Page 14 - Bigyan Katha June 2021

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Ryb 2021
(a) ‡_‡K cvB, a─n nj a-Gi A‡b ̈vb ̈‡Ki n-msL ̈K ̧Ydj| (a) ‡_‡K cvIqv hv‡e:
Rb ̈ cÖgvb Kiv hv‡e| Avgv‡`i we‡kl fv‡e g‡b ivL‡Z n‡e ‡h ‡Kv‡bv abvZ¥K c~Y©msL ̈v n-Gi Rb ̈ 0─n msÁvZ bq|
4) Gici Avgiv ap/q-Gi msÁv ‡`e, ‡hLv‡b a n‡”Q 0 Qvov
‡h‡Kv‡bv ev ̄ÍemsL ̈v, p/q n‡”Q jwNô AvKv‡i cÖKvwkZ GKwU g~j`
msL ̈v Ges q abvZ¥K c~Y©msL ̈v| ap/q-‡K msÁvwqZ Kie ‡mB
ev ̄ÍemsL ̈v x wnmv‡e ‡hLv‡b xq = ap; Avi ejv n‡e, x (A_©vr ap/q)
n‡jv a-Gi p Nv‡Zi q-Zg g~j (qth root of pth power of a)
hvq, Zv nj ·qv`k kZvãxi ms ̄‹iY, g~j cvÐywjwci 1000 eQi c‡ii| d‡j wPý ̧wj w`IdvšÍv‡mi wb‡Ri e ̈eüZ wPý wKbv Zv wbwðZ fv‡e ejv hv‡ebv|
mP~ K msμvšÍ wPý e ̈env‡ii ‡ÿ‡Î wewfbœ mg‡q wewfbœ MwYZÁ wewfbœ Dcvq Aej¤b^ K‡i‡Qb| wewfbœ mgq e ̈eüZ mP~ K wPý GLv‡b ‡gvUvgwy U Kvjvbμy wgKfv‡e ‡`evi ‡Póv Kiv n‡jv| 1484 wL÷« v‡ã divwm MwYZÁ wb‡Kvjvm P ̈‡y Kv‡qU 12, 12x, 12x2-Gi Rb ̈ e ̈envi K‡iwQ‡jb h_vμ‡g 120, 121, 122| 1544 wL÷a v‡ã Rvgv© b MwYZÁ gvB‡Kj w÷‡dj Zviu ÔGwi_‡gwUKv Bw›UMvÖ Õ kxlK© M‡Ö š’ AÁvZivwk Zvi eM© Zvi Nb I PZ_z N© v‡Zi Rb ̈ ‡h wPý ̧wj e ̈envi K‡i‡Qb ‡m ̧wj nj h_vμ‡g A, AA, AAA, AAAA. 1572 wL÷a v‡ã BZvjxq MwYZÁ ivdv‡qj ‡ev‡¤w^ j AÁvZivwk, Zvi eM,© Nb, PZ_y N© v‡Zi Rb ̈ ‡h wPý ̧wj e ̈envi K‡i‡Qb ‡m ̧wj nj h_vμ‡g | 1586 wL÷a v‡ã AÁvZivwki ci ci mP~ ‡Ki Rb ̈ WvP MwYZÁ mvBgb ‡÷wf‡bi e ̈eüZ wPý ̧wj nj,,,, ...| 1591 wL÷a v‡ã divwm MwYZÁ d ̈« v‡1⁄4v wf‡q‡Z Gme wP‡ýi cwieZb© K‡i A-Gi ci ci NvZ wb‡`k© Kivi Rb ̈ e ̈envi K‡ib h_vμ‡g A, Aq (quad), Acub(cubus),Aqq(quad)(quad),...|1631wL÷a v‡ãBs‡iR
M w Y Z Á _ g v m n ̈ v w i q U G m e w P ‡ ý i D b w œ Z K ‡ i A a by v e ̈ e ü Z a , a 2 , a3, a4, ... Gi Rb ̈ e ̈envi K‡ib h_vμ‡g a, aa, aaa, aaaa, ...| Avi 1634 wL÷a v‡ã divwm MwYZÁ wc‡q‡i ‡nwi‡Mvb Zviu Ôμmz vm g ̈v‡_‡gwUKvmÕ kxlK© M‡Ö š’ a, a2, a3, a4, ... Gi cwie‡Z© e ̈envi K‡ib a, a2, a3, a4, ...| Z‡e eZg© v‡b e ̈eüZ wPýmgn~ A_v© r a, a2, a3, a4, ... cfÖ w... Zi ceÖ Zb© K‡ib divwm MwYZÁ ‡i‡b ‡`Kv‡Z© 1637 wL÷a v‡ã| wZwb ‡Kej gvÎ cY~ m© sL ̈K NvZ e ̈envi K‡iwQ‡jb| 1655wL÷a v‡ãBs‡iRMwYZÁRbIqvwjmwewfbœivwkiFYvZ¥K
cY~ m© sL ̈v I fMvœ sk m¤w^ jZ mP~ ‡Ki h_vh_ e ̈vL ̈v I e ̈envi K‡ib| Zviu e ̈eüZ wPý AvR cPÖ wjZ| ‡hgb x─1 = 1/x, x─n = 1/xn, x1/2
n a─n = a1
(a ≠ 0) (b)
GLb (b) Gi mvnvh ̈ wb‡q m~P‡Ki wbqg ̧wj FYvZ¥K c~Y©msL ̈vi
Ges Avgiv wjLe x = q ap . √
hw` p = 1 nq , Z‡e x-‡K ejv n‡e a-Gi q Zg g~j|
g‡b ivL‡Z n‡e, hw` a < 0 Ges q ‡Rvo msL ̈v nq, Zvn‡j a1/q-Gi mg ̄Í gvb ev ̄Íe bq| ‡mRb ̈ ap/q ‡K msÁvwqZ Kivi mgq a-Gi Ici Av‡ivwcZ kZ© n‡jv a ≥ 0.
Gic‡i wbav‡bi Dci cÖ‡qvRbxq kZ© Av‡ivc K‡i m~P‡Ki wbqg- ̧wj‡K g~j` msL ̈vi Rb ̈ cÖgvY Ki‡Z ‡Kv‡bv Amywe‡a ‡bB|
5 ) G B c mÖ ‡ 1⁄2 w k ÿ v _ x © ‡ ` i e j ‡ Z n ‡ e , x 1 / 2 - ‡ K A v g i v √ x w n m v ‡ e wj‡L _vwK| g‡b ivL‡Z n‡e hLb x = a2 (a GKwU ev ̄ÍemsL ̈v), ZLb
√x = √a2 = a
D‡jøL ̈, a nj a-Gi cig gvb (absolute value), hv
msÁvwqZ nq bx‡Pi g‡Zv:
a = a,hLba≥0;
 {─ahLba<0} A_©vr 5 = 5 Ges ─7= 7.
ZvQvov g‡b ivL‡Z n‡e,
√a.√b , hLb a, b Df‡qB abvZ¥K;
x, x1/q = qa, xp/q = qap| Z‡e n-Gi ‡h ‡Kv‡bv ev ̄Íe gv‡bi √√√
√ab = {√a.√b hLb a, b Df‡qB FYvZ¥K }
=
Rb ̈ an RvZxq mvaviY mP~ ‡Ki ceÖ Zb© N‡U m ̈vi AvBR ̈vK wbDU‡bi gva ̈‡g, hLb wZwb wjewb‡Ri wPwVi Dˇi wØNvZ Dccv‡` ̈i e ̈vcv‡i wPý ̧wj e ̈envi K‡ib|
Avi hw` a Ges b Gi g‡a ̈ GKwU abvZ¥K I AciwU FYvZ¥K, ZLb √ab Avi ev ̄ÍemsL ̈v _v‡K bv|
bx‡Pi Abywm×všÍ ̧wjI wkÿv_©x‡`i g‡b ivLv cÖ‡qvRb:
1)' a hw` 1 Qvov ‡h ‡Kv‡bv ev ̄ÍemsL ̈v nq, Ges ax = 1 nq, Zvn‡j x = 0|
2)' hw` x ≠ 0 Ges ‡h ‡Kv‡bv ev ̄ÍemsL ̈v a Gi Rb ̈ ax = 1 nq, Zvn‡j a = 1. n
mP~ K welqK Av‡jvPbvq QvÎQvÎx‡`i ‡evSv‡Z n‡e am Ges (am)n Gi g‡a ̈ cv_K© ̈ Av‡Q| hw`I 222 = (22)2 ‡h‡nZz 222 I (22)2 cwÖ ZwU 16 Gi mgvb wKš‘ 554 ≠ (55)4; KviY 554 = 5625 Ges (55)4 = 520.
(3)
Gev‡i m~P‡Ki aviYv I m~PK wP‡ýi weKv‡ki HwZnvwmK ‡cÖÿvcU wb‡q Av‡jvPbv| GKwU AÁvZivwk‡K bvbv m~P‡K ev Nv‡Z
DbœxZ Kivi aviYv I wP‡ýi e ̈envi cÖvPxbKv‡jB cÖPwjZ wQj| hvui ‡jLvq G wel‡q me©vwaK Av‡jvPbv ‡`Lv hvq, wZwb n‡jb Av‡jKRvw›`Öqvi w`IdvšÍvm, weL ̈vZ ÔGwi_‡gwUKvÕ MÖš’ hvui ‡jLv| AvbygvwbK 275 wLa÷v‡ã Zvui ‡jLvq AÁvZivwki Rb ̈ bvbv m~P‡Ki e ̈envi ‡`Lv hvq| eZ©gv‡b cÖPwjZ AÁvZivwk x Gi Rb ̈ wewfbœ Nv‡Zi wPý x0, x1, x2, x3, ... Gwi_‡gwUKv‡Z h_vμ‡g M0, S, Dγ, Kγ ....wnmv‡e ‡jLv n‡q‡Q| Z‡e ‡h eB‡Z GB wPý ̧wj cvIqv
m~P‡Ki aviYvi weKvk cÖm‡1⁄2 fviZxq MwYZ we‡kl K‡i ‰Rb MwY‡Zi K_v Aek ̈B D‡jø‡Li `vwe iv‡L| 300 wLa÷c~e©v‡ã ev Zvi Av‡M hLb m~PK msμvšÍ aviYv M‡o I‡Vwb, ‡mB mgq ‰Rb‡`i DËiva ̈qb m~‡Î ‡Kv‡bv msL ̈vi wØZxq NvZ‡K eM©, Z...Zxq NvZ‡K Nb, PZy_© NvZ‡K eM© eM©, BZ ̈vw` ejv n‡q‡Q| Avi Aby‡hv‡Mv×vi m~‡Î
( cÖvq 100 wLa÷c~e©vã) ‡Kv‡bv msL ̈vi wØZxq NvZ, PZz_© NvZ, Aóg NvZ‡K ejv n‡Zv h_vμ‡g cÖ_g eM©, wØZxq eM©, Z...Zxq eM©| A_©vr msL ̈vwU a n‡j AvR‡K cÖZx‡K ‡jLv a2-‡K, (a2)2 A_©vr a4-‡K, (a4)2 A_©vr a8-‡K ejv nZ h_vμ‡g a-Gi cÖ_g eM©, wØZxq eM©, Z...Zxq eM© BZ ̈vw`| mvaviY fv‡e (a2)n ‡K ejv nZ a-Gi n-Zg eM©| Abyiƒcfv‡e eM©g~‡ji ‡ÿ‡Î a1/2, a1/4, a1/8-‡K h_vμ‡g
ejv nZ a-Gi cÖ_g eM©g~j, wØZxq eM©g~j, Z...Zxq eM©g~j| Avi mvaviYfv‡e a1/2n-‡K ejv n‡Zv a-Gi n-Zg eM©g~j|
Aby‡hv‡Mv×vi m~‡Î GK RvqMvq hv ejv n‡q‡Q Zv AvR‡Ki cÖZx‡K wjL‡j cvIqv ‡hZ
a1/2.a1/4 = (a1/4)3; a1/4.a1/8 = (a1/8)3
Avgiv we ̄§qvf~Z nB hLb ‡`wL ‰Rbiv m~P‡Ki mnR wbqg [‡hgb
am × an = am+n ; (am)n = amn] m¤^‡ÜI AwfwnZ wQ‡jb| •
‡jLK wewkó MwYZÁ I weL ̈vZ RbweÁvb cPÖ viK| B‡gj: satyabachisar@gmail.com