÙ]
Gi‘⁄ÿ˚\. CYt˚\d œ±j’ ˙U¤m TX ÆYWeL˚[f ˙NL¨j’ YœlT˚\´p
˘Y∞´P‹m.
©\ Gi‘⁄ ÿ˚\Ls G˚Y?
Gi L¶Rm (Arithmetic)
L¶RÆVp ÿtLÙX GiL∞u Rj’Yj˚Rf (Theory of numbers) —h”°u\ Arithmetic Gu\
’˚\Ÿm LQd°”m L˚X (Art of Calculating) G]f —h”°u\ Legistic Gu\ ’˚\Ÿm
TVuTÙh•p C⁄k’s[]. Tß]Ù\Ùm Ët\Ùi•p Af—ÿ˚\ Li”©•lT’ Y˚W C˚Y
C⁄ °˚[L[ÙL˙Y Y[okR]. ©u]o C⁄°˚[L˚[Ÿm œ±l©P A¨j˘Uh•d Gu\
˘NÙp˚Xl TVuT”jR BWm©jR]o. ˙Yfl TX ˘NÙtL˚[l ˙TÙu˙\ Arithmetic Gu\
˘NÙp≠¤m TX UÙt\eLs ®Lrk’ Cu˚\V ®˚X ˘Ttfls[’. J⁄ U≤Ru
A±kß⁄dL˙Yi•V 3 R L∞p Ju\ÙL (Rithmetic) Gi L¶Rm A±VlTh•⁄kR’. Read-
ing, Writing GuT] Ut±Wi”. Cß≠⁄k’ GiL¶Rjßtœ A∞j’s[ ÿd°Vj’Yj˚Rl
◊¨k’˘LÙs[XÙm.
Bp¥lWÙ (Algebra)
GiL¶Rjßu ˘TÙ’Y•Ym; G”j’dLÙhPÙL 3 + 2 =5, 4 + 8 =12, G‡m GiL¶R
YÙd°VeL∞u ˘TÙ’Y•Y˙U CVtL¶Rjßu a + b =c Gu\ YÙd°Vm. a + b =c Gu\
YÙd°Vjßp a,b,c GuT]Ytfldœ A∞d°u\ Ußl◊L∞p £X œ±l©hP Ußl◊ A∞dL
°˚Pdœm YÙd°VeL˙[ 3 + 2 =5, 4 + 8 =12 GuT]. CVtL¶Rjßp GiLfidœl TßXÙL
ÿR≠p G›j’dL˚[l TVuT”jßVY¨p ÿRu˚UVÙ]Yo ©WÙe˙LÙnhÆt\Ù Gu\
©Wg— L¶R˙U˚R. Au]Ùo UÙ±Lfidœ E´˘W›j’dL˚[Ÿm UÙ±≠Lfidœ
˘Un˘V›j’dL˚[Ÿm TVuTPjß]Ùo.
GiLfidœl TßXÙL ˘R¨VÙR UÙ±L˚[ ˚LVÙs°u\ L¶RÆVp ’˚\˙V Bp¥lWÙ
Gufl ·\XÙm. CkR A±ÆVp’˚\ Gl˙TÙ’, Ge˙L, BWm©jR’ G]j ˘R∞YÙLd ·flRp
L•]UÙœm. Cu˚\V Bp¥lWÙ æo‹ LÙi°u\ £X ©Wf£˚]Lfidœ Au˙\ æo‹
Li•⁄l˚Rd L⁄jßp ˘LÙiPÙp Bp¥lWÙ ˘RÙPe°V’ °±v’tœ H\dœ˚\V 1800
Bi”Lfidœ ÿu]o G]d ·\XÙm. B]Ùp æo‹ LiP’ Cu˚\V Bp¥lWÙ
ÿ˚\´XÙL C⁄dL˙Yi”m Gu±p˚X. F°j˙RÙ GiL¶R A˚PVÙ[eLs
TVuT”jß˙VÙ B´⁄dLXÙm. GiLfidœl TßXÙL ˘R¨VÙR UÙ±L˚[ TVuT”jRj
˘RÙPe°V LÙXj˚Rj ’p≠VUÙLd LQd°”R¤m £WUUÙ]’. °±v’‹dœ ÿu]o
H\dœ\V 1800˛1600 dÎ≤TÙm L¶Rl Th•VpL∞p NUUÙ] NU]TÙ”LsTVuT”jßVRÙL
LY≤dLlTh”s[’. °±v’Ætœ ÿu H\dœ˚\V 1550 Cp G›RlThP AdUv
Tl˚TW£p UÙ±Ls Es[ NUuTÙ”Ls TVuT”jRlTh”s[].
30
GiLs EhT”°u\ £X ◊ßoLfidœj æo‹LÙQj ˘RÙPe°V˙RÙ” Bp¥lWÙ
˘RÙPdL™hP’ G]d ·\XÙm (L¶Rl ◊ßoLs LÙQ‹m). S˚Pÿ˚\ R˚Y Gu\ ®˚X´¤m
˙L∞d˚L Gu\ ®˚X´¤m CjR˚LV L¶Rl◊ßoLs EX°u ˘T⁄mTœßL∞¤m
™LlTi˚PVLÙXjßtœ ÿu]˙W ◊ZdLjßp C⁄kR]. CYt±u æo‹L∞p Cu˚\V
Bp¥lWÙÆu L⁄j˚Rd LÙQXÙm.
CkßVÙÆp B¨VThP¨u B¨VThºVm, ©WmUœlR¨u ©WmUv◊P£jRÙkRm, ULÙ≈W¨u
L¶RNÙW Ne°WLm, TÙvLWÙfNÙ¨V¨u ƒXÙY’ G‡m ËpLs Bp¥lWÙ˚Yl
TVuT”jߟs[]. CkßVd L¶R A±OoLs Cj’˚\dœj ßhPUÙ] ˘TVo
A∞dLÆp˚X. œhPLm, AqVdRL¶Rm, CVtL¶Rm ˙TÙu\ TX ˘TVoL˚[
A∞jß⁄kR]o. L¶R ˙U˚RVÙ] ˙L[pÌo øXLiP ˙NÙUVÙ¥´u ‘RkßWNe°WLm’
˙_xP˙RY¨u ‘ŸdßTÙ`Ù’ G‡m ËpLs CVtL¶Rj˚Rl TVuT”jߟs[].
C⁄UÙ±Ls Es[ NUuTÙ”Ls, Tj’ÿ˚\L∞XÙLj æo‹ LÙQ CV¤m YØÿ˚\Ls
B°VYt˚\ CkËpL∞p LÙQXÙm.
Bp¥lWÙ Gu\ L¶RÆVp ’˚\˚V Y[olTßp A˙WÙ©VoLs ÿd°Vl Teœ Y°jR]o.
S≈] ˙U˚XSÙh” EX°tœ Bp¥lWÙ ˙LÙhTÙ”Ls A∞jRYo A˙W©VoLs.
A˙W©VdL¶R ˙U˚RL∞p ™Ll ◊Lr˘Tt\ ÿLmUw ©uÍNÙ Ap ˘LÙWÙv™ G›ßV
°RÙl Ap_ToYApÿLÙYX Gu\ Ëp Bp¥lWÙ Tt± ÿ›˚UVÙL G›RlThP ÿRp
Ëp. Cßp Es[ Ap_To Gu\ ˘NÙp≠≠⁄k’ Bp¥lWÙ Gu\ ˘NÙp E⁄YÙdLm ˘Tt\’.
Ï©VÙj Gu\ ◊Lr˘Tt\ LÙÆVjßu T˚PlTÙ∞VÙ] EUoLnVÙÿm Bp¥lWÙÆu
Y[of£dœl ˘T⁄m Su˘LÙ˚PLs YZe°Ÿs[Ùo.
Y•ÆVp (Geometry)
L¶RÆV≠u J⁄ ÿd°Vd °˚[˙V, ÿd˙LÙQm, YhPm, SÙtLWm, ˙LÙ[m, ThPLeLs,
·m◊Ls G‡m Y•YeL∞u Bn‹ G]f —⁄dLUÙLd ·\XÙm. ◊Æ Gufl ˘TÙ⁄sR⁄m
Geo A[‹ Gufl ˘TÙ⁄sR⁄m metry Gu\ ˘NÙtLs E⁄YÙdLm ˘Ttfl Geometry Gu\
A±ÆVp ’˚\ ˘RÙPeLlThP’ G°lßp Bœm Gufl SmTlT”°\’. Y•ÆV˚X
˚SpSß´u ˘LÙ˚P G] ˘a˙WÙ˙PÙhPv œ±l©hPÙo. Ti˚PV UÙ¶P SÙL√LeLs
SßdL˚WL∞p ApXYÙ ˙RÙu±]. TÙWRjßu £k’SßdL˚W SÙL√Lÿm ∫]ÙÆu
˘aÙVÙe˙LÙ SßdL˚W SÙL√Lÿm ◊Lr˘Tt\˚Y. SßdL˚WLs Y[m ˘Tt±⁄kR˚UVÙp
Ce˙L UdLs ßW[ÙL Y£jR]o.
G°lßu ˚SpSß U˚ZdLÙXeL∞p ˘T⁄° J›° SßdL˚WL∞p ˘T⁄m £˚R‹L˚[
Æ˚RjR]. L˚W´p Y£lTYoLs ReL∞u T´¨PeL˚[Ÿm ®XeL˚[Ÿm Æh”
ÆX°f˘Nufl Y£jR]o. U˚ZdLÙXm LØkR ©u]o ¡i”m L˚W˚V ˙SÙd°f ˘NpYo.
Jq˘YÙ⁄Y¨u E˚P˚U´p Es[ ®XeL∞u Gp˚XLfim AØk’˙TÙ´⁄dœm. AYt˚\
Ufl∫W˚UdL T⁄m ’uTeLs A‡TÆjR]o. TX˙Y˚[L∞p AYoL∞˚P˙V LXLeLfim
˙RÙu±]. È™´u A[‹m Y•Yÿm Tß‹ ˘Nnß⁄kRÙp Gp˚XL˚[ G∞RÙL
A˚Uj’d˘LÙs[XÙm Gufl Li”©•jR]o. CqYÙfl Jq˘YÙ⁄Y¨u ®X E˚P˚ULs
Tt±V A[‹m Y•Yÿm œ±dLlTh•⁄kR Tß‹L˙[ Y•ÆVp G] A±VlThP’.
ÿd˙LÙQm, ˘NqYLm, N¨YLm, SÙtLWm ˙TÙu\ Y•YeL˙[ È™´u A[‹L[ÙL
31
A˚Ukß⁄kR]. T•lT•VÙL Y•ÆV¤dœl È™ŸPu Es[ ˘RÙPo◊ TX≈]U˚PkR’Pu
Y•YeL∞u Lt\XÙL Y•ÆVp E⁄l˘Tt\’. È™ A[‹Pu ˘RÙPo◊s[ ’˚\VÙL Cuflm
®XA[≈h•p ©uTt\lT”°\’.
G°l’ ©W™”Ls ™Ll◊Lr˘Tt\˚Y. CkRl ©W™”L∞u E⁄YÙdLjßp TVuT”jßV
˘RÙØp÷hTm —h•dLÙh”Y’ AYoLs AdLÙXjßp Y•ÆV≠p ß\u ˘Tt\YoL[ÙL
C⁄kR]o GuTRÙœm. ˘R´pv Ru≤Pm L¶Rm LtLYkR ˚TR˙LÙW≥˚] ‘G°l’dœf
˘Np’ Gufl A±‹fljß]Ùo.
Y•ÆV≠u Rk˚R G] A±VlT”m °˙WdLd L¶R A±Oo Îd≠h Aufl Y˚W
Li”©•dLlThP Y•ÆVp L⁄j’dL˚[ CVu\ A[Æp ˙R•j ˘RÙœj’
‘G≠˘Uuhv’Gu\ Ëp CVt±]Ùo. BodL™ºv, ˚TR˙LÙWv, Al˙TÙ˙[Ù¶Vv
˘T˙WÙi, ∂lTÙoLv ˙TÙu\ TX L¶R A±OoLs °˙WdL Y•ÆVp Y[oYRtœl ˘T⁄m
Teœ Y°jR]o.
L¶RÆVp YWXÙtflPu C˚QkR ˙U¤m TX ÆYWeL˚[j ßWhP‹m, L⁄jRWe°p
˘Y∞´P‹m.
Ußl¿”
L¶RÆVp YWXÙtfld L⁄jRWeLm, ˘Y∞¬”, A±d˚L
L¶RÆVp A±OoLfim AYoL∞u Su˘LÙ˚PLfim
øpLiP ˙NÙUVÙ¥ (1465˛1545) (Neelkanda Somayaji)
CYo ◊Lr ˘Tt\ ˙LW[dL¶R A±O⁄m ˙NÙßP Yp¤]⁄m BYo. ßϨu A⁄LÙ˚U´p
Es[ ß⁄dLi•Î¨p ˙LpÌo (˙LW[˘SpÌo) U˚]´p ©\kRÙo. Rk˚RVÙo _ÙR˙RYo,
Rm© NeLWu Gu\ L¶R A±Oo. ’oL¶Rl T˚PlTÙ∞. YP˙N¨ TW˙U—YW⁄m Au]Ù¨u
UL]Ùo YP˙N¨ RÙ˙UÙRW⁄m B£¨VoL[ÙL C⁄kR]o. B¨VThºVjßu Æ[dL E˚WVÙ]
L¶RTÙRm ÿd°V ËXÙœm. RkßWNe°WLm (˙NÙßPÆVp), °WLQ ®oQVm, ˙LÙ[NÙWm,
NkßWNÙVÙL¶Rm, °WLT√hNÙ°WUm, —kRWWÙ_ ©Wf˙]ÙjRWm GuT] ©\T˚Pl◊Ls. ""˚T''
J⁄ Æ°Rÿ\Ù Gi Gufl EXœdœ ÿR≠p ˘R¨ÆjR’ 1671 ˛Cp XÙmToh To≠u Bœm.
ARu C⁄ Ët\Ùi”Lfidœ ÿu]o ˙NÙUÙVÙ¥ G›ßV L¶Rjßp C’
œ±l©PlTh”s[’. RkßWNe°WLjßp ÷iL¶Rm A•lT˚PVÙ] ˙LÙhTÙ”Ls
Æ[dLUÙL ÆYÙßdLlTh”s[]. ©u]o Ët\Ùi”Ls ˘Nu\ ©u]˙W ®ÎhP‡m
˘XT≤—m J⁄Y⁄d˘LÙ⁄Yo A±VÙ’ J˙W LÙXjßp LÙpœXv Li”©•jR]o.
˙NÙUÙVÙ¥´u NULÙXjRYWÙ] ◊’U˚] ˙NÙUÙߨŸm ÷iL¶Rd ˙LÙhTÙ”L˚[
Æ[d°Ÿs[Ùo.
TÙvLWÙfNÙ¨VÙo
Cl˘TV¨p C⁄ A±OoLs YÙrkß⁄kR]o. Cßp ÿR≠p Es[Yo TÙvLWÙfNÙ¨VÙo,
Ju\Ùm B¨VThP⁄dœl ©u]o œfl°V LÙXjßtœ Es[ÙL˙Y ©\kRÙo. L¶RÆV≠¤m
YÙ]ÆV≠¤m CYo ˙U˚RVÙLj ßLrkRÙo. CWiPÙm TÙvLWfNÙ¨VÙo TÙvLWÙfNÙ¨Vo II
32
Gu\ ˘TV¨p A±VlThPÙo. TÙWRm LiPYoL∞p ™Ll ◊Lr˘Tt\ L¶R A±Oo CY˙W.
CY˚Wd œ±j’ Ce˙L Æ[dLlT”°\’.
AD 1114˛Cp CWiPÙm TÙvLWÙfNÙ¨VÙo ©\kRÙo. TZ˚UYÙnkR J⁄ Ti•Rd œ”mT˙U
AY⁄˚PV’. AY¨u Rk˚R´u ˘TVo U˙L—YWo. ˚U„o UÙ®Xjßp ©\kR TÙvWfNÙ¨VÙo
Ew_´≤´u YÙ]ÆVp ˚UVjßp T¶◊¨kRÙo. TX £\kR T˚Pl◊L˚[ G›ßŸs[Ùo.
£jRÙkR£˙WÙU¶, ƒXÙYß, ˙LÙ[ÙjVÙVÙ, °WLL¶Rm GuT] TÙvLWÙfNÙ¨VÙ¨u
T˚Pl◊Ls. Cßp £kRÙkRd L‹Uß´u SÙuœ CVpLs Uh”˙U °˚Pj’s[]. ƒXÙYß
Gu\ ˘TVo „h•V˚U, Ru UL∞u ®˚]YÙp Gufl ˘NÙpXlT”°\’. ƒXÙYß´p 278
—˙XÙLeLs Es[].
L¶RÆV¤dœm YÙ]ÆV¤dœ˙U TÙvLWÙfNÙ¨VÙo ™œkR Su˘LÙ˚PLs A∞j’s[Ùo.
AYt±p £XYt˚\l Tt± Uh”m Ce˙L œ±l©PXÙm.
ÈwVjßu Ußl◊ Gp˚XVt\’ G]d Li”©•jRYo TÙvLWÙfNÙ¨VÙo. LP‹s
Gp˚XVt\Yo GuT˚Rl ˙TÙu˙\ ÈwVjßu Gp˚X´u˚U´˚] AYo ®flÆ]Ùo.
L¶RÆV≠u ™Ll˘T⁄m TVuTÙ˙P CkRdLi”©•l◊.
L¶Rf NVpTÙh” UiPXjßp AYo T˚PjR∞jR LÙhTÙ”Ls ™Ll˘T⁄m TflL[Ùœm.
TWlT[‹, L]A[‹ GuT] Li”©•dL AYo E⁄YÙd°V „jßWeLfidœl ˘T⁄m Teœ
Es[]. dΩd Cd˙Y`‡Lfim ˚TœYÙhWÙt±d Cd˙Y`‡Lfim Au]Ù¨u ËpL∞p
TVuT”jRlTh”s[]. £jRÙkR £˙WÙU¶´u CWiPÙm CV≠p ∏˙ZRWlTh”s[
ÿ˚\´p Es[ ©Wf£˚]Ls ˙NodLlTh”s[].
1) x4 + 12x3+6x2 + 35
2) x3-2x2 – 400x = 9999
Au]Ùo Cj’˚\´p ™œkR A±‹Pu Æ[e°]Ùo G] C’ ®fl‹°u\’. A˚Rl˙TÙufl
˙YflThP ÷iL¶Rjߤm (Differential calculas) AYo ™œkR A±‹s[YWÙLj ßLrkRÙo.
°WLeL∞u CVdLm, ÿd˙LÙQÆVp GuT]Yt±¤m TÙvLWÙfNÙ¨VÙo ◊Lr˘Tt±⁄kRÙo.
ÿd˙LÙQÆV≠u Sin (A+B) = Sin A cos B +CosA Sin B Gu\ YÙnlTÙ˙P TÙvLWÙfNÙ¨VÙ¨u
‘NÙT˙VÙ’Gu\ ˙LÙhTÙ”. L¶RÆV≠u ÷hTUÙ] ©Wf£˚]L˚[ Au]Ùo ˘NnŸs
ÿ˚\´p —˙XÙL Y•Æp Y∞´h”s[Ùo. Au]Ùo Y∞´hP ©Wf£˚]Ls A˚]j’˙U
AZ°Vp EQo‹m ®˚\kR˚Y. G”j’dLÙh” TÙodL‹m.
Aof—]u Ru T˚LY]Ù] LoQ˚]d ˘LÙpX AmT\ÙjÁ¶´≠⁄k’ Am◊Ls Gnß]Ùu
AYt±u TÙß Am◊L[Ùp AYu Ru GßWÙ∞˚Vj R”j’ ®fljß]Ùu. AmT\Ùj’¶´p
Es[ Am◊L∞u Gi¶d˚L´u YodLÍXm ˘LÙi” LoQ≤u œß˚WL˚[d ˘LÙu\Ùu.
3 Am◊L[Ùp Aof—]u LoQ≤u ˘LÙ•˚VŸm œ˚P˚VŸm Æp˚XŸm ÆZf˘NnRÙu.
1 AmTÙp LoQ≤u R˚X˚Vd ˘LÙnRÙu GjR˚] Am◊Ls Aof—]≤u
AmT\Ùj’¶´≠⁄k’ GnVlThP]?
È™´u ˙LÙ[Y•Ym, ARu Dol◊Æ˚N Tt±V A±‹ TÙvLWÙfNÙ¨VÙ¨Pm A˚Ukß⁄kR’.
◊ÆDol◊Æ˚Ndœ Au]Ùo A∞jR ˘TVo ‘RÙo¶LÙjULNdß’ Gu\Ùœm. A˚]j’l
˘TÙ⁄hLfim È™´p Æ›Y’ RÙo¶LÙjUL NdßVÙp Gufl TÙvLWÙfNÙ¨VÙo ®flÆ]Ùo.
33
CqYÙfl A±ÆVp ’˚\´¤m L¶RÆVp ’˚\´¤m TÙvLWÙfNÙ¨VÙo ®LrjßV
Bn‹L˚[Ÿm ˘NVpTÙ”L˚[Ÿm Ußl©”Y’ ™Lf £WUUÙ]’.
Îd≠h
™Ll TZeLÙXjßp ◊Lr˘Tt\ L¶R˙U˚R˙V Îd≠h. Au]Ù¨u ©\kR Fo, LpÆ
GuT] Tt±V ˘R∞YÙ] Tß‹Ls Cp˚X. °˙WdLSÙh” SLWUÙ] ˘ULW´p Au]Ùo
©\kRÙo G] SmTlT”°\’. B]Ùp, ˘ULW´p YÙrkR Îd≠h J⁄ Rj’Yf£kR˚]VÙ[o
G] Cl˙TÙ’ ·\lT”°\’. L¶R A±OWÙ] Îd≠h ˙PÙX™VÙp ®flYlThP
AXdNÙi•¨VÙÆu ˙WÙVp Ts∞d·Pjßp L¶R B£¨VWÙLl T¶VÙt±Ÿs[Ùo. C’
°.ÿ.300 Cu A”jRÙœm. ˙PÙX™, AXdNÙiP⁄dœl ©u˙RÙu±VYo. BRu£p Es[
©˙[h˙PÙv ALÙP™´p Îd≠h LpÆ Lt\Ùo G]d L⁄RTl”°\’.
Îd≠•u ™Ll˘T¨V Su˘LÙ˚P ‘G≠˘Uuhv’ Gu\ ™Ll◊Lr YÙnkR ËXÙœm.
C˚Rl˙TÙu\ J⁄ Ë˚Xl T˚PjRp ©\WÙp CVXÙR’. AjR˚LVf £\l◊ ™dL Ëp C’.
I˙WÙlTÙÆp CkR Ëp Tt±V A±‹ °˙WdL ˘UÙØ´p Es[ ˚L˘V›j’l ©Wß
YÙ´XÙLl ˘T\lTPÆp˚X. UÙ\ÙL A˙W©V ˘UÙØ˘TVol©≠⁄kRÙœm. A˙W©V
˘UÙØ˘TVol©≠⁄k’ A.D. 1120 ˛Cp ARpaÙoh Gu\ Be°˙XVo CXjßu ˘UÙØ´p
G›ß]Ùo. CRuYÙ´XÙL˙Y I˙WÙl©Vo CkËp Tt± A±k’ ˘LÙiP]o. 1570˛Cp CRu
Be°X ˘UÙØ˘TVol◊ ˘Y∞YkR’. ˚T©fidœl ©u]o ˘T⁄U[Æp ÆtT˚]´p
NÙR˚] ˘Tt\ Ëp ‘G≠˘UuPv’ Bœm.
L¶RÆVp ˙LÙhTÙ”Ls G≠˘Uuhv Gu\ Ë≠p Es[]. NoYNUj’Ym, C˚QLWm,
˚TR˙LÙWv ˙LÙhTÙ” CVtL¶Rd L⁄j’dLs, YhPeLs, GiL∞u Ti˚PV YWXÙfl
Y•ÆVp L¶R Y•YeLs G] CqYÙfl L¶RÆVp NÙokR Tp˙Yfl L⁄jRÙdLeLs
G≠˘Uuhv Ë≠p Es[]. Y˚Wÿ˚\ŸPu C˚Y A˚]j˚RŸm Au]Ùo
œ±l©h”s[Ùo. C˚RjRÆW L¶RÆVp, CVt©Vp GuT]Yt±¤m Îd≠h ËpLs
T˚Pj’s[Ùo.
Cmÿ˚\´p L¶RÆV¤dœ A¨V TX Su˘LÙ˚PLfim Îd≠h A∞jRÙo. Cu˚\V
Y•ÆVp Yp¤SoLs AY¨u £X ˙LÙhTÙ”L˚[ ÆUo£d°u\]o. B]Ù¤m Y•ÆVp
L¶Rj˚Rj ˙RÙtflÆjRY⁄m Y[ojRY⁄m AY˙W GuTßp L⁄j’˙YflTÙ” Cp˚X.
˙U¤m, TX L¶R˙U˚RL∞u YÙrd˚L YWXÙflL˚[Ÿm Su˘LÙ˚PL˚[Ÿm ßWhP‹m.
˙LW[m NÙokR Ne°WÙU UÙRYu, ˙NÙUÙߨ, TW˙U—YWo......................©\ TÙWRjß]o B¨VThPo,
CWÙUÙ‡_o, LÙl˙WdLo........................˙U˚X SÙh•]WÙ] ˚TR˙LÙWv, RdLÙoj˘R, ©l˙]Ù£..........
CYoL∞u L¶R Su˘LÙ˚PL∞u ˘NpYÙdœ Sm L¶RlTÙP HtTÙ•p GkR A[Æp
Es[’ ÆYÙßdL‹m, œ±l◊Ls RVÙWÙdL‹m.
JlT˚Pl◊
˙U¤m L¶R˙U˚RL∞u (˙LW∞Vo, CkßVo, ˘Y∞SÙh•]o) YÙrd˚L YWXÙflm
AYoL∞u Su˘LÙ˚PLfim.
$ CWÙUÙ‡_u
$ ©jR˙LÙWv
34
$ LÙp ©P¨d ˙LÙv
$ Bod°™ºv
$ B¨VThPu
$ ©l˙]Ùf£
$ NeLU°WÙU UÙRYu
$ •.Bo LÙl©˙WdLo
$ ˘W˙] ˘PdLÙoj˙R
$ ÿLmU’ ©u Í^ Ap œYÙ¨v™
$ ˚a˙T µVÙ
$ ˘a˙WÙu
$ ˙NÙUÙߨ
$ TW˙UvYWu
$
$
35
AXœ ˛ 3
L¶Rm Lt\p A‘œÿ˚\Ÿm Lt\p ˛ Lt©jRpÿ˚\Lfim EjßLfim
ÿu‡˚W
A±ÆVp NÙok’m A±‹⁄YÙdLm NÙok’m Lt©jRp ™L G∞RÙLd œZk˚RLfidœ
R]RÙdL CV¤m J⁄ TÙP˙U L¶Rm. ˘TÙ⁄jRUt\ Lt©jRp ÿ˚\Lfim Li˙QÙhPjßu
œ˚\TÙ”Lfim TX˙Y˚[L∞p L¶Rj˚R J⁄ L•]l TÙPUÙL UÙt±´⁄d°\’. A±ÆVp
NÙok’ ˘TÙ⁄jRUÙ] BoYm Fh”m ÿ˚\´p L¶Rj˚R GqYÙfl Lt©dLXÙm Gu\
L⁄jß˚] ˙SÙd° B£¨V UÙQYoL˚[d ˘LÙi” ˘Np¤m ÿVt£ ˙Ut˘LÙs[lT”°\’.
ÿd°VUÙL 3 UiPXeL[ÙL CkR AXœ ÆY¨dLlTh”s[’. L¶Rm Lt\p A‘œÿ˚\,
L¶Rm Lt\p ˛ Lt©jRp ÿ˚\, L¶Rd Lt\p EjßLs GuT˚Y˙V Bœm. C˚Y
Jq˘YÙuflm ˙YflThP Lt\p A˚P‹LfiPu ˘RÙPo◊lT”jß˙V ˘NVpTÙ” NÙokR
ÿ˚\´p Æ[dLlTh”s[’.
Lt\p A˚P‹Ls
L¶Rm Lt\p A‘œÿ˚\Ÿm Lt\p ˛ Lt©jRp ÿ˚\Lfim EjßLfim
a) L¶Rm Lt\p A‘œÿ˚\
—tflf„Zp NÙokR’, ˘NVpÿ˚\ NÙokR’, ˘NVpTÙ” NÙokR’, £kR˚]˚Vd œ±j’s[
£kR˚], ©WfN˚]˚Vl TœjRÙWÙnRp, ˘TÙ’˚UT”j’Rp, TuÿL ˙SÙdœ,
U]dLQdœ.
$ Tp˙Yfl E[ÆVp ˛ LpÆ £kR˚]VÙ[oL∞u L¶Rd Lt\p Li˙QÙhPeLs.
$ lÏQ¨u L⁄j’L˚[ Eh˘LÙsfim ®˚XLs
$ ¨fNoh Bo v˘Lml Intelligent Learning
$ L¶Rm Lt\p A‘œÿ˚\Ÿm, Lt\p ˛ Lt©jRp ÿ˚\Lfim, EjßLfim (ELPS)
b) L¶Rm Lt\p ˛ Lt©jRp ÿ˚\Ls
$ ÆßY⁄ÿ˚\, ÆßÆ[dLÿ˚\
$ TœjRÙWÙnRp ˛ Eh°W°jRp ÿ˚\Ls
$ ˘NVpßhP ÿ˚\.
c) L¶Rm Lt\p ˛ Lt©jRp EjßLs
$ R≤STo ˛ œ›f ˘NVpTÙ”Ls
$ L⁄jRWeœLs, JlT˚Pl◊Ls, L¶RÆ˚[VÙh”Ls
d) U˚\ÿL TÙP HtTÙ” (TÙ≠] NUj’Ym, NÍL øß)
36
L¶Rm Lt\p A‘œÿ˚\
˙R∫V TÙP HtTÙ” NhPLjßu TœßVÙL, L¶Rm Lt\≠u ÿd°Vd œ±d˙LÙ[ÙLd
·±´⁄lT’ £kR˚]˚V L¶RUVUÙdLp GuTRÙœm. ˘R∞kR £kR˚], A•lT˚Pd
˙LÙhTÙ”L∞≠⁄k’ A±lÈoYUÙL ÿ•‹L˚[ A˚PŸm ß\u, ™Lÿd°V
÷iL⁄j’dL˚[d ˚LVÙfim ß\u, ©Wf£˚]L˚[ ÿ˚\VÙLl TœlTÙn‹ ˘NnY’Pu
æo‹LÙ‘m ˘TÙfll◊Qo‹ GuT]Yt˚\˙V NCERT CRtLÙLl T¨k’˚Wj’s[’.
Au\ÙPl ©Wf£˚]L∞u L¶Rÿm NÙRÙWQ L¶Rjßu ÷hTlTœlTÙn‹m Ju\ÙL
C˚Q°u\ —RkßWÿm A±‹lÈoYÿUÙ] GpXÙf £kR˚]L˚[Ÿm Ae∏L¨dœm
TVu™dL ÆYÙReL∞u „ZpLs YØ˙V L¶Rm Lt\p S˚P˘T\ ˙Yi”m. CRtœ HtT
÷hTÿm ÷‘dLÿm A±lÈoYÿUÙ] L¶RdL⁄j’L˚[ UÙQYoLs A˚PVf
˘NnRYtœl Tp˙Yfl ÿ˚\Lfim EjßLfim S˚Pÿ˚\´p Es[]. Ti˚PV LÙXeL∞p
L¶Rm LtL SPj˚Rd˙LÙhTÙ” NÙokR ÿ˚\˙V TVuT”jRlTh•⁄kR’ G≤‡m, Cufl
ÿt±¤m A±‹⁄YÙdL ÿ˚\˙V TVuT”jRlT”°\’. S≈]UVUÙ]’m A±ÆVp
NÙokR’UÙ] A±‹⁄YÙdL ÿ˚\´p L¶Rm Lt\p S˚P˘TflYRtLÙ] ÿ˚\L˙[
L¶RdLt\p A‘œÿ˚\´u TœßVÙL Cufl ©uTt\lT”°\’
A‘œÿ˚\ GuT’ J⁄ L⁄j˚Rl Tt±V Sm Li˙QÙhP˙U Bœm. L¶Rd Lt\p
A‘œÿ˚\ GuT’, L¶RdLt\p GqYÙfl C⁄dL˙Yi”m Gu\ S≈]d Li˙QÙhP˙U
Bœm. A’ ÿt±¤m œZk˚RL∞Pm —VUÙL A±‹⁄YÙdLm S˚P˘Tflm Y˚L´p
A˚URp ˙Yi”m. L¶Rd LpÆ´u A‘œÿ˚\ GuT’ Gu] Gufl Eh˘LÙs[d·•V
£X Li˙QÙhPeLs C≤ Æ[dLlT”°u\].
©VÙ˘` (A±‹⁄YÙdL YÙRm) Congnitive Constructivisim
A±‹⁄YÙdLYÙRm ˘RÙPoTÙ] L⁄j’Ls
$ LpÆ CVpTÙ]’m E´¨Vp NÙokR’UÙ] ˘NVXÙœm.
$ Ru —tflf„Z¤Pu TZœm˙TÙ’ E⁄YÙœm RLY˚Ul◊ (Adaptation) TX]ÙLd Lt\p
S˚P˘Tfl°\’..
$ æo‹ LÙQlTP˙Yi•V A±ÆVp NÙokR NU®˚X´u˚U˙V Lt\˚X ˙SÙd°
YØSPj’°\’.
$ S˚Pÿ˚\´¤s[ A±ÆVp Æßÿ˚\LfiPu Jj’l˙TÙLÙR GkR A±ÆVp
LÙW¶Ÿm ˘TÙ⁄[t\RÙ°Æ”m.
$ Es[ÙoYm Lt\˚X ˙SÙd° YØSPj’m J⁄ LÙW¶VÙœm.
$ A‡TYeLs YÙ´XÙL A±‹ E⁄YÙdLlT”°\’.
$ œZk˚Rdœf —RkßWUÙL‹m —VUÙL‹m LtTRtLÙ] YÙnl◊L˚[ E⁄YÙdœY˙R
B£¨V¨u LP˚U.
G.LÙ. J⁄ SÙtLWjßu ˙LÙQeL∞u A[‹L∞u ˘RÙ˚L GqY[‹? Gu\ ◊ßV
©Wf£˚]˚Vd œZk˚R Nkßd°\’. G]d L⁄R‹m.
37
$ CkRl©Wf£˚] œZk˚R´Pm CVpTÙL NU®˚X´u˚U˚V E⁄YÙdœ°\’.
$ ÿu]o Lt\ A±ÆVp NÙokR ·flLfiPu (v∏UÙ) UÙQYu ◊’l ©Wf£˚]˚Vj
˘RÙPo◊lT”j’°\Ùu (SÙtLWj˚R CWi” ÿd˙LÙQeL[ÙL UÙt\XÙm Guflm J⁄
ÿd˙LÙQjßu ˙LÙQeL∞u A[‹L∞u ˘RÙ˚L 180 •°¨ Guflm °˚PjR
ÿu]±‹) ©Wf£˚]dœj æo‹ LÙi°\’..
$ C’ YØ ◊ßV A±˚Y A˚P°u\]o (assimilation)
$ ◊ßV A±Ætœ (SÙtLWjßu ˙LÙQeL∞u A[‹L∞u ˘RÙ˚L 360•°¨) Ru
A±ÆVp TÙo˚Y´p K¨Pm ˘LÙ”dLlT”°\’. C’˙Y C˚NRp (accomodation)
$ CjR˚LV C˚NRp ÍXm œZk˚R´Pm A˚Uk’s[ A±ÆVp TÙo˚Y
ƨY˚P°\’.
˚Y˙LÙh≥ (NÍL A±‹⁄YÙdL YÙRm)
NÍL A±‹⁄YÙdL YÙRj’Pu ˘RÙPo◊˚PV L⁄j’Ls:
$ Lt\¤m Y[of£Ÿm NÍLm NÙok’m TiTÙ” NÙok’m YÙrRp YØVÙL S˚P˘Tfl°\’.
$ ˘TÙ⁄s ˘N±kR NÍLf „ZpLfidœd LpÆ´p ™Lÿd°V CPm Ei”.
$ œZk˚RLs B£¨V¨u ERÆŸPu A±YÙdLm ˘Tfl°u\]o.
$ ·h”\‹Qo‹m Jtfl˚UŸQo‹m NÙokR ÿ˚\L˙[ LpÆ˚Vl TX‡s[RÙL
A˚Ud°\’.
$ Jq˘YÙ⁄ œZk˚R´‡˚PV‹m ZPD˚Vd (Zone of Proximal Development) L⁄jßt
˘LÙi” ©\¨u ERÆŸPu CVu\ A[Æp EVo®˚X˚V ˙SÙd°d œZk˚R˚Vd
˘LÙi” ˘NpX˙Yi”m.
$ ˙R˚YVÙ] „ZpL∞p B£¨Vo ©u’˚Q (Scaffolding) A∞j’ T•lT•VÙLf
—VdLt\˚X ˙SÙd°˙V œZk˚R˚V YØSPjR˙Yi”m.
„Zp NÙokR’
NÙRÙWQ L¶Rf £kR˚]Ls, ˙LÙhTÙ”Ls G]j ˘RÙPe° £dLXÙ] L¶Rm ˙SÙd°l
˙TÙYRtœl TßXÙL œZk˚RL∞u —tflf„ZpL∞p LÙQlT”m NÙRÙWQ ANÙRÙWQ
„ZpL˚[d L¶Rf ˘NVpTÙ”L[Ùd° A∞j˙R L¶Rm Lt©jRp S˚P˘T\ ˙Yi”m.
≈”, ≈h•p Es[ ˘TÙ⁄hLs, —tfll◊\m, Au\ÙPf ˘NVpTÙ”Ls, Æ˚[VÙh”Ls G]d
œZk˚RL∞u Lt©jRp UiPXjßtœl ˘TÙ⁄jRUÙ] ©\ —tflf„Zp NÙokR ˘TÙ⁄hLs
GpXÙm L¶Rm Lt©jRp ˘NVpÿ˚\dœ E¨V A•lT˚Pd LÙW¶Ls BLXÙm.
˘NVpÿ˚\ NÙokR’
L¶Rjßp ˘NVpÿ˚\ NÙo ß\uLfidœ ÿd°Vj’Ym A∞j’ Jq˘YÙ⁄ L¶Rf
˘NVpTÙ˚PŸm RVÙ¨j’ ˘Y∞´P ˙Yi”m. SÙuœ L¶Rf ˘NVpLs, Y•ÆVp
˘NVpÿ˚\Ls GuT]Yt˚\ Etfl˙SÙdœRp, F°jRp, AhPY˚QVÙdœRp, Jl¿”
˘NnRp, A±YÙokR TÙo˚Y, ÿ•‹L˚[ E⁄YÙdœRp, ˘TÙ’˚Ul T”j’Rp ˙TÙu\˚YŸm
L¶Rf ˘NVpTÙ”L∞p EhT”jRlTh•⁄dL ˙Yi”m.
38
˘NVpTÙ” NÙokR’.
Gi A±‹, SÙuœ L¶Rf ˘NVpLs, Y•ÆVp, A[‹Ls ˙TÙu\ Jq˘YÙ⁄
UiPXj˚RŸm ˘TÙ⁄jRUÙ] Lt\p ˘NVpTÙ”Ls YÙ´XÙLd œZk˚Lfidœ
A±ÿLlT”jR˙Yi”m. B£¨Vo Æ[dœY’m œZk˚RLs LY]UÙLd
˙Lh”d˘LÙi•⁄lT’UÙ] ÿ˚\ L¶Rm Lt©jR≠p SmUÙp Htfld˘LÙs[lTPÆp˚X.
A±‹⁄YÙdL YÙRjßp Htfld˘LÙs[d·•V ÿ˚\´p Es[ ˘NVpTÙ”L˚[ Jq˘YÙ⁄
L¶Rd L⁄jßtœm Li”©•j’d ˘LÙ”dLlTP˙Yi”m.
£kRR˚]˚Vd œ±j’s[ £kR˚] (Meta Thinking)
˘NVpTÙ”Ls YØ˙V ˘Nufl ©Wf£˚]Lfidœj æo‹ LiP J⁄ œZk˚R RÙu ˘Nu\
YØÿ˚\L˚[Ÿm ®˚XL˚[Ÿm Ußl©”Y˙RÙ” ˙R˚YVÙ] ß⁄jReL˚[Ÿm ˘NnŸm
˘NVpÿ˚\˙V £kRR˚]˚Vd œ±j’s[ £kR˚].
GkR J⁄ L¶Rf ˘NVpTÙ”m £kR˚]˚Vd œ±j’s[ £kR˚]dœ EhT”jRlTP˙Yi”m.
CRu ÍXm R]’ A±YÙokR £kR˚]Ÿm U]dLQdœm RYflL˚[l TœjR±Ÿm ß\‡m
Y[o°u\]. C’ L¶R CWN˚]dœm AlTÙp ARu AZ°Vp EQo˚Yl ◊¨k’ LÙsY’Pu
◊ßV Es[ÙokR TÙo˚Y (insight learning) E⁄l˘T\‹m œZk˚RLfidœ ER‹°\’.
©Wf£˚]˚Vl TœjRÙWÙnRp
Jq˘YÙ⁄ L¶Rl ©Wf£˚]˚VŸm œZk˚RLs A±‹lÈoYUÙL‹m A±ÆVp NÙok’m
A‘L˙Yi•Ÿs[’. ©Wf£˚] Gu]˘Yuflm GqYÙfl æo‹ LÙQ˙Yi”˘Uuflm,
GjR˚LV RW‹Ls ˙R˚Y Guflm, ARtLÙL GjR˚LV RLYpLs Æ]ÙÆp RWlTh”s[]
Guflm YÙnlTÙ”L˙[Ù, æo‹LÙ‘m ˘NVpÿ˚\˙VÙ G˚R G”j’d˘LÙs[˙Yi”
˘Uuflm œZk˚R —VUÙLl TœjR±V˙Yi”m. CqYÙfl Jq˘YÙ⁄ L¶Rl©Wf£˚]˚VŸm
A±ÆVp ÿ˚\´p A‘° E¨V æo‹LÙ‘m ÿ˚\˚V Y[odL‹m œZk˚RL˚[j
ß\‡˚PVYoL[ÙdL ˙Yi”m. Au\ÙP YÙrÆu ©Wf£˚]L˚[f NkßlT’Pu
TœjR±k’ æoUÙ]m G”dL‹m ©Wf£˚], TœlTÙn‹ ˚UV L¶Rm Lt\p ÿ˚\
œZk˚RLfidœ ER‹m. CRtœ, LQd°”Rp, (Quantification), ˘T⁄m ©Wf£˚]L˚[f £fl
©Wf£˚]L[ÙdœRp, F°jRp, ÿ•‹ G”jRp, ˙NÙßj’l TÙojRp, R≤ STo Bn‹ Bn‹
ÿ˚\ ˙TÙu\ EVo £kR˚]j ß\uLs ˙R˚YlT”m £±V, ˘T¨V L¶Rf „ZpL˚[
B£¨Vo E⁄YÙd°d ˘LÙ”dL˙Yi”m.
˘TÙ’˚UlT”jRp
L¶Rd LpÆ´u ÿd°Vd œ±d˙LÙsL∞p Jufl ˘TÙ’˚UlT”j’m ß\u ˘TflRp.
G”j’dLÙh”Ls YØ ˘TÙ’˚UlT”j’R˚X A˚PŸm TX L¶Rf „ZpL˚[ B£¨Vo
Li”©•j’d ˘LÙ”dL˙Yi”m. G∞RÙ] G”j’dLÙh”Ls ÿRp £dLXÙ]
L¶Rl˘TÙ’˚Ul T”j’Rp Y˚W Tp˙Yfl Yœl◊L∞p ˘NnV˙Yi•V˚Y Es[].
G.LÙ. 1+2+3 = 2×××××3 = 6
2+3+4 = 3×××××3 = 9
4+5+6 = 5×××××3 = 15
39
˘RÙPof£VÙ] 3 GiQp GiL∞u ˘RÙ˚L S”Ƥs[ Gi¶u 3 UPeLÙœm.
Cß≠⁄k’ 3 GiQp GiLfidœl Tßp GjR˚] GiQp GiLs Es[] G≤‡m
˘RÙ˚L LÙiTRtLÙ] ÿ˚\˚Vd œZk˚RLs ˘TÙ’˚UlT”jRh”m.
AßL G”j’dLÙh”L˚[ øeLs Li”©•dL‹m.
TuÿLf £kR˚] (Divergent Thinking)
£kR˚]˚V ˘Yq˙Yfl ÿ˚\L∞¤m ß˚NL∞¤m ˘LÙi” ˘NpYRtLÙ] FdLm A∞dœm
ÿ˚\´p L¶RdLpÆ UÙ\˙Yi”m. ß\kR Æ]ÙdLs ˙TÙu\ œ±l◊Ls, ◊ßoLs,
˘NVpßhPeLs, Bn‹Ls ˙TÙu\ YØÿ˚\L˚[ CRtLÙLl TVuT”jRXÙm. Y•ÆVp,
Gi A˚Ul◊Ls ˙TÙu\˚YŸm TVuT”jRXÙm. ÿRp AX°p RWlTh”s[ ®ß´u
◊ß⁄dœ SÙuœ ApX’ Ik’ ÿ˚\L∞p øeLs æo‹ LÙQXÙm.
EeLs B£¨Vo Utflm SiToL∞u ERÆŸPu ˙YflThP ÿ˚\L∞p ◊ßoL˚[d
Li”©•j’j æo‹LÙQ‹m. TuÿLf £kR˚]´u Jq˘YÙ⁄ L⁄jßtœm ˘TÙ⁄jRUÙ]
G”j’dLÙh”Lfim ©Wf£˚]Lfim AYt±u æo‹Lfim LÙQXÙm ApXYÙ.
U]dLQdœ (Mental Maths)
GkR J⁄ L¶Rf ˘NV˚XŸm ÿR≠p U]dLQdLÙL A∞dL˙Yi”m. N¨VÙ] Æ˚P˚V
A˚PV˙YÙ, Æ˚P˚V ˘S⁄eL˙YÙ, Æ˚P˚V A˚PYRtLÙ] N¨VÙ] YØÿ˚\L˚[
A˚PV˙YÙ U]dLQdœ ER‹°\’. ˘T⁄dLp YœjRpL˚[f ˘NÙkRUÙdL‹m U]dLQdœ
ER‹°u\’.
©Wf£˚]j æo‹ LÙiTRu ÿRp ®˚XVÙL Æ]ÙlTœlTÙn‹m ©u]o U]dLQdœm
TVuT”jR˙Yi”m. U]dLQdœ TVuT”jRÙUp Ap˙LÙ¨Rm Uh”m TVuT”jß
˘NVp˘NnŸm œZk˚RLfidœl ©˚ZLs Y⁄m YÙnl◊Ls AßLm. Gi EQo‹
Y¤l˘T\‹m U]dLQdœ ER‹°u\’.
G”j’dLÙhPÙL, 824 YœjRp 8 Gu\ ˘NV˚X A∞j’ Ap˙LÙ¨Rm ÿ˚\´p ˘NnRÙp
AßLUÙ] UÙQYoLfim 13 G] G›’°u\]o. C˙R ˘NV˚X U]dLQdLÙLf ˘NnRÙp
A˚]Y⁄dœm 103 Gu\ N¨VÙ] Æ˚P °˚Pd°\’.
—VUÙL G”j’dLÙh”Ls Li”©•j’ C’ N¨VÙ Gufl øeLs ˙NÙßj’l TÙodL‹m.
Tp˙Yfl E[ÆVp LpÆ £kR˚]VÙ[oL∞u L¶RdLt\pLi˙QÙhPeLs
Tp˙Yfl E[ÆVp A±OoL∞u ˙LÙhTÙ”Ls, TÙo˚Y ËpLs,
œ›dLXk’˚WVÙPpLs B°VYt±u ÍXm Li”©•d°u\]o. Jl◊˚U ˘Nn’
A±d˚L RVÙ¨d°u\]o. Jq˘YÙ⁄ ˙LÙhTÙ”m L¶RdLt\p A‘œÿ˚\l
Tß‹Pu GqYÙfl ˘TÙ⁄k’°\’ Gufl Li”©•j’d œ±l◊ RVÙ¨dL‹m.
lÏQ¨u Li˙QÙhPm
UÙQYoLs S˚Pÿ˚\´¤s[ A±Æu A•lT˚P´p ◊’dL⁄j’dL˚[Ÿm
A±‹L˚[Ÿm —VUÙL E⁄YÙdœ°u\]o. RLYpLs ˙Ro‹ ˘NnRp, T¨UÙt\j’dœ
40
EhT”j’Rp, æoUÙ]eLs G”jRp, L⁄’˙LÙsL˚[ E⁄YÙdœRp, A±‹Ls, A‡TYeLs
GuT]Yt˚\ A•˘VÙt± ◊ßV OÙ]m ˘TflRp B°V˚Y Lt\p ˘NVpÿ˚\´u TœßVÙL
S˚P˘Tfl°u\].
Íufl ®˚XL[ÙL A±‹ ˘TflRp S˚P˘Tfl°u\’ GuT˙R lÏQ¨u L⁄j’.
1. ˘NVpTÙh” ®˚X (Enactive Stage)
œZk˚RLs ˙SW•VÙ] A‡TYeLfidœd LÙWQUÙ°u\ ˘NVpTÙ”Ls YØVÙL ÿRp
®˚X´p ˘TÙ⁄hL˚[Ÿm L⁄j’L˚[Ÿm ◊¨k’˘LÙs°u\]o. C’ Eh°W°dœm Y[of£
ARÙY’ A±‹Y[of£dœd ˘LÙi” ˘Np°\’. C’ YØVÙLd œZk˚RL∞Pm E[ÆVp
UÙt\m S˚P˘Tfl°\’. CkR ®˚X˚Vf ˘NVpTÙh” ®˚X Gufl ·fl°u\]o.
2. ©mT ®˚X (Iconic Stage)
˘NVpTÙh” ®˚X YÙ´XÙLd °˚PjR E[ÆVp UÙt\m, ˘NVpTÙ”L∞‡˚PV˙YÙ,
˘TÙ⁄hL∞‡˚PV˙YÙ, E[ÆVp NÙo ©mTm œZk˚RL∞Pm E⁄l˘T\ ER‹°\’. Im◊Xj
ß\uL∞u Y[of£ CkR ®˚X˚V FdœÆd°\’. Im◊XuLs YÙ´XÙLd °˚Pdœm
©mTeLs Gu\ ˘TÙ⁄∞p CkR ®˚X˚Vl ©mT ®˚X Gufl ·fl°u\]o.
3. œ±¬h” ®˚X (Symbolic Stage)
©mT ®˚X YØVÙL E⁄YÙdLlT”m A±‹Lfim L⁄j’Lfim L¶l◊Lfim Y¤l˘TflY’
CkR ®˚X´XÙœm. L⁄j’⁄YÙdLjßu EVo ®˚X˙V C’.
˘NVpTÙh” ©mT œ±¬h”
®˚X ®˚X ®˚X
G.LÙ:
$ J⁄ ˘LÙj’l ÈdLs, ˘TÙ⁄hLs, E´¨Ls GuTYt±≠⁄k’ œZk˚RLfidœ ˙SW•
A‡TYm °˚Pd°\’.
$ C˚Y CpXÙR®˚X´p CYt˚\l Tt±V ©mTm E[Y[of£˚Vj R⁄°\’.
$ CYt±u Gi¶d˚L˚Vd œ±dœm GiLs, A˚PVÙ[eLs, œ±¬”Ls GuTYt±u
Y[of£ŸPu A±‹⁄YÙdLm ®˚\‹ ˘Tfl°\’.
L¶Rm Lt\≠p LY≤d ˙Yi•V˚Y.
ELPS (Experience, Language, Picture, Symbol)
A‡TYm, ˘UÙØ, TPm, œ±¬”)
L¶Rm NÙo L⁄j’⁄YÙdLjßu SÙuœ ™L ÿd°V ®˚XL˙[ E, L, P, S GuT˚Y.
E - Experience with Physical Objects
L - Language Spoken that describes the experience
P - Pictures that represent experience
S - Symbols used that generate the experience
41
Au\ÙPm LÙ‘m ˘TÙ⁄hLs ApX’ NÙRÙWQ „ZpLs TVuT”jß ˙SW• A‡TYm
E⁄YÙdœR˙X A‡TYm (Experience) YÙ´XÙL GßoTÙodLlT”°\’. ˘TÙ⁄hLs, L⁄ÆLs,
Bn‹Ls, ˘NVpTÙ”Ls ˙TÙu\Yt˚\ CRtLÙLl TVuT”jRXÙm.
˘Tt\ A‡TYeL˚[d LtL‹m —VUÙL Æ[dœYRtLÙL B£¨Vo ˙YflThP ÿ˚\L∞p
ÆY¨dL‹m YÙnl◊L˚[ E⁄YÙdœm ˙TÙ’ ˘UÙØ (Language) Gu\ CWiPÙm ®˚X
ÿ›˚U A˚P°\’.
°˚PjR A‡TYeL˚[l TPeLs, UÙߨLs Y˚WTPeLs TPÆ[dLeLs, A˚PVÙ[eLs
˙TÙu\Yt˚\l TVuT”jß œ±l©”m ˙TÙ’ TPm (Picture) Gu\ ®˚X ÿ›˚UV˚P°\’.
°˚PjR A‡TYeL˚[l ˘TÙ⁄jRUÙ] ˘UÙØ, TPm GuTYt±tœ AlTÙp ˘TÙ⁄jRUÙ]
L¶Rf £u]eLs, CXdLeLs, UÙ±Ls, ©\ A˚PVÙ[eLs GuT˚Y TVuT”jß
œ±¬PÙLd œ±l©”YRtLÙ] ß\u ˘TflY˚R˙V œ±¬” (Symbol) GuTRu YÙ´XÙL
GßoTÙodLlT”°\’.
C˚Q˙LÙ”Ls Gu\ L⁄j˚Rd œZk˚RL∞Pm E⁄YÙdL, C˚QlTdLeLs Es[
˘TÙ⁄hLs, L⁄ÆLs GuT] ˚LVÙ[‹m LÙQ‹m A‡TYjßtœ EhT”jR‹m ˘Nn°u\
TX YÙnl◊L˚[ A∞dL˙Yi”m.
˙U˚N´u TdLeLs, YÙNp, ˘Tg—, CW´p RiPYÙ[eLs, ◊jRLjßu Æ∞m◊Ls
B°V˚Y.
A‡TYj˚Rf ˘NÙkRUÙd°V œZk˚RL˚[ ˘UÙØ NÙokR ˘Y∞¬”, Æ[dLm, Jl◊˚U,
R≤jRu˚ULs, Li”©•jRp ˙TÙu\ ˘NVpTÙ”L∞p D”T”jR˙Yi”m.
C˚QlTdLeLs, C˚Qd˙LÙ”Ls CYt±u TPeLs RVÙ¨dœm NVpTÙ”L˚[˙V ©u]o
˘NnV˙Yi”m. C˚Qd˙LÙ”Ls EhT”°u\ TX C⁄T¨UÙQ/ ÿlT¨UÙQ/ Y•ÆVp
TPeLs Y˚WYRu YÙ´XÙL CYt˚\l Tt±V L⁄j’⁄YÙdLÿm Y¤l˘Tfl°\’.
C˚Qd˙LÙ”L˚[d œ±l©”°u\ œ±¬”Ls (//ARÙY’ C˚Qd˙LÙ”Ls), ©\ TPeLs
˘R¨k’ ˘LÙsY’m A‡TYeLs YØVÙLl ˘Ttfld˘LÙsYRu YÙ´XÙLd L⁄j’⁄YÙdLm
ÿ›˚U ˘Tfl°\’.
—VUÙL AßL G”j’dLÙh”L˚[d Li”©•j’ ELPS Y¨˚N LÙh”m
˘NVpTÙh”d œ±l◊Ls RVÙ¨dL‹m.
L¶Rm Lt\p ÿ˚\Ls
L¶Rm Lt\p ÿ˚\Ls Íu\ÙL Y˚LlT”jRXÙm.
a) L⁄j’dLs ˙LÙhTÙ”Ls, Y˚WV˚\Ls ˙TÙu\Yt±u Lt©jR¤dœl TVuT”j’m
ÿ˚\Ls (G.LÙ. ÆßY⁄ÿ˚\, Æß Æ[dL ÿ˚\)
b) ©Wf£˚]j æo‹ LÙiTRtœl TVuT”j’m ÿ˚\Ls (G.LÙ. TœjRÙWÙnRp /
˘RÙœjRÙWÙnRp ÿ˚\)
42
h h
b b
c) £dLXÙ]’m ƨYÙ]’UÙ] L¶RdL⁄j’L˚[ E⁄YÙdL‹m æo‹LÙQ‹m
TVuT”j’m ÿ˚\Ls. (G.”. Bn‹ ÿ˚\, ˘NVpßhPÿ˚\, ˙NÙßjR±Ÿm ÿ˚\)
Tp˙Yfl L¶Rm Lt©jRp ÿ˚\Ls L¶Rd L⁄j’l T¨UÙt\m SPj’YRtœ ER‹°u\].
ÿd°Vd L⁄j’dLs
a) ÆßY⁄ÿ˚\ L¶Rd˙LÙhTÙ”Ls, L⁄j’dLs, Y˚WV˚\Ls, ÆßÆ[dLÿ˚\
„jßWeLs GuT˚Y E⁄YÙdœYRtœ
b) TœjRÙWÙnRp ˛ ˘RÙœjRÙWÙnRp ÿ˚\
c) ˘NVpßhPÿ˚\
$ ˙NÙßjR±Ÿm ÿ˚\
$ Bn‹ ÿ˚\
(i) ÆßY⁄ÿ˚\ (Inductive Method)
NÙRÙWQ˙UÙ ANÙRÙWQ˙UÙ B] G”j’dLÙh”Ls YØ˙V ˘Nufl ˘TÙ’YÙ] ÿ˚\L˚[
A˚PY˙R ÆßY⁄ÿ˚\´u R≤f£\l◊. J⁄ L⁄j˙RÙ, ˘TÙ’d˙LÙhTÙ˙PÙ, Y˚WV˚\˙VÙ,
„jßW YÙd°V˙UÙ E⁄YÙdœYRtœ CjR˚LV ˘NVpTÙ”Ls ˘TÙ⁄jRUÙ]˚YVÙœm.
G.LÙ. (1) ÿd˙LÙQjßu ˙LÙQeL∞u ˘RÙ˚L 180 •°¨ BL C⁄dœm
œZk˚RLfidœ 4 ApX’ 5 ˙YflThP ÿd˙LÙQeLp A∞dLl T”°u\]. Jq˘YÙ⁄
ÿd˙LÙQj߇˚PV‹m Íufl ˙LÙQeL∞u A[˚Yd œZk˚RLs A[k’ G›ß ˘RÙ˚L
LÙQh”m. ˘RÙ˚L 180 •°¨ Gufl AYoLs —VUÙLd Li”©•d°u\]o.
C˚QVÙ] ˘NVpTÙ”L˚[ ¡i”m ˘Nn°u\]o. CjR˚LV ˘NVpTÙ”L∞≠⁄k’
UÙQYoLs GkR J⁄ ÿd˙LÙQjߤm ˙LÙQeL∞u A[‹L∞u ˘RÙ˚L 180 •°¨ BL
C⁄dœm G]l ◊¨k’ ˘LÙs°u\]o.
G.”. 2 ˘Ne˙LÙQ ÿd˙LÙQeL∞u TWlT[‹ A=1/2 bh
Jq˘YÙ⁄ ˘Ne˙LÙQ ÿd˙LÙQj߇P‡m A˙R A[‹s[ ˙Y˘\Ù⁄ ˘Ne˙LÙQ
ÿd˙LÙQj˚Rf ˙Noj’ ˚YjRÙp ˘NqYLm Bœm. ˘Ne˙LÙQ ÿd˙LÙQjßu
TWlT[‹ = × ø[m ALXm
2
-= ½ bh
43
(ii) ÆßÆ[dLÿ˚\ (Deductive Method)
˘TÙ’d˙LÙhTÙ”, Y˚WV˚\, „jßWm ˙TÙu\ L¶RdL⁄j’dL˚[ ÿR≠p ·±, A’ N¨
Gufl G”j’dLÙh”Ls YÙ´XÙLd Li”©•lT˙R ÆßÆ[dL ÿ˚\´u £\l◊
˙U˙X RWlTh”s[ G”j’dLÙh”L˚[ ÆßÆ[dL ÿ˚\´p ˘NVpTÙ”L[ÙLj
RVÙ¨dL‹m.
ÆßY⁄ÿ˚\, ÆßÆ[dL ÿ˚\Ls YØ˙V Li”©•dLd ·•V ˙Yfl
G”j’dLÙh”L˚[ øeLs Li”©•dL‹m.
ÆßY⁄ÿ˚\/ ÆßÆ[dLÿ˚\L∞u ®˚\Lfim œ˚\Lfim
C⁄ ÿ˚\Lfim A±‹⁄YÙdL YÙRj’Pu ˙Nok’ ®tT’m ˘NVpTÙ” NÙokR’m Bœm.
B]Ùp, C˚Y Jq˘YÙuflm ˘TÙ⁄jRUÙ] ÿ˚\´p TV‡s[RÙL ˘Y∞´PlTP˙Yi”m.
C˚Y Jq˘YÙu±u ®˚\Lfim œ˚\Lfim ∏˙Z RWlT”°u\].
ÆßY⁄ÿ˚\´u ®˚\Ls
$ L⁄j˚R EsYÙeœRp ÿ›˚UVÙL ®Lr°\’
$ œZk˚RL∞u TeL∞l◊
$ Bn‹ U]lTÙu˚U
$ A±YÙojR £kR˚] NÙokR’
$ ˙SW•VÙ] Etfl˙SÙdLp
$
$
œ˚\Ls
$ ˙SW ÆWVm
$ GpXÙf „ZpL∞¤m ˘TÙ⁄jRUt\’
$
$
ÆßÆ[dL ÿ˚\´u £\l©Vp◊Ls
$ ˘TÙ’d˙LÙhTÙ”L∞≠⁄k’ R≤lThP G”j’dLÙh”L˚[ ˙SÙd°
$ NÙRÙWQ ®˚XÆ≠⁄k’ ANÙRÙWQUÙ] ®˚X˚V ˙SÙd°.
$ J⁄ ˙LÙhTÙh•≠⁄k’ ˙Y˘\Ù⁄ ˙LÙhTÙh•˚] ˙SÙd°
G.LÙ. ˘NqYLjßu TWlT[‹ = ø[m x ALXm Gu\ „jßWjß]Ùp TVuT”jßl TWlT[‹
Li”©•d°u\]o.
44
®˚\Ls
$ —⁄dLÿm ˙SW CXÙTÿm
$ ®˚]YÙt\˚X Y[od°\’.
$ ©Wf£˚]j æo‹ LÙiTßp ˘TÙfll◊Qo‹m ˙YLÿm
$
$
œ˚\Ls
$ E[ÆVp NÙokR’ ApX
$ ®˚]YÙt\¤dœ AßL ÿd°Vj’Ym
$ œZk˚RL∞u ÿ›l TeL∞l◊ Cp˚X.
$
$
ÆßY⁄ÿ˚\, ÆßÆ[dL ÿ˚\ GuT˚Y ˙YflThP Lt\p ÿ˚\Ls G≤‡m
Ju˙\Ù˘PÙufl C˚QkR˚Y Bœm.
ÆßY⁄ÿ˚\dœm, ÆßÆ[dL ÿ˚\dœm L¶R TÙPl◊jRLm TœlTÙn‹
˘Nn’ G”j’dLÙh”Ls Li”©•dL‹m.
II TœjRÙn‹ ˛ ˘RÙœjRÙn‹ ÿ˚\Ls (Analytic - Synthetic Methods)
L¶Rd ˙LÙhTÙ”Lfidœj ˘R∞‹ ˘LÙ”lTRtœm ©Wf£˚]˚V Ht˘\”dL‹m ©Wf£˚]j
æo‹ LÙiTRtœm NÙRÙWQUÙLl TVuT”jRY’ TœjRÙn‹, ˘RÙœjRÙn‹ ÿ˚\L[Ùœm.
J⁄ ©Wf£˚]˚V Ht˘\”dœm ˙TÙ’ Al©Wf£˚] £±V LLÙW¶L[ÙLl TœdLlT”°\’.
AkRl ©Wf£˚]´u æo‹ LÙQ ER‹m ÿ˚\´p Aßp APe°´⁄dœm LÙW¶Ls
TœlTÙn‹ ÍXm ©¨dLlT”°u\]. CqYÙfl ©Wf£˚]´u æo˚Yl◊¨k’˘LÙi” æo‹
LÙQ CV¤°\’.
TœlTÙn‹ Gu\Ùp ©Wf£˚]˚V TœjR¤m, ˘RÙœjRÙn‹ Gu\Ùp ©Wf£˚]dœj æo‹m
Bœm. ÆßY⁄ÿ˚\, ÆßÆ[dLÿ˚\ YØ˙V œZk˚RLs ÿu]o Lt\ L⁄j’dLs,
˙LÙhTÙ”Ls, Y˚WV˚\Ls, „jßWeLs ˙TÙu\Yt˚\ Ceœl TVuT”jR˙Yi”m.
æo‹LÙQ ˙Yi•V L¶Rl ©Wf£˚]L˚[ ÷hTUÙLl TœlTÙn‹ ˘Nn’ æo‹LÙ‘m
˘NVpL˚[ ˙SÙd°f ˘NpY˙R CRu £\l©Vp◊. TœlTÙn‹ ®˚X´p Æ]ÙÆp
RWlTh”s[˚Y G˚Y? Li”©•dL˙Yi•V˚Y G˚Y? ˙U¤m ˙R˚YVÙ] RLYpLs
G˚Y? „jßW YÙd°VeL˚[l TVuT”jR ˙Yi”UÙ? ˙U¤m Li”©•dL ˙Yi•V
RW‹Ls G˚Y? TÙu\Yt˚\d œZk˚RLs ◊¨k’˘LÙs[ Yi”m. RÙœjRÙn‹ ®˚X´p
˙U˙X Li”©•jR RLYpL˚[l TVuT”jß ˘TÙ⁄jRUÙ] ˘NVp ÿ˚\L˚[Ÿm
„jßWYÙd°VeL˚[Ÿm E˙T˙VÙ°j’ ©WNf£˚]j æo‹/ Æ˚P Li”©•dL˙Yi”m.
45
G.LÙ. Ts∞d·Pjßp UÙQYoL∞u ˙UXÙi˚U´p SPdœm ·h”\‹f NeLd L˚Pdœl
˘TÙ⁄hLs YÙeL 10,000 ÏTÙn Es[’. ˘UÙjR ÆtT˚] L˚P´u Æ˚XjRLYp
AhPY˚Q ∏˙Z RWlTh”s[’. GpXÙ Y˚Ll ˘TÙ⁄hL˚[Ÿm EhT”jß,
Jq˘YÙ⁄Y˚LŸm Gi¶d˚L´p œ˚\kR A[Æp 100 Y⁄m T•VÙL J⁄ ˘LÙsÿRp
Th•Vp RVÙ¨dL‹m, (10,000ÏTÙn ÿ›Y’m ˘NXÆPlTP˙Yi”m)
Æ˚X ÆYW AhPY˚Q
Y˚L Æ˚X
1. ˙SÙh”l◊jRLm 10.00
2. ˙T]Ù 3.00
3. ˘Tu£p 1.00
4. v˘L´p 2.00
5. AØlTÙu 1.00
6. L¶Rl˘Th• 20.00
7. ®XlTPeLs 12.00
8. T˚N 7.00
9. LXo ˙T]Ù 5.00
10. v˘Lf ˙T]Ù 10.00
11. UÙodLo 5.00
12. Æ[dLlTPd LÙ°Rm 3.00
CkRl ©Wf£˚]dœj æo‹LÙQ ÿR≠p TœlTÙn‹ Æ]ÙdLs RVÙ¨dL˙Yi”m.
1. Li”©•dL ˙Yi•V˚Y G˚Y? / RVÙ¨dL˙Yi•V˚Y G˚Y?
2. Gu˘]u] RLYpLs RWlTh”s[]?
$ ˘TÙ⁄hLs YÙeœYRtœ E¨V ˘RÙ˚L GqY[‹?
$ Jq˘YÙ⁄ Y˚L´¤m GjR˚] ≈Rm YÙeL˙Yi”m?
$ ˘UÙjRm YÙeL˙Yi•V Y˚LLs GjR˚]?
$ Æ˚X ÆYW AhPY˚Q RWlTh”s[RÙ?
3. ˘LÙsÿRp Th•V˚X GqYÙfl RVÙ¨dLXÙm?
4. Jq˘YÙ⁄ Y˚L´¤m 100 ≈Rm YÙe°]Ùp ˘RÙ˚L ¡ß Y⁄UÙ? G≤p GqY[‹?
5. ¡ß Y⁄m ˘RÙ˚Ldœ Gu˘]u] ˘TÙ⁄hLs GjR˚] ≈Rm YÙeLXÙm?
6. Cl˙TÙ’ ˘RÙ˚L 10,000 BœUÙ?
7. AlT•VÙ]Ùp ˘LÙsÿRp Th•Vp GqYÙfl RVÙ¨dLXÙm?
46
˘TÙ⁄jRUÙ] Th•Vp RVÙ¨j’ 10,000 ÏTÙndœ ˘LÙsÿRp Th•Vp
RVÙ¨dL‹m. Jq˘YÙ⁄ TœlTÙn‹ Æ]ÙÆtœ ˙SWÙL‹m Æ˚P G›R‹m
˙U¤m TX ©Wf£˚]L˚[d Li”©•j’ TœlTÙn‹, Æ]ÙdLs
G›ßjæo‹ LÙiL.
TœlTÙn‹ ÿ˚\´u £\l©Vp◊Ls
$ A±‹lÈoYUÙ]’
$ ˘NVp ß\uL∞u Y[of£
$ ˙RPp U]lTÙu˚U
$ ©Wf£˚]˚V Gßo ˘LÙsYRtLÙ] Ru]m©d˚L
$ ◊’f „ZpL∞p TVuT”j’m ß\u
$
$
˘RÙœjRÙn‹ ÿ˚\´u £\l©Vp◊Ls
$ ˘R¨kRYt±≠⁄k’ ˘R¨VÙRYt˚\ ˙SÙd°
$ ©Wf£˚]j æo‹dœ ÿ›˚U A∞d°\’.
$ ÿ˚\VÙLl Tß‹ ˘Nn°u\].
$
$
(III) ˘NVpßhP ÿ˚\, ˙NÙßjR±Ÿm ÿ˚\, Bn‹ÿ˚\
™L ß\˚UVÙ]’m £dLpLs ®˚\kR’m YœlT˚\dœ ˘Y∞˙VŸm ˘Np°u\ ˘T¨V
L¶Rl ©Wf£˚]L∞u æo‹Ls LÙiTRtœ CkR ÿ˚\Ls TVuT”jRlT”°u\].
TœlTÙn‹, Eh°W°jRp ÿ˚\Ls YœlT˚\´p æo‹LÙQ CVXÙR ©Wf£˚]Lfidœ C˚Y
YÙ´XÙLj æo‹ LÙQXÙm. £X ˘NVpßhPf ˘NVpTÙ”Ls ∏˙Z RWlTh”s[].
G.LÙ. 1. ˙LW[ÙÆu Tp˙Yfl UÙYhPeL∞p Nu\ 10 Bi”L∞p TnR U˚Z´u NWÙN¨
Li”©•j’ Y˚WTPjRÙ∞p TPÆ[dLUÙLj RW‹m.
2. ÆtT˚]dœ Y⁄m Tp˙Yfl ˙NÙl◊L∞u AXœ Æ˚X (°WÙ™tœ GqY[‹) G]d
Li”©•j’ Jl◊˚U ˘NnV‹m.
3. NÙ˚XdLh”UÙ]jßtLÙLd œÆj’ ˚YdLlTh”s[ Np≠ ·hPeL∞u L]A[‹
LÙQ‹m (˘NqYLl ThPLm, ©W™”)
47
˘NVpßhP ®˚XLs
ßhP™Pp
$ ©Wf£˚] A‡TYlT”Rp/ ©Wf£˚] ˘Y∞¬”
$ ©Wf£˚]j æoÆu YØÿ˚\Ls
$ ÆYWf˙NL¨l©u YÙnl◊Ls
$ ÆYWf ˙NL¨l◊d L⁄Æ RVÙ¨jRp
$ ÆYWf ˙NL¨l◊
$ TœlTÙn‹m L⁄j’ ÿ•‹m
$ A±d˚L RVÙ¨jRp
$ ˘Y∞¬”, LXk’˚WVÙPp
˘NVpßhPjßu Ußl¿”, ˙Ut·±V Jq˘YÙ⁄ ®˚X´¤m ®Lr°u\’.
˘RÙPdL®˚X Yœl◊L∞p ˘LÙ”dLd·•V ˘NVpßhPl
©Wf£˚]L˚[d Li”©•dL‹m. ˘TÙ⁄jRUÙ] ˘NVpßhPm ˘Nn’
A±d˚L RVÙ¨dL‹m.
˙NÙßjR±Ÿm ÿ˚\ (Experimental method)
YœlT˚\´˙XÙ L¶R BnYLjß˙XÙ ˚Yj’ ˙NÙR˚]f ˘NVpTÙ”L∞p D”Th”
◊’A±‹⁄YÙdLm ˘T\˙YÙ L¶Rl ©Wf£˚]Lfidœj æo‹ LÙQ˙YÙ ˘NnY˙R
˙NÙßjR±Ÿm ÿ˚\.
˙NÙßjR±Ÿm ÿ˚\ ÆßY⁄ÿ˚\ ˛ ÆßÆ[dL ÿ˚\L˚[ÆP EVokR ®˚X´p Es[’m
TœlTÙn‹, ˘RÙœlTÙn‹ ÿ˚\L˚[ÆP ÷hTUÙ]’m BZÿs[’m Bœm.
£±V Yœl◊L∞p H\dœ˚\V Æ[dLeLfim £dLpLfim œ˚\kR ˘NVpTÙ”L˚[˙V
˙Ro‹ ˘NnVlTP˙Yi”m.
G.LÙ. 1. J⁄ ≠hPo ˘LÙs[[‹ Es[ Tp˙Yfl ˘NqYLl TÙjßWeLs RVÙ¨dL‹m.
CkRf ˙NÙR˚]dœj ˙R˚YVÙ] L⁄ÆLs ˘NVpTÙh” ÿ˚\,
˙NÙßjR±Ÿm TX˚]l Tß‹ ˘NnYRtœ E¨V AhPY˚QLs
RVÙ¨dL‹m.
$ CkRf ˙NÙR˚]´≠⁄k’ œZk˚RLs Eh°W°dLd ·•V £X TXuLs.
$ J⁄≠hPo ˘LÙs[[‹ Es[ Tp˙Yfl ˘NqYLl TÙjßWeL∞u ø[m, ALXm, EVWm
GuT]Yt±u C˚P˙VŸs[ ˘RÙPo◊. Tp˙Yfl Y˚Ll TÙjßWeLs RVÙ¨dœm ˙TÙ’
™Ldœ˚\YÙ] ÍXl˘TÙ⁄s GkRl TÙjßWjßtœ Gufl ©¨jR±Y’Pu AkRl
TÙjßWjßu Y•Yÿm A±°u\]o.
48
$ Lh”UÙ]f ˘NVpTÙ”Ls YØ˙V ˘Tfl°u\ ÷hTm, ’p≠Vm, ÿ›˚U, E⁄YÙdL
EQo‹ B°V˚Y.
˙NÙßjR±Ÿm ÿ˚\´u ®˚\Ls
$ œZk˚RL∞Pm BoYÍh”Y’Pu B]kRÿm A∞d°u\’.
$ ˙LÙhTÙ”L˚[f —VUÙLd Li”©•d°u\]o
$ L⁄j’ Y[of£ ÿ›˚UVÙL S˚P˘Tfl°\’.
$ ˘NVpTÙ”Ls YØ Lt\p S˚P˘Tfl°\’.
$ Bn‹ U]lTÙu˚U Y[o°\’.
$ œZk˚RL∞u ÿ›lTeL∞l◊
$ L⁄ÆL˚[d ˚LVÙfim ß\u ˘Tfl°u\]o.
$
$
˙NÙßjR±Ÿm ÿ˚\´u œ˚\Ls
$ ˙SWm AßLm
$ ˘NX‹ AßLm
$ A±YÙokR £kR˚]dœ ÿd°V™u˚U
$ GpXÙl TÙPlTœßLfidœm CkRÿ˚\ YÙ´XÙL YÙnl◊Ls Cp˚X
$ B£¨V¨u ERÆ/ YØÿ˚\Ls Cu±V˚UVÙR’.
˙NÙßjR±Ÿm ÿ˚\dœ ·”RXÙ] G”j’d LÙh”L˚[ Li”©•dL‹m.
Bn‹ ÿ˚\ (Heuristic method)
˘NVpßhPm, ˙NÙßjR±Ÿm ÿ˚\ GuTYt˚\ ÆP EVo ®˚X´p Es[’ Bn‹ ÿ˚\,
CkR ÿ˚\´u ®˚XLs, œ±d˙LÙsLs, TXuLs GuT˚Y ˘NVpßhP Bn‹
ÿ˚\Lfidœ JlTÙ]˚Y. B]Ùp A[‹, £dLp, ˙SWm GuT˚Y AßLm. ◊’d
L⁄j’dLfim A±‹Lfim E⁄YÙdLlT”YRtœm L¶Rl ©Wf£˚]L∞u æo‹ LÙiTRtœm
Bn‹ ÿ˚\ ER‹m. ˘TÙ’YÙL EVo Yœl◊L∞p CRu YÙnl◊L˚[ AßLUÙLl
TVuT”jRXÙm. I find, I discover GuT˚Y˙V ∂Ψv•d Gu\ ˘NÙp≠u ˘TÙ⁄s.
œZk˚RL∞u YV’, ÿu]±‹ GuT] A•lT˚PVÙLd ˘LÙiP Jq˘YÙ⁄ Yœl©tœm
˘TÙ⁄jRUÙ] Bn‹ÿ˚\f ˘NVpTÙ”L˚[d Li”©•dL˙Yi”m.
G.LÙ. 1. AßL ˙_Ù• CQdL GiL˚[d Li” ©•dL‹m (Amicable Numbers)
220dœ 220 RÆW ©\ LÙW¶L∞u ˘RÙ˚L 284, 284dœ 284 RÆW ©\ LÙW¶L∞u
˘RÙ˚L 220. G]˙Y, 220, 284 B°V] CQdL GiLs. CjR˚LV ˙_Ù•L˚[d
Li”©•dL Ko Bn‹f ˘NVpßhPUÙL A∞dLXÙm.
49
G.LÙ.2. CkßVÙÆp ˙YflThP ˘UÙØL∞p TVuT”j’m L¶R CXdLeL∞u G›j’
E⁄dL˚[d Li”©•dL‹m.
G.LÙ.3. ˚TR˙LÙ¨Vu ͢YiLs Li”©•dL CVtL¶R ÿ˚\ E⁄YÙdL‹m.
˘NVpßhPm, ˙NÙßjR±Ÿm ÿ˚\, Bn‹ ÿ˚\ GuTYt±u Ti◊Ls, LÙW¶
B°V] J˙W ˘NVpTÙh˚P˙V ˘Yq˙Yfl Yœl◊L∞p ˘NVpßhPUÙL˙YÙ
BnYÙL˙YÙ, ˙NÙßjR±Ÿm ÿ˚\VÙL˙YÙ ˘LÙ”dLXÙm. Jq˘YÙu±tœm E¨V
Lt\pÿ˚\, Lt\p EjßLs, RLYp ˙NL¨l◊ ÿ˚\Ls B°V˚Y RVÙ¨dL
B£¨Vo ˙R˚YVÙ] ÿu BVjReLs ˘NnV˙Yi”m.
Bn‹ ÿ˚\´u ®˚\Ls
$ Lt\p ˘NVpÿ˚\´p œZk˚RL∞u ÿ›l TeL∞l◊
$ A±˚Yf —VUÙLl ˘Tfl°u\]o
$ ◊’f „ZpL∞p TVuT”jR CVp°\’.
$ —VUÙLd Li”©•d°u\]o Gu\ U] ®˚\‹.
$ ˘TÙ⁄h ˘N±‹Pu Lt°u\]o.
$ ˘RÙPo LpÆdœ FdLm A∞d°\’.
œ˚\Ls
$ GpXÙl TÙPlTœßLfidœm ˘TÙ⁄kßV’ ApX.
$ ˙SWm AßLm
$ GpXÙd œZk˚RL˚[Ÿm BnYÙ[o Gu\ ®˚X´p ˙UmT”jR CVXÙ’.
$
$
Bn‹ ÿ˚\dœ ÿRXÙL G”j’d LÙh”Ls Li”©•dL‹m.
Tp˙Yfl Lt\p ˛ Lt©jRp EjßLs
˙YflThP Lt©jRp ÿ˚\Lfidœ AlTÙp L¶R Yœl©p TVuT”jRd ·•V £X
œ±l◊L˙[ Lt\p ˛ Lt©jRp EjßLs.
$ R≤STo ˛ œ›d LXk’˚WVÙPpLs
$ JlT˚Pl◊
$ ÆYÙRm
$ ˘NVp ßhPm
$ L¶Rf ˙NL¨l◊
$ L¶R Æ˚[VÙh”
50
$ L¶R Y¨˚NLs
$ L¶Rd L˚R, LÆ˚R
$ L¶Rl ◊ßoLs
$ UÙߨLs ˘NnRp/ LÙh£lT”j’Rp
$ I.£.•.NÙokR EjßLs, ˘Y∞¬”Ls, T¨UÙt\f ˘NVpLs, Al˘Xh”Ls.
$ UÙߨYœl◊Ls.
Jq˘YÙ⁄ Ejßdœm ˘TÙ⁄jRUÙ] Lt\p A˚P˙YÙ L⁄j˙RÙ Li”©•j’ ®Lr‹Ls /
˘NVpTÙ”Ls / G”j’dLÙh”Ls RVÙ¨dL‹m. Jq˘YÙu±tœm E¨V Y˚WV˚\Ls
RVÙ¨j’ œ›Æp / Yœl©p ˘Y∞´h” ˙UmT”jß Tßl◊Ls RVÙ¨dL‹m.
Jq˘YÙ⁄ Lt\p Lt©jRp Ejß˚VŸm ˘RÙPdL®˚X L¶Rl TÙP HtTÙ•≠⁄k’ RVÙ¨j’
Yœl◊L∞p ˘Y∞´P˙Yi”m.
˙YflThP œ›dL∞u L⁄jRWeLm, ÆYÙRm, L¶R Æ˚[VÙh”, A˚Ul◊Ls ˙TÙu\
Tp˙Yfl EjßLs TVuT”j’m˙TÙ’ LY≤dL˙Yi•V]Yt˚\d LXk’˚WVÙPp ÍXm
J›eœT”jRXÙm ApXYÙ?
U˚\ÿL TÙPHtTÙ”
TÙP HtTÙ” T¨UÙt\m ÍXm UÙQYoLs ˘R¨VÙUp A˚P°u\ Ußl¿”Ls, A‡TYeLs,
U]lTÙu˚ULs, ß\uLs, NÍLd·h”\‹Qo‹, GuT] U˚\ÿL TÙPHtTÙ” UiPXeLs.
TÙPHtTÙ” E⁄YÙdœTYoLs ßhP™h” EhT”j’m CjR˚LV LÙW¶Ls Lt˙TÙo
A±VÙU˙X Tp˙Yfl T¨UÙt\ EjßLs ÍXm Lt˙TÙ¨Pm ˘Nufl ˙No°u\].
Bi ˘Ti NUj’Ym, R≤d LY]m ˙R˚YlT”m UÙQYoL∞Pm NÙRÙLUÙ] U]®˚X,
NÍL øß, NUj’m, YÙnl◊L∞p NUj’Ym, R≤STo ˙YflTÙ” ˙TÙu\Yt˚\ Ae∏L¨jRp,
A±ÆVp Li˙QÙhP˚R Y[Wf ˘NnRp, —tflf „Zp TÙ’LÙl◊ ˙TÙu\ ˙YflThP
L⁄j’L˚[ U˚\ÿLTÙP HtTÙh•p EhT•jRXÙm. ˘TÙ⁄jRUÙ] TÙPlTœßLs, L˚R
UÙkRoL∞u TVoLs, A\d L⁄j’Ls APe°Vd L˚RLs, ULÙuLfi˚PV A‡TYeLfim
YÙrd˚LŸm, AW£Vp A˚Ul◊d ˘LÙs[˚LLs, A±ÆVp Li˙QÙhPm, Bi ˘Ti
L˚R UÙkRoL∞u Ti◊ SXuLs, Tp˙Yfl ThP URlTiTÙh”, Ußl◊Ls ˙TÙu\Yt˚\j
˙R˚YVÙ] ÿ˚\´p TÙPl TœßL∞p ˙NolT’ U˚\ÿ TÙP HtTÙh•u TœßVÙœm.
Tp˙Yfl Yœl◊L∞u Es[ TÙPl◊jRLeL˚[f ˙NÙR˚]´h” U˚\ÿL TÙP
HtTÙh•p Jq˘YÙ⁄ UiPXj’Pu ˘RÙPo◊˚PL ®Lr‹L˚[Ÿm NÙuflL˚[Ÿm
Li” ©•j’ œ±l◊ RVÙ¨dL‹m. AYt˚\ Yœl©p ˘TÙ’d LXk’˚WVÙP¤dœl
TVuT”jR‹m.
$ TÙ≠]f NUj’Ym
$ R≤dLY]m ˙R˚YlT”TYoL∞Pm NÙRLUÙ] U]®˚X
$ ˙YflThP Ußl◊Ls
51
$ NÍL Ußl◊Ls
$ ˘Ti NUj’Ym
$ NU YÙnl◊Ls
$ A±ÆVp Li˙QÙhPm
$ E´¨]eL∞Pm L⁄˚Q LÙh”Rp
$
$
52
AXœ ˛ 4
©Wf£˚]jæo‹ L¶RdLpÆ´p
ÿu‡˚W
©Wf£˚]˚Vl TœlTÙn‹ ˘NnŸm ß\‡m LQdœd ·h”YRtLÙ] ß\‡m (computation
skill) ˙NokR˙R L¶Rd ß\u G] G∞˚UVÙL‹m —⁄dLUÙL‹m ·\XÙm. S≈] LÙXjßp
LQdœd ·h”YRtœ CVkßW A˚Ul◊L˚[ ERÆdœ SÙPXÙm Gu\Ù¤m ©Wf£˚]dœj
æo‹ LÙQ ˙Yi•VYoLs ©Wf£˚]´p Es[YoLs ARÙY’ UÙQYoL˙[. BRXÙp
L¶Rd Lt\≠u YØVÙL ˙SÙdL™”Y’ ©Wf£˚]˚Vl TœlTÙn‹ ˘NnYRtœm
©Wf£˚]dœj æo‹ LÙiTRtœm Es[ ß\˚]d ·h”Y˙R Bœm. ©Wf£˚]dœj æo‹
LÙiTRtLÙL Htfld ˘LÙs[d·•V EjßL˚[Ÿm YØÿ˚\L˚[Ÿm ◊¨k’ ˘LÙiPÙp
Uh”˙U CR˚]f ˘NÙkRUÙdL CV¤m. AkR ®˚X´p L¶Rm Lt©jR≠p Es[
©Wf£˚]dœj æo‹ LÙiTßu ÿd°Vj’Yj˚Rl ◊¨k’ ˘LÙsY˙R CkR AX°u
œ±d˙LÙ[Ùœm.
Lt\p A˚P‹Ls
©Wf£˚]dœj æo‹ LÙiTßu Tp˙Yfl ®˚XL˚[ A±k’ ˘LÙsfiRp.
©Wf£˚]dœj æo‹ LÙiTßu Tp˙Yfl EjßL˚[ A±k’ ˘LÙsfiRp.
ÿd°Vd L⁄j’Ls
$ ©Wf£˚]dœj æo‹ LÙiTRtLÙ] Tp˙Yfl ®˚XL˚[d LiP±Rp.
$ ©Wf£˚]˚Vl ◊¨k’ ˘LÙsfiRp (understanding the problem)
$ ©Wf£˚]dœj æo‹ LÙiTRtLÙ] ßhPeL˚[j RVÙ¨jRp (design a plan for problem
solving)
$ ßhPeL˚[l ˘TÙfll˙Ttfl SPj’Rp (carryout the plan)
$ °˚PjR Æ˚P˚V ¡sTÙo˚Y ˘NnRp (look back and examine the solution obtained)
$ Make a table
$ Make an organised List
$ Draw a graph/diagram
$ Look for a pattern
$ Look backward
$ Guess & Check
$ Solve a simple or similar problem.
53
CWÙÿ U∞˚LdL˚P´p 78 ÏTÙndœ A¨£ YÙe°]Ùu. LÙnL±dL˚P´p 45 ÏTÙndœ
LÙnL±Ÿm YÙe°]Ùu. CqÆ⁄ TÙ⁄hL˚[Ÿm Noj’ YÙeL GqY[‹ ÏTÙn NXYÙœm?
CkRl ©Wf£˚]dœ Æ˚P LÙ‘Rp UÙQY‡dœ G∞RÙœm. CRu LÙWQm Æ]ÙÆp
RWlTh”s[ ‘˙Noj’’ GuT’ ·hPp ˘NVpTÙh˚Pf —h”°\’.
˙Y˘\Ù⁄ ©Wf£˚]˚Vf ˙NÙßj’lTÙol˙TÙm.
CWÙÿÆPm 15 ÏTÙn Es[’. 45 ÏTÙndœl ˘TÙ⁄hLs YÙeL CWÙÿ Cu‡m GjR˚]
ÏTÙn ˙Noj’d˘LÙs[˙Yi”m?
ÿRp Æ]ÙÆp Es[ ‘˙Noj’’ Gu\ ˘NÙp˙X Ceœm TVuT”jRlTh”s[’. B]Ùp
©Wf£˚]dœj æo‹ LÙiTRtœd LØjRp ˙R˚YlT”°\’.
˙Y˘\Ù⁄ ©Wf£˚]˚VŸm TÙol˙TÙm.
NÙ˚X˙VÙW Gp˚XdLp≠u C⁄TdLÿm G›ßV˚Rl TÙodL‹m.
˙LÙØd˙LÙ” TnVÚo
128 18
°.¡ °.¡
CYt˚\f ˙NÙßj’l TÙoj’d ˙LÙØ˙LÙh•≠⁄k’ TnVÚodœ Es[ ÁWj˚Rd LQd°P
CV¤UÙ?
CkRl ©Wf£˚]´p ˘NVpTÙh˚Pd œ±j’s[ GkRÆRd œ±l◊Lfim RWlTPÆp˚X.
Gp˚Xd Lp≠p ˙LÙØ˙LÙ” 128°.¡., TnVÚo 18°.¡ GuTß≠⁄k’ SÙm ◊¨k’˘LÙiP’
Gu]?
TnVÚo 18°.¡ G]d LiPÙp UflTdLm ˙LÙØ˙LÙ” GqY[‹ °˙XÙ¡hPWÙL C⁄dœm
G]d LÙi©dL‹m. C˙R ÿ˚\´p £kßjRÙp °˙XÙ¡hPo Gp˚XdLp≠p C⁄
TœßL∞¤m Es[ °˙XÙ¡hPoL∞u ˘RÙ˚L˙V CkR CWi” CPeLfidœm C˚P˙V
Es[ ÁWm.
Íufl ˘Yq˙Yfl ÿ˚\L∞XÙ] ©Wf£˚]L˚[ SÙm Ce˙L Gßo˘LÙi˙PÙm. ÿRp
©Wf£˚]´p N¨VÙ] ˘NVpœ±l◊ RWlTh•⁄kR’. CWiPÙY’ ©Wf£˚]´p ˙SW•VÙ]
œ±l◊ RWlTPÆp˚X. Íu\ÙY’ ©Wf£˚]´p ˘NVpTÙ” œ±l◊ RWÙUp æo‹
LÙQlTP˙Yi•´⁄kR’. ˙Y˘\Ù⁄ ÿ˚\´p ·±]Ùp ÿRp ©Wf£˚]´≠⁄k’
Íu\ÙY’ ©Wf£˚]dœ Y⁄m˙TÙ’ A±‹ÈoYUÙ] ÿ˚\´p Y[of£ ˘RuT”°\’.
J⁄ ©Wf£˚] Gu\Ùp Gu]?
17 Cp C⁄k’ 9 I œ˚\jRÙp GqY[‹? GuT˚R 7Bm Yœl◊ UÙQYo J⁄ ©Wf£˚]VÙL
EQW YÙnl©p˚X. B]Ùp, CWiPÙm Yœl◊ UÙQYo A˚R J⁄ ©Wf£˚]VÙL EQW
YÙnl◊s[’ ApXYÙ, CRtœj æo‹ LÙiTRtœ ˘Yq˙Yfl YØÿ˚\L˚[ UÙQYo BWÙV
˙Yi•Ÿs[’. Ceœl ©Wf£˚]j æo‹ GqYÙfl S˚P˘Tfl°\’?
54
17I 10+7 G] UÙt\XÙm.
10 ˛ Cp C⁄k’ 9 ˚Rd œ˚\jRÙp Jufl °˚Pdœm. 7 ˚Zd ·h•]Ùp 8 G] Æ˚P
°˚Pdœm.
Cu˘]Ù⁄ ÿ˚\´p £kßjRÙ˙XÙ?
17 C≠⁄k’ 10 Id LØjRÙp 7 °˚Pdœm. C⁄l©‡m LØdL˙Yi•V’ 9 ApXYÙ. AR]Ùp
7 EPu Ju˚\f ˙Noj’d ·hP˙Yi”m. AqYÙfl 8 Gu\ Æ˚P °˚Pdœm.
CqYÙfl YflThP ÿ˚\L∞p CkRl ©Wf£˚]dœj æo‹ LÙQXÙm. B]Ùp, 7˛Bm Yœl◊
UÙQYo CkRl ©Wf£˚]dœ EP]•VÙL Æ˚P LÙQ CV¤m. LÙWQm C˚Rl˙TÙu\
HWÙ[UÙ] ©Wf£˚]Lfidœj æo‹ LiP A‡TYm UÙQY¨Pm Es[’.
˙Y˘\Ù⁄ ©Wf£˚]˚Vl TÙol˙TÙm
MA +
Cßp Jq˘YÙ⁄ G›j’m 0 ÿRp 9 Y˚W Es[ 10 CXdLeL∞p Ju\Ùœ˘U≤p GiLs
G˚Y?
CWi•XdL Gi¶tœm Ko CXdL Gi¶tœm C˚P˙V Es[ ·hPp ˘R¨kR J⁄
UÙQY⁄dœ CkRl©Wf£˚] GqYÙfl A‡TYlT”°\’?
A Cu CPjßp Y⁄m CXdLm G’?
5, 6, 7, 8, 9 GuT]Yt±p H˙R‡m Ju\Ùœm. (LÙWQm Gu])
11 EPu Jufl ·h•V˙R A (GR]Ùp?)
CkR CWi” ÆßL∞u T• G”dLd·•V CWi•XdL GiLs G˚Y˘VpXÙm?
45, 56, 67, 78, 89 GuT]Yt±p ˘TÙ⁄jRUÙ]’ G’ 89
Al˘TÙ›’ Gi 89
C’ ·hPp ˘NVpTÙ” Su\ÙLj ˘R¨kR UÙQY⁄dœm J⁄ ©Wf£˚]VÙLj ˙RÙuflm.
LÙWQm, ˘Yflm ·hP¤dœm AlTÙp A±‹ NÙokR ˙Y˘\Ù⁄ ®˚X Ceœj ˘RuT”°\’.
CVkßW ÿ˚\´XÙ] ˘Yflm L¶Rf ˘NVpTÙ”Ls ApXÙUp, Jq˘YÙ⁄ UÙQY⁄m
AYWYo Lt\p ß\‡dœ HtT A±‹lÈoYUÙLf £kßj’ RW‹L∞p Ju˙\Ù˘PÙufldœ
Es[ ˘RÙPo◊L˚[d Li”©•j’ æo‹Ls LÙQ˙Yi•V]˙Y ©WNf£˚]Ls.
©Wf£˚]j æo‹ ®˚XLs
$ ©Wf£˚]˚Vl ◊¨k’˘LÙs[p (Understanding the problem)
$ ©Wf£˚]j æo‹dœ E¨V ßhPm RVÙWÙdœRp (Design a plan for problem solving)
$ ßhPj˚R Ht˘\”jRp (Carryout the plan)
55
$ Æ˚P˚V Etfll TÙojRp (Look back the solution obtained)
$ CR˚]f —⁄dLUÙL ˙Y˘\Ù⁄ ÿ˚\´¤m ·\XÙm.
See plan Do cheek
J⁄ L˚P´p BWg— Ju±u Æ˚X 4 ÏTÙn Bœm. B]Ùp, 10 ÏTÙndœ Íufl BWg—
°˚Pdœm. ß˙]Nu 13 BWg—L˚[ YÙe°]Ùu G≤p GjR˚] ÏTÙn ˘LÙ”dL ˙Yi”m.
CkRl ©Wf£˚]´u æo˚Y GqYÙfl LÙQXÙm.
1. ©Wf£˚]˚Vl ◊¨k’˘LÙs[p
$ J⁄ BWg£u Æ˚X
$ 10 ÏTÙndœd °˚Pdœm BWg—L∞u Gi¶d˚L
$ YÙe°V ˘UÙjR BWg—L∞u Gi¶d˚L.
$ ™Ld œ˚\YÙL GjR˚] ÏTÙn ˘LÙ”dL ˙Yi”m?
2. ©Wf£˚]j æo‹dœ E¨V ßhPm RVÙWÙdœRp
$ 13 BWg—Lfidœ GjR˚] ÏTÙn ˘LÙ”dL˙Yi”m G]d
Li”©•dL˙Yi”m.
$ J⁄ BWg—dœ 4 ÏTÙn ≈Rm 13 BWg—Lfidœ 13x4=52 ÏTÙ˙V
™Ld·•V Æ˚X.
$ 10 ÏTÙn dœ 3 BWg—Ls °˚PlTRÙp ™Ld·•V A[Æp 3 Cu
˘RÙœl◊L[ÙL UÙt\˙Yi”m.
ßhP™”Rp
$ 13 I 3 Cu ˘RÙœl◊L[ÙdœYRtœ 13I 3Bp YœdL ˙Yi”m.
13 ÷ 3 = 4
×
3 + 1
4
×
10 + 4 (4 ˘Ntfl 10 ÏTÙn ≈Rm, 1 ˘Ntfl 4 ÏTÙn)
40 + 4 = 44
Æ˚P˚V EtfllTÙojRp
$ ˘UÙjRm ˘LÙ”dL˙Yi•V’ 44 ÏTÙn.
$ 40ÏTÙndœd °˚Pdœm BWg—Ls 4 x 3= 12
$ 4ÏTÙndœ 1 BWg—
$ 44 ÏTÙndœ 12 + 1 = 13 BWg—Ls
56
˙U¤m ˙Y˘\Ù⁄ Æ]Ù˚Yl TÙol˙TÙm.
J⁄ N’Wÿm ˘Ne˙LÙQÿd˙LÙQÿm ˙NokR ®Xjßu TPm ∏˙Z LÙQlT”°\’.
CRu TWlT[‹ GqY[‹?
CkRl ©Wf£˚]dœj æo‹ LÙiTRtœ E¨V Tp˙Yfl ®˚XL˚[ Eh˘LÙi” Æ˚P˚Vd
Li”©•dL‹m.
1 ÿRp 5 Y˚W Es[ Yœl◊L∞u TÙPl◊jRLeL˚[ Su\ÙL BWÙnk’ H˙R‡m 10
©Wf£˚]L˚[d Li”©•j’ AYt±u ©Wf£˚]j æo‹ LÙ‘m Tp˙Yfl ®˚XL˚[
A•lT˚PVÙLd ˘LÙi” ©Wf£˚]Lfidœj æo‹ LÙiL.
©Wf£˚]j æo‹LÙ‘m Tp˙Yfl EjßL˚[l TœjR±Rp.
Au\ÙP YÙrd˚L´p ©Wf£˚]j æo‹ LÙiTRtœ E¨V J⁄ L⁄Æ˙V L¶Rm. C’
˘NVpTP ˙Yi”˘U≤p CjR˚LV ©Wf£˚]L∞u æo‹Lfidœ YœlT˚\´p YÙnl◊Ls
E⁄YÙL ˙Yi”m. L¶Rl©Wf£˚]L˚[ Eh˘LÙsYRtœm Æ˚P˚V ˙SÙd° Yk’ ˙NW‹m
Yœl©p UÙQYoLfidœ YÙnl◊Ls °˚PdL˙Yi”m. Jq˘YÙ⁄ ©Wf£˚]j æo‹dœm
£X EjßL˚[l TVuT”jR ˙Yi”m. CkR EjßL˚[d ∏rdLÙ‘m ÿ˚\L∞p ˘RÙœj’
YZeLXÙm.
Ø A˚Ul◊ E⁄YÙdœRp
Ø A˚Ul◊ Li”©•jRp
Ø ÆßY⁄m ÿ˚\´p ˙LÙhTÙh˚P E⁄YÙdœRp
Ø F°jRp
Ø AhPY˚QVÙdœRp
Ø LÙWQ LÙ¨Vj ˘RÙPo˚Td Li”©•jRp
Ø TPUÙdœRp.
Ø
S˚Pÿ˚\l ©Wf£˚]j æo‹ LÙiTRtœ E¨V ©Wf£˚]l TœlTÙn‹Æ]ÙdL˚[j RVÙWÙdœRp GqYÙfl?
J⁄ S˚Pÿ˚\l ©Wf£˚]j æo‹dœ AkRl ©Wf£˚]˚V Eh°W°dL˙Yi”m. CkR
Eh°W°jR¤dœ ER‹m Æ]ÙdL˚[ ©Wf£˚]l TœlTÙn‹ Æ]ÙdLs G]d ·flY’i”.
TœlTÙn‹ Æ]ÙdL∞u £\l©Vp◊Ls
$ ˙SW•VÙL Æ˚P˚V ˙SÙd° YØSPjRd·PÙ’.
$ ÆYWeL˚[f ˙NL¨dL‹m ©Wf£˚]˚Vl TœlTÙn‹ ˘NnV‹m ERY ˙Yi”m.
$ œZk˚R´u £kR˚]˚V FdœÆd°\’.
50 ¡hPo30 ¡hPo
57
$ œZk˚R´u T˚PlTÙt\˚X S≠‹T”jRd·PÙ’.
$ Æ]ÙdLfidœ Y¨˚NVÙokR A˚Ul◊ C⁄dL˙Yi”m.
Ts∞d·Pjßu ÿu Tœß´p J⁄ LÙnL±j˙RÙhPm A˚UdL˙Yi”m. 36 ¡hPo
—t\[Æp ˙RÙhPj˚R E⁄YÙdL˙Yi”m. ˙RÙhPm ˘NqYL Y•Yjßp A˚UV˙Yi”m.
AßLUÙLd LÙnL±L˚[j ˙RÙhPjßp SP˙Yi”˘U≤p ˙RÙhPjßu ø[ÿm ALXÿm
GqY[YÙL C⁄dL˙Yi”m?
CkR Æ]Ù˚Y Eh˘LÙsYRtœm / TœlTÙnYRtœm E¨V Æ]ÙdLs G˚Y?
$ Li” ©•dL ˙Yi•V˚Y Gu]?
(™Ld·”Rp TWlT[‹ Es[ ˘NqYLjßu ø[ÿm ALXÿm)
$ ø[ÿm ALXÿm Li”©•dœm ˘NqYLjßu £\l©Vp◊Ls G˚Y?
(™Ld ·”Rp TWlT[‹ Es[RÙL C⁄dL˙Yi”m)
$ ø[ÿm ALXÿm Li”©•dL Gu] RWlTh”s[’?
(˘NqYLjßu —t\[‹)
$ ˘NqYLjßu —t\[‹ Li”©•lT’ GqYÙfl?
(ø[ÿm ALXÿm ·h• 2 Bp ˘T⁄dL˙Yi”m)
$ —t\[Æ≠⁄k’ ø[j˚RŸm ALXj˚RŸm GqYÙfl Li”©•dLXÙm? ø[j˚RŸm
ALXj˚RŸm ·h•]Ùp GqY[‹ °˚Pdœm? GR]Ùp?
(18 °˚PdLm. —t\[Æu TÙßVÙœm)
$ ø[j˚RŸm ALXj˚RŸm GjR˚] ˙YflThP ÿ˚\L∞p AhPY˚QlT”jRXÙm?
[(17,1), (16,2), (15,3), (14,4), (13,5), (12,6), (11,7), (10,8), (9,9)]
$ AhPY˚Q´≠⁄k’ TWlT[˚Y GqYÙfl Li”©•dLXÙm?
(ø[j˚RŸm ALXj˚RŸm ˘T⁄d°)
$ ™Ld ·”RXÙ] TWlT[‹ G’?
81
$ ø[ÿm ALXÿm GqY[‹?
øeLs RVÙWÙd°V AhPY˚Q´≠⁄k’ ˙Yfl GjR˚LV ÿ•‹LfidœYk’˙NWXÙm?
˙U¤m, ˙Y˘\Ù⁄ Æ]Ù˚YŸm BWÙn˙YÙm.
˙YflThP 4 TLÙ GiL∞u ˘RÙ˚L 43. CYt±p Ko Gi 13 G≤p H˚]V GiLs G˚Y?
$ Li”©•dL ˙Yi•V’ Gu]?
$ TLÙ GiLs Gu\Ùp Gu]?
2, 3, 5, 7................
$ ©Wf£˚]j æo‹dœ GkR RW‹Ls Ceœj RWlTh”s[’?
( 4 TLÙ GiL∞u ˘RÙ˚L ˛ 43, J⁄ Gi ˛ 13)
$ 3 TLÙ GiL∞u ˘RÙ˚L GqY[‹?
(43 ˛ 13 = 30)
58
$ H˙R‡m Íufl ˘Yq˙Yfl TLÙ GiL∞u ˘RÙœl˚T G”j’ ·h•l TÙodL‹m.
$ EeLfidœd °˚PjR Æ˚P´u £\l©Vp◊Ls G˚Y?
(GpXÙm Jt˚\ GiL[Ùœm)
$ ÿR≠p °˚PjR Æ˚Pdœm CkR Æ˚Pdœm C˚P˙V B] ÆjßVÙNm Gu]?
$ 3 TLÙ GiL∞u ˘RÙ˚LL∞u Æ˚P GR]Ùp CWh˚P Gi B]’?
(2 Gu\ TLÙ Gi YkRR]Ùp)
$ G≤p H˚]V CWi” TLÙ GiLs G˚Y?
TÙPl◊jRLjß≠⁄k’ Æ]ÙdL˚[d Li”©•j’ TœlTÙn‹ Æ]ÙdL˚[Ÿm GßoTÙodœm
Æ˚PL˚[Ÿm Li”©•dL‹m.
ß\kR Æ]ÙdLs
J⁄ Æ]ÙÆtœ J⁄Æ˚P J⁄ YØÿ˚\ GuTß≠⁄k’ ÆX° ˘Yq˙Yfl Æ˚PLfim
Tp˙Yfl YØÿ˚\Lfim Es[]˙Y ß\kR Æ]ÙdLs. Jq˘YÙ⁄ UÙQY⁄m ReLfidœ
ELkR YØÿ˚\L∞p Æ˚PL˚[d LÙiTRu YØ˙V TuÿLf£kR˚] (Divergent thinking)
A±YÙokR £kR˚], £kR˚]˙VÙhPm Tt±V £kR˚] GuT]Yt±tœm YØLÙh”°\’.
Tp˙Yfl ÿ˚\L∞p BWÙnRp, ˘TÙ’UVUÙdœRp, Eh°W°jRp ˙TÙu\ ˘NVpß\uLs
ƨYÙdLm ˘Tfl°u\].
∏˙Z RWlTh”s[ C⁄ Æ]ÙdL˚[d LY¶dL‹m.
1.
1.5 ¡
2.50 ¡
10 ¡
9 ¡
59
˙U˙X RWlTh”s[ ≈h•u Y˚WTPj˚Rl TÙodL‹m. CRu TWlT[‹ GqY[‹ N’W ¡hPo?
2. AlTÙ, AmUÙ, C⁄ œZk˚RLs TÙh• B°˙VÙo Y£lTRtœl TÙ⁄jRUÙ] J⁄ ≈h•u
Y˚WTPj˚R Y˚WV‹m. øeLs Y˚WkR Y˚WTPjßtœ GqY[‹ N’W¡hPo TWlT[‹
Es[’?
ÿRp Æ]Ù TœlTÙn‹ YÙnl◊L∞u± ˘Yflm TWlT[‹ LÙiTRtLÙ] J⁄ Æ]ÙYÙœm.
CWiPÙY’ Æ]ÙÆp ©Wf£˚]˚V Eh˘LÙsYRtLÙ] J⁄ RW‹m RWlTPÆp˚X. Ceœ
UÙ]Yo ©Wf£˚]˚V Eh˘LÙi” ˘TÙ⁄jRUÙ]’m ˙R˚YVÙ]’UÙ] RW‹L˚[d
LÙQ˙Yi”m. AR]Ùp ÿRp Æ]Ù˚Y J⁄ Í•V Æ]Ù˘Yuflm CWiPÙY’ Æ]Ù˚Y
J⁄ ß\kR Æ]Ù G]‹m ·\XÙm. CkR C⁄ Æ]ÙdL∞u Ru˚UL˚[d ∏rdLÙ‘m
ÿ˚\L∞p AhPY˚QlT”jRXÙm.
Í•V Æ]Ù
1. Well Structured Bœm
2. ˙R˚YVÙ] RW‹Ls Æ]ÙÆp
LÙQlT”m
3. ©Wf£˚]dœ E¨V YØÿ˚\L∞p
˘NpX˙Yi”m
4. A˚]Y¨u N¨ Æ˚P Ju\ÙL˙Y
C⁄dœm
ß\kR Æ]Ù
$ Structured Bœm
$ RW‹L˚[d Li”©•j’, AX£
BWÙnk’Æ˚P˚Vd
Li”©•lTYoL˙[ Ht˘\”dL
˙Yi”m.
$ ©Wf£˚]j æo‹LÙQ Tp˙Yfl
YØÿ˚\Ls Es[].
$ œ±l©hP Æ˚P Cp˚X / TX
Æ˚PL˚[l Tp˙Yfl L¶Rf
˘NVpTÙ”Ls ÍXUÙLd LÙQ
CVp°\’.
$ —RkßWf £kR˚]dœ YÙnl◊ Es[’.
$ CVpTÙ] £kR˚]f ˘NVpTÙh•tœ
FdLm A∞d°\’.
$ A±YÙokR ÿ˚\´p ®fl‹m
EQo‹m T¨UÙt\j ß\‡m
Y[o°\’.
$ LtT˚] E⁄YÙdLjßtœm
CWN˚]dœm CPm °˚Pd°\’.
ß\kR Æ]ÙÆtœ E¨V Ko G”j’dLÙh˚Pl TÙol˙TÙm.
$ EeLs ≈h•u UÙRY⁄UÙ]m GqY[‹? CRtœl TÙ⁄jRUÙ] ÿ˚\´p J⁄ UÙRjßu
œ”mT YW‹ ˛ ˘NX‹j ßhPj˚Rj RVÙ¨dL‹m.
60
˙Yfl £X Æ]ÙdL˚[Ÿm LY¶dL‹m.
$ J⁄ N˚UVp A˚\dœ 12 N’W¡hPo TWlT[‹ Es[’. ARu ø[ÿm ALXÿm
GqY[‹?
$ 1000 ≠hPo RiΩo ®WlTd·•V J⁄ RiΩoj ˘RÙh•´u ø[m, ALXm, EVWm
GuT] GqY[‹?
CkR C⁄ Æ]ÙdL∞¤m £X RW‹Ls RWlTh•⁄kRÙ¤m A±‹lÈoYUÙ] Ju±tœ UtThP
Æ˚PLs Es[]. £X Æ˚PL˚[ ˙YiPÙm Gufl ·flYRtœ E¨V „ZpL˚[l TœjR±V
˙Yi”m. AR]Ùp CqY˚L Æ]ÙdL˚[Ÿm ß\kR Æ]ÙdL[ÙL L⁄jßp ˘LÙs[XÙm.
Í•V Æ]ÙdL˚[j ß\kR Æ]ÙdL[ÙL UÙt±]Ùp œZk˚R´u T˚PlTÙt\¤m,
A±‹lÈoYUÙ] £kR˚]Ÿm Y[⁄m ApXYÙ? AR]Ùp TÙPl◊jRLjßp Es[ Í•V
Æ]ÙdL˚[j ß\kR Æ]ÙdL[ÙL UÙtflYRtœ E¨V T´t£ ˘T\˙Yi”m.
20 ˘N¡ ø[ÿm 10 ˘N¡ ALXÿmU 5 ˘N¡ EVWÿm Es[ J⁄ TÙjßWjßp GjR˚] ≠hPo
RiΩo ®WlT ÿ•Ÿm?
C’ Jo Í•V Æ]Ù ApXYÙ, CR˚] GqYÙfl ß\kR Æ]ÙYÙL BdLXÙm?
1 ≠hPo RiΩo ®WlTd·•V J⁄ TÙjßWjßu ø[m, ALXm, EVWm GkR A[Æp C⁄dœm?
C˚Rl ˙TÙufl TÙPl◊jRLj˚Rl TœlTÙn‹ ˘Nn’ ß\kR Æ]ÙdL[ÙL BdLXÙm.
£X G”j’dLÙh”L˚[ G›R‹m.
61
AXœ ˛ 5
˘RÙØp ÷hTm L¶Rd LpÆ´p
ÿu‡˚W
L⁄j’l T¨UÙt\m ™Ll ˘TÙ⁄[ÙokRRÙL A˚UV˙Yi”˘U≤p RLYp ˘RÙPo©u Teœ
Cu±V˚UVÙRRÙœm. Li”m ˙Lh”m EQok’m LtTRtLÙ] YÙnl˚T A’ LtTYoLfidœ
YZeœ°\’.
L¶RdL⁄j’L˚[ ™Lf£\kR ÿ˚\´p T¨UÙt\m ˘NnV Cu˚\VLÙX A[Æp ˙YflThP
RLYp ˘RÙPo◊ ÿ˚\Ls TVuT”jRlT”°u\]. L¶≤, ˚L˙T£ ˙TÙu\˚Y L⁄j’
T¨UÙt\jßp HtT”jßV UÙt\eLs ◊Wh£LWUÙ]˚YVÙœm. B£¨V UÙQYoLs
Cmÿ˚\L˚[ A±k’˘LÙs[‹m AYt˚\ YœlT˚\´p ˘NVpT”j’YRtœm E¨V
ß\˚]l ˘T\ ˙Yi”m.
L¶Rd LpÆŸPu ˘RÙPo◊s[ Rt˙TÙ˚RV A˚Ul◊L˚[l Tt±Ÿm AYt±u YœlT˚\
YÙnl◊L˚[l Tt±Ÿm Es[ L⁄j’dLs CkR AX°p Æ[dLlT”°u\].
¥˙VÙ¥lWÙ
L¶RdLpÆ´p UÙQY⁄dœm B£¨V⁄dœm J˙W ˙TÙp ERÆ˘NnŸm Ko CVeœ
˘Uu˘TÙ⁄s ¥˙VÙ¥lWÙ Bœm. Y•ÆVp Y•YeLs Y˚Wk’ ˘RÙPeœm ˘RÙPdL®˚X
YœlT˚\L∞p BWm©j’ Bn‹ UÙQYoLfidœ Y˚W TVuT”m C’ —RkßWUÙLd
°˚Pd°u\ J⁄ ˘Uu˘TÙ⁄[Ùœm.
Lt\p A˚P‹Ls
$ L¶Rd LpÆ´p ˘RÙØp ÷hTj-
߇˚PV YÙnl◊L˚[d LiP˚PRp
$ L¶Rd LpÆdœ TVuT”j’°u\
˘Uu ˘TÙ⁄hL˚[d œ±j’
L⁄j’L˚[ A˚PRp.
ÿd°Vd L⁄j’Ls
$ L¶lTÙu, L¶≤, A˚X˙T£
˙TÙu\Yt±tœ L¶Rd LpÆ´p
Es[ Teœ.
$ L¶Rd LpÆdœ ˘TÙ⁄jRUÙ]
Tp˙Yfl Al∞d˙L`uLs.
$ L¶Rd LpÆdœj ˙R˚YVÙ]
Tp˙Yfl ˘Uu˘TÙ⁄hLs.
$ ¥˙VÙ ¥lWÙ
$ °d
$ ˘_ ©WÙbu Bn‹d ·Pm.
62
Cßp L⁄ÆlTh˚P´≠⁄k’ L⁄ÆL˚[j ˙Rok˘R”j’ °WÙ©d LÙh£´p Y•ÆVp
TPeL˚[ Y˚WVXÙm. CqYÙfl Y˚WŸm A˚Ul◊L∞u CVtL¶R Y•Yj˚R Bp¥lWÙ
LÙh£VÙLd LÙiTRtœ CVp°\’. AR]Ùp C˚P®˚X Yœl◊L∞p L¶Rd Lt\¤dœ
Bp¥lWÙ LÙh£ ˙R˚Y´p˚X. C˚R U˚\j’ ˚YlTRtœ Bp¥lWÙ LÙh£´u ˙U˙X
YXlTdLjßp Es[ x A˚PVÙ[jßp °∞d ˘NnRÙp ˙TÙ’UÙ]’. A˚Rl˙TÙufl °WÙ©d
LÙh£´p Es[ G›j’dLfim Cl˙TÙ’ ˙R˚Y´p˚X. °WÙ©d LÙh£´p Right click
˘NnŸm˙TÙ’ °˚Pdœm NÙ[Wjßp Axes GuTßp °∞d ˘Nn’ G›j’dL˚[ ødLXÙm.
GeoGebra Tools
12 ˘RÙœl◊L[ÙL ¥˙VÙ¥lWÙÆp L⁄ÆLs Y¨˚NlT”jRlTh”s[].
¥˙VÙ¥lWÙ ß\dœm˙TÙ’ ∏˙ZLÙiT’ ˙TÙu\ J⁄ NÙ[Wm °˚Pdœm.
63
L⁄ÆlTh˚P´p LÙQlT”m L⁄ÆL∞u A˚PVÙ[jßp ∏˙Z YXlTdLm LÙQlT”m
ÿd˙LÙQY•ÆXÙ] £±V A˚PVÙ[jßp °∞d˘NnRÙp AkRj ˘RÙœl©p Es[ GpXÙd
L⁄ÆLfim °˚Pdœm.
Jq˘YÙ⁄ ˘RÙœl©¤m Es[ L⁄ÆL∞p C˚P®˚X Yœl◊Lfidœj
˙R˚YVÙ]Yt˚\ A±k’˘LÙs˙YÙm.
SET -1 -Movement Tools
Y˚WkR TPeL˚[ CVdœYRtœj ˙R˚YVÙ] L⁄ÆLs
Move
Graphic View Es[ ◊s∞L˚[˙VÙ Y•YeL˚[˙VÙ
CVdœYRtœl TVuT”°\’.
G”j’dLÙh”. ÿd˙LÙQm ABC I CVdœYRtœ Move
L⁄Æ´p °∞d˘NnR©u ÿd˙LÙQjßu Es˙[
°∞d˘Nn’ ©•j’d˘LÙi” ˘U¸˚N CVdL‹m.
ÿd˙LÙQjßu H˙R‡m Ko Ef£´p Es[ ◊s∞´p °∞d˘Nn’˘LÙi” ˘U¸˚N
CVd°]Ùp AkRl ◊s∞Ÿm ARtœ HtT Aߤs[ TdLeLfim UÙflm. AkRl◊s∞˚V
CVdœYRtœd ∏˙TÙo•p Es[ Arrow Key L˚[Ÿm Move L⁄ÆŸPu TVuT”jRXÙm.
◊s∞´p °∞d˘NnR©u Arrow Key L∞p H˙R‡m Ju˚\ A›jßl©•jRÙp AkRj
ß˚N´p ◊s∞ CVeœm.
SET -2 -Point Tools
◊s∞L˚[ A˚PVÙ[lT”j’YRtœ E¨V L⁄ÆLs CkRj ˘RÙœl©p Es[].
New Point
◊ßV ◊s∞L˚[ A˚PVÙ[lT”j’YRtœ.
New Point Il TVuT”jß Graphic view Cp °∞d ˘NnRÙp J⁄ ◊ßV ◊s∞ °˚Pdœm.
J⁄ ˙LÙh•˙XÙ YhPjß˙XÙ °∞d ˘NnRÙp °˚Pdœm ◊s∞ AkRd ˙LÙh•˙XÙ
YhPjß˙XÙ Es[ ◊s∞ Bœm. Move Tool TVuT”jß AkRl TÙ⁄∞u YÙ´XÙL Uh”˙U
◊s∞˚V CVdL ÿ•Ÿm.
A B
C
64
œ±l◊: LoNo NÙRÙWQUÙL + A˚PVÙ[jßp LÙQlT”m. B]Ùp, H˙R‡m J⁄
˘TÙ⁄∞u ¡’ Y⁄m˙TÙ’ A’ ˙Up˙SÙd°V Am◊dœ± TPm A˚PVÙ[UÙL
UÙflm.
Point on Object
J⁄ ˘TÙ⁄∞p Es ◊s∞˚V A˚PVÙ[lT”j’YRtœ.
G”j’dLÙh”. J⁄ ÿd˙LÙQjßu H˙R‡m J⁄ TdLjßp °∞d ˘NnRÙp
ÿd˙LÙQjßu TdLeL∞u YÙ´XÙL Uh”m CVdLÿ•°u\ ◊s∞ °˚Pdœm.
ÿd˙LÙQjßu Es˙[ °∞d ˘NnRÙp TdLeL∞u YÙ´XÙL‹m Es˙[Ÿm CVdL
ÿ•°u\ ◊s∞ °˚Pdœm
Attach/Detach point
J⁄ ◊s∞´p °∞d˘NnR ©u J⁄ Y•Yjßp °∞d ˘NnRÙp ◊s∞ Y•Yj’Pu
˙Noj’ ˚YdLlT”m.
Intersect Two Objects
˙LÙ”Ls, Æ∞m◊Ls, Y•YeLs ˙TÙu\˚Y ˘Yh•d˘LÙsfim ◊s∞˚V
A˚PVÙ[lT”j’YRtœ, ˘Yh•d˘LÙsfim ◊s∞˚V A˚PVÙ[lT”jR ˙Yi•V
Y•YeL∞p Ju\u ©u Ut˘\Ùu\ÙL °∞d ˘NnV˙YÙ ˘Yh•d˘LÙsfim ◊s∞´p
˙SW•VÙL˙YÙ °∞d˘NnVXÙm.
Midpoint or Center
J⁄ LÙh•u / CWi” LÙ”L∞u UVl◊s∞˚V A˚PVÙ[lT”jR
˙LÙh•p / ◊s∞L∞p °∞d ˘NnV‹m.
SET -3 -line Tools
˙LÙ”Ls Y˚WYRtœj ˙R˚YVÙ] L⁄ÆLs.
line through Two Points
CWi” ◊s∞L∞u YØVÙLf ˘Np¤m ˙LÙ” Y˚WYRtœd
L⁄Æ˚Vl TVuTPjßl ◊s∞L∞p °∞d ˘NnV‹m.
Segment between Two Points
CWi” ◊s∞L˚[ C˚Qdœm ˙LÙ” Y˚WYRtœ
L⁄Æ˚Vl TVuT”jßl ◊s∞L∞p °∞d ˘NnV‹m.
Segment With given length From Point
œ±l©hP ø[m Es[ ˙LÙ” Y˚WYRtœd
L⁄Æ˚Vl TVuT”jß J⁄ ◊s∞´p °∞d ˘NnV‹m. ˘RÙPok’ Y⁄m NÙ[Wjßp
˙LÙh•u ø[m A∞j’ ok °∞d˘NnV‹m.
65
Ray through Two Points
J⁄ ◊s∞´p ˘RÙPe° ˙Y˘\Ù⁄ ◊s∞ YØVÙLf ˘Np¤m ˙LÙ” Y˚WYRtœ
L⁄Æ˚Vl TVuT”jßl ◊s∞L∞p °∞d ˘NnV‹m.
SET -4 -Special line Tools
£X £\l◊jRu˚ULs Es[ ˙LÙ”Ls Y˚WYRtœ
Perpendicular line
J⁄ ˙LÙh•u J⁄ ◊s∞ YØVÙLf ˘Neœj’ Y˚WYRtœ
˙LÙh•¤m ◊s∞´¤m °∞d ˘NnV‹m.
Parallel line
J⁄ ˙LÙh•tœ C˚QVÙL ˙Y˘\Ù⁄ ˙LÙ” Y˚WYRtœ
L⁄Æ˚Vl TVuT”jßd ˙LÙh•¤m ◊s∞´¤m °∞d ˘NnV‹m.
Perpendicular Bisector
˙LÙh•u ˘Neœj’ C⁄NU˘Yh• Y˚WYRtœ
˙LÙh•p °∞d ˘NnV‹m.
Angle Bisector
˙LÙQjßu C⁄NU˘Yh• Y˚WYRtœ
˙LÙQm ÿ•‹˘Nn°u\ CWi” ˙LÙ”L∞p / Íufl ˙LÙ”L∞p °∞d ˘NnV‹m.
SET -5 -Polygon Tools
TX˙LÙQeLs Y˚WYRtœ
Polygon
TX˙LÙQm Y˚WV
TX˙LÙQjßu Ef£L[ÙL YW˙Yi•V ◊s∞L∞p Y¨˚NVÙLd °∞d ˘NnV‹m,
˘RÙPe°V ◊s∞´p ¡i”m Yk’ ˙NW˙Yi”m.
œ±l◊: Gßo L•LÙWj ß˚N´p °∞d ˘NnY’ SpX’. L•LÙWjß˚N´p
°∞d˘NnRÙp Angle Tool TVuT”j’m˙TÙ’ TX˙LÙQjßtœ ˘Y∞˙V A˚UkR
˙LÙQeLs RÙu °˚Pdœm.
Regular Polygon
J›eœ TX˙LÙQm Y˚WV
66
L⁄Æ˚Vl TVuT”jß CWi” ◊s∞L∞p °∞d ˘NnV‹m. ˘RÙPok’ Y⁄m NÙ[Wjßp
TX˙LÙQjßu TdLeL∞u Gi¶d˚L˚V A∞j’ OK °∞d ˘NnV‹m.
SET -6 -Circle and Arc Tools
YhPeLfim YhPl TÙLeLfim Y˚WYRtœ
Circle with center through Point
J⁄ ◊s∞˚V ˚UVUÙL‹m ˙Y˘\Ù⁄ ◊s∞´u YØVÙL‹m ˘Np¤m
YhPm Y˚WV
YhP˚UVjߤm YhPm ˘Np°u\ ◊s∞´¤m °∞d ˘NnV‹m.
Circle with center and radious
œ±l©hP BWjßp Es[ YhPm Y˚WV
YhP ˚UVjßp °∞d ˘NnV‹m. ˘RÙPok’ Y⁄m NÙ[Wjßp YhPjßu BWm
A∞j’ °∞d OK ˘NnV‹m.
Compass
J⁄ YhPjßu A˙R A[‹ Es[ ˙Y˘\Ù⁄ YhPm Y˚WV
Rt˙TÙ’ Y˚Wk’s[ YhPjßp °∞d ˘NnV‹m. LoN˚W SLojß Y˚WV˙Yi•V
YhPjßu ˚UVUÙL YW˙Yi•V ◊s∞´p °∞d ˘NnV‹m.
Circle through Three Points
Íufl ◊s∞L∞u YØVÙLf ˘Np¤m YhPm Y˚WV.
L⁄Æ˚Vl TVuT”jß Íufl ◊s∞L∞p °∞d ˘NnV‹m.
SemiCircle through Two Points
A˚WYhPm Y˚WV
L⁄Æ˚Vl TVuT”jß CWi” ◊s∞L∞p °∞d ˘NnV‹m.
Circular Arc with center between Two Points
Æp Y˚WV
˚UVjߤm CWi” ◊s∞L∞¤m °∞d NnV‹m. L•LÙW Gßoß˚N´p Æp Y˚WV CV¤m.
Circular Arc through three Points
Íufl ◊s∞L∞u YØVÙLf ˘Np¤m Æp Y˚WV
L⁄Æ˚Vl TVuT”jß Íufl ◊s∞L∞p °∞d ˘NnV‹m.
Circular Sector with center between Two Points
YhPlTœß Y˚WV
˚UVjߤm CWi” ◊s∞L∞¤m °∞d ˘NnV‹m.
67
Circular Sector through three Points
Íufl ◊s∞L∞u YØVÙLf ˘Np¤m YhPlTœß Y˚WV.
L⁄Æ˚Vl TVuT”jß Íufl ◊s∞L∞p °∞d ˘NnV‹m.
SET -7 Conic Section Tool
øsYhPm, TWÙ˙TÙXÙ ˙TÙu\˚Y Y˚WYRtœj ˙R˚YVÙ] L⁄ÆLs.
SET -8 Measurement Tools
A[‹LfiPu ˘RÙPo◊s[ L⁄ÆLs.
Angle
˙LÙQm A[lTRtœ
˙LÙQm A A[lTRtœ A, B,CGu\ ◊s∞L∞p Y¨˚NVÙLd °∞d ˘NnV‹m
Cp˚X˘V≤p AB, ACGu\ ˙LÙ”L∞p Y¨˚NVÙLd °∞d
˘NnV‹m.
œ±l◊.
$ Gßo L•LÙWj ß˚N´p ˙LÙQeLs A˚PVÙ[lT”jRlTh•⁄dœm. AR]Ùp
°∞d ˘NnŸm Y¨˚N UÙ±]Ùp A˚PVÙ[lT”jRlT”m. ˙LÙQÿm UÙflm.
$ Gßo L•LÙWj ß˚N´p Y˚WkR J⁄ TX˙LÙQjßu Es˙[ CkRd
L⁄ÆL˚[l TVuT”jßd °∞d ˘NnRÙp TX˙LÙQjßu GpXÙd
˙LÙQeL˚[Ÿm A˚PVÙ[lT”jR CV¤m.
Angle with Given size
œ±l©hP A[Æp ˙LÙQm A˚UlTRtœ
G.LÙ. B=40O BœmT•VÙL ˙LÙQm ABC A˚UdL˙Yi”˘U≤p AB
Gu\ LÙ” Y˚Wk’ Segment between Two Points L⁄Æ˚Vl TVuT”jß
A,B GuTYt±p Y¨˚NVÙLd °∞d ˘NnV‹m. ˘RÙPok’ Y⁄m NÙ[Wjßp
˙LÙQjßu A[YÙL 40O G] A∞j’ OK °∞d ˘NnV‹m. Cl˘TÙ›’ A Gu\ ◊ßV
◊s∞ °˚Pdœm. Segment between Two Points L⁄Æ˚Vl TVuT”jß BA Y˚WV‹m
GßoL•LÙWj ß˚N´p ˙LÙQj˚R A˚PVÙ[lT”jßVRÙp AB Cu ∏˙Z A °˚PjR’.
˙U˙X °˚PdL˙Yi”˘U≤p A, B Gu\ ◊s∞L∞p 40O G] A∞j’ Clock wise GuTßp
°∞d ˘NnR©u OK °∞d ˘NnV‹m.
Distance or length
CWi” ◊s∞Lfidœ C˚P˙V Es[ ÁWm, ˙LÙh•u ø[m, —t\[‹
˙TÙu\˚Y A[k’ A˚PVÙ[lT”j’YRtœ.
$ ÁWm A[dL ˙Yi•V ◊s∞L∞p °∞d ˘NnV‹m.
$ ø[m A[dL ˙Yi•V ˙LÙh•p °∞d ˘NnV‹m.
A B
C
A
A′
B40o
68
$ TX˙LÙQjßu Es˙[ °∞d ˘NnV‹m.
$ YhPjßp °∞d ˘NnV‹m.
Area
TWlT[‹ A[lTRtœ
TX˙LÙQjßu Es˙[ °∞d ˘NnV‹m.
YhPjßp °∞d ˘NnV‹m.
SET -9 -Transformation Tools
Reflect Object about line
Reflect Object about Point
Reflect Object about Circle
J⁄ Y•ÆVp Y•Yjßu ©Wß©mTm E⁄YÙdœYRtœ CkR Íufl
L⁄ÆLfim Y•ÆVp Y•Yjßp °∞d˘NnR ©u J⁄ ˙LÙh•˙XÙ ◊s∞´˙XÙ
YhPjß˙XÙ °∞d˘NnV‹m.
Rotate Object around Point by Angle
J⁄ ˘TÙ⁄˚[d œ±l©hP ˙LÙQ A[Æp —ZtflYRtœ
—Zt\ ˙Yi•V ˘TÙ⁄∞¤m J⁄ ◊s∞´¤m °∞d ˘NnV‹m. ˘RÙPok’ Y⁄m
NÙ[Wjßp ˙LÙQ A[‹ A∞dL‹m.
Translate Object by Veetor
Y•Æ´p Y•Yj˚Rd œ±l©hP ÁWjߤm ß˚N´¤m SLoj’YRtœ
Y•Yjßp °∞d˘NnR©u C⁄ ◊s∞L∞p °∞d ˘NnV‹m. ◊s∞L∞u C˚P˙V Es[
ÁWjßp, ÿRp ◊s∞´≠⁄k’ CWiPÙY’ ◊s∞ Es[ ß˚N˚V ˙SÙd° Y•Ym SL⁄m.
Dilate Object from Point by Factor
J⁄ Y•Yj˚Rd œ±l©hP UPeœ ˘T¨VRÙdL‹m / œ±l©hP TÙLm
£±VRÙdœYRtœm.
Y•Yjߤm J⁄ ◊s∞´¤m °∞d ˘NnV‹m. ˘RÙPok’ Y⁄m NÙ[Wjßp Ko Gi
A∞dL‹m. A∞dœm Gi 1 IÆPl ˘T¨V˘R≤p Y•Yÿm AqY[‹ UPeœ ˘T¨VRÙœm,
1 I ÆPf £±V’ G≤p ARtœf NUUÙ] TÙLm Bœm.
69
SET -10 -Special Object Tools
G›j’dLs, TPeLs ˙TÙu\˚Y EhT”j’YRtœ
Insert Text
G›j’dLs EhT”j’YRtœ
L⁄Æ˚Vl TVuT”jß °WÙ©d LÙh£´p °∞d˘NnV‹m. RÙPok’ Y⁄m NÙ[Wjßp
˙R˚YVÙ]Yt˚\j RhPf— ˘NnV‹m. ˘RÙPok’ OK °∞d ˘NnV‹m.
Insert Image
TPeLs EhT”j’YRtœ
L⁄Æ˚Vl TVuT”jß °WÙ©d LÙh£´p °∞d ˘NnV‹m. ˘RÙPok’ Y⁄m
NÙ[Wjß≠⁄k’, L¶≤´p N™j’˚YdLlTh”s[ TPeL˚[j Ro‹ NnVXÙm.
Pen Tool
°WÙ©d LÙh£´p ˙SW•VÙL G›R˙YÙ Y˚WV˙YÙ ˘NnYRtœ
L⁄Æ˚Vj ˙Ro‹ ˘NnR ©u]o ˘U¸£p °∞d ˘Nn’ ©•j’d˘LÙi” °WÙ©d
LÙh£´p TPeLs Y˚WVXÙm.
˘U¸£p YX’ TÙjRÙ˚] A›jßl ©•j’d˘LÙi” AØjRÙp Y˚WkR LÙh˚P
AØdLXÙm.
SET -11 -Action Object Tools
Slider
¥˙VÙ¥lWÙ Y•YeL˚[ A˚NVf ˘NnYRtœ C’ ¥˙VÙ¥lWÙÆp ™L‹m
NdßYÙnkR L⁄Æ G]d ·\XÙm. CWi” GiLfidœ C˚P˙V Es[ GkR
Ußl˚TŸm Htfld˘LÙs[d·•V J⁄ UÙ±VÙL (Variable) £˚XP˚Wd LQd°PXÙm.
Number Slider, Angle Slider, Integer Slider G] CqYÙfl Íufl Y˚Lf £˚XPoLs
Es[] G]d ·\XÙm.
Number Slider
£˚XPo L⁄Æ˚Vl TVuT”jß, °WÙ©d LÙh£´p °∞d ˘NnV‹m. ∏˙Z LÙiT’ ˙TÙp
Slider Gu\ ˘TV¨p J⁄ NÙ[Wm °˚Pdœm.
70
Name GuTßp £˚XP⁄dœ A∞dL Æ⁄m◊m ˘TV˚W A∞dLXÙm. (CkRl ˘TVo
ÿd°VUÙ]RÙœm, £˚XP˚Wl TVuT”j’m GkR E⁄YÙdœR¤dœm C’ ˙R˚YVÙ]RÙœm.)
C˙R NÙ[Wjßp £˚XP¨u ™Ld œ˚\kR Ußl◊ (Min) ™Ld·•V Ußl◊ (Max), AßRÆ˚X
(Increment) ˙TÙu\Yt˚\ Y¨˚NlT”jR ˙Yi”m. ˙R˚YVÙ] UÙt\eL˚[f ˘NnR©u]o
Apply °∞d ˘NnV‹m.
Angle Slider
°WÙ©d LÙh£´p °∞d ˘NnŸm˙TÙ’ °˚Pdœm NÙ[Wjßp Angle GuTRu ˙SWÙLd °∞d
˘NnV‹m. Min, Max C˚Y 0 •°¨ dœm 360 •°¨ dœm C˚P´p Es[ A[‹L˚[˙V
A∞dL˙Yi”m.
Integer Slider
°WÙ©d LÙh£´p °∞d ˘NnŸm˙TÙ’ °˚Pdœm NÙ[Wjßp Integer GuTRtœ ˙SWÙLd
°∞d ˘NnV‹m.
£˚XPoL∞u TVuLs
¥˙VÙ¥lWÙÆu E´oSÙ• £˚XPo G]d ·\XÙm. H˚]V L⁄ÆLfiPu £˚XP˚Wl
TVuTPjR ˙Yi”m. £X G”j’dLÙh”Ls.
$ Name: a. min: o, max:5 Y⁄mT•VÙL J⁄ £˚XP˚W E⁄YÙdL‹m, Circle with center and
radius L⁄Æ˚Vl TVuT”jß J⁄ ◊s∞´p °∞d ˘NnV‹m. ˘RÙPok’ °˚Pdœm
NÙ[Wjßp a (£˚XP¨u ˘TVo) G] A∞dL‹m. CqYÙfl E⁄YÙdœm YhPjßu
BWj˚Rf £˚XP˚Wl TVuT”jß UÙt\XÙm.
Segment with given length from point L⁄Æ˚Vl TVuT”jß J⁄ ◊s∞´p °∞d
˘NnV‹m. RÙPok’ Y⁄m NÙ[Wjßp LÙh•u ø[m a G] A∞j’ ok °∞d NnV‹m.
CqYÙfl °˚Pdœm C⁄ ◊s∞L˚[l TVuT”jß J⁄ N’Wm E⁄YÙdL‹m (Regular
Polygon L⁄Æ TVuT”jRXÙm.) £˚XP¨u Ußl◊ UÙflYRtœ HtT N’Wjßu A[‹
UÙflY˚Rd LÙQXÙm.
œ±l◊: £˚XP¨u Ußl◊ UÙtflYRtœf £˚XP¨p °∞d ˘Nn’ ©•j’d˘LÙi” ˘U¸˚N
SLojR‹m/ £˚XP¨p °∞d ˘NnR ©u Arrow key I A›jR‹m / £˚XP¨p Right Click
˘NnŸm˙TÙ’ °˚Pdœm NÙ[Wjßp Animation on GuTßp °∞d ˘NnV‹m.
$ Name: n. Min:3, Max:50 Y⁄mT•VÙL J⁄ Integer Slider E⁄YÙdL‹m. Regular Polygon
L⁄Æ˚Vl TVuT”jß CWi” ◊s∞L∞p °∞d ˘NnV‹m. ˘RÙPok’ °˚Pdœm
NÙ[Wjßp TX˙LÙQjßu TdLeL∞u Gi¶d˚L n G] A∞dL‹m. £˚XP¨u
Ußl◊ UÙflYRtœ HtT TdLeL∞u Gi¶d˚L UÙflY˚Rd LÙQXÙm.
$ Name: α, min:0o, Max:360o Y⁄mT•VÙL J⁄ Angle Slider E⁄YÙdL‹m. Angle with
given size L⁄Æ˚Vl TVuT”jßd LÙQm E⁄YÙdœm NVpTÙh•p LÙQ A[YÙL
α G] A∞dL‹m. £˚XP¨u Ußl◊ UÙflYRtœ HtT ˙LÙQ A[‹ UÙflY˚Rd
LÙQXÙm.
71
Check Box to show / Hide Objects
E⁄YÙdLeL˚[ U˚\j’ ˚YdL‹m ˙R˚YVÙ] ˙SWeL∞p LÙiTRtœm.
L⁄Æ˚Vl TVuT”jß, °WÙ©d LÙh£´p °∞d ˘NnV‹m. ∏˙Z LÙiT’ ˙TÙu\ J⁄
NÙ[Wm °˚Pdœm.
Caption Gu\ Th•´p check box Cu TVo
A∞dLXÙm. ARu A”jRRÙL C⁄lT’ J⁄
Drop down box Bœm. CRu YXlTdLm
Es[ A˚PVÙ[jßp °∞d ˘NnRÙp
C’Y˚W E⁄YÙdLlThP˚YL∞u Ko
AhPY˚Q °˚Pdœm. Aß≠⁄k’ Check
Box Cp EhT”jR ˙Yi•VYt˚\j ˙Ro‹
˘NnVXÙm.
˙Y˘\Ù⁄ ÿ˚\´¤m E⁄YÙdLeL˚[ Check
Box Cp EhT”jRXÙm. CRtLÙL E⁄YÙdLjßp Right Check Cp ˘NnŸm˙TÙ’ °˚Pdœm
NÙ[Wjß≠⁄k’ Object Properties → Advanced Gu\ ÿ˚\´p °˚Pdœm NÙ[Wjßp Check
Box Cu ˘TVo A∞dL‹m. Caption BL A∞jR ˘TV˚W Ceœl TVuT”jR ˙YiPÙm.
˘TV˚Wj ˘R¨k’˘LÙs[ Check Box Cp Right °∞d ˘NnV‹m. Al˙TÙ’ °˚Pdœm
NÙ[Wjßp ™L‹m ˙Up Tœß´p Boolean Value GuTRu ©u]o Es[ G›j˙R Check
Box Cu ˘TVWÙœm.
Insert Input Box
£˚XP¨u Ußl˚Tj RhPf— ˘Nn’ A∞lTRtœ
L⁄Æ˚Vl TVuT”jß, °WÙ©d LÙh£´p °∞d ˘NnV‹m. Al˙TÙ’ °˚Pdœm NÙ[Wjßp
Caption A∞dLXÙm. Linked Object Gu\ ˘Th•´≠⁄k’ £˚XP¨u ˘TV˚Wj
˙Rok˘R”j’ Apply °∞d ˘NnV‹m. CqYÙfl °˚Pdœm ˘Th•´p £˚XP¨u
Ußl˚Tj RhPf— ˘NnV˙Yi”m.
SET -12 -General Tools
Move Graphic View
Graphic View A˚NVf NnYRtœ
L⁄Æ˚Vl TVuT”jß °WÙ©d ÆÎÆp °∞d ˘Nn’ ©•j’d˘LÙi” ˘U¸˚N
A˚NdL‹m.
Zoom In
Zoom Out
72
E⁄YÙdLeL∞u A[‹ AßL¨dL‹m / œ˚\dL‹m.
L⁄Æ˚Vl TVuT”jß °WÙ©d LÙh£´p °∞d ˘NnV‹m/ Mouse Wheel —Zt\‹m
Show / Hide Object
L⁄Æ˚Vl TVuT”jß LÙQ˙Yi•V / U˚\dL˙Yi•V ˘TÙ⁄∞p °∞d ˘NnV‹m. C≤
˙Yfl HRÙY’ L⁄Æ˚Vl TVuT”j’m˙TÙ’ CkRl ˘TÙ⁄hLs U˚\k’ C⁄dœm. C˚R
¡i”m LÙQ˙Yi”m G≤p C˙R L⁄Æ´p °∞d ˘NnRÙp ˙TÙ’UÙ]’.
Show / Hide Label
˘TVo LÙiTRtœ/U˚\j’ ˚YlTRtœ
L⁄Æ˚Vl TVuT”jß, TVo LÙQ˙Yi•V / U˚\dL˙Yi•V TÙ⁄∞p °∞d NnV‹m.
Copy Visual Style
J⁄ ˘TÙ⁄∞u œQeL˚[ ˙Yfl ˘TÙ⁄fidœ TLoj’YRtœ
L⁄Æ˚Vl TVuT”jß ÿRp ˘TÙ⁄∞¤m ˘RÙPok’ Ut\ ˘TÙ⁄hL∞¤m °∞d ˘NnV‹m.
Delete Object
L⁄Æ˚Vj ˙Rok˙R”jR ©u ødL˙Yi•V ˘TÙ⁄hL∞p °∞d ˘NnV‹m.
ON RIGHT CLICK
°WÙ©d LÙh£´p E⁄YÙdLeL∞p Right °∞d ˘Nn’˘LÙi” AYt±u Ru˚UL∞p
UÙt\eLs ˘NnVXÙm.
G”j’dLÙhPÙL J⁄ YhPjßp Right °∞d ˘NnŸm ˙TÙ’ C˚Rl ˙TÙu\˘RÙ⁄ NÙ[Wm
°˚Pdœm
Cßp ÿRp Y¨˚N´p YhPjßu ˘TV⁄m Æ[dLeLfim Bœm ˛ C Gu\ YhPm, B Gu\
◊s∞˚V ˚UVUÙLd ˘LÙi” 3 AXœ BWjßp Y˚WkR’ Bœm. A”jR Y¨˚N´p
YhPjßu NUuTÙPÙœm. C’ Rt˘TÙ›’ SUdœj ˙R˚Y Cp˚X. Íu\ÙY’ Y¨˚N´p
Es[ Check Box I TVuT”jß YhPj˚R U˚\j’ ˚YdLXÙm. A”jR Y¨˚N´u Check
Box I TVuT”jß YhPjßu ˘TV˚W G›ßd LÙi©dL˙YÙ U˚\j’ ˚YdL˙YÙ
˘NnVXÙm. A”jR Y¨˚N´u Trace on ™L‹m ˘NVpTÙh” YÙnl◊s[RÙœm.
73
CR˚]dœ±j’l ©u]o ƨYÙLd ·\XÙm.
Copy to Input Bar Rt˙TÙ’ SUdœj ˙R˚Y Cp˚X.
Rename - ˘TVo UÙtflYRtœ °∞d ˘NnŸm˙TÙ’ °˚Pdœm NÙ[Wjßp ◊ßV ˘TVo
A∞dL‹m.
Delete GuTßp °∞d ˘NnŸm ˙TÙ’ YhPj˚R ødLXÙm.
Object Properties GuTßp °∞d˘NnŸm ˙TÙ’ C’˙TÙu\ J⁄ NÙ[Wm °˚Pdœm.
Basic:
˘TVo, YW˚V˚\ ˙TÙu\Yt±p UÙt\eLs ˘NnYRtœ
Color :
˙R˚YÙV] ®\m A∞dL
Style :
˙LÙh•u R•Uu, Y•Y˚Ul◊, ®\m ˙TÙu\Yt˚\ J›eœ T”jR.
Algebra :
Rt˙TÙ’ ˙R˚Y´p˚X
Advanced :
$ Condition to show object.
YhPj˚Rj ˙R˚YVÙ] ˙TÙ’
U˚\j’ ˚YlTRtœm / LÙiT-
Rtœm CRtLÙLf (˙NÙR˚]l-
˘Th•)dœl (Check Box)˘TVo
A∞dL˙YÙ £˚XP¨u Ußl◊
A∞dL˙YÙ ˘NnVXÙm. eg: a Gu\
£˚XP˚W E⁄YÙd° Condition to
show Object Box Cp a>1 G]
A∞jRÙp a Cu Ußl◊ Ju˚\ÆP
AßLUÙœm˙TÙ’ Uh”˙U
YhPj˚RdLÙQ ÿ•Ÿm.
74
$ Dynamic Colours. YhPjßu ®\m, ÿ›˚U B°VYt˚\f £˚XPoL˚[l TVuT”jß
UÙt\XÙm. Min:0 Max:1 BœmT•VÙL SÙuœ £˚XPoL˚[ E⁄YÙdL‹m. Red, Green,
Blue, Opacity C˚Y Jq˘YÙu±¤m J⁄ £˚XP¨u ˘TVWÙLd ˘LÙ”j’ Close °∞d
˘NnV‹m. ®\eL˚[d œ±dœm £˚XPoL∞u Ußl◊ UÙflYRtœ HtT YhPjßu ®\m
UÙflY˚Rd LÙQXÙm. Opacity œ±l©”m £˚XP¨u Ußl◊ UÙflYRtœ HtT ®\jßu
ÿ›˚U UÙflY˚Rd LÙQXÙm. £˚XPoLfidœ Animation A∞jRÙp C˚Y˘VpXÙm
RÙUÙL˙Y UÙflm.
Scripting:
Rt˙TÙ’ ˙R˚Y´p˚X.
Trace On:
Y•ÆV≠p £X L⁄j’dL˚[ A˚NVf˘Nn’ LÙi©dL‹m, AZLÙ] TPeL˚[
Y˚WYRtœm CkRd L⁄Æ˚Vl TVuT”jRXÙm. £X G”j’dLÙh”Ls.
$ 1 AXœ BWjßp J⁄ YhPjßp Y˚WV‹m.
YhPjßtœ Trace A∞dL‹m (YhPjßp Right Check
˘Nn’ Trace GuTßp °∞d ˘NnV‹m) Move, Tool,
Arrow Key B°VYt˚\l TVuT”jß YhP
˚UVj˚R A˚Nj’ CkRl TPj˚R E⁄YÙdLXÙm.
$ AI ˚UVUÙLd ˘LÙi” BCu YØVÙLf ˘Np¤m J⁄
YhPm Y˚WV‹m. YhPjßtœ Trace A∞dL‹m.
Move, Tool, Arrow Key B°VYt˚\l TVuT”jß
YhP ˚UVj˚R A˚NVf ˘NnR C’˙TÙu\ J⁄
TPj˚R E⁄YÙdLXÙm.
$ J⁄ Angle Slider ααααα E⁄YÙdL‹m. Angle with Given Size
L⁄Æ˚Vl TVuT”jß A, B Gu\ ◊s∞L∞p °∞d
˘NnŸm˙TÙ’ °˚Pdœm NÙ[Wjßp ˙LÙQjßu
Ußl◊ ααααα Gufl A∞dL‹m. J⁄ ◊ßV ◊s∞ A'
°˚Pdœm. A' UnVUÙd° A YØVÙLf Np¤m YhPm
Y˚WV‹m. YhPjßtœ Trace A∞dL‹m. £˚XP⁄dœ
Animation A∞dL‹m. ◊s∞L˚[Ÿm ˙LÙQj˚RŸm
U˚\j’ ˚YdLXÙm.
75
$ J⁄ Angle Slider ααααα E⁄YÙdL‹m. Angle with Given Size
L⁄Æ˚Vl TVuT”jß A, B Gu\ ◊s∞L∞p °∞d
˘NnŸm˙TÙ’ °˚Pdœm NÙ[Wjßp ˙LÙQjßu Ußl◊
ααααα Gufl A∞dL‹m. J⁄ ◊ßV ◊s∞ A' °˚Pdœm. C˙R
L⁄Æ˚V ¡i”m TVuT”jß A, B Gu\ ◊s∞L∞p
Y¨˚NVÙLd °∞d ˘Nn’ ˙LÙQjßu UßlTÙL 2ααααα
Gufl A∞dL‹m. J⁄ ◊ßV ◊s∞ A'1 Em °˚Pdœm.
A', A'1 B°VYt˚\ C˚Qj’ J⁄ ˙LÙ” Y˚WV‹m.
˙LÙh•tœ Trace A∞dL‹m. £˚XP⁄dœ Animation
A∞dL‹m.
$ ÿu ˘NVpTÙ”L∞p ˙LÙQjßu A[‹ 2α
GuTRtœl TßXÙL 4α G] A∞jRÙp CkRl TPj˚R
E⁄YÙdLXÙm.
Note: Y˚WkR TPeL˚[ AØlTRtœ Key Box Cp Ctrl, FB°VYt˚\f ˙Noj’ ©•j’
A›jR‹m.
˘Nn’ TÙodLXÙm.
1. 5 ˘N.¡ TdLm Es[ J⁄ N’Wm ¥˙VÙ¥lWÙ TVuT”jß E⁄YÙdL‹m.
2. Y˚W ˙_Ù•L∞u ˘RÙ˚L 180 •°¨ BL C⁄dœm Gufl ¥˙VÙ¥lWÙ
TVuT”jßj ˘R∞‹T”jR‹m.
3. 50 ˘N.¡ TWlT[‹ Es[ J⁄ N’Wm Y˚WV‹m.
Æ˚[VÙh”l ˘Th• 5, 6, 7 Yœl◊L∞p I.£.• TÙPl◊jRLUÙ] C˛LpÆ
B°VYt˚\f ˙NÙßj’l TÙoj’ ·”Rp ˘NVpTÙ”L˚[d Li” ©•dLXÙm.
°d (KIG)
L¶Rm LtTßp Y•ÆVp E⁄YÙdLeLfidœl TVuT”m ˙Y˘\Ù⁄ —RkßW ˘Uu˘TÙ⁄s
°d (KIG) Bœm. I•@v·p ETi” BT˙W•q £vPm Es[ L¶≤´p C˚Rd
∏rdLÙ‘m ÿ˚\´p ß\dLXÙm.
Application →Education →KIG
Cl˘TÙ›’ ß\k’ Y⁄m NÙ[Wm °d°u (KIG) Yod v˙Tv Bœm. ¥˙VÙ¥lWÙ˚Yl TÙufl
Y•ÆV≠p TX TÙPlTœßL˚[Ÿm LtTRtœ¨V J⁄ ˘Uu˘TÙ⁄˙[ °d. Tp˙Yfl Y•Y´p
Y•YeL˚[ G∞RÙL Y˚WYRtœm TX Y•ÆVp ˙Rt\eL˚[j ˘R∞‹T”jR‹m CkR
76
˘Uu˘TÙ⁄˚[l TVuT”jRXÙm. PÙdPo¥˙VÙ, ˙L.¥˙VÙ, LÙl¨ ˘RÙPe°V
˘Uu˘TÙ⁄hLfim C˙R ÿ˚\´p L¶Rm LtTßp TVuT”jRXÙm.
J. Freetion Lab
J. Freetion Lab GuT’ L¶Rm LtTRtœ ERÆ◊¨Ÿm Ut˘\Ù⁄ TV‡s[ —RkßW
˘Uu˘TÙ⁄[Ùœm. ˘RÙPdL®˚X Yœl◊L∞p Æ˚[VÙh”l ˘Th•l ◊jRLjßp CRu
˘NVpÿ˚\ Æ[dLlTh”s[’. ˙R˚YVÙ] ˙RPpLs SPjß ˘NVpTÙ”L∞p D”T”≈oLs
ApXYÙ?
77
AXœ ˛ 6
˘RÙPdL®˚X Yœl◊L∞p L¶Rm ˛ Es[PdLTœlTÙn‹
ÿu‡˚W
˘RÙPdL ®˚X Yœl◊L∞u Es[PdLÿm ARu Gp˚XŸm Jq˘YÙ⁄ B£¨V
T´t£VÙ[⁄m ˘R¨k’˘LÙs[ ˙Yi”m. Jq˘YÙ⁄ Es[PdLl ©¨‹m Yœl◊L∞p
GqYÙfl Æ≤˙VÙLm NnVlTh”s[’ Guflm AYt±u Y[of£Ÿm RÙPof£Ÿm GqYÙfl
Es[’ Gufl T¨˙NÙßj’ ◊¨k’˘LÙs[ ˙Yi”m. TÙPl◊jRLeL∞u ÷iTÙo˚Y (scan-
ning) YØVÙL C˚R S˚Pÿ˚\lT”jRXÙm. Gi A±‹, SÙpY˚Lf ˘NVpLs, A[‹Ls,
Y•ÆVp, ˙SWm B°V Es[PdLl ©¨‹L˚[f —Zp Hflÿ˚\´p Æ[d°d˘LÙi”
Tp˙Yfl Yœl◊L∞XÙL Jl¿”T”jRlTh”s[˚Rl TœjR±Ÿm ˙TÙ’ Lt\p Lt©jRp
˘NVpÿ˚\´u Es[PdLjßu GkR ®˚XY˚W ˙TÙL ˙Yi”m GuT˚Rl ◊¨k’ ˘LÙs[
CV¤m.
L¶R L⁄j’Ls, J⁄ Ne°≠´u ˘RÙPo˙TÙX Jufld˘LÙufl ˘RÙPo◊˘LÙi”s[’ Gufl
◊¨k’ ˘LÙsYRtœm, Gi A±‹, SÙpY˚Lf ˘NVp, A[‹Ls, Y•ÆVp, ˙SWm ˙TÙu\
Tp˙Yfl ©¨‹L∞u Lt\p A˚P‹Ls, L⁄j’dLs, Lt\p L⁄ÆLs B°VYt˚\
TœjR±YRtœm, ˘RÙPdL®˚X Yœl◊L∞p Es[ Tp˙Yfl ©¨‹Ls L⁄j’ T¨UÙt\
ÿ˚\˚V L¶RLt\p A‘œÿ˚\´u A•lT˚P´p ˘R¨k’˘LÙs[‹m CkR AXœ
YØVÙL CV¤°\’.
Gi A±‹, GiL∞u CPUßl◊ Æ[dLm, TVuTÙ” Gu\ L⁄j’Lfidœ Ceœ
ÿd°Vj’Ym A∞dLlT”°\’. B]Ùp Jq˘YÙ⁄ Yœl©¤m Lt\p A˚P‹Lfidœ HtT
GiL˚[ CXdLjߤm G›jߤm G›’YRtœm, Æ[dœYRtœm ˘RÙPdL ®˚X
Yœl◊L∞p œZk˚RLfidœl T´t£ A∞dLj ß\u ˘T\ ˙Yi•VYoL˙[ B£¨V
UÙQYoLs. ·hPp, LØjRp, ˘T⁄dLp, YœjRp ˙TÙu\ L⁄j’L˚[Ÿm ˘NVpL˚[Ÿm
TVuT”jß S˚Pÿ˚\l©Wf£˚]L˚[l TœlTÙnk’ æo‹ LÙiTRtLÙ] ˘NVpTÙ”L˚[
˘R¨k’˘LÙsYRtœl Ts∞d·P TÙP HtTÙh˚P £\kR ÿ˚\´p TVuT”jR ˙Yi”m.
A[‹LfiPu ˘RÙPo◊T”jß YÙrd˚L „ZpL∞u YÙ´XÙL ø[jßu Tp˙Yfl AXœL˚[
œ±j’m, TWlT[Æu Tp˙Yfl YÙnl◊L˚[d œ±j’m A[‹L∞u Tp˙Yfl AXœL˚[
œ±j’m B£¨V UÙQYoLs ß\u˘Ttfld˘LÙs[ ˙Yi”m. A’ YØVÙL ˘Yq˙Yfl
AXœL∞u TWvTW ˘RÙPo◊, Jl◊˚UT”jRp, Æ[dLm A∞jRp, AhPY˚QlT”jRp,
©WfN˚] TœlTÙn‹, Tp˙Yfl YØL∞p R”Rp, F°j’d ·flRp, ’p≠Vl T”jRp TÙu\
˘NVpß\uLfidœ HtT S˚Pÿ˚\l ©Wf£˚]L˚[ E⁄YÙdL ˙Yi”m.
Y•ÆVp Gu\ ©¨‹Pu ˘RÙPo◊T”jß ˘NqYLm, ÿd˙LÙQm, YhPm Gu‡m Y•ÆVp
Y•YeL˚[j ˘R¨k’˘LÙsYRtœm CkR Y•YeL˚[f ˙Noj’˚Yj’ ˘Yq˙Yfl
Y•YeLs E⁄YÙdœYRtœm A˚PYRtœUÙ] ß\u Ts∞lTÙP HtTÙh•p C⁄k’ °˚PdL
˙Yi”m.
78
—Yod L•LÙWm, LdL•LÙWm, B°VYt˚\l TÙoj’ SWm ·flYRtœm SÙhLÙh•´p C⁄k÷
RYYpL˚[d Li”©•lTRtœm LÙXm Gu\ L⁄j’dœ ˘Yq˙Yfl Yœl◊L∞p Es[
YÙnl◊L˚[ TœjR±YRtœm Be°X ˛ R™r UÙReL∞u ˙RßL˚[d Li”©•lTRtœm
CYt±tœ C˚P˙VVÙ] TWvTW ˘RÙPo˚Td Li” ©•lTRtœm Ts∞lTÙP HtTÙh•p
Tp˙Yfl Yœl◊L∞p Es[ ˘NVpTÙh” ÿ˚\L˚[Ÿm T¨UÙt\ EjßL˚[Ÿm
˘R¨k’˘LÙsfiRp GuT’ TÙP HtTÙh•u Cu±V˚UVÙR LÙW¶VÙœm.
Lt\p A˚P‹Ls
$ ˘RÙPdL ®˚X´u Tp˙Yfl UiPXeLfiPu ˘RÙPo◊ T”jß L⁄j’L˚[d
L¶RdLt\p A‘œÿ˚\ŸPu ˘RÙPo◊ T”jß Æ[dLU∞jRp.
$ L⁄j’L∞u ˘RÙPof£Ÿm Y[of£Ÿm TWvTW ˘RÙPo◊m ˘R¨k’˘LÙs[p.
ÿd°V L⁄j’Ls
$ Gi A±‹
$ SÙpY˚Lf ˘NVpLs
$ A[‹Ls
$ Y•ÆVp
$ LÙXm
$ L⁄j’L∞u ˘RÙPof£, Y[of£, TWvTW ˘RÙPo◊
˘RÙPdL ®˚X´p Tp˙Yfl UiPXeLfiPu RÙPo◊˚PV L⁄j’Lfim L¶RLt\p
A‘œÿ˚\Ÿm
Gi A±‹
˘T¨V’, £±V’, AßLm, œ˚\‹ Gk\ ÿu Gi L⁄j’Ls (Pre-number concepts)
ÆjßVÙNUÙ] A‡TYeL∞u YØVÙL °˚PdLl ˘Tt\ ©u Gi A±Ætœ LPk’
˘Np°\’.
Ko Gi¶u Tp˙Yfl YÙnl◊L∞u A‡TYm œZk˚Rdœd °˚PdLl˘Tflm ˙TÙ’ Gi
A±‹ Es[’ G]d ·\XÙm.
Gi A±‹Pu ˘RÙPo◊ E˚PV Lt\p A˚P‹L˚[ Jq˘YÙ⁄ Yœl©¤m C⁄k’
˙Rok˘R”j’ ˘TÙ⁄jRUÙ] Lt\p A˚P‹L˚[ Y¨˚NlT”jß YZeL ˙Yi”m.
GiQp EjßLs
1. œ±l©hP Gi¶d˚L´XÙ] ˘TÙ⁄hL˚[ Jq˘YÙu\ÙL Gi¶, Gi¶d˚L
GqY[‹ Gufl Li”©•dL‹m.
2. J⁄ œ±l©hP Gi¶p C⁄k’ ˘RÙPe° ˙Y˘\Ù⁄ Gi Y˚W GiQ ˙Yi”m.
3. J⁄ œ±l©hP Gi¶p C⁄k’ ˘RÙPe° ∏r ˙SÙd° GiQ ˙Yi”m. G.LÙ (15, 14, 13, ........)
4. J⁄ œ±l©hP Gi˚Q Æh”Æh” GiQ‹m.
Gi A±˚Y Ttfld˘LÙsYRtœf NVpTÙ”L˚[ TÙ⁄hLfiPu RÙPo◊T”j’Y’ TÙp
YÙrd˚Lf „ZpLfiPu ˘RÙPo◊ T”jR‹m CVX ˙Yi”m. CqYÙfl Gi A±˚Yl ˘Tt\
79
œZk˚Rdœ GiL∞u Æ[dLjßp G∞RÙ] L⁄j’ E⁄YÙ°\’. ˘Yq˙Yfl ÿ˚\´p
GiL˚[ Æ[dœYRtLÙ] G”j’dLÙh” ∏˙Z RWlTh”s[’.
BßLÙX U≤Ru Gi¶ EflßT”j’Y’Pu ˘RÙPo◊T”jß Jufld˘LÙufl
˘TÙ⁄jRlT”m ÿ˚\˚Vl TVuT”jß]Ùu. °.ÿ. 9 Bm Ët\Ùi•p UÙvXm (Mos-
lem) Bh£VÙ[oL[Ùp ReL∞u ˙TWW˚N ˙UtLjßV SÙPÙ°V v˘T´≤tœm,
°ZdLjßV SÙPÙ°V CuP≥≠tœm ƨYÙd°V ˙TÙ’ "∂k’ AW©d' GiLs Gu\
˘TVo TWYj ˘RÙPe°V’. Cufl SÙm TVuT”j’Y’ CkR Giÿ˚\VÙœm.
Giÿ˚\´u £\l©Vp◊Ls ∏˙Z ˘LÙ”dLlTh”s[].
1. 0, 1, 2, ......................... 9 Y˚W Es[ 10 CXdLeL˚[ TVuT”jß Cu˚\V Giÿ˚\
E⁄YÙdLlTh”s[’. B]Ùp C’ TjR•UÙ] Giÿ˚\ Gu\ TV¨p A±VlT”°\’.
2. CPUßl◊ Gu\ L⁄j˚R T¨∫X˚] ˘Nn’ GqY[‹ ˘T¨V Gi˚QŸm, ARu
CPUßl©tœ HtT œ±l©P CV¤°\’.
3. "ÈwVj˚R' J⁄ CXdLUÙLl TVuT”jR CV¤°\’.
CPUßl©u L⁄j’⁄YÙdLm N¨VÙ] ÿ˚\´p SPdL ˙Yi”m G≤p Íufl®˚XLs,
T¨∫X˚] ˘NnV ˙Yi”m.
1. T⁄l˘TÙ⁄hL˚[l TVuT”jß L⁄j’⁄YÙdLm (Lm◊Lh”dLs, ÿj’dLs......)
2. T⁄l˘TÙ⁄hLs ÿ˚\´XÙ] ®Lrj’Rp (TPeLs, ATÙdLv.......)
3. œ±¬”L˚[l TuT”jß ®Lrj’Rp.
ÿRp CWi” ®˚XL˚[d LPk’ ˘Nu\ ©u Íu\ÙY’ ®˚X´u ®Lrj’Rp SP˘T\
˙Yi”m.
SÙpY˚Ld ˘NVpLs
SÙpY˚Lf ˘NVpLfiPu ˘RÙPo◊˚PV Lt©jRp CXdœLs
$ ·hPp, LØjRp, ˘T⁄dLp, YœjRp GuT]Yt±u L⁄j˚R ˘R¨k’˘LÙsfiRp.
7
5 + 2
7 JuflLs ˙NokRÙp
1 + 1 + 1 + 1 + 1 + 1 + 1
6 + 1
2 Íuflm Juflm
˙NokRÙp
3 CWi”m Juflm
˙NokRÙp
8 - 1
80
$ 4 ˘NVpÿ˚\L˚[Ÿm ˘Yq˙Yfl ÿ˚\L∞p ◊¨k’˘LÙs[‹m ˙R˚Ydœ HtT
˘TÙ⁄jRUÙ] ÿ˚\˚V ˙Rok˘R”dL‹m CV¤°\’.
$ ˘NVpTX˚] Ußj’d ·flYRtLÙ] ß\u °˚Pd°\’.
$ S˚Pÿ˚\ ©Wf£˚]Lfidœj æo‹LÙQ CV¤°\’.
$ SÙpY˚Lf ˘NVpLfiPu ˘RÙPo◊˚PV ◊ßV S˚Pÿ˚\l ©Wf£˚]L˚[ E⁄YÙdL
CV¤°\’
SÙpY˚Lf ˘NVpLs
·hPp
S˚Pÿ˚\f „ZpL∞u YÙ´XÙL "·hPp' Gu\ L⁄j˚R G”j’˚WdL ˙Yi”m.
˘TÙ⁄hLs, TPeLs YØVÙL ·hPp, ◊¨k’˘LÙiP©u, GiL˚[ TVuT”jߟs[ ·hPp,
G”j’˚WdL ˙Yi”m.
0 0 0 0 + 0 0 0 = 0 0 0 0 0 0 0
4 + 3 = 7
CWi” CWi•XdL GiL[Ù]Ù¤m CWi” Íu±XdL GiL[Ù]Ù¤m ·hPp
˘NVpÿ˚\ ˘NnYRtœ Tp˙Yfl YØÿ˚\Ls ˙Rok˘R”dLlT”°\’ GuT’
◊¨k’˘LÙs[ ˙Yi”m.
G.LÙ: 234 + 156
(F°j’ N¨VÙ] Tß˚Xd LÙQ ˙Yi”m)
ÿ˚\ 1 ÿ˚\ 2 ÿ˚\ 3
234 = 200 + 30 + 4 234 + 156 = 230 + 150 + 4 + 6 234 + 156 = 234 + 6 + 150
156 = 100 + 50 + 6 = 380 + 10 = 240 + 150 = 390
= 300 + 80 + 10 = 390 =====
= 390 ======
======
·hPp ˘NVpL˚[f ˘NnŸm ˙TÙ’ œZk˚RLs ◊¨Vd·•V RYflL˚[j ˘R¨k’˘LÙi”
æo‹ LÙiTRtLÙ] YØÿ˚\L˚[ A±‹fljR ˙Yi”m.
LØjRp
˘TÙ⁄hL˚[l TVuT”jߟm TPeL∞u YÙ´XÙL‹m LØjRp ˘NVpÿ˚\ ◊¨V ˙Yi”m.
G.LÙ. 7 - 3 = 4
ôôôôôôô - ôôô = ôôôô
7 - 3 = 4
81
LØjRp ˘NVpL∞p ◊¨Vd·•V RYflLs G˚Y GpXÙm?
65 - 65 - 65 -
26 26 26
--- --- ---
49 41 30
== == ==
CkR RYflL˚[j ß⁄j’YRtœl ˘TÙ⁄jRUÙ] ÿ˚\´XÙ] Tp˙Yfl Lt\p ˘NVpTÙ”L˚[
ßhP™Pp ˙Yi”m.
˘T⁄dLp
¡i”m ¡i”m ·h”Y˙R ˘T⁄dLp Gu\ L⁄j˚R Tp˙Yfl S˚Pÿ˚\f „ZpLs
YÙ´XÙL G”j’˚WdL ˙Yi”m.
J⁄ ˘TÙ⁄∞u 4 ≈Rm ·hPeLs ˘T⁄dLp „ZXXÙL G”j’˚WdLlTh”s[’.
˘TÙ⁄hLs ·hPp Ei˚U ˘T⁄dLp Ei˚U
$ $ $ $ 4 1
×
4 = 4
$ $ $ $
$ $ $ $ 4+4 = 8 2
×
4 = 8
$ $ $ $
$ $ $ $
$ $ $ $ 4+4+4 = 12 3
×
4 = 12
$ $ $ $
$ $ $ $
$ $ $ $ 4+4+4+4 = 16 4
×
4 = 16
$ $ $ $
CqYÙfl ˘T⁄dLp Ei˚UL˚[l Tp˙Yfl GiLfiPu ˘RÙPo◊T”jß E⁄YÙdœYRtLÙ]
YÙnl◊Ls °˚PdL ˙Yi”m.
AhPY˚Q˚V E⁄YÙdœY’ AYt±u Y¨˚NŸm A˚Ul◊m ˘R¨k’˘LÙs[‹m,
Ei˚UL˚[d LiP±k’ ˘NVpT”j’Y˚R G∞RÙdœYRtœUÙœm. ˘T⁄dLp
Ei˚UL˚[ ®fl‹YRtœ Gu˘]u] ˘NVpTÙ”L˚[ A∞dLXÙm.
¡i”m ¡i”m ·hP≠u YÙ´XÙL AhPY˚Q E⁄YÙdœRp.
AhPY˚Q´p C⁄k’ NUUÙ] ˘T⁄dLp Ei˚UL˚[d Li”©•jRp
( 2
×
4 = 8 , 4
×
2 = 8 )
82
2
×
9 6
×
3
18
3
×
6 9
×
2
$ Jq˘YÙ⁄ ˘T⁄dLp Ei˚U´u ÿu]Ù¤m ©u]Ù¤m Es[Yt˚\d LiP±Rp.
G.LÙ. 5 x 4 = 20, G®p 4 x 4 =?, 6 x 4 = ?
$ A•lT˚P ˘T⁄dLp Ei˚UL˚[d LiP±Rp
(A•lT˚P ˘T⁄dLp Ei˚UL˚[ AhPY˚Q´p C⁄k’ LiP±Rp)
CjR˚LV ˘NVpTÙ”L˚[ AhPY˚QŸPu ˘RÙPo◊ T”jß ˘NnR©u H˚]V
˘NVpTÙ”L[Ù] Æ˚[VÙh”Ls, TÙPpLs, L˚RLs, S˚Pÿ˚\l ©Wf£˚]Ls CYt˚\f
˘NnV ˙Yi”m.
˘T⁄dLp ˘NVpLs
˘T⁄dL≠u ˘Yq˙Yfl YØÿ˚\L∞u YÙ´XÙLl ˘T⁄dLp TXuLÙQ, Jq˘YÙ⁄
©¨f£˚]dœm ˘TÙ⁄jRUÙ] ˘NVpÿ˚\˚Vj ˙RWk˘R”dL ˙Yi”m.
ÿ˚\ 1
25 x 12
10 2
20 200 40
5 50 10
250 50 300
ÿ˚\ 2
25 x 12 = 25 x 4 x 3 = 100 x 3 = 300
ÿ˚\ 3
25 x 12 = 25 x 2 x 6 = 50 x 6 =
= 50 x 2 x 3 =
= 100 x 3 = 300
ÿ˚\ 4
25 x 12 = (10 + 2) 25 = 250 + 50 = 300
YœjRp
¡i”m ¡i”m LØjR˙X YœjRp Gu\ L⁄j˚R G”j’˚WlTRtœf NUUÙL Teœ˚YjRp
Gu\ L⁄j˚Rl ˘T\ ˙Yi”m.
83
GLÙ: 18 ◊jRLeLs 3 SToLfidœ NUUÙL Te°PXÙm.
18 -
3
15 -
3
12 -
3
9 - 18 ÷ 3 = 6
3
6 -
3
3 -
3
0 -
¡i”m ¡i”m LØjR≠u YÙ´XÙLl ˘T⁄dLp Gu\ L⁄j’ G”j’˚WdLlT”°\’
YœjRp APVÙ[m (÷) A±ÿLlT”°\’. YœjR˚X ˘T⁄dLp Ei˚ULfiPu ˘RÙPo◊
T”jR‹m ˘Yq˙Yfl Lt\p ˘NVpTÙ”L∞u YÙ´XÙLf ˘NVpT”jR‹m CV¤m.
6 x 3 = 18 Cp C⁄k’ 18 ÷ 3 = 6,
18 ÷ 3 = 3
YœjRp NVp◊¨Ÿm TÙ’ ®LZd·•V RYflLs, G”jRdLÙh”L∞u YÙ´XÙLl TœlTÙn‹
˘Nn’ æo‹ LÙiTRtLÙ] YØÿ˚\L˚[d LXk’˚WVÙPXÙm.
13 3100 1003
7 721 7 721 7 721
7 21 700
021 700 21
21 700 21
0 0 0
Jq˘YÙ⁄ YœjRp ˘NV≠¤m Ykß⁄dœm YœjRp TX≤u £\l◊Ru˚UL˚[ ˘Yq˙Yfl
©Wf£˚]L∞u YÙ´XÙL G”j’˚WdL ˙Yi”m.
©u] GiLs
©u] GiL∞u Lt\¤Pu ˘RÙPo◊˚PV Lt\p CXdœLs
$ ©u] GiLs Gu\ L⁄j˚R ˘Yq˙Yfl ®˚XL∞p ˘R¨k’˘LÙsYRtœ.
84
$ ©u] GiL˚[d œ±l©”YRtLÙ] ˙Y˘\Ù⁄ ÿ˚\ RNU Y•Ym G]l
◊¨k’˘LÙsYRtœ
©u] Gi L⁄j˚R 3 ˘Yq˙Yfl ®˚XL[ÙL YZeœRp
1. Ju±u TÙLUÙL J⁄ ˘TÙ⁄˚[f NU TÙLeL[Ùd°, J⁄ œ±l©hP Tœß˚V œ±l©”Rp.
2. ©u] Gi˚Q YœjR≠u TÙLUÙL (3/4 Gu\ ©u] Gi˚Q 3 ÷ 4) œ±l©”Rp.
3. ©u]j˚R ·hPjßu TÙLUÙL œ±l©”Rp.
AXœL˚[ Eh°W°j’ ˙U¤m G”j’dLÙh”L˚[Ÿm ˘NVpTÙ”L˚[Ÿm
˙NL¨dL‹m.
RNU GiLs
©u] Gi˚Qœ±l©”YRtLÙ] ˙Y˘\Ù⁄ ÿ˚\˙V RNU GiLs.
$ 0.5 GuT’ 5/10, 1/2 Gu\ ©u] GiL˚[d œ±l©”RYtLÙ] ˘Yq˙Yfl
YØÿ˚\L[Ùœm. B]Ùp RNU©u]m 10, 100, 1000 TÙu\Yt±u RÙœßVÙL Uh”˙U
œ±l©PlT”°\’.
$ Ju˚\ 10 NUTœßL[Ùd° °˚Pdœm ©u] GiL˚[d œ±j’s[ L⁄j’ RNU
Gi˚Q G”j’˚WdL ERÆ◊¨°\’.
$ 45.36 Gu\ RNU Gi 45 ÿ› Gi‘m, Tjßp JuflLs Íuflm, ˱p JuflLs
Bflm, ˙NokRRÙœm.
45.36 = 45 + + = 45 + 0.3 + 0.06
RNU GiL∞p œZk˚RLs ˘NnVd ·•V RYflL˚[d ∏˙Z RWlTh”s[ ©Wf£˚]L∞u
YÙ´XÙLd LXk’˚WVÙP‹m.
1) 35.46 = 35.5 2) 23.35 + 2.4 = 5.39
3) = 0.5 4) 0.005 =
A[‹Ls
$ A[‹LfiPu ˘RÙPo◊˚PV Lt\p CXdœLs
$ A[lTRu ˙R˚Y˚V EQoj’Rp
$ GYt˚\˘VpXÙm GiQ ˙Yi”m? GYt˚\˘VpXÙm A[dL ˙Yi”m? GuT˚R
©¨jR±YRtœ
$ Tp˙Yfl A[‹L˚[j ˘R¨k’ ˘LÙsYRtœ (LÙXm, ø[m, G˚P ˘LÙs[[‹)
S˚Pÿ˚\ YÙrd˚L´p, Gi¶d˚L˚V ˙TÙufl A[lT’m ™L‹m ÿd°Vj’Ym
YÙnkR’ Bœm.
85
∏˙Z RWlTh”s[ ·tflL˚[d LXk’˚WVÙP‹m
Gi‘YRtœm A[lTRtœm C˚P˙V Es[ ÆjßVÙNm Gu]?
(˘TÙ⁄jRlT”j’R¤m, Jl◊˚U ˘NnR¤m)
Gl˙TÙ˘RpXÙm A[dL ˙S¨”m?
Gi‘YRtœ CVp°u\Yt˚\ Uh”UÙ A[lT’
Gi‘YRtœ CVXÙRYt±tœ G”j’dLÙh” LÙiL.
(’¶, CPm, Ëp.........)
ø[m
$ ø[m Gu\ L⁄j’
$ ø[j˚R œ±dœm ˙Yfl ˘NÙtLs (ALXm, BZm, EVWm, —t\[‹)
$ ø[jßu G”jR˚WjRp ˛ Ae∏LÙWm CpXÙR JWX°p ˘RÙPe° Ae∏LÙWm Es[ßtœ.
$ ø[jßu NoY˙RN AXœ.
$ ø[jßu TZeLÙX AXœLs (NÙu, —Y”, ÿZm UÙfl, To˙XÙe, ˚Up)
$ £±V A[‹Lfidœf £±V KWXœ ˙R˚YVÙœm
$ ø[m GqY[‹ G] F°j’d ·flRp, N¨VÙ] A[‹Pu Jl◊˚U ˘NnRp)
˘LÙs[[‹
$ ˘LÙs[[‹ Gu\ L⁄j’
$ ˘LÙs[[ÆtLÙ] KWXœ
$ ˘LÙs[[Ætœm L] A[Ætœm C˚P˙V Es[ ˘RÙPo◊.
$ ˘LÙs[[ÆtLÙ] SÙh”l◊\ A[‹Ls (°[Ùv, Ll, œl©)
$ Tp˙Yfl A[ÆXÙ] TÙjßWeL˚[j RVÙo ˘NnRp.
G˚P (TÙWm)
$ G˚P Gu\ L⁄j’
$ RWÙ— CpXÙUp G˚P˚V LÙ‘Rp
$ RWÙ— E⁄YÙdœRp
$ G˚P´u KWXœLs, TWvTW ˘RÙPo◊
$ G˚P˚V F°j’d ·flRp
$ 1 ≠hPo RiΩ¨u G˚P
˙SWm
$ ˙SWjßu JWXœLs, TWvTW ˘RÙPo◊
$ L•LÙWm E⁄YÙdœRp, L•LÙWj˚Rl TÙoj’ ˙SWm ·flRp
$ am, pm, 24 U¶d·o ˙SWm.
$ YÙWm, UÙRm, Y⁄Pm CYt˚\d œ±j’s[ A±‹, TWvTW ˘RÙPo◊.
$ LÙX A[‹ Gu\ L⁄j’ (Æ]Ù•, ®™Pm, U¶d·o, SÙs, YÙWm, UÙRm, Y⁄Pm)
$ LÙX AhPY˚Q (LÙXd˙LÙ” E⁄YÙdœRp, J⁄ ST¨u YÙrd˚L ®˚XLs, SÙh•p
86
S˚P˘Tflm ÿd°V ®Lr‹Ls, YWXÙtfl Ei˚ULs CYt˚\d LÙXd ˙LÙh•p
œ±l©”Rp.
$ ˙SWj˚R F°j’d ·flRp.
$ ƒl Y⁄Pm
$ ÁWj˚R ˙SWj’Pu ˘RÙPo◊T”jßd ·flRp.
$ ®Z¤dœm ˙SWjßtœm Es[ ˘RÙPo◊
ƒl Y⁄Pm (Leap Year)
J⁄ Y⁄Pm GuT’ 365.24219 SÙhLs Bœm. NÙRÙWQUÙL J⁄ SÙh LÙh•´p J⁄
Y⁄Pm 365 SÙhL[ÙLj ßhPl T”jRlTh”s[’. Al˙TÙ’ ¡ßŸs[ 0.24219 SÙhL˚[d
·h• 4˛Cp J⁄ Y⁄Pj˚R 366 SÙhL[ÙdLlTh”s[’. 3 Y⁄Pm ©l⁄Y¨dœ 28
SÙhLs Gu\Ùp 4 ˛ Bm Y⁄Pm 29 SÙhLs. 366 SÙhLs Y⁄°u\ Y⁄PUÙœm ƒl
Y⁄Pm. CkR ÿ˚\´p ˘RÙPokRÙp 400 Y⁄Pm ˘Np¤m ˙TÙ’ J⁄ SÙs AßLUÙL
Y⁄m. AR]Ùp 400 Bp YœdLd ·•V Ët\Ùi”L˚[ ƒl Y⁄PUÙLd LQd°PXÙm.
G”j’dLÙhPÙL 2100 ˛ Cp ©l⁄Y¨dœ 28 SÙhLs C⁄dœm. B]Ùp 2000 ˛ Cp
©l⁄Y¨dœ 29 SÙhLs Ei” (LÙWQj˚Rd Li” ©•dL‹m)
SÙhLÙh•´u YWXÙfl
Ët\Ùi”Lfidœ ÿu]˙W J›eLÙL BLÙV Etfl˙SÙdL˚X SPjß RLYpL˚[l
Tß‹ ˘Nn°u\ ÿ˚\ EXLjßu A˚]j’ CPeL∞¤m C⁄kR’. CkßVoLfim
CjR˚LV Etfl ˙SÙdLpLs YØVÙL ShNjßWeLs, °WLQeLs ˙TÙu\Yt˚\R
’p≠VUÙL ·flYRtœj ˙R˚YVÙ] SÙhLÙh•dœ Y•Ym A∞jR]o. TÙ©˙XÙ≤VoLs
NkßW≤u Y•YeL˚[ A•lT˚PVÙLd ˘LÙi” SÙhLÙh•˚Vj RVÙ¨jR]o.
˘T¸oQ™ ÿRp ˘TÙoQ™ Y˚W 29.53 SÙhLs B]RÙp AYoLfi˚PV UÙReLfidœ
Ju±˚PÆh” 29˛EU 30 Em SÙhL[ÙL C⁄kR’. Y⁄Pjßp 365.24 SÙhL[Ù]RÙp
AYoL∞u Y⁄PeLfidœf £X˙Y˚[L∞p 12˛ Em £X˙Y˚[L∞p 13˛Em UÙRes
C⁄kR].
°±v’ Y⁄Pm 1582 ˛ Cp 13 ˛ BY’ ˙TÙl B] °¨L¨ ˘Y∞´hP SÙhLÙh•˙V
°¨˙LÙ¨Vu SÙhLÙh•.
UXVÙ[m SÙhLÙh•
°±v’ Y⁄Pm 825 BLv•p ˘LÙpX Y⁄Pm Gu\ ˘TVo Es[ UXVÙ[m SÙhLÙh•
BWmTUÙ]’, ˘LÙpX Y⁄Pjßp WÙ£L∞u ˘TVoLs UÙReLfidœ A∞dLl
Th”s[]. „¨Vu J⁄ WÙ£´p ˘NpY’ ÿRp AkR WÙ£´u C⁄k’ ˘Y∞˙V
˘Np°u\ SÙs Y˚WVÙœm UÙRjßu LÙX A[‹. „¨Vu A˚]j’ WÙ£L˚[Ÿm
LPk’ ˘NpYRtœ G”j’d ˘LÙsfim ˙SWm NUUt\RÙp U˚XVÙ[ UÙReLfidœ 28
ÿRp 32 SÙhLs Y˚W LÙX A[‹ Ei”.
87
Y•ÆVp
$ Y•ÆV≠u ˙RÙt\j˚RŸm Y[of£˚VŸm A±k’ ˘LÙsYRtœ.
$ Tp˙Yfl Y•ÆVp Y•YeL˚[ A±k’ ˘LÙsYRtœ.
$ Tp˙Yfl Y•ÆVp Y•YeL˚[ E⁄YÙdœYRtœ.
$ Tp˙Yfl Y•ÆVp Y•YeL∞u £\l©Vp◊L˚[ A±k’ ˘LÙsYRtœ.
$ Y•ÆVp Y•YeL˚[ ˙Noj’ ˚Yj’ Tp˙Yfl Y•YeL˚[ E⁄YÙdœYRtœ.
Y•ÆV≠u Es[PdLm
—tfll◊\eL∞u Es[ TÙ⁄hL˚[ Etfl SÙdL‹m £\l©Vp◊L∞u A•lT˚P´p Jl¿”
˘Nn’ Y˚LlT”jR‹m UÙQYoLfidœ YÙnl◊ °˚Pd°u\’.
$ ˙SÙh”l ◊jRLeLs
$ SÙQVeLs
$ ™ßYi•f NdLWm
$ Tp˙Yfl Y•YeL∞p Es[ LÙ°Rm.
$ Tp˙Yfl ø[eL∞p Es[ L´fl, Ëp..............
C˚Rl ˙TÙu\ ˘TÙ⁄hL˚[f £\l©Vp◊L∞u A•lT˚P´p Y˚LlT”jß J˙W
Ti◊˚PVYt±u ˘TVoL˚[ A±k’ ˘LÙi” ˘TÙ⁄hL˚[f ˙NL¨jRp, Y˚WRp,
E⁄YÙdœRp, ˙Noj’ ˚Yj’ Y•YeLs E⁄YÙdœRp ˙TÙu\ ˘NVpTÙ”L˚[f ˘NnV
UÙQYoLfidœ CVX ˙Yi”m.
È™ Gufl ˘TÙ⁄sT”°u\ "¥˙VÙ' Gu\ ˘NÙp¤m A[‹ Gufl ˘TÙ⁄sT”°u\
"Uß' Gu\ ˘NÙp¤m ˙Nok’ Y•ÆVp Gu\ ˘NÙp Y•Ym ˘LÙiP’. È™´p
A[‹L˚[d œ±l©”°u\ A±ÆV˙X Y•ÆVp.
TÙPl◊jRLm, B£¨Vo L˙V” CYt˚\l TVuT”jß Lt\p A˚P‹Ls, L⁄j’Ls,
Lt\p L⁄ÆLs, T¨UÙt\ ÿ˚\ TÙu\ Yt±u AhPY˚Q˚Vj RVÙ¨dL Yi”m.
Gi A\‹: Yœl◊ 1 .....................................................
Lt\p A˚P‹ ÿd°Vd L⁄j’Ls T¨UÙt\ ÿ˚\ Lt\p L⁄ÆLs
1 ÿRp 5 Y˚WŸs[, Tp˙Yfl Yœl◊L∞u TÙPl TœßL˚[ YflThP UiPXeL∞u
A•lT˚P´p ˙U˙X RWl Th”s[ AhPY˚Q´u UÙߨ´p ®WlT ˙Yi”m.
88
G”j’dLÙh”
UiPm: Y•ÆVp Yœl◊ : 3
UiPXm : Gi A±‹ Yœl◊ : 3
Lt\p A˚P‹
—tfll◊\jßp Es[
Y•YeL∞p C⁄k’
˘NqYLm, ÿd˙LÙQm,
YhPm CYt˚\
A±k’ ˘LÙs°u\]o.
˘NqYLm, ÿd˙LÙQm,
YhPm B°VYt˚\l
TVuT”jß ˘Yq˙Yfl
Y•YeL˚[ E⁄YÙd°
Y˚WV‹m ˘Nn°u-
\]o.
ÿd°Vd
L⁄j’Ls
$ ˘NqYLm
$ ÿd˙LÙQm
$ YhPm
$ ˘NqYLm
$ ÿd˙LÙQm
$ YhPm
T¨UÙt\ ÿ˚\Lt\p Lt©jRp
˘NVpÿ˚\
—tfll◊\jßp C⁄k’,
˘NqYLm, ÿd˙LÙQm,
YhPm B°V Y•Ye-
Ls ˙NL¨dLl-T”°\’.
CkR Y•Ye-L∞p C⁄k’
˘NqYLm, ÿd˙LÙQm,
YhPm B°V˚Y
Y˚LlT”jRlT”°u\].
˘NqYLm, ÿd˙LÙQm,
YhPm ˙TÙu\
Y•YeL˚[f ˙Noj’
˚Yj’ AZ°V
Y•YeL˚[ E⁄YÙd°
Y˚W°u\]o.
Lt\p LڮLs
˘NqYLm
ÿd˙LÙQm
YhPm (Tp˙Yfl
A[‹Ls
E˚PV˚Y)
˘NqYLm
ÿd˙LÙQm
YhPm
B°VYt±u
Tp˙Yfl
Y•YeLs.
Lt\p A˚P‹
999 Y˚WŸs[ Gi
L˚[ YÙ£d°u\]o.
ÿd°Vd
L⁄j’Ls
$ 99 ˛ EPu 1 I
˙Nodœm ˙TÙ’
100 °˚Pd°\’.
$ 10 Tj’Ls Nok-
RÙp 100, 100
ËflLs ˙Nok-
RÙp 1000
$ £±V Íufl
CXdL Gi 100
$ ˘T¨V Íufl
CXdL Gi 999
T¨UÙt\ ÿ˚\Lt\p Lt©jRp
˘NVpÿ˚\
Æ˚[VÙh”l ◊jR-Le-
L˙[Ù, TPeL˙[Ù, Gi
Ah˚PL˙[Ù TVu-T”jß
99 EPu 1 I ˙NojRÙp 100
Bœm Guflm ˘RÙPok’
Y⁄m GiLs 101, 102, ..........
GuT]Yt˚\ YÙ£dL‹m
G›R‹m ˘Nn°u\]o.
˘Yq˙Yfl ˘NVpTÙ”Ls
YØVÙL J⁄ Íufl CXdL
Gi¶u ÿu]⁄m
©u]⁄m Es[ Gi-
L˚[d Li”©•j’ YÙ£d-
°u\]o.
Lt\p
LڮLs
Æ˚[VÙh”
ÏTÙn
˙SÙh”Ls,
TPeLs, Gi
Ah˚PLs.
89
Ußl©”Rp
Gi A±‹, SÙpY˚Lf ˘NVpLs, A[‹Ls, Y•ÆVp, ˙SWm B°V ©¨‹L∞p
Tp˙Yfl Yœl◊dL∞u Lt\p A˚P‹Ls, L⁄j’Ls, Lt\p Lt©jRp ˘NVpÿ˚\Ls,
Lt\p L⁄ÆLs, GuT]Yt±u œ±l◊ RVÙo ˘NnRp.
JlT˚Pl◊
£X UiPXeL∞u Lt\p A˚P‹Ls ∏˙Z ˙NodLlTh”s[].
Gi A±‹ Std - 3
SÙpY˚Lf ˘NVpLs
·hPp Std - 3
Gi A±‹
◊ßoLs, L¶R
Æ˚[VÙh”dLs
TœjR±‹
(A˚Ul◊dLs)
999 Y˚W Es[
GiL˚[ YÙ£jRp
Íufl CXdL
GiL∞u CPUßl◊
999 Y˚W Es[
GiL∞u Æ[dLeLsÍufl CXdL GiL∞p
C⁄k’ ˘T¨V’, £±V’
©¨j’ G›’°u\]o. Íufl
CXdLeLs TVuT”jß
˙YflThP Íufl CXdL
GiL˚[ E⁄YÙdœRp
Íufl CXdL GiL∞u
Hfl, C\eœ Y¨˚N
·hPp
Íufl CXdL GiL˚[
GiL∞u ·h”j ˘RÙ˚LVÙL
˘Yq˙Yfl ÿ˚\L∞p
CWi” Íufl CXdL
GiL∞u ·h”j˘RÙ˚L
·hPp ˘NVp TVuT”j’m
S˚Pÿ˚\l ©Wf£˚]
E⁄YÙdLm
·hPp ˘NVp EhT”m
˘Yq˙Yfl TœjR±‹l
©Wf£˚]Ls
·hPp ˘NVp
TVuT”j’m S˚Pÿ˚\l
©Wf£˚]j æo‹
90
LØjRp Std - 3
˘T⁄dLp Std - 3
LØjRp
J⁄ Íufl CXdL Gi¶p C⁄k’
˙Y˘\Ù⁄ Íufl CXdL Gi˚Qd
œ˚\jRp
LØjRp ˘NVp
TVuT”jRlT”°\’
S˚Pÿ˚\l ©Wf£˚] E⁄YÙdLm
LØjRp ˘NVp TVuT”j’m
S˚Pÿ˚\l ©Wf£˚]j æo‹
GiL∞u £\l©Vp◊L˚[d
œ±jR ÿ•‹L˚[
E⁄YÙdœRp
Gi Y¨˚NL∞u Ru˚U˚V
Æ[dœRp
˘T⁄dLp ˘NVp
TVuT”j’m Gi
˘RÙPo◊L∞u Ru˚U˚V
Æ[dœRp
˘T⁄dLp
˘T⁄dLp Ei˚UL˚[
E⁄YÙdœRp
˘T⁄dLp Ei˚ULs
TVuT”jßV
©Wf£˚]jæo‹
˘T⁄dLp Ei˚ULs
TVuT”jߟs[ ©Wf£˚]
E⁄YÙdLm
˘T⁄dLp ˘NVp œ±jR Gi
˘RÙPo◊L˚[d
Li”©•jRp
91
L⁄j’dL∞u, Y[of£Ÿm ˘RÙPof£Ÿm TWvTWj˘RÙPo◊m
L¶Rjßu L⁄j’Ls Ju˙\Ù˘PÙufl ˘RÙPo◊ E˚PV]YÙœm. Gi A±‹ ˘T\ÙR
UÙQYoLfidœ GiL∞˚P˙V Es[ ˘NVpL˚[f ˘NnYßp £WUm HtT”°\’. L¶Rd
L⁄j’dLs Ju˙\Ù˘PÙufl ˘RÙPo◊ ˘LÙi”s[]. G]˙Y GkR J⁄ L⁄j˚RŸm E⁄YÙdL
˙Yi”m Gu\Ùp Aj’Pu ˘RÙPo◊˚PV ˙Yfl£X L⁄j’L˚[Ÿm ˘Ttfld˘LÙs[
˙Yi”m. Aj’Pu L⁄j’L∞u RÙPof£VÙLd L⁄j’ UiPXeL˚[d Li” ©•lTRtœm
L⁄j’dLfid°˚P˙V Es[ TWvTWj ˘RÙPo˚Td Li”©•dL‹m CVX˙Yi”m. L¶Rm
Lt\˚X SpXÿ˚\´p YœlT˚\´p S˚Pÿ˚\lT”j’YRtœ L¶Rd
L⁄j’dLLfid°˚P´XÙ]l TWvTWj ˘RÙPo˚Td Li”©•lT’ ˙R˚YVÙ]RÙœm.
Gi A±‹, SÙpY˚Lf ˘NVpLs, Y•ÆVp Y•YeLs, A[‹Ls, ˙SWm. B°VYt±p
Jq˘YÙ⁄ ©¨‹Lfidœm Es[ Y[of£˚Vd Li”©•dL ˙Yi”m.
Jufl ÿRp Ik’Y˚WŸs[ TÙP◊jRLeLs, B£¨Vo ˚L˙V” B°VYt±u ’˚QŸPu
Jq˘YÙ⁄ ©¨‹L∞u ˘RÙPof£Ÿm Y[of£Ÿm Lt\p A˚P‹L˚[ A•lT˚PVÙd°
AYt±u Gp˚X˚Vl TœlTÙn‹ ˘Nn’ œ±l◊ RVÙ¨d°u\]o.
Jq˘YÙ⁄ ©¨Æ¤m Es[ Lt\p A˚P‹L∞u Y[of£˚V ˘Yq˙Yfl AXœL∞p
—Zt£ÿ˚\´p ◊¨k’˘LÙi” RVÙ¨dL ˙Yi”m.
Ju\Ùm Yœl◊ ÿRp IkRÙm Yœl◊ Y˚WŸs[ Gi A±Æu Y[of£ ˙LÙ• Y˚W´XÙ]
GiL˚[ A˚PY’ IkRÙm Yœl©XÙœm.
Std I - (1 -10 Y˚W)
Std II - (1 -99 Y˚W)
Std III - (1 -999 Y˚W)
Std IV - (1 -9999 Y˚W)
Std V - (XhNm, Tj’XhNm, ˙LÙ•)
©\ UiPXeL∞u Lt\p A˚P‹L∞u Y[of£˚VŸm ˙U˙X ˘LÙ”dLlT”s[˚Rl
˙TÙufl TœlTÙV‹ ˘Nn’ Li”©•dL ˙Yi•VRÙœm.
ß\uL∞u TWvTWj ˘RÙPo◊
Jufl ÿRp Ik’Y˚W Yœl©p ˘Yq˙Yfl L¶Rj ß\uLs TX UiPXeL[ÙL
©¨dLlTh”s[]. Jufl, CWi”, Yœl◊L∞p J⁄e°˚QkR A‘œÿ˚\ Es[’.
L⁄j’L∞u TWvTWj ˘RÙPo˚TŸm Jufl ÿRp Ik’Y˚W Yœl◊L∞p Es[ Lt\p
A˚P‹L˚[Ÿm L⁄jßp ˘LÙs[ ˙Yi”m. Gi A±‹ Gu‡m ©¨Æp Yœl◊ 1 ÿRp
Yœl◊ 5 Y˚WŸs[ ˘RÙPo◊Ls ∏˙Z ˘LÙ”dLlTh”s[].
Class - I 1˛10 GiLs
˘T¨V’, £±V’
HflY¨˚N
C\eœ Y¨˚N
92
Class - II 1 ÿRp 99 Y˚WŸs[ GiLs
GiL∞u CPUßl◊
HflY¨˚N
C\eœY¨˚N
GiL∞u Æ[dLeLs
Class - III 1 ÿRp 999 Y˚WŸs[ GiLs
GiL∞u CPUßl◊
HflY¨˚N
C\eœY¨˚N
GiL∞u Æ[dLeLs
Class - IV 1 ÿRp 9999 GiLs
GiL∞u CPUßl◊
HflY¨˚N
C\eœ Y¨˚N
GiL∞u Æ[dLeLs
Class - V XhNm, Tj’XhNm, ˙LÙ•
˙TÙu\ ˘T¨V GiL˚[
CXdLeL∞u Gi¶d˚L A•lT˚P´p
YÙ£lTRtœm G›’YRtœm CV¤°\’.
˘T¨V GiL∞u HflY¨˚N
C\eœ Y¨˚N
GiL∞u Æ[dLeLs.
Jq˘YÙ⁄ Yœl©¤m A±‹fljRlThP GiL˚[ G›’Ytœm, YÙ£lTRtœm AYt˚\
TVuT”j’YRtœm HWÙ[UÙ], YÙrd˚L A‡TYeLfim T´t£Ÿm ˘NnRÙp Uh”˙U
L⁄j’⁄YÙdLj˚R N¨VÙ] ÿ˚\´p A˚PV ÿ•Ÿm. GiL∞p ˘T¨V˚R ◊¨k’˘LÙi”,
EVWkR ß\uLs ˘T⁄YRtLÙ] ˘NVpTÙ”L∞u YÙ´XÙL XhNm, Tj’XhNm, ˙LÙ•, ˙TÙu\
˘T¨V GiL˚[l Tt±V L⁄j’ E⁄YÙ°\’. ˘T¨V GiLs EhT”m NkRoTeLs YØ CP
Ußl©u A•lT˚P´p GiL˚[ G›’YRu YÙnl˚T A±V‹m. L¶R WN˚]´¤m
EVokR ®˚X˚V A˚PV‹m CV¤m.
ß\uL∞u ˘RÙPof£Ÿm Y[of£Ÿm.
Jufl ÿRp Ik’ Y˚W Yœl◊L∞¤s[ Tp˙Yfl L¶R ß\uL˚[ ©¨‹L[ÙL ©¨dœm
˙TÙ’ AYt±u ˘RÙPof£Ÿm, Y[of£Ÿm GkR[‹ Ju˙\Ù˘PÙufl ˘RÙPo◊ ˘LÙi”s[’
GuT˚R A±k’ ˘LÙs[XÙm. (ERÙWQUÙL) Gi A±‹ Gu\ ©¨Æ¤s[ ß\uL˚[l
TœlTÙn‹ ˘Nn’ ˙U˙X œ±l©PlTh”s[’.
Std I - 1 - 10
Std II - 1 - 99
93
Std III - 1 - 999
Std IV - 1 - 9999
Std V - XhNm, Tj’XhNm, ˙LÙ• Y˚W
C’˙TÙufl ©\ ©¨‹L˙[Ù” ˘RÙPo◊˚PV ß\uLs, Jq˘YÙ⁄ Yœl©u Lt\p A˚P‹Ls
T¨˙NÙßj’ TœlTÙn‹ ˘NnV ˙Yi”m.
Lt\p A˚P‹L∞u „Zt£
L¶Rd Lt\≠p Lt\p A˚P‹Ls —Zt£ A‘œÿ˚\´u A•lT˚P´XÙœm. Gi A±‹
Gu\ ©¨Æp Uh”m ÿRp, CWiPÙY’ AXœL∞p SÙpY˚L NVpLs, _Ù™ßV Y•YeLs,
A[‹Ls ˘RÙPe° GpXÙ ©¨‹L∞¤m Lt\p A˚P‹Ls Jq˘YÙ⁄ Yœl©¤m ˘Yq˙Yfl
AXœL∞p, œ±l©PlTh”s[]. Lt\p A˚P‹Ls L•] Ru˚Ud ˙LtT TX AXœL∞p
EhT”jRl Th”s[RÙp L¶Rd Lt\p £\l◊˚PVRÙ°\’.
Std 3 - Cp ˙YflThP ©¨‹L∞p Lt\p A˚P‹Ls ∏˙Z ˘LÙ”dLlTh”s[’. Gi
A±‹ (AXœ 1)
Ø 999 Y˚WŸs[ GiL˚[ YÙ£d°u\]o.
Ø 999 Y˚WŸs[ GiL˚[ Tp˙Yfl ÿ˚\L∞p Æ[dœ°u\]o.
Ø Íu±XdL GiL˚[ CPUßl©t˙LtT ©¨j˘R›’°u\]o.
Ø Íufl CXdLeL˚[ ET˙VÙ°j’ Tp˙Yfl Íu±XdL GiL˚[ E⁄YÙdœ°u\]o.
Ø Íu±XdL GiL∞p ˘T¨V’, £±V˚Rj Li”©•d°u\]o.
Ø Íu±XdL GiL˚[ HflY¨˚N, C\eœY¨˚N´p G›’°u\]o.
Ø CPj˚Rd œ±l©P GiL˚[ ET˙VÙ°d°u\]o.
Ø Íu±XdL GiL˚[ ET˙VÙ°j’ GiLs A•lTP´XÙ] S˚Pÿ˚\l ©Wf£˚]dœ
æo‹ LÙi°u\]o.
Ø Gi Y¨˚N´u Ejß˚V Æ[dL‹m ◊ßV GiY¨˚N˚Vd LiP˚PV‹m ˘Nn°u\]o.
SÙpY˚L NVpLs
·hPp (AXœ 2)
Ø J⁄ Gi˚Q ˙YflThP GiL∞u ˘RÙ˚LVÙL Tp˙Yfl ÿ˚\L∞p G›’°u\]o.
Ø J⁄ ÿu±XdL Gi˙QÙ” ˙Y˘\Ù⁄ Íu\XdL Gi˚Q ·h”YRtLÙL ÿ˚\L˚[
Æ[dœ°u\]o.
Ø S˚Pÿ˚\ ©Wf£˚]Lfidœj æo‹LÙQ ·hPp ˘NV˚X ET˙VÙ°d°u\]o.
Ø S˚Pÿ˚\ ©Wf£˚]L∞u æo˚Y ˙YflThP ÿ˚\L∞p Æ[dœ°u\]o.
Ø Gi Y¨˚N´u Ru˚U˚V Æ[dœ°u\]o.
94
LØjRp (AXœ 3)
Ø J⁄ Gi¶≠⁄k’ ˙Y˘\Ù⁄ Gi˚Q LØdœm ÿ˚\˚V Æ[dœ°u\]o.
Ø S˚Pÿ˚\ ©¨f£˚]Lfidœj æo‹ LÙQ LØjRp ˘NV˚X ET˙VÙ°d°u\]o.
Ø ©¨f£˚] æo‹dLÙL, Tp˙Yfl ÿ˚\L˚[ Æ[dœ°u\]o.
Ø LØjRp ˘NVpL∞u Æ˚P N¨VÙ GuT˚R T¨˙NÙßlTRtLÙ] ÿ˚\˚V ·fl°u\]o.
Ø GiL∞u R≤jRu˚U˚V Tt± ÿ•‹Ls E⁄YÙd°u\]o.
Ø GiY¨˚N´u Ru˚U˚V Æ[dœ°u\]o.
˘T⁄dLp (AXœ 5, 7)
Ø S˚Pÿ˚\ „ZpL∞p ˘T⁄dLp Ei˚UL˚[ E⁄YÙdœ°u\]o.
Ø ©Wf£˚] æo‹dLÙL ˘T⁄dLp Ei˚UL˚[ TVuT”j’°u\]o.
Ø ©Wf£˚] æo‹dLÙL ˘T⁄dLp ˘NVp ET˙VÙ°d°u\]o.
Ø ˘T⁄dLp Ei˚ULs TVuT”j’m S˚Pÿ˚\ ©Wf£˚]Ls E⁄YÙdœ°u\]o.
Ø ˘T⁄dL˙XÙ” ˘RÙPo◊˚PV Gi ˘RÙPo◊Ls Li”©•j’ Æ[dœ°u\]o.
YœjRp (AXœ 9)
Ø J⁄ œ±l©hP Gi¶d˚L˚V NUUÙ] ·hPeL[Ùd° UÙtflYRtœ YœjRp ˘NV˚X
ET˙VÙ°d°u\]o (˘RÙPof£VÙL LØjRp, ¡i”m ¡i”m)
Ø YœjRp ˘NV˚X Tp˙Yfl ÿ˚\L∞p Æ[dœ°u\]o. (¡i”m ¡i”m LØjRp
·hPeL[ÙdœRp, NUUÙL Teœ˚YjRp)
Ø YœjRp ˘NVpLs ˘NnYRtLÙLl Tp˙Yfl YØL˚[ Æ[dœ°u\]o.
Ø S˚Pÿ˚\ ©Wf£˚]Lfidœj æo‹LÙQ YœjRp ˘NV˚X ET˙VÙ°d°u\]o.
Ø YœjRp ˘NV˙XÙ” ˘RÙPo◊˚PV Gi˘RÙPo◊Ls Li” ©•j’ Æ[dœ°u\]o.
_Ù¡ßV Y•YeLs (AXœ 4)
Ø —t±¤ÿs[ Y•YeL∞≠⁄k’ N’Wm, ÿd˙LÙQm, YhPm, CYt˚\l ©¨jR±°u\]o.
Ø N’Wm, ÿd˙LÙQm, YhPm CYt˚\ ET˙VÙ°j’ Tp˙Yfl Y•YeLs E⁄YÙdL‹m,
Y˚WV‹m ˘Nn°u\]o.
Ø ÷hTUÙL‹m, ’p≠VUÙL‹m A[‹L˚[ ˚LVÙfi°u\]o.
Ø PÙs°WÙm ’i”Ls ET˙VÙ°j’ Y•YeLs E⁄YÙdœ°u\]o.
Ø ÿlT¨UÙ] Y•YeLs E⁄YÙdœ°u\]o.
˙SWm (AXœ 6)
Ø YÙhf, L•LÙWm CYt˚\l TÙoj’ ˙SWj˚R ·fl°u\]o.
Ø ˙SWm LQd°”YRtLÙ] J⁄ AX˙L U¶d·o.
Ø œ±l©hP ˙SWeLfid°˚P˙VŸs[ ˙SWÆjßVÙNeL˚[d LQd°”°u\]o.
95
Ø U¶d·o, ®™Pm CYt±u Ju˘\Ù˘PÙu\Ù] ˘RÙPo˚T Æ[dœ°u\]o.
Ø ˙SWj˙RÙ” ˘RÙPo◊T”jß ˘NVpTÙ”Ls ßhP™P‹m LÙX AhPY˚Q, LÙXiPo
E⁄YÙdœ°u\]o.
Ø LÙXiP˚W TÙoj’ Be°X, U˚XVÙ[, R™r UÙReL∞u ˙RßLs, Æ˙N` SÙhLs
Li” ©•d°u\]o.
A[‹Ls (AXœ 8)
ø[m
Ø ø[j˚R A[dœm AX˙L ¡hPo.
Ø £±V ø[j˚R A[dL ˘Nu±¡hPo ET˙VÙ°d°˙\Ùm.
Ø ø[jßu AXœLfidœ C˚P´XÙ] ˘RÙPo˚T Æ[dœ°u\]o.
Ø ¡hPo, ˘Nu»¡hPo, ˙Pl/v˘L´p ET˙VÙ°j’. ø[jß˚] N¨VÙL A[d°u\]o.
Ø ø[m, A[˙YÙ” ˘RÙPo◊˚PV S˚Pÿ˚\ ©Wf£˚]Lfidœj æo‹ LÙi°u\]o.
˘LÙs[[‹
˘LÙs[[Æu AXœL∞d°˚P˙VŸs[ ˘RÙPo˚T Æ[dœ°u\]o. (™p≠¡hPo, ≠hPo)
A[‹ TÙjßWm ET˙VÙ°j’ ˘LÙs[[˚Y ’p≠VUÙL A[d°u\]o.
˘LÙs[[˙YÙ” ˘RÙPo◊˚PV S˚Pÿ˚\ ©Wf£˚]Lfidœj æo‹ LÙi°u\]o.
G˚P (AXœ 10)
Ø G˚P Li”©•dœm Tp˙Yfl L⁄ÆL˚[ A±ÿLm ˘Nn°u\]o.
Ø G˚P´u AXœLfidLfid°˚P´XÙ] ˘RÙPo˚T Æ[dœ°u\]o. (°WÙm/°˙XÙ°WÙm)
Ø RWÙ— ET˙VÙ°j’ ’p≠VUÙL G˚P˚V A[d°u\]o.
Ø G˚P˙VÙ” ˘RÙPo◊˚PV S˚Pÿ˚\ ©Wf£˚]dœj æo‹ LÙi°u\]o.
Jq˘YÙ⁄ Yœl©¤m ˙YflThP ©¨‹L∞u ˘RÙPof£Ÿm, Y[of£Ÿm, Lt\p A˚P‹L˚[
T¨˙NÙßj’d Li” ©•dLXÙm. Lt\p A˚P‹L∞u A•lT˚P´p, L⁄j’ TœlTÙn‹
SPjß L¶Rd Lt\p A‘œÿ˚\d˙LtT Lt\p ˘NVpTÙ”L˚[j RVÙWÙd° T¨UÙt\m
˘NnV ˙Yi”m.
![DIPLOMA IN ELEMENTARY EDUCATION D.El.Ed....Untπma C≥ Fen-sa‚dn FPyp-t -j≥ DIPLOMA IN ELEMENTARY EDUCATIOND.El.Ed. t]∏¿ ˛ 108 Ie, {]hr-Øn-hn-Zym-`ymkw ˛kao-]\hpw {]tbm-Khpw](https://img.pdfslide.us/doc/110x75/60e42ec476f5ed651d7d8989/diploma-in-elementary-education-deled-untma-ca-fen-saadn-fpyp-t-ja.jpg)
