Page 13 - Bigyan Katha June 2021

Ryb 2021 13 wkÿYZË¡ I BwZnv‡mi Av‡jv‡K mP~ K
mZ¨evPx mi
(1)
a─n-‡K msÁvwqZ Kiv nq bx‡Pi g‡Zv:
(i) bs m~‡Î m = ─n ewm‡q cvB
an × a─n = an─n = a0 = 1 [(I)-Gi mvnv‡h¨] ∴ a─n = 1/an (II)
wKš‘ ‡h Abywm×všÍ (I)-G MvwYwZKfv‡e DcbxZ nIqv hvq wb, ‡mLv‡b Zvi Dci wbf©i K‡i Abywm×všÍ (II) MvwYwZKfv‡e ï×iƒ‡c we‡ewPZ n‡Z cv‡i bv|
Gici p, q-‡K abvZ¥K c~Y©msL¨v a‡i ap/q-Gi A_© bx‡Pi g‡Zv K‡i ‡`Iqv nq|
(i) m~·K evi evi cÖ‡qvM K‡i cvB,
ap/q × ap/q × ap/q × ... × ap/q (q msL¨K evi )
= ap/q+p/q+p/q+...+p/q (q msL¨K evi ) = ap
∴ (ap/q)q = ap ∴ ap/q = qap (III)
wKš‘ ATM¢yZ e¨vcvi, q ap -Gi ‡h Aw¯ÍZ¡ eZ©gvb, ‡m m¤^‡Ü Avgiv
we`¨vjq MwY‡Z ‡h me ¸iæZc¡ Y~ © aviYv cov‡bv nq, mP~ K nj Zv‡`i g‡a¨ GKwU|GBaviYviDc¯v’cbviRb¨we‡klh‡Ziœ c‡ÖqvRb|KviYwKQy wKQy cvV¨eB‡Z ev ‡kYÖ xK‡ÿ mP~ K I mP~ K msμvšÍ mÎ~ BZ¨vw` ‡hfv‡e cwi‡ewkZ nq, Zv wkÿY-ZË¡ Abmy v‡i (pedagogically) ÎæwUg3y bq|
‡Kv‡bv GKwU wbw`©ó msL¨v, aiv hvK 3-‡K ci ci cvuPevi ¸Y Kiv n‡j cvIqv hv‡e 3.3.3.3.3, hv‡K ms‡ÿ‡c 35 wnmv‡e ‡jLv nq| 35-GB MvwYwZK wP‡ý 3-‡K ejv nq wbavb (base) Ges 5-‡K ejv nq m~PK (index) ev NvZ (power)| Abyiƒcfv‡e, 74 = 7.7.7.7 Ges 74 cÖZxKwU‡Z 7 nj wbavb Ges 4 nj m~PK| mvaviYfv‡e ejv hvq, hw` a GKwU ev¯ÍemsL¨v‡K wb‡`©k K‡i Ges n GKwU abvZ¥K c~Y©msL¨v (positive integer) nq, Zvn‡j n msL¨K a-Gi ¸Ydj A_©vr a.a.a.....a (n msL¨K evi)-‡K ‡jLv n‡e an wnmv‡e Ges ejv n‡e, an GB MvwYwZK wP‡ý a n‡”Q wbavb I n n‡”Q m~PK ev NvZ| GBfv‡e m~PK msÁvwqZ nIqv DwPZ| wKš‘ wKQy ‡ÿ‡Î ‡`Lv hvq n-‡K abvZ¥K c~Y©msL¨v aiv n‡jI a-Gi m¤^‡Ü ‡Kv‡bv wKQy ejv nqbv| hw`I wkÿv_©x‡`i Rvbv GKvšÍ cÖ‡qvRb, a Kx ai‡Yi msL¨v|
Gici m I n-‡K ‡h ‡Kv‡bv `ywU abvZ¥K c~Y©msL¨v a‡i m~P‡Ki wbqg¸wj bx‡Pi g‡Zv K‡i wee„Z Kiv nq:
(i) an.am = an+m
(ii) an ÷ am = an─m (n > m)
(iii) (am)n = amn
(iv) (ab)n = an.bn
cieZ©x ch©v‡q n Ges m-‡K abvZ¥K c~Y©msL¨v a‡i msÁvi
mvnvh¨ wb‡q m~θwj cÖgvb Kiv nq| D‡jøL¨, (i)-G ewY©Z wbqg‡K m~P‡Ki ‡gŠwjK wbqg ejv nq, ‡h‡nZz evwK¸wj (i)-‡K AbymiY K‡i cvIqv hvq| Z‡e m~Î (ii)-Gi ‡ÿ‡Î g‡b ivL‡Z n‡e a n‡”Q k~b¨ Qvov ‡h‡Kv‡bv ev¯ÍemsL¨v|
Zvici a0, a─3, a2/3, a─5/6 RvZxq cÖZxK¸wji A_© ‡`Iqv n‡q _v‡K, me ai‡bi msL¨v m, n-Gi Rb¨ (i) wbqgwU mZ¨ a‡i wb‡q| wKš‘ msÁv ‡`evi mgq m I n-‡K ‡KejgvÎ abvZ¥K c~Y©msL¨v aiv n‡qwQj; me ai‡bi msL¨v m I n-Gi Rb¨ (i)-‡K cÖwZôv Kiv nqwb| d‡j Avgiv wkÿY-ZË¡ welqK (pedagogical) ÎæwUi wkKvi nB|
Av‡iv AevK n‡Z nq, hLb ‡`Lv hvq (i)-‡K mZ¨ a‡i a0-GB cÖZx‡Ki A_© bx‡Pi g‡Zv K‡i ‡`Iqv n‡”Q:
m = 0, a‡i (i) ‡_‡K cvIqv hv‡e an .a0 =an+0
eva0 =an/an, ∴a0 =1
wKš‘ Avgv‡`i g‡b ivL‡Z n‡e, (I) m¤úK©wU ZLbB mZ¨ e‡j
M„nxZ n‡e hLb an MvwYwZKfv‡e A_©c~Y©| Avgiv wKš‘ me ai‡bi msL¨v n-Gi Rb¨ an A_©c~Y© n‡e wKbv ej‡Z cvwibv; KviY ïaygvÎ abvZ¥K c~Y©msL¨v n-Gi Rb¨ an-‡K msÁvwqZ Kiv n‡q‡Q| d‡j hyw3c~Y©fv‡e Abywm×všÍ (I)-G Avgiv DcbxZ n‡qwQ, Zv ejv P‡j bv| g‡b ivL‡Z n‡e, GKwU MvwYwZK wee„wZi cÖwZwU c` msÁvZ I A_©en n‡jB wee„wZwU MÖnY‡hvM¨|
√
√
(I)
wK K‡i wbwðZ njvg?
Z‡e GBme wkÿY-ZË¡¡ welqK ÎæwU Avgiv mn‡RB `~i Ki‡Z
cvwi, hw` bx‡Pi g‡Zv K‡i m~PK msμvšÍ ¸iæZ¡c~Y© w`K¸wj wkÿv_©x‡`i mvg‡b Zz‡j awi|
(2)
1) cvÖ _wgKfv‡e an-Gi msÁv ‡`evi mgq a Ges n ‡Kvb ai‡bi msL¨v‡K wb‡`k© Ki‡Q, Zv ¯úó K‡i ej‡Z n‡e| Av‡MB ejv n‡q‡Q, ‡h ‡Kv‡bv ev¯ÍemsL¨v a Ges ‡h ‡Kv‡bv abvZ¥K cY~ m© sL¨v n-Gi Rb¨ an n‡”Q n msL¨K a-Gi ¸Ydj| g‡b ivL‡Z n‡e, a-Gi m‡1⁄2
(n ─ 1) msL¨K evi a-‡K ¸Y K‡i an cvIqv n‡q‡Q| [a-Gi m‡1⁄2 n msL¨K evi bq; GUv D‡jLø Kijvg, wKQy Bs‡iwR eB‡Z GUv ejv nq|]
an-‡K msÁvwqZ Kivi mgq n-‡K abvZ¥K cY~ m© sL¨v aiv n‡q‡Q Ges ejv n‡q‡Q an nj n msL¨K evi a-Gi ¸Ydj| g‡b ivL‡Z n‡e, n abvZ¥K cY~ m© sL¨v Qvov Ab¨ ai‡bi msL¨v n‡j A_v© r FYvZ¥K cY~ m© sL¨v, kb~ ¨, ev fMvœ sk n‡j an cZÖ xKwU ‡Kv‡bv A_© enb Ki‡e bv| KviY Ô3-Gi ─6 evi msL¨K evi ¸YdjÕ ev Ô5 Gi 3/4 msL¨K evi ¸YdjÕ evK¨vsk¸wj A_n© xb| hvB ‡nvK Zvici Av‡M D‡jwø LZ mP~ ‡Ki wbqg¸wj weeZ„ K‡i msÁvi mvnvh¨ wb‡q ‡m¸wj cgÖ vb Kiv n‡e| Gic‡i wewfbœ mP~ K I wewfbœ wbavb wb‡q bx‡Pi mÎ~ ¸wj weeZ„ Kiv n‡e:
(i)' am.an.ar = am+n+r
(ii)' {(am)n}r = amnr
(iii)' (abc...)n = an.bn.cn ...,
‡hLv‡b a, b, c, ... nj ev¯ÍemsL¨v Ges m, n, r, ... nj abvZ¥K c~Y©msL¨v|
2) Gic‡i a-‡K kb~ ¨ Qvov ‡h‡Kv‡bv ev¯ÍemsL¨v we‡ePbv K‡i Avgiv msÁvwnmv‡eMnÖ YKiea0 =1;a0 =1wnmv‡ecgÖ vYKiv‡PóvKie bv| Av‡MB ‡`‡LwQ, a0 = 1 wnmv‡e cgÖ vY Kivi ‡Póv MvwYwZK-fv‡e A‡hŠw3K I MÛ‡Mv‡ji| Avgiv Aek¨B g‡b ivLe 00 AmsÁvZ|
3) Gev‡i a-‡K k~b¨ Qvov ‡h ‡Kv‡bv ev¯ÍemsL¨v a‡i Ges n-‡K GKwU abvZ¥K c~Y©msL¨v a‡i a─n Gi msÁv ‡`Iqv n‡e bx‡Pi g‡Zv
Av‡iv Avð‡h©i e¨vcvi, Abywm×všÍ (I)-Gi Ici wbf©i K‡i
a─n = 1a . 1a . 1a ..... 1a (n msL¨K evi) (a)