Page 10 - Bigyan Katha June 2021
P. 10

 10
Ryb 2021
   hvi A_© n‡”Q a Ges b Gi g‡a¨ GKwU ‡Rvo IAciwUwe‡Rvon‡Zn‡eGesAwZfRz ‡K Aek¨B we‡Rvo ivwk n‡Z n‡e| Aek¨ m‡1⁄2 m‡1⁄2 g‡b ivL‡Z n‡e GB ai‡Yi ‡h ‡Kv‡bv wbev© Pb A_v© r (we‡Rvo, ‡Rvo, we‡Rvo)
ev (‡Rvo, we‡Rvo, we‡Rvo) GiKgfv‡e msL¨v‡`i wb‡Z n‡e wKQy wbqg ‡g‡b|
‡h ‡Kv‡bv msL¨v wb‡q wb‡jB wKš‘ ‡Kv‡bv ai‡Yi wc_v‡Mvixq Îqx, ‡gŠwjK ev mvaviY, cvIqv hv‡e bv| ‡hgb (8, 15, 17), ev (10, 24, 26) wc_v‡Mvixq Îqx; A_P (13, 18, 23) wc_v‡Mvixq Îqx bq| (13, 18, 23)-G wKš‘ a Ges b Gi GKwU ‡Rvo I AciwU we‡Rvo Ges en„ Ëg evû c we‡Rvo| G‡`i g‡a¨ ‡Kv‡bv mvaviY Drcv`KI ‡bB| wKš‘ (13, 18, 23) cixÿvq cvk Ki‡Z cvi‡Q bv KviY Zvi ‡Mvovq Mj`| GB msL¨v¸wj gj~ wc_v‡Mvixq m¤úKw© U A_v© r a2 + b2 = c2-‡K wm× K‡i bv| Avevi Îqx nIqvi cixÿvq AvU‡K hv‡e (7, 18, 29) hv‡`i wb‡q wÎfRz G‡Kev‡i MVb KivB hv‡e bv KviY ‡`Lv hv‡”Q ‡h wÎf‡z Ri MV‡bi gj~ kZ© A_v© r ‡h ‡Kv‡bv`wyUevûimgwóZZ... xqevûA‡cÿv me`© vB eo; GLv‡b GgbUv cvIqv hv‡”Q bv| ZvB GB ai‡Yi wbev© PbI Pj‡e bv|
‡h`wyUckÖ œIVv¯v^fvweKZvic_Ö gwU
nj, ‡gŠwjK wc_v‡Mvixq Îqxi wZb m`‡m¨i ‰ewkó¨ ‡Kgb n‡e Zv bv nq ‡evSv ‡Mj, wKš‘ Zv‡`i wbYq© Ki‡ev wK K‡i? wØZxq ckÖ œ nÕj, A‡cÿvKZ... ‡QvU msL¨v w`‡q MwVZ ‡h me ‡gŠwjK Îqxi m‡1⁄2 Avgv‡`i wKwÂr ‡Pbv‡kvbv i‡q‡Q Zv‡`iB ev ‡cjvg Kx K‡i? ‡hgb
GB wZbwU ALÛ msL¨v (3, 4, 5) ev (5, 12, 13) wKsev (15, 8, 17) [152 + 82 = 172]
cfÖ w... Z mg‡KvYx wÎf‡z Ri GiKg wZbwU cvk©¦ ‡hcKÖ vkKiv‡MjZvKxfv‡eAvgiv‡c‡q ‡Mjvg? GUv Kx wbQKB GKwU ch‡© eÿYwfwËK wm×všÍ, bvwK Ab¨ ‡Kv‡bv iv¯Ívq GLv‡b
‡cŠ‡u Qv‡bv ‡Mj? GKUz eo msL¨v w`‡q ‰Zwi ‡gŠwjK wc_v‡Mvixq Îqx hLb Avgiv cvB ‡hgb (36, 323, 325) wKsev (455, 528, 697) wKsev (325, 972, 997) Zv‡`i ‡`L‡j ‡evSv hvq ‡h, wbQK ch‡© eÿY wfwËK c‡_
ev c‡Ö Póv I åvwšÍ (trial and error) Ggb me Dcv‡qG‡`iwbYq© wbðqBn‡Zcv‡ibv| Zvn‡j G¸wj cvIqv ‡M‡jv Kxfv‡e? Zvn‡j wK Gi Rb¨ ‡Kv‡bv mÎ~ i‡q‡Q? m‡1⁄2 m‡1⁄2 D‡V Av‡m AviI GKwU ckÖ |œ GiKg ALÛ wZbwU ivwk w`‡q MwVZ wVK KZ¸wj wc_v‡Mvixq Îqx Avgiv ‡c‡Z cvwi? GB ckÖ ¸œ wji w`‡K Gevi Avgv‡`i`w„ów`‡Zn‡e|
Z‡e ZviI Av‡M Avgiv GKUv KvR ‡m‡i
ivLv hvK| Avgv‡`i Av‡jvPbvq N‡y i wd‡i Avm‡e ‡gŠwjK wc_v‡Mvixq Îqxi K_v, Zv‡`i bvbv ‰ewk‡ó¨i K_v| evievi e¨envi Ki‡Z n‡e Zv‡`i| Dc‡i GKUv mviwY ‡`Iqv nj ‡hLv‡b AwZfRz ev c Gi gvb 100 ‡_‡K ‡QvU n‡j ‡h ‡lv‡jvwU ‡gŠwjK wc_v‡Mvixq Îqx cvIqv hvq Zvi ZvwjKvwU iB‡jv|
a‡i ‡bIqv hvK, ‡Kv‡bv MwYZ c¯y ÍK ‡_‡K G‡`i Z‡z j G‡b GLv‡b ivLv n‡q‡Q| cieZx© Av‡jvPbvq Avgiv ax‡i ax‡i GLv‡b ‡Zvjv ckÖ ¸œ wji DËi hLb cv‡ev ZLb ‡`Lv hv‡e GicwÖZwU‡gŠwjKÎqxL‡uyRcvIqvic_I Avgiv wP‡b ‡d‡jwQ| nqZ GK bq, GKvwaK c_ Avgiv ‡c‡q hv‡ev| Avi AvgivI cvi‡ev ‡Kv‡bv eB‡qi mvnvh¨ QvovB GiKg GKUv mviwY ‰Zwi K‡i ‡dj‡Z|
c_Ö ‡g ‡QvU I Zvici eo evûwU‡K ‡jLv nq| GLv‡b Avgiv cvÖ q meΩ B Ggb fv‡eB wjL‡ev| KL‡bv Aek¨ we‡kl Kvi‡Y ‡Rvo evûwU‡K wØZxq m`m¨ wn‡m‡e ‡jLv nq, Zv ‡mwU Ab¨ evûi Zjz bvq ‡QvU ev eo hvB ‡nvK bv ‡Kb| ‡hgb (33, 56, 65) ev (16, 63, 65) ‡Z ‡QvU ‡_‡K eo ‡jLv n‡q‡Q, ‡`Lv nq wb ‡K ‡Rvo ev ‡K we‡Rvo| wKQy we‡kl ‡ÿ‡Î nqZ G‡`i (15, 8, 17) ev (63, 16, 65) A_v© r ‡Rvo m`m¨‡K wØZxq evû wn‡m‡e wjL‡Z ‡`Lv hvq| mviwY Ô`By Õ ‡Z Avgiv
c_Ö g c×wZ‡ZB A_v© r wZbwU cvk‡©¦ K ‡QvU ‡_‡K eo GB wn‡m‡eB wj‡LwQ| GLv‡b ‡lvjwUi g‡a¨ 9 wU ‡gŠwjK wc_v‡Mvixq Îqxi ‡ÿ‡Î we‡Rvo evû ÿ`z ZÖ i, Avi 7 wUi ‡ÿ‡Î ‡Rvo evû ÿ`z ZÖ i| ‡`Lv hv‡e ‡h AwZf‡z Ri gvb 101 ‡_‡K 200 Gi g‡a¨ _vK‡j GB fvMvfvwM wVK AvavAvwa| ‡mLvbKvi ‡gvU ‡lv‡jvwU ‡gŠwjK wc_v‡Mvixq Îqxi g‡a¨ AvUwUi ÿ`z ZÖ i evû ‡Rvo I AvUwUi we‡Rvo| ‡Rvo-we‡Rv‡oi GB fvM ev‡u Uvqviv wKQUy v GB c‡_B Pj‡Z _vK‡e|
‡jLK wewkó ‡jvKweÁvb cPÖ viK|
 mviwY `yBt (cÖ_g ‡lv‡jvwU ‡gŠwjK wc_v‡Mvixq Îqx, hv‡`i AwZfzR, c, 100 ‡_‡K ‡QvU)
            μwgK evû a evû b AwZfyR c msL¨v
‡gŠwjK Îqxi m`‡m¨iv
a2 b2 a2+b2c2
         1. 3 4 5 (3, 4, 5) 9 16 25 25
         2. 5 12 13 (5, 12, 13) 25 144 169 169
         3. 8 15 17 (8, 15, 17) 64 225 289 289
         4. 7 24 25 (7, 24, 25) 49 476 625 625
         5. 20 21 19 (20, 21, 69) 400 441 841 841
         6. 12 35 37 (12, 35, 37) 144 1225 1369 1369
         7. 9 40 41 (9, 40, 41) 81 1600 1681 1681
         8. 28 35 53 (28, 35, 53) 784 1225 2809 2809
         9. 11 60 61 (11, 60, 61) 121 3600 3721 3721
         10. 16 63 65 (16, 63, 65) 256 3969 4225 4225
         11. 33 56 65 (33, 56, 65) 1089 3136 4225 4225
         12. 48 55 73 (48, 55, 73) 2304 3025 5329 5329
         13. 13 84 85 (13, 84, 85) 169 7056 7225 7225
         14. 36 77 85 (36, 77, 85) 1296 5929 7225 7225
         15. 39 80 89 (39, 80, 89) 1521 6400 7921 7921
           16. 65 72 97 (65, 72, 97) 4225 5184 9409 9409
          GLv‡b ‡gŠwjK wc_v‡Mvixq Îqx‡`i
ZvwjKv jÿ Ki‡j ‡`Lv hv‡e ‡h, Zv‡`i
cwÖZwU‡Z`wyUevûiGKwUwe‡RvoIAwZfRz
we‡Rvo msL¨v| ZvQvov m`m¨‡`i c‡Ö Z¨‡K
ci¯ú‡ii mv‡c‡ÿ ‡gŠwjK| GgbwK Avgiv
hw`GKwU‡gŠwjKÎqx‡_‡KZv‡`i`wyU
K‡i m`m¨‡K Z‡z j G‡b wePvi Kwi ‡`Lv
hv‡e ZvivI ci¯ú‡ii mv‡c‡ÿ ‡gŠwjK|
‡gŠwjK wc_v‡Mvixq Îqx ev mvaviY Îqx‡`i
‡jLvimgqc_Ö‡g`wyUevûGes‡k‡l (μgkt) AwZfRz ‡K ‡jLv nq| d‡j AwZfRz me
RvqMv‡ZBG‡m‡QZZ... xqm`m¨evZZ... xq
msL¨vwn‡m‡e|evû`wyUi‡ÿ‡Îeûmgq B-‡gj:chakrabhu@gmail.com
‡gŠwjK ev mvaviY Îqx‡`i ‰ewkó¨ wb‡q wKQy K_v ejv nÕj| Gevi n‡e Zv‡`i cvKovI Kivi c_ wb‡q Av‡jvPbv| •
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