Page 6 - Bigyan Katha Mar 21

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gv©P 2021
wewfbœ ai‡bi msL ̈vi DTM¢‡ei‡cÿÖ vcU
MmZ ̈evPx mi
wY‡Zi ïiæ ‡h me aviYv‡K wfwË K‡i Zv‡`i g‡a ̈ msL ̈vi aviYv GK ̧iæZc¡Y~©aviYv,hvimP~bvcvÖ‰MwZnvwmKh‡yM|gvblyhLbcïcvjb Ki‡Z wkLj ZLb cï‡`i wn‡me ivLvi c‡Ö qvRb Abfy Z~ nj|
cv_‡i ev Mv‡Q `vM w`‡q, A_ev bwy o ev cv_iL‡Ûi mvnvh ̈ wb‡q, A_ev `wo- RvZxq wRwb‡m wMU w`‡q G‡`i m‡1⁄2 cï‡`i GK-GK m¤úK© (one-to-one correspondence) ̄v’ c‡bi gva ̈‡g ïiæ nj ̧b‡Z ‡kLv| GK-GK m¤úK© ̄v’ c‡bi GB Avwe®‹vi MwY‡Zi weKv‡ki ‡ÿ‡Î GK ewjô c`‡ÿc| ̧b‡Z ‡kLv I msL ̈v aviYvi mP~ bv nevi A‡bK c‡i msL ̈vi bvgKiY ïiæ nq| Gi`xNK© vjcimsL ̈v‡jLviRb ̈wPýe ̈eüZnj|e ̈vwejbxq,wgkixq, gvqvb, fviZxq, ‰PwbK cfÖ w... Z cvÖ Pxb mf ̈Zvq wewfbœ iKg wjwc e ̈envi Kiv nj I wewfbœ iKg msL ̈v c×wZ Avwe®‹Z... nj| G‡`i g‡a ̈ cvÖ Pxb fviZxq‡`i Avwe®‹Z... msL ̈vc×wZ nj mev© ‡cÿv wbLZuy I KvhK© ix msL ̈vc×wZ| Avi GUv m¤e¢ nj kb~ ̈ I ̄v’ bxqgv‡bi aviYv Avwe®‹v‡ii d‡j| msL ̈v‡K cKÖ vk Kivi ‡ÿ‡Î GUv nj GK `ß„ c`‡ÿc| 1,2,3,4,5,6,7,8,9,0 GB `kwU msL ̈vwPý evA1⁄4e ̈enviK‡i‡h‡Kv‡bvmsL ̈vcKÖ vkKivm¤e¢ n‡jv|
‡h‡nZy fviZxq msL ̈vc×wZ‡Z A‡1⁄4i msL ̈v `k Ges wbavbI `k, ZvB GwU `kwgK ̄v’ bxqgvb msL ̈vc×wZ wn‡m‡e cwiwPZ| GB c×wZ‡Z GKK ‡_‡K ïiæ K‡i ci ci ̄v’ bxqgvb `k ̧b ew„ × ‡c‡q h_vμ‡g `kK, kZK, nvRvi ev mnm,Ö `k nvRvi, jÿ BZ ̈vw`‡Z cwiYZ nq| GB c×wZ‡Z wjwLZ ‡h ‡Kv‡bv msL ̈vi Wvb cvÖ wšÍK A1⁄4 nj GKK| GKK ‡_‡K ïiæ K‡i ciciAew ̄Z’ evgw`‡Ki ̄v’b ̧wjigvb`k ̧bK‡i‡e‡ohvq|D`vniY wn‡m‡e ejv hvq, ÔwZb jÿ m‡Z‡iv nvRvi cvPu ‡kv mvZvbeŸBÕ msL ̈vwU‡K A‡1⁄4 ‡jLv n‡j cvIqv hv‡e Ô317597Õ, hvi we ̄ZÍ... AvKvi njÑ 317597=3×100000+1×10000+7×1000+5×100+9×10+7
hvB ‡nvK, `kwgK ̄v’ bxqgvb msL ̈v c×wZ‡Z wewfbœ ai‡Yi msL ̈v Av‡Q|Aviwewfbœai‡YimsL ̈viDTM¢en‡q‡Qwewfbœ‡cÿvc‡U|Gev‡i ‡mB ̧iæZc¡ Y~ © welq wb‡q Av‡jvPbv| Avgiv Rvwb, msL ̈v ‡kLvi cvÖ _wgK chv©‡qwkÿv_x©iv1,2,3,4,5,...cfÖ w...ZmsL ̈vwk‡L_v‡K|G‡`i ̄v^fvweK msL ̈v (Natural Numbers) ev abvZ¥K cY~ © msL ̈v (Positive Integers) ejvn‡q_v‡K|ckÖ œnj,G‡`i ̄v^fvweKmsL ̈vejvnq‡Kb?
Avgvi g‡b nq, ‡h‡nZz msL ̈v mg‡~ ni g‡a ̈ G ̧wj cvÖ PxbZg, Avgiv
̄v^ fvweK fv‡e (in a natural way) G‡`i m‡1⁄2 cwiwPZ n‡qwQ Ges G‡`i MVb Kivi Rb ̈ Ab ̈ ‡Kv‡bv cqÖ v‡mi Øvi ̄’ n‡Z nqbv, ZvB G‡`i ̄v^ fvweK msL ̈v ejv nq| ‡mRb ̈ MwYZÁ wjI ‡cvì ‡μv‡bKvi (1823-1891) e‡j‡Qb, ÒGod made positive integers, all else is due to man.Ó
GLb Avgiv ‡`Le, ‡Kb Ab ̈ ai‡Yi (all else) msL ̈v MV‡bi
c‡Ö qvRb Abfy Z~ nj| ckÖ œ nj, ïagy vÎ ̄v^ fvweK msL ̈vi ‡mU A_v© r N = {1,2,3,4,....} Rvbv _vK‡j me iK‡gi MYbvi KvR ev ‰`bw›`b Rxe‡bi me mgm ̈vi mgvavb Avgiv Ki‡Z mÿg ne? Gi DËi n‡”Q Ñ bv|
wewfbœ iKg mgm ̈vi mgvavb Kivi Rb ̈ Avgv‡`i mgxKiY MVb Ki‡Z nq Ges Zv‡`i mgvavb Ki‡Z nq| ïay ̄v^ fvweK msL ̈v Rvbv _vK‡j Ges m, n ‡h ‡Kv‡bv `wy U ̄v^ fvweK msL ̈v‡K wb‡`k© Ki‡j Avgiv ‡KejgvÎ x = m + n Ges x = m . n ai‡Yi mgxKi‡Yi mgvavb
Ki‡Z cvi‡ev; wKš‘ x + n = m RvZxq mgxKi‡Yi memgq mgvavb Kiv m¤¢e n‡e bv| hw` n > m nq, Z‡e mgvavb wnmv‡e cvIqv (m Ñ
n) msL ̈vwU ̄v^ fvweK msL ̈vi ‡m‡U cvIqv hv‡e bv| ‡hgb, x + 8 = 3 n‡j,x=3Ñ8Ges(3Ñ8) ̄v^fvweKmsL ̈vi‡Mvôxf3y bq|d‡jm, n ‡h ‡Kv‡bv `wy U ̄v^ fvweK msL ̈v n‡j x + n = m RvZxq mgxKi‡Yi mgvavb m¤¢e Kivi Rb ̈ c‡Ö qvRb nj ‡mU N-Gi cwieab© ; c‡Ö qvRb, Z = {.......,Ñ3,Ñ2,Ñ1,0,1,2,3,......} ev cY~ © msL ̈vi ‡mU wbgv© ‡Yi|
GLb ckÖ œ nj, cY~ © msL ̈vi ‡mU MV‡b wK Avgv‡`i me mgm ̈vi mgvavbn‡e?GiDËiIn‡eÑbv|KviYm,n‡h‡Kv‡bv`wyUcY~© msL ̈v‡K wb‡`k© Ki‡j mx = n RvZxq mgxKiY ZLbB mgvavb‡hvM ̈ hLb n n‡e m-Gi ̧wYZK| 5x = 8 A_ev 2x = Ñ3 RvZxq mgxKiY ‡`Iqv _vK‡j cvIqv hv‡e x = 8/5 A_ev x = Ñ3/2| wKš‘ 8/5
ev Ñ3/2 Ggb ‡Kv‡bv cY~ m© sL ̈vi Aw ̄ÍZ¡ ‡bB| ZvB G mgm ̈v ‡_‡K DËi‡Yi Rb ̈ c‡Ö qvRb cY~ © msL ̈vi ‡m‡Ui cwieab© Ges gj~ ` msL ̈vi ‡mUQ-Giwbgv©Y|m,n‡h‡Kv‡bv`wyUgj~ `msL ̈vn‡jGesm≠0 n‡j mx = n RvZxq mgxKi‡Yi mgvavb memgq m¤¢e| D‡jLø ̈, gj~ ` msL ̈v n‡”Q p/q AvKv‡ii msL ̈v ‡hLv‡b p, q cY~ m© sL ̈v Ges q ≠ 0|
jÿYxq, ci ci cwieab© K‡i gj~ ` msL ̈vi ‡mU Q = {x : x = p/q,
p, q cY~ m© sL ̈v, q ≠ 0 } MVb K‡iI Avgv‡`i me mgm ̈vi mgvavb n‡e
bv| hw` GKwU mg‡KvYx mgwØevû wÎfRy _v‡K hvi mg‡KvY msjMœ cwÖ ZwUi
evûi ‰`N ̈© 1 GKK, Zvn‡j Zvi AwZf‡y Ri ‰`N ̈© x GKK n‡j Avgiv cve
x2=2,wKš‘gj~ `msL ̈vi‡m‡UGRvZxqmgxKi‡Yimgvavbm¤¢ebq|
GBAmwyeav‡_‡K‡ewi‡qAvmviRb ̈c‡ÖqvRbQ-Gicwieab©|gj~`
msL ̈vi m‡1⁄2 Agj~ ` msL ̈v wgwk‡q ev ̄Íe msL ̈vi ‡mU R wbgv© Y Ki‡Z nq|
cgÖ vY Kiv hv‡e ‡h, Ggb ‡Kv‡bv gj~ ` msL ̈v ‡bB hvi eM© 2| hvi eM© nj 3
2,‡mGKwUAgj~ `msL ̈vGesG‡KwPýZKivnq√2w`‡q|GiKg√4, 4
√125,.............njAgj~ `msL ̈v‡`i`‡j|Z‡eGBAgj~ `msL ̈v ̧wj‡K gj~ `msL ̈vmnMm¤w^jZeûc`ivwkgvjvwewkómgxKi‡YiexRwnmv‡e cvIqv hv‡e| G‡`i ejv nq exRMvwYwZK (algebric) msL ̈v| wKš‘ Avi GK ai‡YiAgj~ `msL ̈vAv‡Qhv‡`igj~ `msL ̈vmnMm¤w^jZeûc`ivwkgvjv wewkó mgxKi‡Yi exR wnmv‡e cvIqv m¤¢e bq| ‡hgb, p, e, pe, G‡`i AexRxq(Transcendental)msL ̈vejvnq|
GLb GK bR‡i wewfbœ ai‡bi msL ̈v ‡Pv‡Li mvg‡b wb‡Pi Q‡K ‡f‡m DV‡e:-
c~Y©msL ̈v
g~j` msL ̈v
fMœvsk
ev ̄Íe msL ̈v
Ag~j` msL ̈v
exRMvwYwZK
AexRxq
FYvZ¥K
k~b ̈
abvZ¥K
FYvZ¥K
abvZ¥K
GK (1)
‡gŠwjK
‡hŠwMK
‡jLK wewkó MwYZÁ I weL ̈vZ RbweÁvb cPÖ viK| B‡gj: satyabachisar@gmail.com