Feb 09

al- Ø mb kn\n- a - I fn- e qsS Hcp bm{X

Akm-[y-amb

x

=+

Nne -{]-iv\-߃

KWnXw

KWn-X-imkv{Xw, At\Iw Ipg-™p-a-dn™ {]ho¨N{µ {]iv\-߃ sIm≠v \q‰m-≠p-I-fmbn a\p-jy-cm-in-bpsS ss[j-Wn-I-Xsb sh√p-hn-fn® imkv{X-im-J-bm-Wv. hnI-kn-®p-sIm-≠n-cn-°p∂ Hcp imJ-bm-b-Xn-\m¬ At\Iw h¿j-ß-fmbn DØ-c-an-√m-Xn-cp∂ ]e {]iv\ßfpw ]pXnb ]T-\-ß-fpsS shfn-®-Øn¬ sXfn-bn-°s∏-´n-´p-≠v. s^¿a-bpsS Ah-km-\- Xn-bdw CXn\v Hcp DZm-l-c-W-am-Wv. Akm- [ y- s a∂v sXfn- b n- ° - s ∏- ´ n- ´ p≈ aq∂v KWnX {]iv\-ß-sf-∏-‰n-bmWv ChnsS hnh-cn-°p-∂-Xv. 1) dqfdpw tImºkpw D]-tbm-Kn®v Hcp tImWns\ Xpey-amb aq∂v tImWp-I-fmbn hn`-Pn-°m≥ km[ya-√. 2) dqfdpw tImº pw D]-tbm-Kn®v Hcp Iyq_ns‚ Cc´n hym]vX-ap≈ as‰mcp Iyq_v \n¿Ωn-°m≥ km[y--a√. 3) dqfdpw tImºkpw D]-tbm-Kn®v Hcp hrØ-Øns‚ Xpey-hn-kvXo¿Æ-ap≈ Hcp NXpcw \n¿Ωn-°m≥ km[n-°-bn-√. Cu aq∂v {]iv\-ß-fnepw dqfdpw tImº pw D]tbm-Kn®v F∂ hmIyw Bh¿Øn-°p-∂Xv {i≤n-°p-at√m. dqf¿ F∂Xv c≠v _nµp-°sf tbmPn--∏n-°p∂ tcJ hc-bv°m≥ am{Xw km[n-°p∂ H∂m-Wv. CXn-¬ kvsIbn-ep-t]mse Af-hp-Iƒ tcJs-∏-Sp-Øn-bn-´p-≠mhn-√. apI-fn¬ sImSpØ aq∂v {]iv\-ßfpw dqfdpw tImº pw D]-tbm-Kn®v I≠p-]n-Sn-°m-\m-hmØ {]iv\-ß-fm-Wv. tImWns\ Xpey Af-hp-I-fp≈ aq∂v `mK-ß-fm°p-∂-Xn\v ]e D]-I-c-W-ßfpw \n¿Ωn-°-s∏-´n-´p-≠v. F¶nepw dqfdpw tImºkpw D]-tbm-Kn®v CXv Akm[y-am-Wv. Ch Akm-[y-am-sW∂v Fß-s\-bmWv sXfnbn-°p-∂Xv?. {ian-®p-t\m-°q. Ref: Fun and Fundamentals of Mathematics.

30 imkv{X-tI-cfw

1 cubic unit

2 cubic unit

Nn{X-Øn¬ Hcp bq\n‰v hym]vX-ap≈ Hcp Iyq_v sImSp-Øn-cn°p∂p. CXns‚ Cc´n hym]vXap≈ as‰mcp Iyq_v \n¿Ωn-°p-tºmƒ AXns‚ hiß-fpsS \ofw F{X-bm-bn-cn-°pw.? Hcp Iyq_ns‚ Cc´n hym]vX-ap≈ as‰mcp Iyq_v \n¿Ωn-°m≥ km[n-°p-I-bn-√. X

X

sK‰nw-Mv tlmw ssN-\o-kv / I-f¿/ 35mm/97an-\p-´v /2007

X/3

Nn{X-Øn¬ x Af-hp≈ Hcp tIm¨ \¬Inbn-cn-°p-∂p. tImº pw dqfdpw D]-tbm-Kn®v x/3Afhp≈ Hcp tIm¨X hc- b v t °- ≠ n- b n- c n- ° p- ∂ p. tImº pw dqfdpw am{Xw D]- t bm- K n®v Hcp tImWns\ Xpey-amb aq∂v `mK-ß-fm°m≥ km[n°p-I-bn-√. r 2

hnP-b-Ip-am¿ ªmØq¿

" √m C-e-Ifpw A-Xn-s‚ th- c p- I - f n- t e- ° v X- s ∂ Xn- c n- ® v t]m-Ipw" F-∂v ]-g-sam-gn-bp≠v.

F

2

r =a

a

r Bc- a p≈ hrØw I≠t√m? CXns‚ XpeyhnkvXo¿Æ-ap≈ Hcp ka-N-Xpcw \n¿Ωn-°m≥ km[n-°p-I-bn-√.

Jayanth-V-Narliker and Mangala Narliker

Po-hn-°m-\m-bn ssN-\-bp-sS H-c‰- Ø p- ≈ sj≥- s k- \ n¬ \n¿ΩmW tPm-en-Iƒ sN-øp-∂ kp-lr-Øp-°-fmb en-bp-hpw km-thmbpw ˛ a-cn-®p t]mbm¬ en-bp-hn-s\ A-bm-fp-sS ho-´n¬ (ssN-\-bp-sS a-t‰ A-‰-Øp-≈ tIzm≥-

Kznw-Kn¬ ) F-Øn-®v kw-kv-I-cn-°m-sa∂v H-cn-°¬ kmthm hm-°p \¬-Inbn-cp∂p. a-≤y-h-b- n-seØn-b Cu kp- l r- Ø p- ° - f p- s S a- Z y- ] m- \ cw- K tØm-sS-bmWv "sK-‰nw-Mv tlmw' F-∂ ssN-\o-kv kn-\n-a Bw-cw-`n-°p∂-Xv . Ip-Sn-®p Ip-Sn-®p en-bp a-cn-®p t]mbn. Xm≥ kp-lr-Øn-\v \¬In-b hm-°p]m-en-°m-\m-bn {Km-ao-W\m-b kmthm en-bp-hn-s‚ ihi-co-cw A-£-cm¿-Y-

Øn¬ Npa-∂v A-bm-fp-sS ho-´n-te-°v \-S-°p-∂ Zo¿-Lam-b bm-{X-bmWv "sK‰nw-Mv tlmw' km-thm-bp-sS Zo¿-L-bm-{X-bn¬ Iq-´v en-bp am-{X-amWv. P-U-am-sW-¶nepw kp-lrØn-t\m-Sv ]-g-b-Xp-t]m-se bm-{Xbn- e p- S - \ o- f w kmthm kw- k m- c n- ° p∂p≠v. hf-sc Zo¿-Lam-b bm-{X-bv-°v hml-\w hmS-I-s°-Sp-°m-\p≈ ]Ww s^{_p-hcn 2009 31

al- Ø mb kn\n- a - I fn- e qsS Hcp bm{X tkm- t hmbp-sS I-øn-en√mbn- c p∂p. AXpsIm-≠p X-s∂ km[m-c-W h-kv-{X-ß-sfm-s° [-cn-∏n®v Hcp Iq-fnw-Kv •mkpw ^n-‰v sN-bv-Xv a-Zy]n-® H-cm-sf F∂ t]m-se _- n¬ ko-‰n-en-cpØn-bm-Wv kmthm en-bp-hpam-bn bm-{X B-cw-`n-°p-∂-Xv. bm-{X-°m¿-°v aq{X-sam-gn-°m-\mbn a-e-ºm-X-bn-sem-cnS-Øv h-≠n \n¿Øp∂p. F√m-h-cpw sh-fn-bn-en-dßn. enbp am-{X-ta _- n-ep≈q. X-߃ aq{Xsam-gn-°p∂-Xv t\m-°n-bn-cn-°p-I-bm-Wv A-bm-sf-∂v ]d-™v kv-{Xo-Iƒ _lfw sh-bv-°p-I-bpw No-Ø ]-d-bp-Ibpw sN-bvXp. AXn-sem-cp-hƒ en-bp A-hfp-sS apJ-Øp t\m-°n-Nn-cn-°p-I-bpw sN- b v - X p F- ∂ p IqSn ]- c m- X ns∏´p. Ip-Sn-®p t_m-[w sI-´ H-cp ]m-h-am-sWs∂m-s° ]d-™v kmthm A-hsc-sbms° k-am-[m-\n-∏n-°p-∂ cw-K-ap≠v. t{]£-Icpw A-dn-bp-∂n√ A-Xv en-bp-hn-s‚ i-h-am-sW∂v. bm-{X-°n-S-bn¬ ssl-sh tam-„m°ƒ h-≠n XS-™v sIm-≈-b-Sn-°m-\mbn _- n¬ I-b-dp∂p. sa-‰¬ Un-‰Œ-sdm-s°-bm-bmWv Hm-tcm-cm-fp-sSbpw Hfn-® ]-Whpw kz¿-Æhp-sam-s° kwL-Ø-e-h≥ X-∏n-sb-Sp-°p-∂Xv. en-bphn-s‚ ]-W-hpw A-bmƒ F-Sp-°m≥ {ian-°p-tºmƒ kmthm X-S-bp∂p. ' kz¿§-Øn-te-bpw `q-an-bntebpw F¥pw Xm≥ tam-„n°pw ' F-∂p ]-d-bp-∂ Abm-tfm-Sv ˛ ' H-cp a-cn-® a-\p-jy-s‚ ap-Xepw \n-߃ F-Sp°ptam?' F-∂p tNm-Zn-°p-∂p km-thm. A-t∏mƒ am{XamWv h-≠n-bn-se bm-{X-°mcpw \-Ωfpw A-dn-bp-∂Xv en-bp a-cn-®-Xm-sW-∂v. kplrØn-t\m-Sp-≈ B-flm¿-Y-X-bp-sS t]-

32 imkv{X-tI-cfw

cn¬ kmthm \-S-Øp-∂ Cu bm-{X-sb°p- d n- ® v a- \ -   n- e m- ° n- b - t ∏mƒ

Hmtcm h-≠n-Iƒ-°v ssI Im-Wn-®v Ah-i-\m-bn \-S-°p-I-bm-Wv km-thm. \-∑-bp-sSbpw Xn-∑-bp-sS-bpw kvt\-l-Øn-s‚bpw h-©-\-bp-sSbpw Aen-hn-s‚bp-sam-s° a-\p-jym-h-ÿ-I-fneq-sS˛ ]p-Xn-b No-\-bp-sS Po-hn-X-ßfn-eq-sS-bp-≈ Zo¿-L-bm-{X-bm-Wv ]n∂oSv.

sIm≈kw-L-Ø-e-h≥ ( A-bm-sfm-cp k-lr-Zb\pw km-[p-hp-amWv) h√m-sX A¤p-X-s∏-Sp-Ibpw ˛ sIm-≈ ap-X-sems° km-thm-bv-°v \¬-In ˛' C-bm-sf°≠v ]Tn-°m≥' a-‰p-≈-h¿-°v H-cp D-]tZ-ihpw \¬-In C-d-ßn-t∏m-hp-∂p. X-ß-fp-sS h-I-I-sf√mw km-thmbp-sS I-øn¬ \n∂pw Xn-cn-s®-Sp-Øv Ign-™-t∏mƒ a-‰p bm-{X-°m-cp-sS `m-hw am-dn. i-h-tØm-sSm-∏w bm-{X sN-øm\m-hn-s√∂v _-l-fw sh-®v hn-P-\-am-b s]-cp-h-gn-bn¬ km-thm-sbbpw en-bp-hns\bpw A-h¿ C-d-°n-hn-´p.

X-s‚ kp-lr-Øn-s‚ i-co-chpw ]pd- Ø v Np- a ∂v , IS- ∂ v t]m- h p- ∂

\n-jv-I-f-¶-\qw \-∑-\n-d-™-h-\pam-Wv kmthm. Cu th-jw A-Wn-™ kmthm s_≥-jm hf-sc kzm-`m-hnIam-b A-`n-\-b-Øn-eq-sS A-Xn-i-tbm‡n-\nd-™ Cu bm-{X A-hn-kv-a-c-Wob-am-°n am‰n. k-m-thm-bp-sS en-bphnt\m-Sp-≈ s]-cp-am-‰-sam-s° t{]-£-Icn¬ Nn-cn-bp-b¿-Øp-∂-XmWv. ]s£ Nn-cn-am-bp-tºm-tg°pw t{]-£-I-cp-sS lr-Z-b-Øn¬ h√m-sXm-cp hn-߬ _m°n-\n¿Øpw C-Xn-se Hm-tcm cw-Khpw bm-{X-bn-se B-I-kv-an-I-X-bm-Wv thsdm-cp {]-[m-\ kwK-Xn . kmthm \nch-[n Po-hn-X-ß-fm-Wv ]-cn-N-b-s∏-SpØp-∂Xv. Im-iv sIm-Sp-°m-\n√m-Ø-Xn\v Xn-∂ `£-Ww ap-gp-h≥ O¿-±n-∏n°p-∂ tlm-´¬ \-S-Øn-∏p-Imc≥ , h-≠n X-≈n Ãm¿-´m-°nbm¬ en-^v-‰v X-cm-sa-∂v ]d-™v h-≠n D-¥n-∏n-®v Ãm¿-´m-°n˛ \n¿-Øm-sX t]m-Ip-∂-bmƒ, Iøn-se C-Øn-cn∏-Ww t]mepw tam- „ n- ° p- ∂ k- { X- Ø n- s e kl Xm-a-k-°m¿, i-º-f-am-bn I-≈-t\m-´v \¬Inb sXm-gn-epS-aIƒ, tcm-K-ap-≈-h-cp-sS c‡w t]mepw hn¬-∏-\ \-S-Øp∂ A-t[m-tem-I߃, ssk°nƒ N-hn-´n Xn-_-t‰m-fw bm{X-sN-øp-∂ kv-t\-lw \n-d-™ k-t¥mj-hm\m-b sN-dp-∏-°mc≥, Im-ap-In \-

„-s∏-´-Xn-\v s]m-´n-°-c-bp-∂ ( km-thmhn-s\ hf-sc Zq-cw h-≠n-bn¬ sIm≠pt]m-b) {S-°v ss{Uh¿, a-c-Wm-\-¥-cN-S-ßp-Iƒ ap≥-Iq-´n kw-L-Sn-∏n-°p-∂ {Km-a-Øn-se [\n-I hr≤≥, ap-Jw hn-

AXpw ]-‰p-∂n√. A-bmƒ-°v sl-∏ss‰-‰n-kv _n tcm-K-ap-≠s{X ˛ ]-s£ G-Xv c-‡hpw hm-ßp-∂ A-\-[nIr-X ª-Uv _m-¶p-Im¿ km-thm-hn-s\ Iq-´ns°m-≠p t]m-Ip∂p. t]m-eo-kv ssd-

I- c n- ° m≥ t]m- e o- k n- s ‚ \n¿- t ±- i Øm¬ kmthm \n¿-_-‘n-X-\m-hp∂p. Nn-Xm-`-kv-a-hp-am-bn H-cp t]m-eo-kpImc-s‚ k-lm-b-tØm-sS en-bp-hn-s‚ ho-S-t\z-jn-®v bm-{X-Xp-S¿-∂p. A-h-km\w ho-Sv I-s≠-Øp∂p. A-t∏m-tg°pw en-bp-hn-s‚ IpSpw- _ w ho- S v D- t ]- £ n- ® v v A- h n- s S \n∂pw t]m-bv-°v-gn-™n-cp∂p. X-I¿∂p In-S-∂ hm-Xn¬-]m-fn-bn-se-gpXn-b ]pXn-b hn-em-khpw tX-Sn bm-{X Xp-Scp-∂ km-thm-bn¬ kn\n-a A-h-km\n-°p∂p.

Ir-Xam-b `mcy-tbm-sSm-∏w Bƒ-∏m¿∏n√m-Ø H-cn-S-Øn¬ tX-\o-® h-f¿-Øep-am-bn Po-hn-°p-∂ H-cp \√ a-\p-jys‚ Ip-Spw-_w Cß-s\ ]-e-X-cw Bƒ°m-sc-bm-Wv kmthm bm-{X-bn¬ I≠p-ap-´p-∂Xv. A-]-cn-NnX-s‚ a-c-Wm-\¥c-N-S-ßp-Iƒ \-S-°p-∂n-S-sØ-Øn ˛ a- ‰ p- ≈ - h ¿- s °m- ∏ w \n- e - h n- f n- ° p- ∂ kmthm , ]m-S-tØ-°v I-tÆdp-tIm-e߃-s°m-∏w Ip-Øn-\n¿Øn-b en-bphn-s‚ i-co-cw cm{Xn h-∂v X-∏p-∂ kmthm, {S-°v S-b-dn-\p-≈n¬ en-bp-hn-s\ a-S-°n-s°-´n D-cp-´n-s°m-≠p t]m-hp-∂ km-thm, a-Sp-∏v tXm-∂n kp-lr-Øn-s‚ i-h-°p-gn-°-cn-In¬ X-s∂ Ip-gn-sbm-cp°n B-´n hn-´ I√p-sIm-≠v X-e-°nSnsIm-≈n-®v Ip-gn-bn-te-°v a-cn-®p ho-gm\p-≈ {i-aw \-S-Øp-∂ km-thm, H-s° lm-ky-tØm-sSm-∏w hf-sc Km-Vam-b k¶-Shpw \-Ωn-ep-W¿-Øpw. ssI-øn-se Im-si√mw Xo¿-∂t∏mƒ kz- ¥ w c- ‡ w hn¬- ° m≥ kmthm Xo- c p- a m- \ n- ° p∂p. ]- s £

Un¬ kmthm I-kv-‰-Un-bn-em-hp∂p. A-hn-sS sh-®v H-cp kv-{Xo-sb ]-cn-Nb-s∏-Sp-∂p. A-h¿ bq-\n-th-gv-kn-‰n-bn¬ ]Tn-°p-∂- X-s‚ aI-\v th-≠n ]-Ww D-≠m-°m≥ ]e tPm-enbpw sN-øp∂-h-cm-Wv ˛ aI-\v X-s‚ ]-Ww am{Xw

bm-Mv jmMvv a-Xn-sb-∂v A-h¿-°-dn-bmw.Po-hn-X-Ønte-°v £-Wn-°m-\m-bn Xm≥ A-h-fpsS A-cn-In-te-°v Xn-cn-®v h-cm-sa-∂v kmthm hm-°p-\¬-Ip-∂p. (]-s£ Ah-fp-sS hn-em-kw hm-ßm≥ kmthm ad-∂pw t]m-bn) No-™p Xp-Sßn-b i-hw kw-kv-

bmw-Mv jm-Mv kw-hn-[m-\w sNbv-Xn-cn-°p-∂ Cu kn-\n-a-°v 2007se Xn-cp-h-\-¥-]pcw IFFK bn¬ t{]-£I-cp-sS h≥ kzo-I-c-W-am-Wv e-`n-®-Xv. ih-sØ tIm-ew sI-´n-®v lmkyw DXv∏mZn-∏n-°p-∂ \n-ch-[n kn-\n-a-Iƒ ae-bm-f-Øn-e-S-°-ap-≠v ({`m-¥∑m-cpw \Ωp-sS kn-\n-a-bv-°v lm-ky-Øn-\p-≈ kzm-`mhn-I I-Ym-]m-{X-ß-fm-Wt√m !) sK‰nwMv tlm-an¬ bm-Mv jmMvv kmthm-bnepw en-bp-hnepw Iq-Sn Po-hn-XØn- s ‚bpw km- a q- l ym- h - ÿ - I - f psSbpw k-¶o¿-ÆXI-sf I-dp-Ø lmky-Øn-eq-sS A-h-X-cn-∏n-°p-I-bm-Wv. H-cp a-\p-jy≥ kp-lr-Øn-s\ Npa∂vv \-S∂p-t]m-hp-tºmƒ ssI \o´o´pw \n¿-Øm-sX t]m-b \n-ch-[n hml-\-߃ , a-e-bn-Sn-®n-en-s\ Xp-S¿-∂v h-gn-b-S™vv tdm-Un¬ \n-c-\n-c-bm-bn \n¿-Øn-bn-´n-cn-°p-∂ H-cp cw-K-ap-≠v Cu kn-\n-a-bn¬.. A-hsc-sbm-s° IS∂p-sIm-≠v en-bp-hn-s\ Np-a-en-se-SpØvv \S-∂p t]m-hp-∂ kmthm ˛ {]Ir-Xn-bp-sS ta¬ a-\p-jy-\pw b-{¥ßƒ-°pw D-≈ \n- m-cam-b A-[n-Imc-ß-sf-bpw, A-l-¶m-c-sØbpw hfsc kq-£v -a-am-bn ]-cn-l-kn-°p-∂p-≠v . s^{_p-hcn 2009 33