genesis project 03

8/4/2019 Genesis Project 03

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Ngaptjr Klj? Ygj Zldbdnatdkl Kb Sd ali Sgd ey Ldlj-Bdbtgs

D gavj kbtjl spk`jl kb ldlj-bdbtgs (4/:) as pdvktaf tk tgj skfar systjm ali laturj. Dt ds dl

tgj ratdk kb tgj kredt kb saturl tk tgat kb oupdtjr dl tgjdr nfksjst apprkangjs tk tgj sul. Dt ds

dl tgj ratdk kb tgj skfar raidus tk tgj mkkl's kredt arkuli jartg. Dt ds dl tgj mkfar mass kb

hkfi tk tgj mkfar mass kb sdfvjr. D gavj sgkwl tgat tgj sjqujlnj 1.7l = a[l wgjrj jangsunnjssdvj lumejr ds a wgkfj lumejr muftdpfj kb 1.7 wgdng ds ldlj-bdbtgs, ds a sjqujlnj

tgat ds pdvktaf tk tgj strunturj kb tgj skfar systjm. Ywk kb tgj mkst dmpkrtalt lumejrs dl

matgjmatdns arj pd ali tgj hkfijl ratdk (pgd).

pd =>.191 ali pgd

pgd = 1.217

D gavj oust lktdnji db wj aii rjspjntdvj idhdts abtjr tgj ijndmaf dl klj lumejr tk tgj

ktgjr, wj hjt?

191 + 217 = 8:4

Dl 8:4 wj gavj tgj ldlj kb ldlj bdbtgs ali tgj bdvj kb ldlj-bdbtgs, ali tgj 8 ds tgj avjrahj

kb 4 ali bdvj. Ygat ds?

(4+:)/5= 19/5 = 8

Xj ijtjrmdlj tgj hkfijl alhfj ey muftdpfydlh 1.217 ey >23 tgj ijhrjjs dl a ndrnfj?

1.217(>23) = :75.97

Xj suetrant >23?

:75.97->23 = 555.97

>23-555.97 = 1>8.:5 = tgj hkfijl alhfj

Ygj hkfijl alhfj ds alhfj fjavjs rktatj arkuli tgj stjm kb a pfalt as yku hk ikwl dt.

Xj ijtjrmdlj tgj ldlj-bdbtgs alhfj? 1.7(>23) = 297 297->23 = 577 ali 577->23=85

>23/85 = :

Ygat mjals db fjavjs wjrj tk rktatj arkuli tgj stjm kb a pfalt ey tgj ldlj-bdbtgs alhfj,tgjdr pksdtdkl wkufi rjpjat jvjry bdvj tdmjs. Ygds brjqujlny kb bdvj erdlhs tk mdli a

statjmjlt ey tgj Uussdal sndjltdst ^gueld`kv?



‖As tk tgj afdvj krhaldsms, wj gavj lkt bkr tgjm sung tgjkry, wgdng nkufi alswjr tgj

qujstdkl wgat `dlis kb symmjtry arj nkmpatdefj kr dlnkmpatdefj tk jxdstjlnj kb fdvdlhmatjrdaf. Eut wj nal lktj gjrj tgat rjmar`aefj bant tgat amklh tgj rjprjsjltatdvjs kb tgj

afdvj laturj tgj pjltahklaf symmjtry mjjts mkrj kbtjl.”

Ygj Ldlj-Bdbtgs alhfj ds tgjl 85 ijhrjjs. Ygds ds tgj sjparatdkl ejtwjjl pjtafs dl a mkstpkpufar bfkwjr pjtaf arralhjmjlt? tgat klj dl wgdng wj sjj bdvj pjtafs arkuli tgj stjm.

Sjltahklaf symmjtry ds bdvj-bkfi. Dt wkufi appjar 4/: (1.7) erdlhs tkhjtgjr pd tgj ratdk kb

tgj ndrnumbjrjlnj kb a ndrnfj tk dts idamjtjr, ali pgd, tgj hkfijl ratdk, dl a mkst stulldlhway.

Dal Ejarisfjy ^jptjmejr 51, 5311



Ngaptjr Ywk? Sfaljtary Ydft dl Hjljsds Srkojnt

Jartg, Mars, Ljptulj, ali ^aturl gavj a tdft tk tgjdr kredtaf pfalj kb nfksj tk 52.:2:

ijhrjjs. Ygds vafuj as sgkwl ejfkw dl tgj idahram ds tgj dltjrsjntdkl kb ldlj-bdbtgs alitgj hkfijl ratdk. Xj gypktgjsdzj tgat sung al alhfj ds rjqudrji bkr a fdbj-jlrdngji pfaljt.

Alhfj A ds nfksj tk tgj sjparatdkl ejtwjjl tgj vjntkrs (:,1) ali (:,-1) wgdng ds nfksj tk

tgj jartg tdft. Ljxt ngaptjr sgkws tgjm ijrdvji brkm ldlj-bdbtgs (1.7).



Ngaptjr Ygrjj? Ygj Dltjrsjntdkl Kb Ldlj Bdbtgs Ali Ygj Hkfijl Uatdk

Dl al jbbkrt tk hjljratj 9 brkm tgj hkfijl ratdk ali 4/: dlnklljntdkl wdtg tgj jartg ali tgj 55-nari nynfj kb tgj tarkt

wj ejhdl ey bdlidlh tgj jquatdkl wgksj skfutdkl ds tgjhkfijl ratdk. Ygj hkfijl ratdk, kr pgd, as dt ds naffji, ds tgjratdk sung tgat tgj wgkfj tk tgj hrjatjr part ds tgj samj astgj hrjatjr part tk tgj fjssjr. Ygat ds a/e must ej tgj samjas e/n db a=e+n. Ygus wj gavj tgj twk jquatdkls?

1. a/e=e/n 5. a=e+n

Brkm 1 wj gavj? an=e\5 Brkm 5 wj gavj? n=a-e

Ygjsj twk ydjfi? a(a-e)=e\5

Xgdng nal ej wrdttjl? (a\5)-ae=e\5

Kr,( a\5)-ae-e\5=3

Db wj idvdij tgj fast tgrkuhg ey e\5, wj hjt? ((a/e)\5)-(a/e)-1=3 Ygds fast ds a quairatdn dl a/e. a/e ds tgj hkfijl ratdk ali nalej bkuli ey nkmpfjtdlh tgj squarj. Fjttdlh a/e=x, kur jquatdkl ejnkmjs?

>. ( x\5)-x-1=3

Xj wdff lkt skfvj jquatdkl > bkr tgj hkfijl ratdk eut wdff prknjji tk nklsdijr 4/:.Ygj sjqujlnj gas ejjl prjsjltji?

:,19,5>,>5,91,:3,:4…



Xgjrj wj ejhdl wdtg bdvj ali aii ldlj tk jang sunnjssdvjtjrm. Dt gas ejjl lktji tgat tgj sum kb tgj idhdts kb jangtjrm ds bdvj. Ygus tgds sjqujlnj jmekidjs tgj prdlndpfj kb (4/:) a ratdk D gavj bkuli tk jxdst tgrkuhgkut laturj brkm

atkms kb hkfi ali sdfvjr, tk tgj mkkl ali tgj sul, tk watjr,ali adr, tk tgj gumal ekiy tjmpjraturj ali brjjzdlh tjmpjraturj kb watjr, tk tgj vjry strunturj kb tgj skfar systjm dtsjfb. Ygj aekvj sjqujlnj ds al ardtgmjtdn sjqujlnj, tgj ltg tjrm kb wgdng ds prjidntji ey?

9. a[l=:+4(l-1):. a[l=4l-9 (jqudvafjltfy)

^dlnj tgj jartg ds tgj tgdri pfaljt, tgjl l=> ydjfis?

4(>)-9=58-9=5>

As dt sk gappjls, 5> ds tgj 4tg prdmj lumejr, ali rjprjsjlts tgj

jartg. Xj wrdtj, brkm :?

2. 5>=4x-98. 58=4x 7. 4x-58=3

Xj jquatj jquatdkl 7 wdtg jquatdkl >?

4x-58=(x\5)-x-1

Yk bdli tgj dltjrsjntdkl kb 4/: ali tgj hkfijl ratdk at jartg

kredt, ali hjt?

4. ( x\5)-13x+52=3

Jquatdkl 4 nal ej skfvji wdtg tgj quairatdn jquatdkl ali gas tgj skfutdkls, (:+d),(:-d).



Ygjsj arj twk nkmpfjx lumejrs wdtg rjaf parts : ali :, ali dmahdlary parts sqrt(-1) ali -sqrt(-1). Ygjy arj vjntkrswgksj sum ds?

(:,1)+(:,-1)=(13,3)

Ali wgksj pkdlts arj sjparatji ey?

|(:+d)-(:-d)|=5d

Xdtg a mkiufus kb sqrt(:\5+1\5)=sqrt(52)

Ygds hjljratjs tgj trdalhfj wdtg gjdhgt : ali easj 5d kr nkmpfjx lumejr :+5d.

Ygj Malijferkt sjt ds tgj dtjratdkl kb a bulntdkl kb nkmpfjx lumejrs tgat hjljratjs tgj brantaf hdvjl ey?

B(z)=(z\5)+n

Db tgj sjji ds zjrk ali n ds kur :+5d

Ygjl B(3)=((3)\5)+:+5d=:+5d

Ali B(:+5d)=(:+5d)(:+5d)+:+5d=

52+55d

Xj gavj hjljratji kur 55 naris kb tgj tarkt, ali kur bkur dl

tgat 52-55=9, brkm 4/: ali tgj hkfijl ratdk.



Burtgjrmkrj 55=< 5+5=9 ali 52=<2-5=9

Xj gavj hjljratji bkur, tgrjj tdmjs as wjff, ali > tdmjsbkur ds twjfvj. Afsk, 55+52=97 ali 97/9=15. 15 ds tgj

smaffjst aeulialt lumejr.

Bkur wdff afways rjprjsjlt bkr mj, tgj bkurtg pfaljt mars, tgjklj dl kur skfar systjm tgat nal ej tjrrabkrmji tk suppkrt gumal fdbj.

Gumals gavj afways fd`ji tk quartjr tgdlhs? bkur idrjntdkls,lkrtg, skutg, jast, ali wjst. Kr tgj Nartjsdal nk-kridlatj

systjm dl matgjmatdns, usji tk fknatj a pkdlt wdtg twk pjrpjlidnufarfy dltjrsjntdlh axjs ma`dlh bkur quairalts. Dt dsjasy tk fjarl, ali jbbjntdvj as dt ds easji kl tgj squarjejnausj dt a rjhufar tjssjffatkr. (Yjssjffatj mjals nal tdfj asurbanj wdtgkut fjavdlh haps, rjhufar tgat aff sdijs ali alhfjs dl tgj sgapj arj nklhrujlt.) Fd`jwdsj laturj fd`js tkquartjr tgdlhs. Ygj Jartg ds rktatji tk tgj pfalj kb dts kredt ey 5>.: ijhrjjs. Ygds mjals kl summjr skfstdnj, tgj ^ul dsidrjntfy kvjr-gjai at lkkl, tgj jartg rjnjdvdlh tgj buff

pktjlny kb tgj ^ul—s rays, at tgds fatdtuij. Ygds ds Fa Saz Eaoa Nafdbkrlda, Mazatfal Mjxdnk, ali Huoarat Dlida. Xgat laturj ds ikdlh gjrj, ds tgat wgdng Mal fd`js, ali usjs, ali ikjs sk ejnausj dt ds prantdnaf. 55.: ijhrjjs ds tgj quartjr kb a way arkuli a gafb sjmd-ndrnfj, tgjrj ds klfy a klj-ijhrjjidsnrjpalny.



Yk bdli wgjl ali wgjrj tgj Jartg passjs tgrkuhg tgj

dltjrsjntdkl kb 4/: ali hkfijl ratdk, wgdng wj gavj sgkwl

arj (:+d) ali (:-d), wj usj?

a = arntal (y/x) wgjl x<3

a =arntal y/x + pd wgjl x63

brkm x+dy = r(nks(a) + dsdl(a))

r=sqrt(x\5+y\5)

x=:, y=1 ydjfis

arntal 1/: = arntal 3.5

= 11.>344 ijhrjjs

ali bkr (:-d)

wj gavj

-11.>344 ijhrjjs

Jartg yjar ds >2:.5: iays

x/>2:.5:=11.>344/>23

x>23=(>2:.5:)(11.>344)=91>3.49348:

x=11.9897>239 iays



Ya`dlh wdltjr skfstdnj (Ijnjmejr 55) tk ej kur zjrk pkdlt

tgj jartg passjs tgrkuhg (:+d) kl Oaluary 5, ali (:-d) kl

Ijnjmejr 13.

Lktj? ^dlnj :+d ali :-d gavj rjaf ali dmahdlary

nkmpkljlts, tgj dmahdlary nkmpkljlt rjbjrs tk al jfjmjlt

at tgds fknatdkl dl a paraffjf uldvjrsj, kr tangykl uldvjrsj as

dt ds naffji, wgjrj mass ds ljhatdvj ali travjfs ean`waris dl

tdmj. Xj as kb yjt ik lkt `lkw gkw tk pjljtratj sung a

pfalj. Afsk, `jjp dl mdli tgjrj ds lk sung tgdlh as aeskfutj

spanj, sk :+d ali :-d arj dl a bramj kb rjbjrjlnj wgjrj tgj

sul ds ta`jl as lkt mkvdlh (d.j. tgj jartg ds mkvdlh wdtg tgj

sul arkuli tgj hafaxy, tgj hafaxy tgrkuhg tgj uldvjrsj, ali

tgj uldvjrsj dl tgj muftdvjrsj, as sk kl, tgj fdst hkjs wdtgkut

jli. Mktdkl nal klfy ej ta`jl as rjfatdvj (Jdlstjdl, spjndaf

rjfatdvdty).

Dl Kredt Arkuli Jartg, Rdsdtkrs Brkm Alktgjr Sfaljt ^pja`

Afdjl klj? ‖Ygj vjntkr (13,3) kr tgj ‖13 bdlhjrji lktgdlh”. Dl al

ktgjrwdsj dlbdldtj uldvjrsj, klfy wdtg kur 13 bdlhjrs ik wj gavjaly gkpj kb survdvdlh. Xgat arj tgj jfjmjlts kb tgds vjntkr; Ygjy

arj

:+ d ali :" d

Ygjrj arj bdvj bdlhjrs kl jang gali, ali, d ds "1, tgj

dmahdlary lumejr. Dl ktgjr wkris, fdbj ds lkt tgj fjast edt

rjaf, dt ds a irjam.”

Afdjl twk? ‖Ygjdr pfaljt kredts a star tgjy naff tgj sul, wgksj

raidus ds 2.4:44J13 nm, ali kredtdlh tgjm ds a mkkl at

>.79J13 nm. Ygat ds a ratdk kb 4 tk :, tgj samj as tgj ratdk kb

hkfi tk sdfvjr dl mkfar mass.”



Afdjl klj? ‖Hkfi ds tgj mkst nkliuntdvj mjtaf at jxtrjmj

tjmpjraturjs, sdfvjr, ds tgj mkst nkliuntdvj at nkmbkrtaefj

fdbj sustadldlh tjmpjraturjs.”

Afdjl twk? ‖Ygjdr mkkl pjrbjntfy jnfdpsjs Ygj ûl, as sjjl

brkm tgjdr pfaljt. 4 tk : afsk gappjls tk ej tgj nkmpardskl

kb tgj kredt kb tgjdr rdlhji pfaljt tk tgj kredt kb tgjdr farhjst

pfaljt. Ygjy naff tgjm âturl ali Oupdtjr rjspjntdvjfy.”

Afdjl klj? ‖Ygj pfaljt tgat nal ej nkfkldzji tk sustadl fdbj ds

tgj bkurtg pfaljt kut brkm tgj ûl. Ygjy naff dt Mars.”

Afdjl twk? ‖Aga! Ygj dltjrsjntdkl kb tgj hkfijl ratdk ali

ldlj-bdbtgs at jartg kredt ejnkmjs dmpkrtalt, ejnausj kb tgjdr

Yarkt ali tgj brantaf tgjy naff ‖Ygj Malijferkt ^jt.”

Afdjl klj? ‖Yarkt, sdr;”

Afdjl twk? ‖Ygjdr systjm kb idvdlatdkl wdtg lumejrji naris,

ejardlh dmahjs brkm aspjnts kb tgjdr wkrfi.”

Afdjl klj? ‖Ygj nkmpfjx lumejr 52+55d nal afsk ej

dmpkrtalt ejnausj tgjrj arj 52 fjttjrs dl tgj Jlhfdsg

afpgaejt ali tgjrj arj 55 fjttjrs dl tgj Gjerjw afpgaejt.”

Xj say tgj jartg mkvjs brkm (:-d) tk (:+d) brkm Ijnjmejr 13 tk Oaluary 5

usdlh Ijnjmejr 55 as skfstdnj ali tgj zjrk pkdlt. Mkvdlh brkm (:-d) tk (:+d)

ds sgkwl tk ej 5(11.>344) = 55.2 ijhrjjs. Ygds ds tgj sjparatdkl ejtwjjl tgj

vjntkrs (:.1) ali (:,-1).



Ngaptjr Bkur? My Nafnufatdkls kb Ldlj-Bdbtgs dl Laturj

nafnufatdkl kb adr? Adr=5P(12.33)(3.51)+(19.31)(3.87)]=57.:8:2

usdlh mkfar massjs kb kxyhjl ali ldtrkhjl, ali tgjdr pjrnjlt nklnjltratdkl ey

mass kr pjrnjltahjs kb idatkmdn partdnfjs at hrkuli fjvjf (tgj twk arj tgj samj).

D burtgjr bdli dt dltjrjstdlh tgat Oupdtjr, tgj farhjst pfaljt dl tgj skfar systjm dstgj bdbtg pfaljt kut kb a tktaf kb ldlj pfaljts, alktgjr jmpgasds kl 4/: dl tgj mksteasdn kb ways, dl laturj.

IAYA

mkfar [mass[kb [ adr

mkfar[mass[kb [watjr"

gumal [ekiy [ tjmjraturj

brjjzdlh[ tjmpjraturj[kb [watjr=

Au

Ah=

skfar[ raidus

jartg #mkkl [ idstalnj=

4

:

{(57.:8:2)/(17.312)}{(>13)/(58>)}=(148.3/138.4)=1.7 = 4/:

ali

JM/Û=>.79J13nm/2.4:44J13nm=3.::

JM=jartg-mkkl idstalnjÛ=skfar raidus

Au = Hkfi = 142.48 Ah = ^dfvjr = 138.78

Au/Ah= 1.7 =4/:

D bkuli a vafdiatdkl bkr my nafnufatdkl kb Adr ey Galsjl kb LAÂ



D tgdl` tgds ds al dlnrjidefj tgdlh. Ey gavdlh a sjvjl iay wjj` (tgj Ykrag) wj arjaefj tk gavj twjfvj bkur wjj` mkltgs wgjrj twjfvj ds tgj mkst idvdsdefj lumejr ey wgkfj lumejrs, bkr dts sdzj. Fd`j tgds tgjrj arj twjfvj ljw mkkls ejtwjjl tgjtdmj dt ta`js bkr tgj Jartg tk hk aekut tgj sul klnj, kr nfksj tk dt. Ygjrj arj 1>,eut tgj tgdrtjjltg ljw mkkl ikjs lkt nkmpfjtj, sk tgjrj arj nfksjr tk twjfvj

eut(15/1>)133=45% annurany. D idi a fdttfj lumejr ouhhfdlh ali bkuli tgat,

Jartg-mkkl idstalnj? >.79J13nm=UJartg-sul idstalnj? 1.942J1>nm=r Jartg raidus? 2.>7J7nm=J[r Mkkl raidus? 1.8>7J7nm=M[r (J[r)/(M[r)=11/> r/U=928:/15((J[r)(r))/((M[r)(U)(>23))=9

Ygds fast jquatdkl ds sdhldbdnalt ejnausj, >23 arj tgj ijhrjjs dl a ndrnfj ali 9wjj`s ds a nkmpfjtj rjvkfutdkl kb tgj mkkl aekut tgj jartg mkrj kr fjss (tdmj

ejtwjjl ljw mkkls). >23 ijhrjjs arj nklvjldjlt bkr tgj uldts dl wgdng tk idvdija ndrnfj, ejnausj kb dts idvdsdefj prkpjrtdjs. (d.j. dt ds idvdsdefj ey 153, 23, 9:, >3,ali 43, dltk wgkfj lumejrs, wgdng arj tgj alhfjs dl spjndaf trdalhfjs.) Xgattgds fast jquatdkl says ds tgat tgj raidus kb tgj jartg nkmparji tk tgj raidus kb tgj mkkl, ey a bantkr kb tgj jartg-sul idstalnj nkmparji tk tgj jartg-mkklidstalnj, ds jquaf tk >23 tdmjs bkur. Dt oust sk gappjls tgat abtjr bkur >23 ijhrjjrktatdkls kb tgj jartg arkuli tgj sul, wj gavj a fjap yjar tk ma`j tgj wgkfjnafjliar wkr`!!!

Afsk, >23 pfus 9 ds nfksj tk tgj Jartg yjar. (>29/>2:)133=44.8% annurany. Db ykunklsdijr tgat tgj Jartg, Mkkl ali ^ul bkrmji brkm a nfkui kb has ali iustulijr dts kwl hravdty kvjr pjrgaps edffdkls kb yjars, tgds ds qudtj al jxtrakridlaryarralhjmjlt tk gavj knnurrji.



Ngaptjr Bdvj? Aekut Ygj âha

Hada Gypktgjsds says tgj pgysdnaf aspjnts kb Jartg dltjrant wdtg

tgj edkfkhdnaf aspjnts kb Jartg tk madltadl sung a systjm tgat ta`jl

as a wgkfj ds Hada. Dt burtgjr tjffs us gumals arj a part kb Hada alidb gumals gavj ejnkmj eai bkr Hada dl skmj way, Hada wdff

jfdmdlatj gumals tk aioust tgj systjm bkr dts kwl hkki. D ejfdjvj,

brkm my wkr` dl ‖Ygj Jxndtdlh Ljptulj âha”, tgat tgj pfaljt

Ljptulj gkfis a nfuj as tk wgat gumals ljji tk ik tk nkultjr tgjdr

ljhatdvj jbbjnts kl Hada.

Ygj Jxndtdlh Ljptulj âha ey Dal Ejarisfjy idbbjrs brkm Ygj

^panj Kiyssjy sjrdjs kb Artgur N. Nfar`j dl tgat dt ds lkt aekut tgjpfaljt Oupdtjr ali tgj edrtg kb a ljw fdbj ejardlh star systjm dl kur

skfar systjm. Dt ds aekut tgj pfaljt Ljptulj ali gkw dt gkfis tgj

`jy tk tgj sunnjss kb gumaldty ali tgj pfaljt Jartg.

Xgjl D sgkw tgat tgj jartg kredts Oupdtjr as a hgkst (HjljsdsSrkojnt 31), tgds ds dltjrjstdlh ejnausj dl 5331 A ^panj Kiyssjy,Iavdi Ekwmal jlnkultjrs a hgkstfy rjprkiuntdkl kb tgj Jartg atOupdtjr, ejbkrj dt nklvjrts dltk a sul naffji Fundpgjr tk dts mkkl

Jurkpa. Oupdtjr ds kbtjl naffji a badfji star, kr badfji sul dl ktgjrwkris.

Ejnkmdlh a spanjbardlh ndvdfdzatdkl ds, dl my pkdlt kb vdjw, mkrj askndaf ali pkfdtdnaf prkefjm tgal a tjnglkfkhdnaf klj.

Db tgj pjkpfj wjrj tk hjt ejgdli tgjdr hkvjrlmjlts wdtg sungaspdratdkls, ali tgj hkvjrlmjlts wjrj tk usj rjskurnjs wdsjfywgdfj malahdlh tgdlhs bkr tgj wjff ejdlh kb tgj pjkpfj, tgj

tjnglkfkhy wkufi nkmj kb dts kwl annkri. Dt ds afrjaiy aivalndlhat astkldsgdlh ratjs.

Dt ds tgj purpksj kb tgds wkr` tk bdli rjaskl tk hk dltk spanj tgatnal ej prjsjltji tk skndjty, tk hjt tgjm ejgdli tgj hkaf kb tgjsjttfjmjlt kb tgj skfar systjm, ali jvjl ejgdli tgj hkaf kb hkdlhtk tgj stars, ektg kb wgdng wkufi gavj dmmjlsj rjturls.



Ygds wkr` afsk strdvjs tk bdli tgj sjnrjt ejgdli fdbj sustadldlh starsystjms sk as tk nrjatj, madltadl, ali rjouvjlatj tgjm. ^ung aljlijavkr nkufi ej naffji a ‖prkojnt hjljsds”.



Ngaptjr ^dx? Ygj Skdltdlh Kb Ygj Sfaljts

D gavj sgkwl gkw Oupdtjr ali âturl pkdlt tk tgj Jartg, ejnausj wdtg âturl

at aekut 4 uldts brkm tgj sul dl dts nfksjst apprkang, ali Oupdtjr at aekut bdvj

uldts wdtg dts nfksjst apprkang tk tgj sul, tgat puts tgj jartg at aekut klj uldt

brkm tgj sul, ali, tgat, burtgjr, ldlj-bdbtgs ds tgj ratdk kb tgj skfar raidus tk

tgj fular kredt kb tgj mkkl arkuli tgj jartg, as wjff as tgj ratdk kb tgj mkfar

mass kb hkfi tk tgat kb sdfvjr. Dt sgkufi afsk ej mjltdklji tgat 4 ali : arj

vkfumjs kb twk kb tgj bdvj rjhufar skfdis dl tjrms kb tgj rjhufar skfdi tgj

tjtragjirkl gavdlh a vkfumj kb klj dl Eun`mdlstjr Buffjr—s ^yljrhjtdns.

D tgjl sgkw tgat tgj jartg, dl turl, pkdlts tk Ljptulj tgrkuhg Rjlus ali

Mars dl tgj Ljptulj Jquatdkl. D wrktj?

Gj wrktj dl gds Okurlaf as bkffkws?

Xj gavj al jquatdkl bkr a sjqujlnj tgat sgkws tgj Jartg straiifji ejtwjjl Rjlus ali Mars. Rjlus ds a badfji Jartg. Mars prkmdsjs tk ej Ljw Jartg. Ygj jquatdkl, tgat D prjsjltji dl my wkr`, ‖YgjIknumjlt”, ds?

P(Rjlus-kredt)/(Jartg-kredt)]P(Jartg-mass)/(Mars-mass)]l „ (Marskredtaf #) = a[l

Ygj Mars kredtaf lumejr ds 9. Db wj walt tk `lkw wgat pfaljt dl tgjskfar systjm gkfis tgj `jy tk tgj sunnjss kb Jartg, kr tk tgj sunnjss kb gumals, wj fjt l => sdlnj tgj Jartg ds tgj tgdri pfaljt kut brkm tgjûl, dl tgj jquatdkl ali tgj rjsuft ds a[l = 18.2. Ygds mjals tgjpfaljt tgat gkfis tgj `jy ds Ljptulj. Dt gas a mass kb 18.5> jartgmassjs, a lumejr vjry nfksj tk kur 18.2. Ygj ljxt stjp ds tk alafyzjLjptulj, dts nkmpksdtdkl ali ktgjr bantkrs. Gj tgkuhgt gj must gavj

ejjl efdli. Lkt klfy was Ljptulj tgj dlidnatji pfaljt, gj bkuli dtgai ljarfy tgj samj surbanj hravdty kb jartg. Ygkuhg dt was mungmkrj massdvj tgal jartg, dt was mung farhjr ali tgjrjbkrj fjss ijlsj.

Ygat was wgy dt namj kut tk gavj tgj samj surbanj hravdty.

Ygj Ljptulj Jquatdkl ds ijrdvji ey nklsdijrdlh 4/:, wgdng ds 1.7. D wrktj tgj bkffkwdlh?

âfvkrjttd gai iklj dt. Ygj strunturj kb tgj skfar systjm ali iylamdnjfjmjlts kb tgj Zldvjrsj ali Laturj dl hjljraf wjrj tdji up dl tgj



twk sjqujlnjs?

:, 19, 5>, >5,…

ali

1.7, >.2, :.9, 8.5,…

Nkufi gj bdli tgj nklljntdkl ejtwjjl tgj twk tk fknafdzj tgj pdvktafpkdlt kb tgj skfar systjm; Tjs gj nkufi, gj tkk` tgjdr idbbjrjlnj,suetrantdlh rjspjntdvj tjrms dl tgj sjnkli sjqujlnj brkm tgksj dl tgjbdrst sjqujlnj tk ketadl tgj ljw sjqujlnj?

>.5, 13.9, 18.2, 59.7,…

Xgdng ds al ardtgmjtdn sjqujlnj wdtg nkmmkl idbbjrjlnj kb 8.5mjaldlh dt ds wrdttjl

8.5l „ 9 = a[l

Ygj a[l ds tgj ltg tjrm kb tgj sjqujlnj, l ds tgj lumejr kb tgj tjrmdl tgj sjqujlnj.

Ygds gj lktdnji nkufi ej wrdttjl?

P(Rjlus-kredt)/(Jartg-kredt)]P(Jartg-mass)/(Mars-mass)]l „ (Marskredtaf #) = a[l

Ygjrj was hkki rjaskl wrdtj tgjsj sjqujlnjs brkm wgat D wrktj gjrj?

Xj krdhdlaffy tgkuhgt tgat 1.7 (4/:) knnurrji dl tgj skfar systjm alilaturj, afklj. Dt ds tgj ratdk kb ^aturl kredt tk Oupdtjr kredt dl tgjdrnfksjst apprkangjs tk tgj sul. Dt ds tgj ratdk kb tgj skfar raidus tk tgjfular kredt. Dt ds ratdk kb tgj mkfar mass kb hkfi tk tgat kb sdfvjr. YgjlD idsnkvjrji tgat apprkxdmatjfy twdnj tgat vafuj (>.8) was rjnurrjlttgrkuhgkut tgj skfar systjm ali laturj. Ywdnj 1.7 ds >.2. D bkuli dtknnurrji dl Oupdtjr, ^aturl, Jartg, Mjrnury, ali Rjlus tk lamj a bjw as >.8. Lkw D gavj bkuli dt dl Ljptulj ali Zralus, prjndsjfy?

(ljptulj kredt)/(Zralus kredt)=

(>3.3:87)/(14.1714)=3.>84:8~3.>7~3.>8~3.>2



3.>8(jartg mass)/(mars mass)=3.>8(13)=>.8

Dt knnurs gjrj tk mj wj arj lkt oust ijafdlh wdtg 1.7 ali >.8, eut aly wgkfj lumejr muftdpfj kb 1.7. Xj arj ijafdlh wdtg? a[l=1.7l

Fjt us try tgrjj tdmjs 1.7. Dt ds :.9. Dmmjidatjfy wj sjj dt dl tgj mkstkevdkus, Oupdtjr. Dt ds tgj farhjst ali mkst massdvj pfaljt dl tgj skfarsystjm. Dts burtgjst idstalnj brkm tgj sul dl dts kredt ds :.9:: AZ ~:.9.

Xj sjj dt dl tgj bkffkwdlh

(saturl mass)/(ljptulj mass)=(4:.198)/(18.5>)=:.:55~:.9

D gavj afrjaiy lktji tgat ^aturl ali Ljptulj gavj sdmdfar

dlnfdlatdkls tk tgjdr kredt tgat arj klfy nkmmkl tk Jartg ali Mars.

Ygus, wj arj ijafdlh gjrj wdtg tgj sjqujlnj?

1.7, >.2, :.9, 8.5,…

Burtgjrmkrj0

P(vjlus kredt)/(jartg kredt)]P(jartg mass)/(mars mass)] = 8.5

^kmj kb tgj mkst dltjrjstdlh knnurrjlnjs kb >.8 dl laturj arj?

P(vkfumj kb saturl)/(vkfumj kb Oupdtjr)](vkfumj kb mars)

=(mjrnury raidus)(jartg raidus)\5

=P(vjlus kredt)/(jartg kredt)(jartg raidus)]\>

=P(mjrnury kredt)/(jartg kredt)](jartg raidus)\>

=3.>8 nuedn jartg raidd

3.>8 nal ej nklvjrtji tk >.8 ey muftdpfydlh dt ey tgj ratdk kb jartg mass tk

mars mass ejnausj dt ds nfksj tk tjl.

(jartg raidus)/(mkkl raidus)=

9(ijhrjjs dl a ndrnfj)(mkkl idstalnj)/(sul idstalnj)



= >.8

Ygjrj arj aekut as maly iays dl a yjar as ijhrjjs dl a ndrnfj.

Xgjl D say tgat tgj jartg pkdlts tk Ljptulj tgrkuhg mars ali Rjlus, wj

mdhgt as wjff say dt pkdlts tk Ljptulj ali Zralus, ejnausj D gavj bkuli

tgjy arj nkupfji, wgdng nal ej sjjl ey tgat wgdng D wrktj gjrj?

D nafnufatj tgat tgkuhg Ljptulj ds mkrj massdvj tgal Zralus, dts vkfumj dsfjss sung tgat tgjdr prkiunts arj nfksj tk jqudvafjlt. Dl matg?

L[v = vkfumj kb LjptuljL[m = mass kb Ljptulj

Z[v = vkfumj kb ZralusZ[m = mass kb Zralus

(L[v)(L[m) = (Z[v)(Z[m)

Ygds ds apprkprdatj, ejnausj tgjsj twk pfaljts arj sdmdfar dl sdzj, mass, alinkmpksdtdkl ali arj lkt oust has hdalts fd`j ^aturl ali Oupdtjr, eut arj afsktgj twk dnj hdalts.



D lkw sdt gjrj rjaidlh tgj sndjlnj bdntdkl wkr` Sfutklda. Laturaffy, D ejhdl tktgdl` aekut tgj pfaljt Sfutk, kr wgat skmj astrklkmjrs tkiay nklsdijr apfaljt, ali tgj klfy spgjrj D gavj yjt tk nklsdijr. Gkwjvjr, rdhgt kbb tgj eatds dt lkt kevdkus tgat Ljptulj ali Zralus pkdlt tk Sfutk db yku `lkw tgjgdstkry kb Sfutk;

Dt ds tgds? Ygj Astrklkmjr, Zreadl Rjrrdjr dl tgj 1793—s prjidntji tgjjxdstjlnj kb Ljptulj ejnausj kb pjrtureatdkls dl tgj Kredt kb Zralus. Mkrjkesjrvatdkls nkmdlh fjai tk tgj dija tgat Ljptulj afklj nkufi lkt ej tgjnausj. A ldltg pfaljt was sjarngji bkr, ali bdlaffy nklbdrmji dl 14>3.

Ygds nkmpfjtjs a okurljy tgrkuhg tgj skfar systjm tgat tjffs a stkry wdtgmjaldlh, eut ds klfy tgj ejhdlldlh kb kur dlvjstdhatdkls, ejnausj tgj snd-bdekk` Sfutklda, ey Rfaidmdr Kerungjv ds aekut al arntdn jxpjidtdkl wgjrjsndjltdsts bdli dl tgj Arntdn al kpjldlh dl tgj jartg tk a warm, fusg wkrfiwarmji ey a eaff kb gjat at tgj njltjr kb tgj jartg naffji Sfutk. D `lkw kb

tgksj wgk ejfdjvj tgat tgj buturj bkr gumaldty ds lkt dl spanj, eut dlskmjtgdlh tgjy naff ‖prkojnt hjljsds” brkm a star trj` mkvdj, wgjrjdl fusg,sustadlaefj jnksystjms arj nrjatji ejljatg tgj jartg ey gkffkwdlh dt kut dlpfanjs. Xgjtgjr kr lkt sung a tgdlh ds bjasdefj, ikjs lkt mattjr tk mj. D bdlimysjfb at prkojnt hjljsds dl tgj sjlsj tgat wj try tk ulijrstali gkw tk nrjatjali madltadl gaedtaefj jnksystjms dlijbdldtjfy.

Dal EjarisfjyAprdf 55, 5311



Ngaptjr ^jvjl? Ygj Idsnkvjrdjs Srjsjltji ^k Bar

>.8 rjnurrjlt tgrkuhgkut laturj

vkfumj[kb [ saturlvkfumj[kb [ oupdtjr

(vkfumj[kb [mars) =

(mjrnury [ raidus)(jartg[ raidus)5=

P(vjlus[kredt /jartg [kredt )(jartg [ raidus)]>=

(mjrnury [kredt /jartg [kredt )(jartg [ raidus)>=

3.>8 nuedn jartg raidd

jartg[ raidus

mkkl [ raidus= 9(ijh rjjs[ dl[ndrnfj)

mkkl [ idstalnj

sul [ idstalnj=

>.8

F = M >.:

F=133 skfar fumdlksdtdjs ds a zjrk mahldtuij star

133 = M >.:

fkh 133 = >.: fkh M

5/>.: =fkh M

3.:8 = fkh M

M = >.8 skfar massjs

A5

lS=

^aturl [kredt

Oupdtjr[kredt =

4

:

l = 9 = mars kredt lumejr



A = arja kb a > ey 9 rjntalhfj

Lumejr kb > ey 9 rjntalhfjs dl a 2 ey 7 rjntalhfj = l = 9S=pjrdmjtjr dlsnrdeji dl a 2 ey 7 rjntalhfj = 53

(>,9,:) ds bdrst Sytgahkrjal trdpfjt

mkfar[mass[kb [ adr

mkfar[mass[kb [watjr"

gumal [ekiy [ tjmjraturj

brjjzdlh[ tjmpjraturj[kb [watjr=

Au

Ah=

skfar [ raidus

jartg #mkkl [ idstalnj=

4

:

3.>8 nal ej nklvjrtji tk >.8 ey muftdpfydlh dt ey tgj ratdk kb jartg mass tk mars mass

ejnausj dt ds nfksj tk tjl.



Ngaptjr Jdhgt? Gkw D Maij My Idsnkvjrdjs

Gkw idi D ijtjrmdlj tgj jquatdkl?

P(vkfumj kb saturl)/(vkfumj kb Oupdtjr)](vkfumj kb mars)

=(mjrnury raidus)(jartg raidus)\5

=P(vjlus kredt)/(jartg kredt)(jartg raidus)]\>

=P(mjrnury kredt)/(jartg kredt)](jartg raidus)\>

=3.>8 nuedn jartg raidd

Xgjrj 3.>8 nal ej nklvjrtji tk >.8 ey muftdpfydlh dt ey tgj ratdk

kb jartg mass tk mars mass ejnausj dt ds nfksj tk tjl.

Dt bjff kl my fap ey nklsdijrdlh bkffkwdlh passahjs D wrktj dl

5338?

Ldlj-bdbtgs (4/:) ds jmekidji ey tgj bamkus Sytgahkrjaltgjkrjm. Dt ds dl tgj bdrst Sytgahkrjal trdpfjt (>,9,:). YgjSytgahkrjal tgjkrjm, wgdng statjs tgj rjfatdklsgdp ejtwjjltwk sdijs kb a rdhgt trdalhfj wdtg dts gypktjlusj ds usji tknafnufatj tgj idstalnj ejtwjjl twk pkdlts dl a pfalj, kr spanj, hdvjl tgjdr nkkridlatjs dl a rjntalhufar nkkridlatjsystjm. Dt statjs tgat klj sdij squarji pfus tgj ktgjr sdijsquarji jquafs tgj gypktjlusj squarji. (>,9,:) ds tgj bdrst

trdpfjt tgat satdsbdjs tgds rjfatdklsgdp. Ygat ds >, squarji pfus9 squarji jquafs : squarji. > squarji ds kur 4, 9 squarjids 12, ali kur : ds squarji tk ma`j 5:. (>\5+9\5=:\5). Ldljds afsk tgj bdvj pfus tgj bkur. Ygds affkws us tk asskndatj alalhfj wdtg ldlj-bdbtgs, ey way kb tgj trdalhfj tgj trdpfjt



ijsnrdejs, ali, tk bdli a hjkmjtrdn, kr pdntkrdaf sdhldbdnalnjkb tgj ratdk.

Ygj alndjlt Jhyptdals, asskndatji mung sdhldbdnalnj wdtg

tgds trdpfjt, nklstruntdlh tgj @dlh—s ngamejr dl tgj HrjatSyramdi wdtg tgjsj prkpkrtdkls.

Hjkmjtrdn ^dhldbdnalnj Kb ldlj-bdbtgs, tgj nksmdn jquatdkl.

Ey tgj Sytgahkrjal tgjkrjm

Xj gavj x\5+y\5=g\5

Ygj bdrst Sytgahkrjal trdpfjt ds (>,9,:)

Ali x=9, y=>, g=:

Db

A= arja kb > ey 9 rjntalhfj

S= pjrdmjtjr dlsnrdeji dl 2 ey 7 rjntalhfj = 53

Ali, l= lumejr kb > ey 9 rjntalhfjs dl a 2 ey 7 rjntalhfj.

Ygjl,

(A\5)/lS = 4/:

Ygat wgdng wj gavj gjrj ds a nksmdn jquatdkl lkt oust bkr tgj skfar systjm, eut tgat gkfis bkr tgj atkmdn wkrfi aliedkfkhdnaf dltrdnandjs kb fdbj. Db wj dmahdlj tgj skfar systjmbkrmji brkm prktkpfaljtary idsn arkuli sul, ekidjs bkrmdlhbrkm partdnfjs ali tgjl gjridlh smaffjr partdnfjs tk ma`jidstrdeutdkl kb pfaljts as wj `lkw tgjm tkiay, D tgdl` tgds



jquatdkl gkfi bkr jvkfutdklary iylamdns kb skfar systjm alilunfjar ngjmdstry idntatji ey stars tk ma`j jfjmjlts. Ygdspdnturj kutfdlji aekvj wdff ydjfi mung iylamdns upklalafysds.

A\5=arja squarji=S*S*S*S=S\9

S=pjrdmjtjr

Lumejr = l

Ygjl sdlnj A\5/S= S\>= vkfumj

Rkfumj/l=4/:

âturl kredt (nfksjst apprkang tk tgj sul) =4 astrklkmdnafuldtsOupdtjr kredt (nfksjst apprkang tk tgj sul)= : astrklkmdnafuldts

9tg pfaljt = mars

Xj gavj?

Rkfumj = (âturl/Oupdtjr)Mars

âturl ali Oupdtjr arj tgj mkst massdvj pfaljts dl tgj skfar systjm. Mars ds tgj pfaljt tgat nal ej nkfkldzji ey gumals.

D gavj sgkwl tgat tgjy rjprjsjlt tgj sanrji vkfumj.

Aprdf 2 5333 tgjsj tgrjj pfaljts usgjrji dl tgj sjnkli mdffjlldum

bkrmdlh a ndrnfj ey tgj mkkl, 4 ijhrjjs anrkss.



Nafnufatdkl kb ânrji Rkfumj

J[r=1= Jartg raidus

M[r=3.:>5= Mars raidus

O[r= 11.58= Oupdtjr raidus

^[r=4.99= âturl raidus

R = 9/> (pd)r\>

= 9/>(>.1)(11.58)\>= :442

=9/>(>.1)(4.99)\>= >:59

=9/>((>.1)(3.:>5)\>= 3.2>38

Rkfumj = (âturl/Oupdtjr)mars

= >:59/:442(3.2>38) = 3.>8 nuedn jartg raidd

Apprkxdmatjfy tgj samj lumejr as mjrnury raidus ali mjrnury

idstalnj brkm sul dl jartg uldts (Jartg raidd ali Astrklkmdnaf

Zldts).



^anrji Rkfumj Alafyzji, ^anrji Fjlhtg Ketadlji, Iylamdn

^trunturj Kb ^kfar ^ystjm Ujvjafji.

D nafnufatji tgj sanrji vkfumj tk ej 3.>8 nuedn jartg raidd, wgdng D

gavj lktdnji nkdlndijs wdtg mjrnury idstalnj brkm tgj sul, alimjrnury raidus dl jartg uldts. ^dlnj vkfumj ds dl nuedn uldt

fjlhtgs, ali raidd ali idstalnjs dl jartg raidd ali jartg-sul

idstalnjs, wj ta`j tgj nuej rkkt kb kur sanrji vkfumj tk hjt tgj

fjlhtg kb klj sdij kb a nuej wdtg tgj sanrji vkfumj. Dt ds 3.85 jartg

raidd. Xj wdff naff dt tgj sanrji fjlhtg, dt ds tgj vjlus-sul idstalnj

dl jartg uldts (astrklkmdnaf uldts), wgjrj vjlus ds tgj pfaljt abtjr

mjrnury, kr tgj sjnkli pfaljt.



ûmmary kb ânrji Rkfumj

D gavj bkuli ldlj-bdbtgs (4/:), wgdng ds jquaf tk 1.7 dl tgj mkst

sanrji aspjnts tk mal, hkfi, sdfvjr, mkkl, sul, watjr, adr, gumal

ekiy tjmpjraturj, ali gavj gavj ijrdvji brkm tgds sanrji ratdk, krnksmdn lumejr, tgj nksmdn vkfumj ey jxprjssdlh dt hjkmjtrdnaffy.

Ygat sanrji vkfumj ds?

(âturl/Oupdtjr)mars=3.>8 nuedn jartg raidd

Xgjrj âturl ali Oupdtjr arj tgj mkst massdvj pfaljts dl tgj skfar

systjm, ali Mars ds tgj pfaljt tgat nal ej nkfkldzji ey gumals.

Ya`dlh tgj nuej rkkt kb 3.>8 hdvjs tgj sanrji fjlhtg, dt ds jquaf tktgj idstalnj kb tgj pfaljt Rjlus brkm tgj sul dl jartg uldts

(astrklkmdnaf uldts), wgdng ds tgj sjnkli pfaljt (3.85

Astrklkmdnaf Zldts). Ygj 3.>8 ds tgj mjrnury-sul idstalnj dl jartg

uldts, ali tgj mjrnury raidus dl jartg uldts. Mjrnury ds tgj bdrst

pfaljt.

âturl ds at 4 jartg uldts (astrklkmdnaf uldts) dl dts nfksjst

apprkang tk tgj sul, Oupdtjr at bdvj.



Ngaptjr Ldlj? Ywk ^tkrdjs Ygat Xalt Yk Ej Ykfi



^tkry 1

Ywk erktgjrs, klj al afngjmdst tgj ktgjr al astrklkmjr, mjt bkr

fulng. Ygj wadtjr sadi tk tgjm, ‖Udhgt tgds way, a taefj bkr twk.”

^yiljy, tgj astrklkmjr, sadi ‖Ygal` yku”, Uafpg, tgj afngjmdst

sadi tk ^yiljy, ‖Tku ikl—t mdli db D sdt ey tgj wdlikw.”

^yiljy sadi, ‖Lkt at aff, abtjr aff yku gavj afways sadi D am dl tgj

iar`.”

Uafpg alswjrji, ‖Tjs, aff kb tgat rjaskl, ali s`jptdndsm, fjavjs lk

rkkm bkr dltudtdkl.”

Ygjy ektg sat ikwl.

Uafpg sadi, ‖^yiljy, D gavj ejjl wkr`dlh kl a ljw prkojnt ali

gavj gai tk ik a fdttfj astrklkmdnaf rjsjarng tk ijvjfkp dt. As yku

`lkw, D ulijrstali tgj prdlndpfjs kb ngjmdstry, as al afngjmdst,

eut tgj idbbjrjlnj ds D am lkt ikdlh sndjlnj wdtg dt, eut am hkdlh

tgrkuhg a prknjss kb spdrdtuaf ijvjfkpmjlt wgjl D mdx ngjmdnafs,yku kl tgj ktgjr gali ik jxpjrdmjlts tk eudfi a ekiy kb

`lkwfjihj tgat mdhgt ej usji tk suppkrt a gypktgjsds aekut

laturj.”

^yiljy alswjrji ‖Tjs, ali as al astrklkmjr D ulijrstali

ngjmdstry, wgy ikl—t yku as al afngjmdst, stuiy astrkfkhy;”

Uafpg sadi, ‖D gavj nklsdijrji dt, lk ejttjr pfanj tk start tgal tk

start pdn`dlh ykur eradl aekut tgj gjavjls.”

^yiljy alswjrji, ‖K.@. Xgat ik yku walt tk `lkw;”



Uafpg put tgj qujstdkl tk gdm, ‖Hafaxdjs arj nkffjntdkls kb stars

tgat bkrmji kut kb hdalt has nfkuis, ali pfaljts arj spgjrdnaf

massjs tgat bkrm arkuli stars. Ygj stars arj vjry bar apart, ali tgjpfaljts kredt nfksj dltk tgj stars, tgj stars arj jxtrjmjfy bar apart,

eut tgj stars ali tgjdr pfaljts arj lkt, ds tgat rdhgt;

^yiljy sadi, ‖Tjs, mkrj kr fjss.”

Uafpg nkltdluji, ‖^k tgj pfaljts kredtdlh tgj ^ul, wgdng ds klj

star amdi tgj edffdkls dl tgj hafaxy, nkufi gavj lk nklljntdkl wdtg

tgj ktgjr stars;”

^yiljy sadi, ‖Ygat ds rdhgt.”

Uafpg wjlt kl, ‖Ygjl ikl—t yku bdli dt bully tgat tgj nfksjst star

tk us, ds tgj tgdri erdhgtjst star dl tgj s`y ali tgj jartg ds tgj tgdri

pfaljt;”

^yiljy alswjrji, ‖Dt must ej a nkdlndijlnj, tgjy gappjl aff tgjtdmj dl sndjlnj.”

Uafpg sadi, ‖Ygjl fjt us hk kl wdtg tgds rjaskldlh. Oupdtjr ds tgj

bdbtg pfaljt brkm tgj sul ali tgj farhjst, mkst massdvj pfaljt, ali,

tgj star ^drdus ds tgj bdbtg ljarjst star, ali afsk tgj erdhgtjst star dl

tgj s`y.”

^yiljy sadi, ‖Ygat ds dltjrjstdlh, D idil—t `lkw tgat.”

Uafpg nkltdluji, ‖Lkw, D am hkdlh tk hjt astrkfkhdnaf. Ygj

nklstjffatdkl Ekktjs ds `lkwl as ‖tgj ekatmal”, ali tgj erdhgtjst

star dl tgat nklstjffatdkl ds Arnturus, wgdng gappjls tk ej tgj

bkurtg erdhgtjst star dl tgj s`y. Ygj bkurtg pfaljt ds Mars.”



^yiljy sadi, ‖Tjs, ali sk;”

Uafpg alswjrji, ‖Xjff, Mars ds tgj pfaljt wj nal tjrrabkrm, tgat ds

ma`j gaedtaefj bkr gumal fdbj ejnausj tgj dlljr pfaljts arj tkk

nfksj tk tgj sul ma`dlh tgjm tkk gkt, ali tgj kutjr pfaljts ejykliMars arjl—t jltdrjfy skfdi, tgjy arj has hdalts, sk wj must, dt wkufi

sjjm, sjt sadf upkl tgj nksmdn knjal bkr Mars.”

^yiljy sadi, ‖Xjff tgat aff skulis vjry mystdnaf, tgkuhg fdvdlh kl

a pfaljt kb fdmdtji rjskurnjs, ali dlnrjasdlh pkpufatdkl, wj must

klj iay nkfkldzj Mars.”

Uafpg sadi, ‖D tgkuhgt yku mdhgt say skmjtgdlh fd`j tgat, ali Dnkufil—t ahrjj mkrj dl my kwl way, eut D gavj bkuli ngjmdnaf

dlidnatdkls tgat wj must gjai bkr Mars. Dt ds as db dt was put tgjrj

bkr us tk skfvj tgj prkefjms wj wkufi jvjltuaffy jlnkultjr.”

^yiljy as`ji, ‖Dl wgat way;”

Uafpg alswjrji, ‖Xjff tgj idbbjrjlt jfjmjlts arj maij kut kb

varydlh lumejrs kb dijltdnaf partdnfjs, tgj lumejr ijtjrmdljs tgjjfjmjlt, ali tgj idbbjrjlt jfjmjlts nal ej nkmedlji dl idbbjrjlt

ways tk ma`j idbbjrjlt nkmpkulis, fd`j tgj jfjmjlts skidum ali

ngfkrdlj nkmedlj tk ma`j saft.”

^yiljy alswjrji, ‖Tjs, ali…”

Uafpg sadi, ‖…Ali wj nal asskndatj wdtg aly jfjmjlt a rjfatdvj

mass ey nkmpardlh tgj samj lumejr kb atkms tgat nklstdtutj tgat

jfjmjlt. Dt ds naffji tgj mkfar mass.”

^yiljy sadi, ‖Udhgt.”



Uafpg wjlt kl, ‖Gjrj ds wgat D gavj bkuli. Jartg adr ds aekut 87%

ldtrkhjl ali 51% kxyhjl. Xj nal trjat ektg as idatkmdn

mkfjnufjs. Ygj mkfar mass kb ldtrkhjl has (L5) ds 5(19.31) alitgj mkfar mass kb kxyhjl has (K5) ds 5(12.33). ^k,” (ali gj wrktj

kl a lap`dl)?

Jartg Adr = 5P(12.33)(3.51) + (19.31)(3.87)] = 57.:8:2

‖Ygj martdal atmkspgjrj ds mkstfy narekl idkxdij (NK5). Ygds

hdvjs?”

NK5 = (15.31) + 5(12.33) = 99.31

‖Ygj avjrahj kredtaf idstalnj kb mars brkm tgj sul nkmparji tk

tgj avjrahj kredtaf idstalnj kb tgj jartg brkm tgj sul ds 1.: ali

99.31 idvdiji ey 57.:8:2 ds 1.:, kr >/5 dl ktgjr wkris. Ds wgat tgds

says ds tgat, tgj mars-sul idstalnj ds tk tgj jartg-sul idstalnj as

martdal adr ds tk jartg adr.”

^yiljy sadi, ‖Dltjrjstdlh, eut a fdttfj mystdnaf bkr mj.”

Uafpg sadi, ‖Mayej dt wdff skuli a fdttfj fjss mystdnaf db D sgkw yku

mkrj.”

^yiljy sadi, ‖Hk kl, D wkufi fd`j tk sjj dt.”

Uafpg sadi, ‖K.@., gjrj ds wgat D gavj hkt. Ygj jartg atmkspgjrj

was klnj mkstfy narekl idkxdij (NK5) fd`j tgj mars atmkspgjrj ds

tkiay, ultdf pfalt fdbj namj afklh ali startji nklvjrtdlh tgj narekl

idkxdij dltk kxyhjl, usdlh fdhgt brkm tgj sul ali dl tgj prknjss

ma`dlh tgj mkst buliamjltaf suhar, hfunksj (wgdng ds at tgj

ekttkm kb tgj bkki ngadl). A prknjss naffji pgktksyltgjsds.”



^yiljy alswjrji, ‖K.@., D am lk edkfkhdst, eut dt skulis hkki tk

mj.”

Uafpg nkltdluji, ‖Xjff hfunksj ds N2G15K2, wgdng hdvjs a mkfar

mass kb,” (ali gj wrktj kl tgj lap`dl)?

N2G15K2 = 2(15.31) + 15(1.337) + 2(12.33) = 173.12

‖Ali watjr (G5K) nkvjrs mkst kb jartg ali ds jssjltdaf tk fdbj, dts

mkfar mass ds?”

G5K = 5(1.337) + (12.33) = 17.312

‖173.12 idvdiji ey 17.312 ds aekut 13 ali tgj jartg ds aekut 13

tdmjs as massdvj as mars. Ygat ds, hfunksj ds tk watjr as tgj jartg ds

tk mars.”

^yiljy sadi, ‖D ikl—t `lkw, tgjrj gas tk ej skmj skrt kb

jxpfalatdkl, tgj bkrmatdkl kb tgj skfar systjm ds nkmpfdnatji ali

gdhgfy tgjkrjtdnaf. Ygjsj rjfatdvj massjs skrtdlh tgjmsjfvjs kut

fd`j tgds must gavj skmjtgdlh tk ik wdtg tgj iylamdns kb nklijlsatdkl ali alhufar mkmjltum. Ygdl` kb a eun`jt kb stkljs,

pjeefjs, ali sali ejdlh swulh dl a ndrnfj ey a rkpj tdji tk tgj

galifj. Ygj njltrdpjtaf bkrnj sjlis tgj mkrj massdvj stkljs tk tgj

ekttkm kb tgj eun`jt, tgj fdhgtjr sali rjsts kl tkp, wdtg tgj pjeefjs

dl ejtwjjl.”

Uafpg erk`j dl, ‖Kr gumal ijstdly ds tdji up dl tgj strunturj kb

laturj.”



^tkry 5

‖Gd Bral`, tgal`s bkr ma`dlh dt ikwl gjrj, D rjaffy apprjndatj yku

puffdlh away brkm ykur wkr` at tgj kesjrvatkry. D am sk hfai yku

nkufi hjt ikwl gjrj sk qudn`, wj rjaffy ljji al astrklkmjr, ali Dtgdl` wgat D am hkdlh tk sgkw yku wdff gavj maij ykur vdsdt

wkrtgwgdfj.”

Bral` as`ji, ‖Xgat ik yku gavj Oalj;”

^gj ejhal, ‖Xj uljartgji tgjsj wrdtdlhs kl nfay taefjts dl a idh dl

Jastjrl Yur`jy. D gavj oust bdldsgji tralsfatdlh dt brkm tgj alndjlt

Aramadn dt was wrdttjl dl, wj tgdl` dt tk ej skmj 5,:33 tk >333yjars kfi. Dt sjjms tk ej astrkfkhdnaf, D gkpj yku erkuhgt afklh klj

kb ykur galiekk`s kb astrklkmy ali astrkpgysdns wdtg yku fd`j D

as`ji.”

Bral` sadi, ‖Tjs D idi, nal D sjj tgj malusnrdpt;”

Oalj sadi, ‖Tjs, kb nkursj, gjrj dt ds.”

Bral` tkk` tgj tralsfatdkl ali ejhal rjaidlh. Dt rjai?

D am ^afhkl, trdeaf astrkfkhjr bkr tgj smaff trdej wgk naff

tgjmsjfvjs Nalad, ali rjsdij dl tgj vaffjy wj naff Gatgaf. Dt ds my

oke tk prkvdij spdrdtuaf ali ahrdnufturaf hudialnj tk my pjkpfj wgk

lumejr 8>5 tgds yjar, bkr tgj ejljbdt kb tgjdr fdbj abtjr tgds wkrfi

ali bkr tgjdr prkspjrdty dl prjsjlt tdmjs. Ygds dlvkfvjs usdlh my

`lkwfjihj kb tgj gjavjls, as wj ejfdjvj kur ijstdldjs nal ej rjai

dl tgj stars, ali wj `lkw tgj sjaskls arj rjfatji tk tgj pksdtdkls kb

tgj stars, sul, mkkl ali pfaljts.

D gavj ta`jl dt upkl mysjfb tk prjidnt tgj sdzjs kb tgj mkkl ali

jartg, tgjdr sjparatdkl, ali, kurs wdtg tgj sul, hdvjl wgat D ejfdjvj

tgjy must ej brkm my nklljntdkl wdtg idvdlj ijsdhl. Ygat ds tgdlhs



must ej a njrtadl way hdvjl tgj nrjatkr wkufi klfy ma`j laturj

annkridlh tgat, wgdng ds idvdlj.

Al jxampfj kb tgds ds tgj mkkl nal rdsj dl tgj jast buff klj ldhgt,

idmdldsg dl sdzj jang ldhgt rdsdlh fatjr, ultdf dt idsappjars, ali tgjl starts hrkwdlh ultdf dt ds buff ahadl, dl nynfjs tgat fast aekut

tgdrty iays. Kl tgj ktgjr gali as tgj sjaskls hk brkm nkfi, tk

warmjr, tgjl tk gkt, ali nkkfjr ahadl ali ean` tk nkfi, tgj sul

mkvjs anrkss tgj s`y brkm skutg tk lkrtg, ali ean` tk skutg ahadl,

dl a nynfj tgat fasts aekut >2: iays.

Ygj >2:-iay nynfj kb tgj sul nkrrjspklis nfksjfy tk 15 kb tgj >3-

iay nynfjs kb tgj mkkl. Ywjfvj ds tk my pjkpfj tgj mkst dmpkrtalt lumejr, dt ds 5 sdx tdmjs, > bkur tdmjs.

D lkw ejhdl my prjidntdkl kb tgj mkkl, jartg ali sul. Ygjy arj

spgjrjs wgksj idstalnj brkm tgjdr njltjrs tk jihjs wdff ej

nklsdijrji, at fjast dl tgj nasj kb jartg ali mkkl. Ygj mkst idvdlj

jxprjssdkl tgat nal ej bkrmji ds a prkpkrtdkl, wgjrj a prkpkrtdkl

ds tgj statjmjlt tgat klj qualtdty ds tk a sjnkli qualtdty as a tgdri

qualtdty ds tk a bkurtg.

D wdff tgjrjbkrj say tgj jartg raidus ds tk tgj mkkl raidus, as tgj

mkkl idstalnj ds tk tgj sul idstalnj, tgj fattjr ratdk dlnrjasji ey

>23 bkur tdmjs.

Xj dlnrjasj ey >23 bkur tdmjs ejnausj wj idvdij tgj ndrnfj dltk

>23 uldts as tgksj arj 23 sdx tdmjs, 23 ejdlh tgj mjasurj dl

sjparatdkl ejtwjjl fdljs kb a trdalhfj kb aff sdijs tgj samj, ali

>23 ejdlh 43 bkur tdmjs, 43 ejdlh tgj sjparatdkl ejtwjjl fdljs dl

tgj squarj wgdng gas aff sdijs tgj samj, as wjff as >23 ds nfksj tk

tgj >2: iay nynfj kb tgj sul.



D say bkur tdmjs >23, ejnausj bkur tdmjs >2: ali klj part dl bkur ds

jxantfy tgj iays kb 9 nynfjs kb tgj sul brkm skutg tk lkrtg ali ean`

skutg ahadl.

D burtgjr wrdtj tgds dl tgj ejfdjb tgat tgj antuaf sdzjs ali idstalnjskb tgj mkkl ali sul wdff klj iay ej `lkwl, ali tgjrjey tgj buturj

wdff vjrdby wgat D gavj ijrdvji easji kl gkw tgdlhs must ej wgjl

tgj gali kb tgj nrjatkr ds at wkr`.

Bral` bdldsgji rjaidlh ali sadi tk Oalj, ‖Xjff Oalj, D wdff wrdtj

ikwl dl tgj bkrm kb a matgjmatdnaf jquatdkl, wgat tgj malusnrdpt

says dl wkris, ali pfuh dl tgj lumejrs. Dt ds?”

(Jartg Uaidus)/(Mkkl Uaidus) = >23(9)(Mkkl Idstalnj)/(^ul

Idstalnj)

‖Ygj jartg raidus tk mkkl raidus ds 11/> ali >23 tdmjs 9 ds klj

tgkusali bkur gulirji ali bkrty. Ygj mkkl idstalnj tk tgj sul

idstalnj ds 15/928:, sk”

(11/>) = (1993)(15/928:) = >.8 = (18573/928:) =>.8

‖Gds rjfatdklsgdp gkfis pjrbjntfy, db yku rkuli tgj lumejrs tk tgj

bdrst pfanj abtjr tgj ijndmaf, wgdng ds rjaffy qudtj rjmar`aefj. Ygj

wrdtjrs rjvjrjlnj bkr laturj sjjms tk ej ijrdvji brkm sdmpfj

sgapjs ali easdn prdlndpfjs as wjrj tk tgj jxtjlt kb wgat was

`lkwl sk fklh ahk, wgjl tgj trutg ds tgat tgj krdhdl kb tgj sdzj kb

tgj mkkl ali jartg, tgjdr sjparatdkl ali kurs wdtg tgj sul, arj tdji

up dl lumjrkus dlvkfvji bantkrs. Ygj mkkl may gavj ejjl a ekiy

napturji ey jartg hravdty, kr, may gavj bkrmji as a ekiy tgat

erk`j kbb tgj jartg jarfy dl dts bkrmatdkl wgjl dt was fjss skfdi.

Ygj kredtaf idstalnjs kb tgj jartg ali mkkl gavj sk maly bantkrs

dlvkfvji, tgj mjngaldns arj sk nkmpfdnatji tgat jvjl kur mkst

skpgdstdnatji matg nallkt prjidnt tgjdr vafujs, jvjl rjaffy ej

mkijfji annuratjfy wdtg nkmputjrs.”



Oalj sadi, ‖Gds rjvjrjlnj bkr tgj lumejr 15 bkr tgj rjaskls gj sadi,

wgat nal yku say aekut tgat;”

Bral` alswjrji, ‖Gj sjjms tk ej gavdlh al dl`fdlh tk tgj bant tgat

15 ds tgj smaffjst aeulialt lumejr. Al aeulialt lumejr ds alumejr wgksj idvdskrs, jxnjpt bkr dtsjfb, aii up tk a vafuj hrjatjr

tgal dt. Ygat ds 15 ds idvdsdefj ey 1, 5, >, 9, ali 2 a wgkfj lumejr

kb tdmjs ali 1+5+>+9+2 = 12 wgdng ds hrjatjr tgal 15.”

Oalj sadi, ‖Xgatjvjr tgj nasj, tgj dmpfdnatdkls kb tgds idsnkvjry

arj mkrj tgal oust dltjrjstdlh, tgjy gavj ijjp pgdfkskpgdnaf

dmpfdnatdkls, brkm wgat yku gavj jxpfadlji tk mj ali D am hkdlh

tk gavj a fkt kb bul trydlh tk bdhurj kut gkw tk prjsjlt tgds.”

Bral` sadi, ‖Db alytgdlh, dt sgkws yku nal wkr` ean`waris ali hjt

skmjwgjrj, tgat ds gkw D wkufi apprkang tgds. Dl ktgjr wkris, tgj

uldvjrsj mjals skmjtgdlh, sk fjts ma`j tgj mkijf kl tgat, ali

vjrdby dt abtjrwaris. Gj sdmpfy attrdeutjs laturj ds jfjhalt tk a

nrjatkr, wgjl rjaffy jfjhalnj may sdmpfy ej wgat wkr`s, tgrkuhg

jvkfutdkl. Ygat ds db dt ikjsl—t wkr`, dt ikjsl—t survdvj, wgat ds fjbt

ds wgat wkr`s, ali dt may ej tgat wgat wkr`s ds wgat wj sjj asjfjhalt.”



Ngaptjr 13? Ngkksdlh Klj ^tar Kut kb Ygj Edffdkls

Fjklari `ljw Afpga Njltaurd was tgj nfksjst star systjm tk jartg, ali tgat dt was a trdpfj

systjm skmj 9.5 fdhgt yjars away. Gj afsk `ljw tgj prdmary, Afpga Njltaurd A, was a

H5R madl sjqujlnj star, wgdng ma`js dt tgj samj `dli kb star as tgj ^ul0 sdmdfar dl sdzj,

fumdlksdty, ali tjmpjraturj. Gj lktji tgat as sjjl brkm tgj Jartg, tgj star gai alapparjlt mahldtuij kb zjrk. Gj gai bkuli tgat stars kb aeskfutj mahldtuij zjrk kl tgj

madl sjqujlnj, wgdng arj efuj ali gkttjr tgal tgj sul, wjrj dmpkrtalt.

Gj `ljw tgat tgj sul gai al aeskfutj mahldtuij kb 9.7>, wgdng was dts erdhgtljss brkmtjl parsjns. Gj `ljw a parsjn was dmpkrtalt ejnausj dt was tgj paraffax kb al keojnt

hdvjl ey tgj jartg-sul sjparatdkl. Dl aiidtdkl, gj `ljw 13 was dmpkrtalt, ejnausj

gumals gavj 13 bdlhjrs, ali tgat prkeaefy gai skmjtgdlh tk ik wdtg tgj easj 13 nkultdlh

systjm. Ahadl, gj ejhal tk sjj tgj dmpkrtalnj dl madl sjqujlnj stars kb apparjltmahldtuij zjrk. Ygjrjbkrj, gj as`ji, gkw bar sgkufi a zjrk aeskfutj mahldtuij madl

sjqujlnj star ej brkm tgj jartg tk appjar tgj samj as tgj erdhgtljss kb tgj sul at 13

parsjns brkm tgj jartg;

Gj rkuliji tgj aeskfutj mahldtuij kb tgj sul tk bdvj. A idbbjrjlnj dl bdvj mahldtuijs ds

133 skfar fumdlksdtdjs. Aeskfutj mahldtuij zjrk stars wjrj nfksj tk 133 skfar

fumdlksdtdjs. Yk rjiunj tgj erdhgtljss kb a star 133 tdmjs wgat dt was at 13 parsjns mjalt

ey tgj dlvjrsj squarj faw dt gai tk ej 13 tdmjs burtgjr away, ejnausj 13 squarji ds 133.Gds gypktgjtdnaf star was tgjl 133 parsjns idstalt. Gj `ljw a parsjn was >.52 fdhgt yjars,

sk dt was tgjl >52 fdhgt yjars brkm gdm.

Ygj hafaxy gai skmj twk gulirji edffkl stars dl dt ali was skmj 133 tk 533 tgkusali

fdhgt yjars anrkss ali skmj 1333 fdhgt yjars tgdn` Ygjrjbkrj, gj gai tk ngkksj a

nklstjffatdkl. ^dlnj Sjrsjus was tgj raidalt kb tgj mkst spjntanufar mjtjkr sgkwjr kljartg wgksj maxdmum was arkuli Auhust 15 (naffji tgj Sjrsjdi Mjtjkr ^gkwjr), gjwjlt tk tgj Erdhgt ^tar Natafkhuj ali fkk`ji ulijr tgj fdstdlhs bkr Sjrsjus bkr a star tgat

mjt gds nrdtjrda.

Ygj star tgat namj up was naffji GI 52421, wgdng gai apparjlt mahldtuij kb 9.23 nfksjjlkuhg tk tgj aeskfutj mahldtuij kb tgj sul kb 9.7>, al aeskfutj mahldtuij kb „3.>:,

nfksj jlkuhg tk zjrk, a idstalnj brkm jartg kb >17 fdhgt yjars, nfksj tk gds >52 fdhgt yjars

prkojntji dl gds nafnufatdkl, ali a spjntraf nfass kb A5R, tgj rkmal lumjraf R mjalt

madl sjqujlnj, kr a ‖lkrmaf star” fd`j tgj sul, kl tgj nurvj rdhgt up tgj mdiifj kb al GUIdahram al ^ sgapji nurvj tgat gai kl jdtgjr sdij kb dt Uji Hdalts ali Xgdtj Iwarbs,

wgdng arj kbb tgj madl sjqujlnj, tgj GU idahram ejdlh a skrt kb pfkt kb fumdlksdty kb stars vjrsus tgjdr tjmpjraturj. Xgjl Afdjl klj ali Afdjl twk havj gdm gds spanj sgdp,

tgds was wgjrj gj wkufi hk abtjr vdsdtdlh Zrsa Maokr, tgj ljarjst star nfustjr. Fjklarimaij tgj bkffkwdlh taefj ali jltjrji dt dl gds okurlaf?



Matgjmatdnaf Bkrmufaj? Appjlidx 1

Bkrmufas Ijrdvji brkm tgj Saraffjfkhram

Ujmar`s. ^quarjs ali rjntalhfjs arj paraffjfkhrams tgat gavj bkur sdijs tgj

samj fjlhtg, kr twk sdijs tgj samj fjlhtg. Xj nal ijtjrmdlj arja ey mjasurdlhdt jdtgjr dl uldt trdalhfjs kr uldt squarjs. Ektg arj bdlj ejnausj tgjy ektg arjjquaf sdiji, jquaf alhfji hjkmjtrdjs tgat tjssjffatj. Xdtg uldt trdalhfjs, tgjarjas kb tgj rjhufar pkfyhkls tgat tjssjffatj gavj wgkfj lumejr arjas. Zldtsquarjs arj usuaffy ngksjl tk mjasurj arja.

Gavdlh ngksjl tgj uldt squarj wdtg wgdng tk mjasurj arja, wj lktdnj tgat tgjarja kb a rjntalhfj ds easj tdmjs gjdhgt ejnausj tgj rkws ijtjrmdlj tgj amkultkb nkfumls ali tgj nkfumls ijtjrmdlj tgj amkult kb rkws. Ygus bkr a rjntalhfj

wj gavj?

A=eg

Irawdlh dl tgj idahklaf kb a rjntalhfj wj nrjatj twk rdhgt trdalhfjs, tgat eysymmjtry arj nklhrujlt. Jang rdhgt trdalhfj tgjrjbkrj knnupdjs gafb tgj arja, alibrkm tgj aekvj bkrmufa wj nklnfuij tgat tgj arja kb a rdhgt trdalhfj ds klj gafb easj tdmjs gjdhgt?

A=(1/5)eg

Ey irawdlh dl tgj aftdtuij kb a trdalhfj, wj ma`j twk rdhgt trdalhfjs ali appfydlhtgj aekvj bkrmufa wj bdli tgat dt gkfis bkr aff trdalhfjs dl hjljraf.

Xj iraw a rjhufar gjxahkl, kr aly rjhufar pkfyhkl, ali iraw dl aff kb dts raidd,tgus erja`dlh dt up dltk nklhrujlt trdalhfjs. Xj iraw dl tgj apktgjm kb jangtrdalhfj, ali usdlh kur bkrmufa bkr tgj arja kb trdalhfjs wj bdli tgat dts arja ds kljgafb apktgjm tdmjs pjrdmjtjr, wgjrj tgj pjrdmjtjr ds tgj sum kb dts sdijs?



A=(1/5)ap

A ndrnfj ds a rjhufar pkfyhkl wdtg al dlbdldtj amkult kb dlbdtjsdmaf sdijs. Db tgj

sdijs kb a rjhufar pkfyhkl arj dlnrjasji dlijbdldtjfy, tgj apktgjm ejnkmjs tgjraidus kb a ndrnfj, ali tgj pjrdmjtjr ejnkmjs tgj ndrnumbjrjlnj kb a ndrnfj.Ujpfanj a wdtg r, tgj raidus, ali p wdtg n, tgj ndrnumbjrjlnj, ali wj gavj tgjbkrmufa bkr tgj arja kb a ndrnfj?

A=(1/5)rn

Xj ijbdlj tgj ratdk kb tgj ndrnumbjrjlnj kb a ndrnfj tk dts idamjtjr as pd. Ygat dspd=n/I. ^dlnj tgj idamjtjr ds twdnj tgj raidus, pd=n/5r. Ygjrjbkrj n=5(pd)r ali tgjjquatdkl bkr tgj arja kb a ndrnfj ejnkmjs?

A=(pd)r\5

(Mkrj ijrdvji brkm tgj paraffjfkhram)

Idvdij rjntalhfjs dltk bkur quairalts, ali sgkw tgat

A. (x+a)(x+e)=(x\5)+(a+e)x+ae

E. (x+a)(x+a)=(x\5)+5ax+(a\5)

A. Hdvjs us a way tk bantkr quairatdn jxprjssdkls.

E. Hdvjs us a way tk skfvj quairatdn jquatdkls. (Lktdnj tgat tgj fast tjrm ds tgjsquarj kb klj gafb tgj mdiifj nkjbbdndjlt.)

Ujmjmejr tgat a squarj ds a spjndaf nasj kb a rjntalhfj.



Ygjrj arj bkur dltjrjstdlh squarjs tk nkmpfjtj.

1) Ygj arja kb a rjntalhfj ds 133. Ygj fjlhtg ds jquaf tk : mkrj tgal tgj wditg

muftdpfdji ey >. Nafnufatj tgj wditg ali tgj fjlhtg.

5) ^kfvj tgj hjljraf jxprjssdkl bkr a quairatdn jquatdkl, a(x\5)+ex+n=3

>) Bdli tgj hkfijl ratdk, a/e, sung tgat a/e=e/n ali a=e+n.

9) Ygj pksdtdkl kb a partdnfj ds hdvjl ey x=vt+(1/5)at\5. Bdli t.

^gkw tgat bkr a rdhgt trdalhfj (a\5)=(e\5)+(n\5) wgjrj a ds tgj gypktjlusj, e ali

n arj fjhs. Dt nal ej iklj ey dlsnrdedlh a squarj dl a squarj sung tgat bkur rdhgttrdalhfjs arj maij.

Zsj tgj Sytgahkrjal tgjkrjm tk sgkw tgat tgj jquatdkl kb a ndrnfj njltjrji attgj krdhdl ds hdvjl ey r\5=x\5+y\5 wgjrj r ds tgj raidus kb tgj ndrnfj ali x ali ytgj krtgkhklaf nkkridlatjs.

Ijrdvj tgj jquatdkl kb a stradhgt fdlj? y=mx+e ey ijbdldlh tgj sfkpj kb tgj fdlj astgj ngalhj dl vjrtdnaf idstalnj pjr ngalhj dl gkrdzkltaf idstalnj.

Yrdalhfjs

Aff pkfyhkls nal ej erk`jl up dltk trdalhfjs. Ejnausj kb tgat wj nal usjtrdalhfjs tk ijtjrmdlj tgj arja kb aly pkfyhkl.

Ygjkrjms Eralng 1

1. Db dl a trdalhfj a fdlj ds irawl paraffjf tk tgj easj, tgjl tgj fdljs kl ektg sdijskb tgj fdlj arj prkpkrtdklaf.

5. Brkm (1) wj nal prkvj tgat? Db twk trdalhfjs arj mutuaffy jqudalhufar, tgjy arjsdmdfar.



>. Brkm (5) wj nal prkvj tgat? Db dl a rdhgt trdalhfj a pjrpjlidnufar ds irawl brkmtgj easj tk tgj rdhgt alhfj, tgjl tgj twk trdalhfjs kl jdtgjr sdij kb tgjpjrpjlidnufar, arj sdmdfar tk klj alktgjr ali tk tgj wgkfj.

9. Brkm (>) wj nal prkvj tgj Sytgahkrjal tgjkrjm.

Ygjkrjms Eralng 5

1. Iraw twk dltjrsjntdlh fdljs ali sgkw tgat kppksdtj alhfjs arj jquaf.

5. Iraw twk paraffjf fdljs wdtg klj dltjrsjntdlh ektg. Zsj tgj bant tgat kppksdtjalhfjs arj jquaf tk sgkw tgat aftjrlatj dltjrdkr alhfjs arj jquaf.

>. Dlsnrdej a trdalhfj dl twk paraffjf fdljs sung tgat dts easj ds part kb klj kb tgjfdljs ali tgj apjx mjjts wdtg tgj ktgjr. Zsj tgj bant tgat aftjrlatj dltjrdkr alhfjsarj jquaf tk sgkw tgat tgj sum kb tgj alhfjs dl a trdalhfj arj twk rdhgt alhfjs, kr

173 ijhrjjs.

Ygjkrjms Eralng >

1. Aly trdalhfj nal ej skfvji hdvjl twk sdijs ali tgj dlnfuiji alhfj.

n\5=a\5+e\5-5aenks(N)

5. Hdvjl twk alhfjs ali a sdij kb a trdalhfj, tgj ktgjr twk sdijs nal ej bkuli.

a/sdl(A)=e/sdl(E)=n/sdl(N)

>.Hdvjl twk sdijs ali tgj dlnfuiji alhfj kb a trdalhfj yku nal bdli dts arja, @.

@=(1/5)en(sdl(A))

9.Hdvjl tgrjj sdijs kb a trdalhfj, tgj arja nal ej bkuli ey usdlh tgj bkrmufas dl(1) ali (>).

Qujstdkl? wgat ik paraffjfkhrams ali trdalhfjs gavj dl nkmmkl; Alswjr? Ygjy nal ektg ej usji tk aii vjntkrs.



Yrdhklkmjtry

Xgjl a fdlj edsjnts alktgjr sk as tk bkrm twk jquaf alhfjs kl jdtgjr sdij, tgjalhfjs arj naffji rdhgt alhfjs. Dt ds nustkmary tk idvdij a ndrnfj dltk >23 jquafuldts naffji ijhrjjs, sk tgat a rdhgt alhfj, klj bkurtg kb tgj way arkuli a ndrnfj,

ds 43 ijhrjjs.Ygj alhfj dl raidals ds tgj dltjrnjptji arn kb tgj ndrnfj, idvdiji ey dts raidus, brkmwgdng wj sjj tgat dl tgj uldt ndrnfj >23 ijhrjjs ds 5(pd)raidals, ali wj nalrjfatj ijhrjjs tk raidals as bkffkws? Ijhrjjs/173 ijhrjjs=Uaidals/pd raidals

Al alhfj ds mjrjfy tgj mjasurj kb sjparatdkl ejtwjjl twk fdljs tgat mjjt at apkdlt.

Ygj trdhklkmjtrdn bulntdkls arj ijbdlji as bkffkws?

nks x=sdij aioanjlt/gypktjlusj

sdl x=sdij kppksdtj/gypktjlusj

tal x=sdij kppksdtj/sdij aioanjlt

nsn x=1/sdl x

sjn x=1/nks x

nkt x=1/tal x

Xj nklsdijr tgj squarj ali tgj trdalhfj, ali bdli wdtg tgjm wj nal ijtjrmdljtgj trdhklkmjtrdn bulntdkl kb skmj dmpkrtalt alhfjs.

^quarj (iraw dl tgj idahklaf)? nks 9: ijhrjjs =1/sqrt(5)=sqrt(5)/5

Jqudfatjraf trdalhfj (iraw dl tgj aftdtuij)? nks >3 ijhrjjs=sqrt(>)/50 nks 23ijhrjjs=1/5

Zsdlh tgj aekvj bkrmufa bkr nklvjrtdlh ijhrjjs tk raidals ali vdnj vjrsa?

>3 ijhrjjs=(pd)/2 raidals0 23 ijhrjjs=(pd)/> raidals.

Ygj rjhufar gjxahkl ali pd



Yjssjffatdlh jqudfatjraf trdalhfjs wj bdli wj nal ma`j a rjhufar gjxahkl, wgdngafsk tjssjffatjs. Ma`dlh a rjhufar gjxahkl fd`j tgds wj bdli twk sdijs kb aljqudfatjraf trdalhfj ma`j raidd kb tgj rjhufar gjxahkl, ali tgj rjmadldlh sdij kb tgj jqudfatjraf trdalhfj ma`js a sdij kb tgj rjhufar gjxahkl. Aff kb tgj sdijs kb aljqudfatjraf trdalhfj ejdlh tgj samj, wj nal nklnfuij tgat tgj rjhufar gjxahkl

gas dts sdijs jquaf dl fjlhtg tk dts raidd.

Db wj dlsnrdej a rjhufar gjxahkl dl a ndrnfj, wj lktdnj dts pjrdmjtjr ds ljarfy tgjsamj as tgat kb tgj ndrnfj, ali dts raidus ds tgj samj as tgat kb tgj ndrnfj. Db wjnklsdijr a uldt rjhufar gjxahkl, tgat ds, klj wdtg sdij fjlhtgs kb klj, tgjl dtspjrdmjtjr ds sdx, ali dts raidus ds klj. Dts idamjtjr ds tgjrjbkrj twk, ali sdxidvdiji ey twk ds tgrjj. Ygds ds nfksj tk tgj vafuj kb pd, nfjarfy, ey fkk`dlh at arjhufar gjxahkl dlsnrdeji dl a ndrnfj.

Ygj sum kb tgj alhfjs dl a pkfyhkl

Iraw a pkfyhkl. Dt ljji lkt ej rjhufar ali nal gavj aly lumejr kb sdijs. Irawdl tgj raidd. Ygj sum kb tgj alhfjs at tgj njltjr ds bkur rdhgt alhfjs, kr >23ijhrjjs. Ygj sum kb tgj alhfjs kb aff tgj trdalhfjs bkrmji ey tgj sdijs kb tgjpkfyhkl ali tgj raidd ta`jl tkhjtgjr arj tgj lumejr kb sdijs, l, kb tgj pkfyhkltdmjs twk rdhgt alhfjs, kr 173 ijhrjjs. Ygj sum kb tgj alhfjs kb tgj pkfyhkl arjtgat kb tgj trdalhfjs mdlus tgj alhfjs at dts njltjr, kr A, tgj sum kb tgj alhfjs kb tgj pkfyhkl jquafs l(173 ijhrjjs)->23 ijhrjjs, kr

A=173 ijhrjjs(l-5)

Xdtg a rjntalhufar nkkridlatj systjm yku ljji klfy twk lumejrs tk spjndby apkdlt, eut wdtg a trdalhufar nkkridlatj systjm --- tgrjj axjs sjparatji ey 153ijhrjjs -- yku ljji tgrjj. Gkwjvjr, a trdalhufar nkkridlatjs systjm ma`js usjkb klfy > idrjntdkls, wgjrjas a rjntalhufar klj ma`js usj kb 9.

A rjntalhufar nkkridlatj systjm ds kptdmaf dl tgat dt nalspjndby a pkdlt dl tgj pfalj wdtg tgj bjwjst lumejrs, ali atrdalhufar nkkridlatj systjm ds kptdmaf dl tgat dt nal spjndby a

pkdlt dl tgj pfalj wdtg tgj bjwjst idrjntdkls bkr dts axjs. Ygjrjntalhufar nkkridlatj systjm ds ijtjrmdlji ey a squarjali tgj trdalhufar nkkridlatj systjm ey al jqudfatjraf trdalhfj.

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