Page 6 - Bigyan Katha_Jan2021

6
Rvbyqvix 2021
eLk& vwj cvÐwz jwc: eMg© ‡~ ji
Avmbgœ vb wbYq©
Gcv_©mviw_ gy‡Lvcva¨vq
B ‡kl wKw¯Íi Av‡jvPbvq Avgiv ïay eMg© ‡~ ji Avmbgœ vb wbY‡© qi mÎ~ m¤‡^ Ü Av‡jvPbv Ki‡ev, hv wbtm‡›`‡n PgKc`Ö Ges hv‡K mß`k kZ‡K Kjbwe`¨v (calculus) e¨envi K‡i cvIqv Newton-
Raphson iteration formula-i m‡1⁄2 wgwj‡q ‡`L‡j we¯§‡q nZevK n‡q ‡h‡Znq,k×Ö vqgv_vbZn‡qAv‡mcvÖPxbfviZxqMwYZgbxlvicwiP‡q|
‡Quov‡Lvuov eL&kvwj cvÛzwjwci GKvwaK As‡k AvswkKfv‡e D‡jøL Kiv GB m~Î Ges Zvi cÖ‡qv‡M Kiv AšÍZ 6wU D`vniY ‡_‡K ‡h g~j m~Î D×vi Kiv wM‡q‡Q Zv nj :
AK...‡Z wkøó K...Zy¨bv ‡kl‡”Q‡`v wØms¸Yt| Z`&eM© `j mswkøó BwZ ïw× K...wZÿqt \
we‡kl‡Áiv Gi gv‡b K‡i‡Qb, Ôhw` (msL¨vwU) cY~ e© M© bv nq, Zvn‡j (Zvi ‡_‡K) wbKUZg cY~ e© M© msL¨v we‡qvM K‡iv; Aewkó (msL¨v)-‡K (H cY~ e© M© msL¨vi) wظY w`‡q fvM K‡iv| (Gfv‡e cvIqv fMvœ sk‡K) eM© K‡i Zvi A‡aK© ‡K H cY~ e© M© msL¨v I GB fMvœ s‡ki ‡hvMdj (A_vr© c_Ö g av‡ci Avmbgœ vb) w`‡q fvM K‡iv Ges G‡K (c_Ö g av‡c cvIqv Avmbœ gvb ‡_‡K) we‡qvM K‡iv; (GUvB n‡e eMg© ‡~ ji wØZxq av‡ci Avmbgœ vb) hv ‡kl av‡ci ïw×Ki‡Yi (refinement)-Gi Rb¨ cvIqv msL¨vi eM© cwigvY Kg [Kw... Z ÿqt, less the square]Õ|
AvR‡Ki exRMwY‡Zi wPý e¨envi K‡i wjL‡j e¨vcviUv GiKg n‡e,
aiv hvK, cY~ e© M© bq Ggb msL¨v q-Gi eMg© j~ Avmbgœ v‡b wbYq© Ki‡Z
n‡e| q-Gi wbKUZg cY~ e© M© msL¨v a2 n‡j, q = a2 + r n‡e| Zvn‡j
√q=√a2 +r =a+ r ,mÎ~ Abhy vqxGUvc_Ö gav‡ciAvmbœgvb| 2a r2
nv‡Zi mvg‡b calculator _vK‡j e¨envi K‡i ‡`Lyb| cvuP Ni ch©šÍ wb‡j GUvB Avmbœ `kwgK gvb!
GB m‡~ Îi c‡Ö qv‡M weivU eoeo msL¨vi Avmbœ eMg© j~ wbYq© Kiv Av‡Q eLk& vwj cvÛwz jwc‡Z| me‡P‡q PgKc`Ö D`vniY nj √336009 ,
hvi fMvœ sk gvb ‡ei Kiv Av‡Q 579 515,225,088 , 777,307,500
‡h gvb AvR‡Ki wn‡m‡e `kwg‡Ki ci 13 Ni chš© Í wVK| ïay ZvB bq, GB MbYv ‡h wØNvZ mgxKi‡Yi mgvavb Ki‡Z wM‡q cvIqv, ÔcZÖ ¨qÕ As‡k ‡mB mgxKi‡Y GB Avmbœ gvb‡K ewm‡q (fMvœ sk AvKv‡i) mgxKiY‡K hvPvB (verify) K‡i ‡`Lv‡bv Av‡Q| ‡mB KvR Ki‡Z wM‡q ‡h me fMvœ sk ‡jLv n‡q‡Q Zv GKUy e— ! bx‡P GKUv wj‡L w`jvg:—
50753383762746743271936 7250483394675000000
jÿ¨ Kiæb, GB ai‡bi msL¨v‡`i wjL‡Z wM‡q evievi k~b¨evPK Ôwe›`yÕ wPý e¨eüZ n‡q‡Q, msL¨vi g‡a¨ Ges G‡Kev‡i Wvbav‡iI| Z‡e GB Ôk~b¨Õ kã Ges Zvi Dc¯’vc‡bi wPý wn‡m‡e we›`y (•) eL&kvwj cvÛzwjwc‡Z e¨eüZ n‡q‡Q AviI GKwU we‡kl Zvrc‡h¨©, evievi; Avi Zv nj AÁvZ ivwk (unknown) ‡evSv‡Z (‡hgbUv AvaywbK exRMwY‡Z e¨eni Kiv nq x wPý‡K)| A_©vr k~b¨ evPK we›`y wPý e¨eüZ n‡q‡Q Ôk~b¨¯’vbÕ ‡evSv‡Z mgm¨vi Ôb¨vmÕ (statement) As‡k, mgvav‡b wKš‘ GB wPý KL‡bv AskMÖnY K‡iwb (‡hgbUv exRMwY‡Zi x me©ÿY K‡i _v‡K)| GKUv D`vniY ‡`Iqv hvK| Folio 22-Gi wcQb w`‡K (verso) ‡h mgm¨v Am¤ú~Y©fv‡e ‡jLv Av‡Q, Zvi ‡_‡K D×vi K‡i we‡klÁiv ej‡Qb, A1⁄4Uv GB iKg; ÔcÖ_g Rb‡K wKQy cwigvY A_© ‡`Iqv nj, wØZxq Rb‡K Zvi wظY, Z...Zxq Rb‡K wZb¸Y Ges PZy_© Rb‡K Pvi¸Y ‡`Iqv nj, hw` ‡gvU 200 ‡`Iqv n‡q _v‡K Zvn‡j cÖ_g R‡bi cÖvß A‡_©i cwigvY Ges Ab¨vb¨‡`i cÖvß A‡_©i cwigvb ej|
r
(
)
() 2a
2
2a
, Ges
2
Gevi m~Î ‡g‡b G‡Mv‡j cici cv‡ev
( r ) a + 2a
( r )2 ‡k‡l a + r − 2a r
, hv m~Î Abyhvqx wØZxq av‡ci A v m b œ g v b | j ÿ ¨ K i æ b , c _Ö g a v ‡ c i A v m b œ g v ‡ b Î w y U ( e r r o r ) - i c w i g v Y
2a
, hv‡K mij Ki‡j cvIqv hv‡”Q,
; m~Î Abyhvqx wØZxq av‡ci
2a 2( a + 2a )
eai‡j‡jLvhvq√q=√a2 +r =a+ r +e;Dfqc‡ÿeM©Ki‡j
) a2+r= r 2
(
2 r2a
a + +e
e + e = – ( )2 2a
1 2(a+ r)
1
∴ √10 = √32 + 1 = 3 + 16 – 36
2 ( 3 + 16 )
= 3.16228
2a
ïw×Ki‡Y e2 -‡K AeÁv K‡i (neglecting) ïay e-‡K ‡bIqv n‡q‡Q| GUvB m‡~ Îi ‡kl As‡ki e3e¨ (Kw... Z ÿq)| KZUv PgrKvi GB mÎ~ ? GKUv mnR D`vniY wb‡Ri gZb wb‡q ‡`Lv hvK| GB mÎ~ Abmy v‡i √10 -Gi gvb KZ n‡e? GLb 10 = 32 + 1, A_v© r a = 3 I r = 1,