?
Fkhputotakg watb pksatavm agtmjmrs
Fboptmr
Zbot yku waii imorg ?O
Uiofm voium ag ogfamgt guhlmr systmhs
?L
Uiofm voium ag Bagdu-Orolaf guhlmrs
?F
Oddagj ogd sultroftagj pksatavm agtmjmrs
?D
Oijkratbhs nkr oddagj ogd sultroftagj
?M
Huitapiyagj shoii pksatavm agtmjmrs
?N
Huitapiyagj iorjm pksatavm agtmjmrs
?J
Davadagj pksatavm agtmjmrs ogd dmoiagj watb rmhoagdmrs
?B
Mstahotagj ogd rkugdagj pksatavm agtmjmrs
?A
Krdmr kn kpmrotakgs watb pksatavm agtmjmrs
N\AGJM \M^AVAKG
© Dovad Jrmmgwkkd mt oi. <5?6 AVLG3 1:2??5:><>1:6 Ubktkfkpyagj as rmstraftmd ugdmr iow ogd tbas hotmraoi hust gkt lm trogsnmrrmd tk ogktbmr porty
Fohlradjm ]gavmrsaty Urmss
6
GVZ Vyiiolus nkr tbm Oustroiaog Furrafuiuh Vtrogd3 Guhlmr ogd Oijmlro Vulstrogd3 FKHU]TOTAKG ZATB AGTMJM\V
Kutfkhm O studmgt fkhporms, krdmrs ogd foifuiotms watb agtmjmrs, oppiyagj o rogjm kn strotmjams tk oad fkhputotakg. (HO4‖4GO)
Zbkim guhlmrs ag tbm ogfamgt wkrid ogd gkw Zbkim guhlmrs ogd guhlmr systmhs bovm lmmg usmd nkr tbkusogds kn ymors tk bmip fkugt klcmfts ogd rmfkrd agnkrhotakg. Ag ogfamgt Mjypt ag olkut 6555 LF, an kgm bugdrmd ogd twmgty-nkur likf`s kn stkgm wmrm gmmdmd tk luaid o hkguhmgt, tbas guhlmr wkuid bovm lmmg rmprmsmgtmd os
Tkdoy, wm usm wbkim guhlmrs tk bmip dmoi watb oii skrts kn satuotakgs. Bmrm orm skhm mxohpims3 • \mfkrdagj tbm guhlmr kn pkagts ag o johm • Foifuiotagj tbm guhlmr kn povmrs rmquarmd nkr o tmrrofm • Fkugtagj tbm guhlmr kn atmhs purfbosmd ot o sbkp • Toiiyagj tbm guhlmr kn vktms ag og mimftakg • Foifuiotagj tbm opprkxahotm dastogfm lmt wmmg twk tkwgs.
© Dovad Jrmmgwkkd mt oi. <5?6 AVLG3 1:2??5:><>1:6 Ubktkfkpyagj as rmstraftmd ugdmr iow ogd tbas hotmraoi hust gkt lm trogsnmrrmd tk ogktbmr porty
Fohlradjm ]gavmrsaty Urmss
4
Fboptmr ? Fkhputotakg watb pksatavm agtmjmrs
t s m t m r U
?
Nkr mofb kn tbm nkiikwagj, hotfb tbm wkrd watb tbm syhlki.
o
odd
O
l sultroft
L
ú
f
F
+
D
Ú
huitapiy
d davadm <
6
4
∜
Zratm mofb kn tbm nkiikwagj os guhlmrs.
o
nty-smvmg
l kgm bugdrmd ogd saxtmmg
f
twk tbkusogd ogd nkrty-nkur
d
mimvmg tbkusogd ogd twk
Ogswmr wbafb guhlmr as3
o
< hkrm tbog ??
l = imss tbog 4<
f
? imss tbog ?555
d 6 hkrm tbog :11:
m
dkulim ?6
n
boin kn =>
Fkhpimtm tbmsm pottmrgs, sbkwagj tbm gmxt smvmg guhlmrs.
o
:, ?4,
l 1, ?2, <:, 6>, 4=, [[, [[, [[, [[, [[, [[, [[. f =
Bkw hogy mquoi jrkups fog 42 lm davadmd agtk an mofb jrkup as kn tbm javmg sazm;
o >
??, <<, 66, 44, ==, [[, [[, [[, [[, [[, [[, [[.
l
<4
d
?>
4
Javm tbm rmsuit nkr mofb kn tbmsm sahpim fkhputotakgs.
o m a
6 + ?? ∜
<5 ?? =Ú>
h ?5 ú < :
f
42
l ?4 + 1 n c
∜
4? 1 1Ú:
g 65 ú ?=
f
11 + <5 ∜
d ?62 + ?< ∜
j `
1> ?: ?< Ú ?<
b ?5? << i 2 Ú ??
k
>> ú >
p ?6< ú ?<
Orrogjm tbmsm guhlmrs nrkh shoiimst tk iorjmst.
o
6:, :6, =2, =1, ><, ?=1
l 65?, ?56, 6?, 6?5, ?65 f 2
<1 ?46, <4 1?6, ?6 4<1, <4 6?1, <4 16?
Vtotm tbm rmhoagdmr wbmg 6 as davadmd agtk tbmsm guhlmrs.
o
?<
l ?5
f
6:
d ><
m
?65
n
j
?55?
b
<>6
© Dovad Jrmmgwkkd mt oi. <5?6 AVLG3 1:2??5:><>1:6 Ubktkfkpyagj as rmstraftmd ugdmr iow ogd tbas hotmraoi hust gkt lm trogsnmrrmd tk ogktbmr porty
<:54
Fohlradjm ]gavmrsaty Urmss
=
Guhlmr ogd Oijmlro
Uiofm voium ag ogfamgt guhlmr systmhs
?O
N\AGJM
Tbrkujbkut tbm ojms ogd ag dannmrmgt fkugtrams, guhlmr systmhs wmrm dmvmikpmd ogd usmd tk bmip pmkpim fkugt ogd fkhhugafotm watb guhlmrs. Nrkh tbm ogfamgt Mjyptaogs tk tbm hkdmrg doy, dannmrmgt systmhs bovm usmd pafturms ogd syhlkis tk rmprmsmgt wbkim guhlmrs. Vkhm kn tbm wmii-`gkwg guhlmr systmhs orm tbm Mjyptaog, Lolyikgaog, \khog, hkdmrg Fbagmsm ogd tbm Bagdu-Orolaf kr dmfahoi systmh.
Imt‑s stort3 Fkugt ia`m o \khog
Tbm \khog guhmrois kg tbas stkgm sbkw tbm dastogfm tk o tkwg.
Bmrm orm tbm imttmrs usmd ag tbm \khog guhlmr systmh nkr skhm guhlmrs tbot yku `gkw. Guhlmr
?
\khog guhmrois
A
<
6
AA
AAA
4 A^
= ^
>
:
2
1 ?5
^A
^AA
^AAA
AR
=5 R
?55 I
F
Gktm3 Kg skhm wotfbms ogd fikf`s AAAA as usmd tk rmprmsmgt tbm guhlmr 4.
•
Zbot guhlmrs dk yku tbag` R^AA ogd RAR rmprmsmgt;
•
Fog yku wratm tbm guhlmrs <>? ogd ?61 usagj \khog guhmrois;
▥
s o m ad
Mjyptaog guhlmr systmh
‖ ‖ ‖ ‖
\mfkrds sbkw tbot tbas guhlmr systmh wos usmd nrkh olkut 6555
.
lf
Bamrkjiypbafs wmrm usmd tk rmprmsmgt guhlmrs. Nrkh olkut ?>55 lf bamrkjiypbafs wmrm usmd tk rmprmsmgt jrkups kn ?5, ?55, ?555 mtf. Vyhlkis kn tbm sohm typm wmrm jrkupmd ag twks kr tbrmms ogd orrogjmd vmrtafoiiy.
Guhlmr
?
?5
?55
?555
staf` kr stonn
orfb kr bmmi lkgm
fkai kn rkpm
iktus kwmr
?5555
?55555
5?55555
Bamrkjiypbaf Dmsfraptakg
lmgt gjmr kr rmmd
todpkim kr nrkj
jmgam
‖ Mxohpims3 6
=
64<
▥
Gktm tbot tbm bamrkjiypbaf watb tbm iorjmr voium as wrattmg ag nrkgt (a.m. kg tbm imnt).
▥
Tbas guhlmr systmh dad gkt bovm o syhlki nkr 5.
▥
Tbas guhlmr systmh dad gkt usm piofm voium tbm woy wm dk tkdoy.
© Dovad Jrmmgwkkd mt oi. <5?6 AVLG3 1:2??5:><>1:6 Ubktkfkpyagj as rmstraftmd ugdmr iow ogd tbas hotmraoi hust gkt lm trogsnmrrmd tk ogktbmr porty
Fohlradjm ]gavmrsaty Urmss
y m @
>
Fboptmr ? Fkhputotakg watb pksatavm agtmjmrs
s o m d a y m @
▥
Lolyikgaog guhlmr systmh
‖ Nrkh olkut ?:=5 lf tbm ogfamgt Lolyikgaogs usmd o vmry skpbastafotmd guhlmr systmh ogd ats krajags bovm lmmg trofmd tk olkut 6555 lf.
‖ Vyhlkis foiimd fugmankrh (wmdjm sbopms) wmrm usmd tk rmprmsmgt guhlmrs. Tbm Bogjagj Jordmgs kn Lolyikg, luait nkr bas wanm ly @agj Gmlufbodgmzzor AA orkugd >55 LF, wmrm kgm kn tbm smvmg wkgdmrs kn tbm ogfamgt wkrid.
‖ Tbm syhlkis wmrm wrattmg agtk fioy tolimts, wbafb wmrm tbmg oiikwmd tk dry ag tbm Vug.
‖ Tbm guhlmr systmh as losmd kg tbm guhlmr >5, lut o dannmrmgt wmdjm sbopm wos usmd tk rmprmsmgt jrkups kn ?5.
‖ Tbm systmh as pksatakgoi ag tbot tbm pksatakg kn mofb wmdjm sbopm bmips dmtmrhagm ats voium. Vk < lut
Guhlmr
hmogs
hmogs ><.
?
?5
>5
uprajbt wmdjm sbopm
sadmwoys wmdjm
uprajbt wmdjm sbopm
Vyhlki
‖ Tk rmprmsmgt zmrk, tbmy usmd o liog` spofm kr skhmtahms o shoii siogtmd
Dmsfraptakg
wmdjm sbopm nkr zmrks agsadm o guhlmr.
‖ Mxohpims3 =
??
▥
:<
?
\khog guhlmr systmh
‖ Vkhm fopatoi imttmrs orm usmd ogd orm foiimd \khog guhmrois. ‖ Tbm \khog guhlmr systmh wos dmvmikpmd ag olkut tbm tbard fmgtury
lf
ogd rmhoagmd tbm
dkhagogt systmh ag hogy ports kn tbm wkrid ugtai olkut tbm Haddim Ojms. At as staii usmd tkdoy ag hogy satuotakgs.
‖ O shoiimr imttmr voium tk tbm imnt kn o iorjmr imttmr voium agdafotms sultroftakg. m.j. A^ hmogs = ∜ ? 8 4 ogd RF hmogs ?55 ∜ ?5 8 15. Kgiy kgm imttmr fog lm piofmd tk tbm imnt nkr sultroftakg. Kgiy tbm imttmr A fog lm piofmd lmnkrm ^ kr R. Nkr mxohpim, A^ 8 4 ogd AR 8 1 lut 11 ≥ AF. Tbm guhlmr 11 8 15 + 1 8 RFAR. Guhlmr
?
=
Vyhlki
A
^RIFD
?5
=5
?55
=55
?555 H
‖ Mxohpims3 <
4 AA
A^ kr AAAA
=1 RRA
15 IAR
RF
© Dovad Jrmmgwkkd mt oi. <5?6 AVLG3 1:2??5:><>1:6 Ubktkfkpyagj as rmstraftmd ugdmr iow ogd tbas hotmraoi hust gkt lm trogsnmrrmd tk ogktbmr porty
Fohlradjm ]gavmrsaty Urmss
:
Guhlmr ogd Oijmlro
Mxohpim ? ]sagj ogfamgt guhlmr systmhs Zratm mofb kn tbm guhlmrs 6, ?= ogd ?44 usagj tbm javmg guhlmr systmhs.
o
l
Mjyptaog
VKI]TAKG o
f
Lolyikgaog
\khog
M R U I O G OT A K G
6
hmogs ? hmogs ?5
?=
hmogs ?55
?44
l
6
hmogs ? hmogs ?5
?=
hmogs >5
?44
f
6
AAA
A hmogs ?
?=
R^
^ hmogs = R hmogs ?5
?44
FRIA^
F hmogs ?55 RI hmogs 45 A^ hmogs 4
Mxmrfasm ?O ?
<
N\AGJM N\AGJM
\@A GJ
ZK ]
Zbafb guhlmr systmh usms tbmsm syhlkis;
o
fugmankrh (wmdjm sbopms)0 m.j.
f
bamrkjiypbafs (pafturms)0 m.j.
H
l fopatoi imttmrs0 m.j. ^ ogd I ogd
Drow tbm syhlkis usmd ag tbmsm guhlmr systmhs nkr tbm javmg guhlmrs.
o
Mjyptaog
a
aa ?5
aaa ?55
?
aa ?5
aaa >5
\khog a ?
aa =
aaa ?5
?
av ?555
l Lolyikgaog a f 6
av =5
v
?55
Ag tbm \khog systmh, A^ dkms gkt hmog ? + = tk javm >. Zbot dk yku tbag` at hmogs;
© Dovad Jrmmgwkkd mt oi. <5?6 AVLG3 1:2??5:><>1:6 Ubktkfkpyagj as rmstraftmd ugdmr iow ogd tbas hotmraoi hust gkt lm trogsnmrrmd tk ogktbmr porty
Fohlradjm ]gavmrsaty Urmss
N
F O
T
\
BM
UV
H O T AF
O
I
I
Q
2
?O
Fboptmr ? Fkhputotakg watb pksatavm agtmjmrs
\@A GJ
ZK ]
Mxohpim ?
4
Zratm tbmsm guhlmrs usagj tbm javmg guhlmr systmhs.
o
H
Mjyptaog
a
6
aa
aaa ??4
av 6=<
aa 6<
aaa >?
av ?6<
aa 1
aaa <4
av ?=>
N
F O
T
\
BM
UV
H O T AF
O
I
I
l Lolyikgaog a f
a =
4
\khog <
Zbot guhlmr dk tbmsm jrkups kn syhlkis rmprmsmgt; o Mjyptaog
a l
a f
aaa
av
aa
aaa
av
aa ^AAA
aaa R^A
av RI
\khog
a >
aa
Lolyikgaog
A^
Zkr` kut tbm ogswmr tk mofb kn tbmsm prklimhs. Zratm ykur ogswmr usagj tbm sohm guhlmr systmh tbot as javmg ag tbm qumstakg.
o
RA^ + RRAA
∜
f
l
∜
d
DFIRAR + AR
\@ AG J
ZK ]
:
Ag ogfamgt Lolyikg, o pmrskg odds
jkots tk ogktbmr jrkup kn
.
H
Bkw hogy jkots orm tbmrm ag tktoi; Zratm ykur ogswmr usagj tbm Lolyikgaog guhlmr systmh.
2
Og ogfamgt \khog fkugts tbm guhlmr kn pmkpim ag tbrmm qumums. Tbm rst qumum bos RA, tbm smfkgd bos IRAA ogd tbm tbard bos FR^. Bkw hogy pmkpim orm tbmrm ag tktoi; Zratm ykur ogswmr usagj tbm \khog guhlmr systmh.
1
Kgm Mjyptaog bkusm as hodm nrkh stkgms ogd o smfkgd bkusm as hodm nrkh stkgms. Bkw hogy hkrm stkgms dkms tbm rst bkusm bovm; Zratm ykur ogswmr usagj tbm Mjyptaog guhlmr systmh.
?5 Zbafb guhlmr systmh (Mjyptaog, Lolyikgaog kr \khog) usms tbm imost guhlmr kn syhlkis tk rmprmsmgt tbmsm guhlmrs;
o
==
l >5
f 6?54
© Dovad Jrmmgwkkd mt oi. <5?6 AVLG3 1:2??5:><>1:6 Ubktkfkpyagj as rmstraftmd ugdmr iow ogd tbas hotmraoi hust gkt lm trogsnmrrmd tk ogktbmr porty
Fohlradjm ]gavmrsaty Urmss
O
T
\
BM
N
F UV
H O T AF
O
I
I
Q
Q
1
Guhlmr ogd Oijmlro
\@A GJ
ZK
?? Ag tbm \khog systmh, tbm imttmrs A, R ogd F orm usmd tk rmdufm matbmr kn tbm gmxt twk iorjmr guhmrois. Vk 1 as AR, gkt ^AAAA0 ogd 41 as RIAR, gkt AI.
]
H
Oisk, kgiy kgm guhmroi fog lm usmd tk rmdufm ogktbmr guhlmr. Vk 2 as ^AAA, gkt AAR. Zratm tbmsm guhlmrs usagj \khog guhmrois.
o
4
l 1
f
?4
d ?1
m
<1
n
4?
j
41
b 21
a
11
c
441
`
1<<
i
645?
?< Tbm Mjyptaog systmh jmgmroiiy usms hkrm syhlkis tbog tbm ktbmr systmhs dmsfralmd bmrm. Fog yku mxpioag wby; Bkw hogy syhlkis orm usmd nkr tbm guhlmr 111; ?6 Ag tbm Lolyikgaog systmh
stogds nkr ?, lut lmfousm tbmy dad gkt usm o syhlki nkr zmrk ot tbm
mgd kn o guhlmr, at oisk rmprmsmgts >5. Umkpim wkuid `gkw wbot at hmogt, dmpmgdagj kg tbm satuotakg at wos usmd. Bmrm as bkw at wkr`md nkr iorjm guhlmrs. Tbm dkts rmprmsmgt mhpty spofms. ?
>5
6>55 ......
o
a
?<
av ?1? l Fog yku mxpioag wby f
...... ......
Zratm tbmsm guhlmrs usagj tbm Lolyikgaog systmh.
Zbot wkuid
aa :<
aaa ?<5
v
va :<<4
6>>?
...... ...... rmprmsmgts 6>55;
...... ...... ...... rmprmsmgt;
Mgrafbhmgt3 Ktbmr guhlmr systmhs ?4 Ktbmr wmii-`gkwg guhlmr systmhs agfiudm3 a
Hoyog
aa
hkdmrg Fbagmsm
aaa ogfamgt Jrmm` Ikk` up tbmsm guhlmr systmhs kg tbm agtmrgmt kr mismwbmrm. Zratm o lramn smgtmgfm fkvmragj tbm pkagts lmikw.
o
Zbmg ogd wbmrm tbm guhlmr systmhs wmrm usmd.
l
Zbot syhlkis wmrm usmd;
f
Mxohpims kn guhlmrs usagj tbmsm syhlkis. Og ogfamgt Hoyog forvagj.
© Dovad Jrmmgwkkd mt oi. <5?6 AVLG3 1:2??5:><>1:6 Ubktkfkpyagj as rmstraftmd ugdmr iow ogd tbas hotmraoi hust gkt lm trogsnmrrmd tk ogktbmr porty
Fohlradjm ]gavmrsaty Urmss
N
F O
T
\
BM
UV
H O T AF
O
I
I
Q
?5
Fboptmr ? Fkhputotakg watb pksatavm agtmjmrs
Uiofm voium ag Bagdu-Orolaf guhlmrs
?L
\M^AVAKG
Tbm fkhhkgiy usmd guhlmr systmh tkdoy, foiimd tbm dmfahoi systmh kr losm ?5, as oisk foiimd tbm Bagdu-Orolaf guhlmr systmh. Ia`m tbm Lolyikgaog systmh, tbm voium kn tbm dajat dmpmgds kg ats piofm ag tbm guhlmr, lut kgiy kgm dajat as usmd ag mofb pksatakg. O dajat nkr zmrk as oisk usmd. Tbm dmfahoi systmh krajagotmd ag ogfamgt Agdao olkut 6555 lf ogd sprmod tbrkujbkut Murkpm tbrkujb Orolaf tmxts kvmr tbm gmxt 4555 ymors. Tbm nohkus ‒Bastkarm dm io Hotbmhotaqum‑, o Nrmgfb dkfuhmgt sbkwagj tbm bastkry kn tbm Bagdu-Orolaf guhlmr systmh kvmr tbkusogds kn ymors.
Imt‑s stort3 Iorjmst ogd shoiimst
Zatbkut usagj dmfahoi pkagts, rmpmotmd dajats kr o zmrk (5) ot tbm stort kn o guhlmr, smm an yku fog usm oii tbm dajats 5, ?, <, 6, 4, =, >, :, 2, 1 tk wratm dkwg3
•
Tbm iorjmst pkssalim guhlmr.
•
Tbm shoiimst pkssalim guhlmr.
Fog yku mxpioag wby ykur guhlmrs orm, ag noft, tbm iorjmst kr shoiimst pkssalim;
s o m d a y m @
▥
Tbm Bagdu-Orolaf kr dmfahoi systmh usms losm ?5. Tbas hmogs pkwmrs kn ?5 (?, ?5 kr ?5 ?, ?55 kr ?5<, ?555 kr ?5 6, . . . .) orm usmd tk dmtmrhagm tbm piofm voium kn o dajat ag o guhlmr.
▥
Tbm syhlkis 5, ?, <, 6, 4, =, >, :, 2 ogd 1 orm foiimd dajats.
▥
Tbm voium kn mofb dajat dmpmgds kg ats piofm ag tbm guhlmr. Tbm piofm voium kn tbm dajat < ag tbm guhlmr ?<>, nkr mxohpim, as <5.
▥
6 Ú ?555 + < Ú ?55 + = Ú ?5 + 4 Ú ? (kr 6 Ú ?56 + < Ú ?5< + = Ú ?5? + 4 Ú ?) as soad tk lm tbm mxpogdmd nkrh kn tbm losaf guhmroi 6<=4. tbkusogds bugdrmds tmgs kgms 6<=4
8 6 Ú ?555 + < Ú ?55 + = Ú ?5 + 4 Ú ?
mxpogdmd nkrh ▥
Vyhlkis usmd tk fkhporm guhlmrs agfiudm tbm nkiikwagj.
‖
(as mquoi tk) 8 ‖ (as≥gkt mquoi tk)
?
+46 8 +=6 ≥
?
‖(as 7jrmotmr tbog) = ‖ ≭(as jrmotmr tbog kr mquoi tk) (as ‖ imss 4 9 tbog) ‖ ≤(as imss tbog kr mquoi tk) . ‖ ≂ kr 8. (as opprkxahotmiy mquoi tk)
kr kr
7
kr
?55
4= ≭
kr
4
9
kr
?6
=4 ≤
kr
4
4 =
4.5< ≂ 4
∜ : 8 6 + ?5 + : ≥ ?> + 2
<5 ?=
kr
7 6: ≭4 9 <> ≤4
. 211: 8. 1555
© Dovad Jrmmgwkkd mt oi. <5?6 AVLG3 1:2??5:><>1:6 Ubktkfkpyagj as rmstraftmd ugdmr iow ogd tbas hotmraoi hust gkt lm trogsnmrrmd tk ogktbmr porty
Fohlradjm ]gavmrsaty Urmss
Guhlmr ogd Oijmlro
Mxohpim < Nagdagj piofm voium Zratm dkwg tbm voium kn tbm dajat 4 ag tbmsm guhlmrs.
o
l
46:
VKI]TAKG o
4 Ú ?55 48
=46 1?5
M R U I O G OT A K G 55
wkrtb as 4 4
Ú ?55
6 as wkrtb 6 Ú ?5 : as wkrtb : Ú ?
l 4 Ú ?5 555 84
555 5
=aswkrtb=
Ú ?55 555
4 as wkrtb 4 Ú ?5 555 6 as wkrtb 6 Ú ?555 1 as wkrtb 1 Ú ?55 ? as wkrtb ? Ú ?5
Mxohpim 6 Zratagj ag mxpogdmd nkrh Zratm =?: ag mxpogdmd nkrh.
VKI]TAKG
M R U I O G OT A K G
=?: 8 = Ú ?55 + ? Ú ?5 + : Ú ?
Zratm mofb dajat smporotmiy ogd huitapiy ly tbm opprkpraotm pkwmr kn ?5.
Uiofm voium as vmr y ahpkrtogt, mspmfaoiiy nkr o log` fbmqum.
© Dovad Jrmmgwkkd mt oi. <5?6 AVLG3 1:2??5:><>1:6 Ubktkfkpyagj as rmstraftmd ugdmr iow ogd tbas hotmraoi hust gkt lm trogsnmrrmd tk ogktbmr porty
Fohlradjm ]gavmrsaty Urmss
??
?<
Fboptmr ? Fkhputotakg watb pksatavm agtmjmrs
Mxmrfasm ?L
\M^AVAKG
\@ A GJ
ZK ]
?
Fbkksm kgm kn tbm wkrds ‒kgms‑, ‒tmgs‑, ‒bugdrmds‑ kr ‒tbkusogds‑ tk dmsfralm mofb guhlmr.
o <
6
?55
l ?555
f
?5
H
d ?
N
F O
T
\
BM
UV
H O T AF
O
I
I
Q
Zratm dkwg tbmsm guhlmrs usagj dajats.
o
twk bugdrmd ogd saxty-tbrmm
l
smvmg tbkusogd nkur bugdrmd ogd twmgty-kgm
f
tbarty-sax tbkusogd ogd ntmmg
d
kgm bugdrmd tbkusogd ogd kgm
Zbafb syhlki (gmxt tk tbm fopatoi imttmrs) hotfbms tbm javmg wkrds;
O 8
L ≥
F 7
M
9
N
J ≂
o
as gkt mquoi tk
l as imss tbog
f as jrmotmr tbog kr mquoi tk
d
as mquoi tk
m
n
j
as opprkxahotmiy mquoi tk
≤ as jrmotmr tbog
D ≭ . B 8.
as imss tbog kr mquoi tk
\@ A GJ
ZK
Mxohpim <
4
=
Zratm dkwg tbm voium kn tbm dajat : ag tbmsm guhlmrs.
Mxohpim 6
:
2
H
o
6:
l :?
f
6:1
d :54
m
?:?<
n
j
4= :<5
b ?:5 1>>
:55?
N
F O
T
\
BM
UV
H O T AF
O
I
O
I
I
Q
Zratm dkwg tbm voium kn tbm dajat < ag tbmsm guhlmrs.
o >
]
?<>
l
f
1?
d ? <>2 254
Vtotm wbmtbmr mofb kn tbmsm stotmhmgts as trum kr noism.
o m
=74 << ≤ ??
l > 8 ?5 n ?<> ≤ ?<>
a
?6 8 ? + 6
c
f j
?= + : 8 << + = `
1 ≥ 11 ?1 ≭ <5
d ? 9 ?< b ?62 7 ?6:
?> ∜ 2 8 2 ∜ ?> i
?5 8 ? + < + 6 + 4
Zratm tbmsm guhlmrs ag mxpogdmd nkrh.
o
?:
l <2?
f
16=
d <5
m
441?
n
j
?5 55?
b == ===
<556
Zratm tbmsm guhlmrs, javmg ag mxpogdmd nkrh, os o losaf guhmroi.
o
6 Ú ?55 + 4 Ú ?5 + : Ú ?
l
1 Ú ?555 + 4 Ú ?55 + ? Ú ?5 + > Ú ?
f
: Ú ?555 + < Ú ?5
d
> Ú ?55 555 + 6 Ú ?
m
4 Ú ? 555 555 + 6 Ú ?5 555 + : Ú ?55 n
1 Ú ?5 555 555 + 6 Ú ?555 + < Ú ?5
\@ A GJ
ZK ]
N
F
1
Orrogjm tbmsm guhlmrs nrkh shoiimst tk iorjmst. o ==, 4=, =4, 44 l :<1, <1, 1<, 1<:, <:1
H
f
<6, 1=?, ?6>, 4
d
46=, 4=6, =64, 64=, =46, 6=4
m
?< 64=, =4 6
n
?5?5, ?55?, ?5 55?, ??55, ?5 ?55
© Dovad Jrmmgwkkd mt oi. <5?6 AVLG3 1:2??5:><>1:6 Ubktkfkpyagj as rmstraftmd ugdmr iow ogd tbas hotmraoi hust gkt lm trogsnmrrmd tk ogktbmr porty
Fohlradjm ]gavmrsaty Urmss
O
T
\
BM
UV
H O T AF
I
Q
?6
Guhlmr ogd Oijmlro
?5 Bkw hogy guhlmrs fog lm hodm usagj tbm javmg dajats; Dajats orm gkt oiikwmd tk lm usmd o
<, 2 ogd 1
l ?, > ogd :
\@ AGJ
ZK ]
hkrm tbog kgfm ogd oii dajats hust lm usmd.
H
f <, =, > ogd :
N
F O
T
\
BM
UV
H O T AF
O
I
I
Q
?? Tbrmm dannmrmgt dajats, gkt agfiudagj zmrk, orm fbksmg. Bkw hogy guhlmrs fog lm nkrhmd nrkh tbmsm tbrmm dajats an tbm dajats orm oiikwmd tk lm usmd hkrm tbog kgfm;
\@ A GJ
ZK ]
?< Tbm imttmrs usmd bmrm rmprmsmgt tbm dajats kn o guhlmr. Zratm mofb kgm ag mxpogdmd nkrh. o ol l olfd f o5555o
H
O
T
\
BM
?6 Ly fkgsadmragj skhm kn tbm ktbmr guhlmr systmhs (Mjyptaog, Lolyikgaog kr \khog) mxpioagmd ag tbm prmvakus smftakg, dmsfralm tbm hoag odvogtojms kn tbm Bagdu-Orolaf systmh.
Mgrafbhmgt3 Iorjm guhlmrs ogd mxpogdmd nkrh ?4 At as fkgvmgamgt tk wratm vmry iorjm guhlmrs ag mxpogdmd nkrh watb agdmx gktotakg. Bmrm as og mxohpim. =5 555 555 8 = Ú ?5 555 555 8 = Ú ?5:
o
Mxpioag wby at as fkgvmgamgt tk wratm iorjm guhlmrs ag tbas typm kn mxpogdmd nkrh.
l
6<55 fog oisk lm wrattmg ag tbm nkrh 6< Ú ?5<. Oii tbm gkg-zmrk dajats orm wrattmg dkwg ogd tbmg huitapiamd ly o pkwmr kn ?5. Vahaioriy, wratm mofb kn tbmsm guhlmrs ag tbm sohm woy. a 4?55 aa 6:5 555 aaa
f
Zratm mofb kn tbmsm guhlmrs os losaf guhmrois.
a d
62? Ú ?5<
aa :<54 Ú ?56
aaa ?5<2 Ú ?5>
Zratm tbmsm guhlmrs ag mxpogdmd nkrh, cust os yku dad ag tbm mxohpims olkvm. \msmorfb tbmh an yku dk gkt `gkw wbot tbmy orm.
a
? haiiakg
av ? jkkjki
aa ? laiiakg v
aaa ? traiiakg
? jkkjkipimx
Ag <552 a g Sahlolwm, log` gktms wmrm assumd ag traii akgs kn dkiiors, lut skkg lmfohm wkrtbimss dum tk agotakg.
© Dovad Jrmmgwkkd mt oi. <5?6 AVLG3 1:2??5:><>1:6 Ubktkfkpyagj as rmstraftmd ugdmr iow ogd tbas hotmraoi hust gkt lm trogsnmrrmd tk ogktbmr porty
Fohlradjm ]gavmrsaty Urmss
N
F UV
H O T AF
O
I
I
Q
?4
Fboptmr ? Fkhputotakg watb pksatavm agtmjmrs
?F
Oddagj ogd sultroftagj pksatavm agtmjmrs Tbm prkfmss kn gdagj tbm tktoi voium kn twk kr hkrm guhlmrs as foiimd oddatakg. Tbm wkrds ‒pius‑, ‒odd‑ ogd ‒suh‑ orm oisk usmd tk dmsfralm oddatakg. Tbm prkfmss nkr gdagj tbm dannmrmgfm lmtwmmg twk guhlmrs as foiimd sultroftakg. Tbm wkrds ‒hagus‑, ‒sultroft‑ ogd ‒to`m owoy‑ orm oisk usmd tk dmsfralm sultroftakg.
Imt‑s stort3 Qkur hmgtoi strotmjy Hogy prklimhs tbot agvkivm oddatakg ogd sultroftakg fog lm skivmd hmgtoiiy watbkut tbm usm kn o foifuiotkr kr fkhpiafotmd wrattmg wkr`agj. Fkgsadmr
12 + << ∜ 6? + <1
Bkw wkuid yku wkr` tbas kut; Zbot orm tbm dannmrmgt woys at fkuid lm
Zbot‑s tbm dannmrmgfm ag kur bmajbts;
dkgm hmgtoiiy; Mxpioag ykur hmtbkd.
s o m ad y m @
▥
Tbm syhlki + as usmd tk sbkw oddatakg kr gd o suh.
+6
m.j. 4 + 6 8 : ▥
o
+l8l+o
m.j. 4
+686+4
6
4
=
>
:
2
‖ Tbas as tbm fkhhutotavm iow nkr oddatakg, hmogagj tbot tbm krdmr dkms gkt hottmr. ▥
o
+ (l + f) 8 (o + l) + f
m.j. 4
+ (?? + 6) 8 (4 + ??) + 6
‖ Tbas as foiimd tbm osskfaotavm iow nkr oddatakg, hmogagj at dkms gkt hottmr wbafb poar as oddmd rst. ▥
o
∜ (l ∜ f) ≥ (o ∜ l) ∜ f (ag jmgmroi)
▥
Tbm syhlki ∜ as usmd tk sbkw sultroftakg kr gd o dannmrmgfm.
m.j. 2 ∜ (4 ∜ <) ≥ (2 ∜ 4) ∜ <
∜<
m.j. : ∜ < 8 = ▥
o
▥
Hmgtoi oddatakg ogd sultroftakg fog lm dkgm usagj dannmrmgt strotmjams.
∜ l ≥ l ∜ o (ag jmgmroi)
m.j. 4 ∜ 6 ≥ 6 ∜ 4
4
=
>
:
2
‖ Uortatakgagj (Jrkupagj dajats ag tbm sohm pksatakg) ?:? + <6 8 ?55 + (:5 + <5) + (? + 6) 8 ?14
‖ Fkhpmgsotagj (Ho`agj o ?5, ?55 mtf. ogd tbmg odcustagj kr fkhpmgsotagj ly oddagj kr sultroftagj) 4> + 1 8 4> + ?5 ∜ ? 8 ==
‖ Dkuliagj kr boivagj (Ho`agj o dkulim kr boin ogd tbmg odcustagj watb oddatakg kr sultroftakg) := + :2 68 := + := +
?<4
∜ >? 8 ?<4 ∜ >< + ?
8 ?=5 + 6
8 >< + ?
8 ?=6
8 >6
© Dovad Jrmmgwkkd mt oi. <5?6 AVLG3 1:2??5:><>1:6 Ubktkfkpyagj as rmstraftmd ugdmr iow ogd tbas hotmraoi hust gkt lm trogsnmrrmd tk ogktbmr porty
Fohlradjm ]gavmrsaty Urmss
?=
Guhlmr ogd Oijmlro
Mxohpim 4 Hmgtoi oddatakg ogd sultroftakg ]sm tbm sujjmstmd strotmjy tk hmgtoiiy wkr` kut tbm ogswmr.
o
?6< + ?=> (portatakgagj)
l
<= + ?1 (fkhpmgsotagj)
f
=> ‖ ?2 (fkhpmgsotagj)
d
6= + 6> (dkuliagj kr boivagj)
VKI]TAKG o
M R U I O G OT A K G
?6< + ?=> 8 <22
l <= 4+ ?1 8
?55
65
+<
+
?55 + =5
+>
<55 +25
+2
+ ?1<= 8 <= + <5 ‖ ?
4
8 4= ‖ ? 8 44
f
=> ‖ ?2 6 8
?2‖=> 8 => ‖ <5 + <
2
8 6> + < 8 62
d 6= :+ 6> 8
+ 6>6= 8 6= + 6= + ?
?
8 :5 + ? 8 :?
Mxmrfasm ?F
\@A GJ
ZK ]
?
o Iast tbrmm wkrds tbot hmog
oddatakg.
H
l Iast tbrmm wkrds tbot hmog sultroftakg. <
6
Zratm tbm guhlmr wbafb as3
o
6 hkrm tbog :
l =2 hkrm tbog ??
f
: imss tbog ?1
d ?6: imss tbog ?=:
o Nagd tbm suh kn tbmsm poars kn guhlmrs. a l
a 4
=
< ogd >
aa ?1 ogd 2
aaa >< ogd :5
Nagd tbm dannmrmgfm lmtwmmg tbmsm poars kn guhlmrs. ?? ogd =
aa <1 ogd ?6
aaa ?5? ogd 16
Vtotm wbmtbmr mofb kn tbmsm stotmhmgts as trum kr noism.
o
4+67>
l ?? + ?1 ≭ 65
f ?6 ∜ 1 9 2
d
<> ∜ ?= ≤ ?5
m
n
?+:∜4≭4
=5 ∜ 9 6=
Javm tbm rmsuit nkr mofb kn tbmsm fkhputotakgs.
o
: pius ??
l << hagus 6
f tbm suh kn ?? ogd
d
?<2 odd ?<
m
n
6> to`m owoy ?=
tbm dannmrmgfm lmtwmmg ?6 ogd 4
© Dovad Jrmmgwkkd mt oi. <5?6 AVLG3 1:2??5:><>1:6 Ubktkfkpyagj as rmstraftmd ugdmr iow ogd tbas hotmraoi hust gkt lm trogsnmrrmd tk ogktbmr porty
Fohlradjm ]gavmrsaty Urmss
N
F O
T
\
BM
UV
H O T AF
O
I
I
Q
?>
?F
Fboptmr ? Fkhputotakg watb pksatavm agtmjmrs
\@A GJ
ZK ]
Mxohpim 4o
>
Hmgtoiiy gd tbm ogswmrs o <6 + 4? d <4> + =5< j ?45 <:6 + <62 4?5
tk tbmsm suhs. Bagt3 ]sm tbm portatakgagj strotmjy. l :? + <> f ?62 + 44? m 16: + ?? n ?654 + 4<16 b 615 44: + <5? ?6< a ?55 55? + ?5? 5?5
:
Hmgtoiiy gd tbm ogswmrs tk tbmsm dannmrmgfms. Bagt3 ]sm tbm portatakgagj strotmjy. o <1 ∜ ?2 l =: ∜ ∜ ?<2= n ?5 ?5? ∜ ?55
Mxohpim 4l
2
Hmgtoiiy gd tbm ogswmrs tk tbmsm suhs. Bagt3 ]sm tbm fkhpmgsotagj strotmjy. o ?= + 1 l >4 + ?? f ?1 + :> d ?2 + ??= m 6? + ?6> n <4= + =<
Mxohpim 4f
1
Hmgtoiiy gd tbm ogswmrs tk tbmsm dannmrmgfms. Bagt3 ]sm tbm fkhpmgsotagj strotmjy. o 6= ∜ ?? l 4= ∜ ?1 f ?=> ∜ 42 d <44 ∜ << m 6:> ∜ =1 n = ∜ ?11
Mxohpim 4d
?5 Hmgtoiiy gd tbm ogswmrs tk tbmsm suhs ogd dannmrmgfms. Bagt3 ]sm tbm dkuliagj kr
boivagj strotmjy. o <= + <> d <45 ∜ ?
l >= + >6 m
42< ∜ <45
H
N
F O
T
\
BM
UV
H O T AF
O
I
I
Q
f ?
?55> ∜ =54
?? Hmgtoiiy gd tbm ogswmrs tk tbmsm fkhputotakgs. o
?? + ?2 ∜ ?:
d ?6> + ?< ∜ ?= j ?5?5 ∜ ?? +
l 6: ∜ ?1 + 1
f ?5? ∜ ?= +
m <2 ∜ ?5 ∜ 1 + = b =∜:+<
n a
61 + :? ∜ ?5 ∜ ?5 ?5 ∜ <= + ?2 \@A GJ
ZK
?< Jory wkr`md : bkurs kg Hkgdoy, = bkurs kg Tumsdoy, ?6 bkurs kg Zmdgmsdoy, ?? bkurs kg
Tbursdoy ogd < bkurs kg Nradoy. Zbot as tbm tktoi guhlmr kn bkurs tbot Jory wkr`md duragj tbm wmm`; ?6 Ag o lottagj aggagjs, Ubai bat ?<>
rugs ogd Horak bat ?1 rugs. Bkw hogy hkrm rugs dad Ubai bat fkhpormd tk Horak;
© Dovad Jrmmgwkkd mt oi. <5?6 AVLG3 1:2??5:><>1:6 Ubktkfkpyagj as rmstraftmd ugdmr iow ogd tbas hotmraoi hust gkt lm trogsnmrrmd tk ogktbmr porty
Fohlradjm ]gavmrsaty Urmss
]
H
N
F O
T
\
BM
UV
H O T AF
O
I
I
Q
?:
Guhlmr ogd Oijmlro
\@ A GJ
ZK
?4 O norhmr rmdufmd bas fottim guhlmrs nrkh 2> tk =4. Bkw hogy fkws wmrm to`mg owoy;
]
H
T
\
BM
?= Loj O bos ?2 horlims ogd loj L bos : nmwmr horlims tbog loj O. Zbot as tbm tktoi guhlmr kn
horlims; ?> Hott bos 6> fords ogd Ogdy bos 6= hkrm fords tbog Hott. An tbmy fkhlagm tbmar fords, bkw hogy
dk tbmy bovm ag tktoi; ?: Mofb sadm kg o hojaf traogjim odds up tk tbm sohm guhlmr, os sbkwg ag tbas
mxohpim watb o suh kn ?< kg mofb sadm. 4 ?<
?< 6
=
< ?
>
?<
o
Uiofm mofb kn tbm dajats nrkh ? tk > ag o hojaf traogjim watb tbrmm dajats oikgj mofb sadm sk tbot mofb sadm odds up tk tbm javmg guhlmr. a 1 aa ?5
l Uiofm mofb kn tbm dajats nrkh ? tk 1 ag o hojaf traogjim watb nkur dajats oikgj mofb sadm sk tbot mofb sadm odds up tk tbm javmg guhlmr. a
<5
aa
<6
© Dovad Jrmmgwkkd mt oi. <5?6 AVLG3 1:2??5:><>1:6 Ubktkfkpyagj as rmstraftmd ugdmr iow ogd tbas hotmraoi hust gkt lm trogsnmrrmd tk ogktbmr porty
Fohlradjm ]gavmrsaty Urmss
N
F O
UV
H O T AF
O
I
I
Q
?2
?F
Fboptmr ? Fkhputotakg watb pksatavm agtmjmrs
\@A GJ
ZK ]
Tbm hmgtoi strotmjy kn portatakgagj as mosy tk oppiy nkr <6 + =4 lut bordmr nkr <6 + =1. Mxpioag wby. l Tbm hmgtoi strotmjy kn portatakgagj as mosy tk oppiy nkr ?=2 ∜ 4> lut bordmr nkr ?=? ∜ 4>. Mxpioag wby.
?2 o
H
T
?1 Fkhpimtm tbmsm guhlmr smgtmgfms an tbm imttmrso, l ogd f rmprmsmgt guhlmrs. o
o
+ l 8 f sk f ∜ [[ 8 o
l
o
+ f 8 l sk l ∜ o 8 [[ ?
<5 Tbas hojaf traogjim usms tbm dajats ? tk >, ogd bos mofb sadm oddagj tk tbm sohm
tktoi. Tbas mxohpim sbkws o sadm tktoi kn 1. o Bkw hogy dannmrmgt sadm tktois orm pkssalim usagj tbm sohm dajats; l Mxpioag ykur hmtbkd.
> <
= 4
6
guhlmrs;
Mgrafbhmgt3 Hojaf squorms << O hojaf squorm bos mvmry rkw, fkiuhg ogd hoag daojkgoi oddagj tk tbm
sohm guhlmr, foiimd tbm hojaf suh. Nkr mxohpim, tbas hojaf squorm bos o hojaf suh kn ?=. Nagd tbm hojaf suhs nkr tbmsm squorms, tbmg ii ag tbm hassagj guhlmrs.
o
l
> : <
=
f
?5 ??
?=
?6
?4
?<
?1
<5
d
?
?=
4
1
<
?=
6
=
:
?=
2
?
>
?=
?=
?=
?=
?=
?=
4
>
1 ??
?6
<
?>
Tbas hojaf squorm wos `gkwg ag ogfamgt Fbago os o ‒Ik Vbu‑ squorm ogd usms kgiy tbm guhlmrs ? tk 1. At as sbkwg ag tbm haddim kn tbas ogfamgt dmsajg os syhlkis kg o turtim sbmii, surrkugdmd ly tbm ogahois wbafb rmprmsmgt tbm trodatakgoi Fbagmsm gohms nkr tbm ymors.
© Dovad Jrmmgwkkd mt oi. <5?6 AVLG3 1:2??5:><>1:6 Ubktkfkpyagj as rmstraftmd ugdmr iow ogd tbas hotmraoi hust gkt lm trogsnmrrmd tk ogktbmr porty
Fohlradjm ]gavmrsaty Urmss
N
F O
\
BM
UV
H O T AF
O
I
I
Q
?1
Guhlmr ogd Oijmlro
?D
Oijkratbhs nkr oddagj ogd sultroftagj At as gkt oiwoys proftafoi tk hmgtoiiy skivm prklimhs agvkivagj oddatakg ogd sultroftakg. Nkr fkhpimx fkhputotakgs o prkfmdurm agvkivagj o guhlmr kn stmps fog lm usmd ogd tbas bmips tk javm tbm ogswmr. Vufb o prkfmdurm as foiimd og oijkratbh. Nkr tbm oddatakg oijkratbh, an twk dajats odd tk hkrm tbog 1, tbmg tbm bajbmr piofm voium dajat ag tbm suh fog lm forramd tk tbm gmxt fkiuhg. Nkr tbm sultroftakg oijkratbh, an twk dajats sultroft tk lm imss tbog 5 tbmg o ‒?‑ fog lm lkrrkwmd nrkh tbm gmxt fkiuhg.
Imt‑s stort3 Tbm hassagj dajats Dasfuss wbot dajats sbkuid jk ag tbm mhpty lkxms. Javm rmoskgs nkr ykur ogswmrs. ?
4
+ 1
=
?
<
▥
▥
▥
=
∜ =
?
4 1
4
Og oijkratbh as o prkfmdurm agvkivagj o guhlmr kn stmps tbot mvmgtuoiiy imods tk tbm ogswmr tk o prklimh. Oddatakg oijkratbh ‖ Orrogjm tbm guhlmrs vmrtafoiiy sk tbot tbm dajats watb sahaior piofm ? 4+< 8> <64 voium orm ag tbm sohm fkiuhg. 6 +1 8 ?< ?1< ‖ Odd dajats ag tbm sohm fkiuhg, stortagj kg tbm rajbt. 4<> + 8 ?+
8
∜
8
∜
▥
8
Mxohpim = ]sagj tbm oddatakg oijkratbh Javm tbm rmsuit nkr mofb kn tbmsm suhs. o
<> +
>>
461
l +
?:<
© Dovad Jrmmgwkkd mt oi. <5?6 AVLG3 1:2??5:><>1:6 Ubktkfkpyagj as rmstraftmd ugdmr iow ogd tbas hotmraoi hust gkt lm trogsnmrrmd tk ogktbmr porty
Fohlradjm ]gavmrsaty Urmss
s o m d ay m @
<5
Fboptmr ? Fkhputotakg watb pksatavm agtmjmrs
VKI]TAKG o
?
M R U I O G OT A K G
Odd tbm dajats vmrtafoiiy. > + > 8 ?<, sk forry tbm ? tk tbm tmgs fkiuhg.
<> >>
+
1<
l
?
+
1 + < 8 ??, sk forry o ? tk tbm tmgs fkiuhg. ? + 6 + : 8 ??, sk forry o ? tk tbm bugdrmds fkiuhg.
?
4 6 1
? : < > ? ?
Mxohpim > ]sagj tbm sultroftakg oijkratbh Javm tbm rmsuit nkr mofb kn tbmsm dannmrmgfms. :4 o
∜ ?=
>
M R U I O G OT A K G
Lkrrkw ? nrkh : tk ho`m ?4 ‖ = 8 1. Tbmg sultroft ? nrkh > (gkt :).
?
: 4
∜?
6<45
∜ <:
VKI]TAKG o
l
=
= 1
l
6
∜<
:
?
6<45
= ?
1
Lkrrkw ? nrkh 4 tk ho`m ?5 ‖ ? 8 1. Vultroft < nrkh 6 (gkt 4). Lkrrkw ? nrkh 6 tk ho`m ?< ‖ : 8 =. Gktm tbot < ‖ < 8 5 ogd yku dk gkt gmmd tk sbkw o 5 lmnkrm tbm =.
Mxmrfasm ?D
\@ A GJ
ZK
?
<
Hmgtoiiy gd tbm rmsuits tk tbmsm sahpim suhs. o 2+1 l 2: + ?4 f ?62 + > m 112 + : n ?1 + ?<4 j ?5< + 11 Hmgtoiiy gd tbm rmsuits tk tbmsm sahpim dannmrmgfms. o ?6 ∜ = l 6> ∜ 1 f := ∜ 2 m 6: ∜ << n ?54 ∜ ?< j 4> ∜ ?:
]
H
d 11 + ?? b =< + ?5=6
d ?55 ∜ ?> b ?55? ∜ <<
© Dovad Jrmmgwkkd mt oi. <5?6 AVLG3 1:2??5:><>1:6 Ubktkfkpyagj as rmstraftmd ugdmr iow ogd tbas hotmraoi hust gkt lm trogsnmrrmd tk ogktbmr porty
Fohlradjm ]gavmrsaty Urmss
N
F O
T
\
UV
B MH OTA
FO
I
I
Q
Guhlmr ogd Oijmlro
\@ A GJ
ZK
6
Zbot as tbm hassagj dajat ag mofb kn tbmsm fkhputotakgs; o l f < : 6 > ? < 6 +6
?
+?
=
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Javm tbm ogswmr tk mofb kn tbmsm oddatakgs. Fbmf` ykur ogswmrs watb o foifuiotkr. :4 o 6> l f d ?: 4: +<=
+=?
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Vbkw ykur wkr`agj tk gd tbm rmsuit nkr mofb kn tbmsm suhs. o 2= + :> l ?6? + 14 f ?:6< + 41:
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Nagd tbm ogswmrs tk tbmsm sultroftakgs. Fbmf` ykur ogswmrs watb o foifuiotkr. =4 4> o l 2= f d 14
∜<6
m
?<=
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2
Vbkw ykur wkr`agj tk gd tbm ogswmr tk mofb kn tbmsm dannmrmgfms. o 6< ∜ ?> l ?<4 ∜ :: f >?6 ∜ ?<2 d ?554 ∜ 262
1
Norhmr Jrmmg kwgs <2: sbmmp, norhmr Lrkwg
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\ TB M
kwgs =<> sbmmp ogd norhmr Jrmy kwgs ?54? sbmmp. Bkw hogy sbmmp orm tbmrm ag tktoi;
Fohlradjm ]gavmrsaty Urmss
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?5 O for‑s kdkhmtmr sbkws ?< ?62 `aikhmtrms ot tbm stort kn o ckurgmy ogd ?< :?4 `aikhmtrms
ZK ]
ot tbm mgd kn tbm ckurgmy. Zbot wos tbm ckurgmy‑s dastogfm;
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Bkw hogy studmgts orm tbmrm ag tktoi; Nagd tbm dannmrmgfm lmtwmmg tbm guhlmr kn studmgts ag tbm sfbkkis.
?< Nagd tbm hassagj dajats ag tbmsm suhs. o
l
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Narst wkr` kut tbm ogswmr tk tbmsm sahpim fkhputotakgs lmnkrm dkagj port l. a <2 + ?2 ∜ ?: aa 6> ∜ ?1 + <5 l Nkr port a olkvm, as at pkssalim tk wkr` kut ?2∜ ?: ogd tbmg odd tbas tktoi tk <2; f Nkr port aa olkvm, as at pkssalim tk wkr` kut ?1+ <5 ogd tbmg sultroft tbas tktoi nrkh 6>; d Fog yku sujjmst o jkkd hmgtoi strotmjy nkr port aa olkvm tbot javms tbm fkrrmft ogswmr;
?4 o
?= o l
Zbot orm tbm hassagj dajats ag tbas suh; Mxpioag wby tbmrm as hkrm tbog kgm pkssalim smt kn hassagj guhlmrs ag tbm suh javmg kppksatm. Javm skhm mxohpims.
H
<
6
+
4
<
?
?> Tbm suh kn twk guhlmrs as 1:2 ogd tbmar dannmrmgfm as :4. Zbot orm tbm twk guhlmrs; ?: Ho`m up skhm kn ykur kwg prklimhs ia`m _umstakg?> ogd tmst tbmh kg o nramgd.
Mgrafbhmgt3 Hkrm hojaf squorms ?2 Fkhpimtm tbmsm hojaf squorms. o
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Fohlradjm ]gavmrsaty Urmss
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Guhlmr ogd Oijmlro
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Huitapiyagj shoii pksatavm agtmjmrs Tbm huitapiafotakg kn twk guhlmrs rmprmsmgts o rmpmotmd oddatakg. Nkr mxohpim, 4Ú < fkuid lm tbkujbt kn os 4 jrkups kn < kr < + < + < + <. ○○○○ ○○○○
4Ú<
Vahaioriy, 4 Ú < fkuid lm tbkujbt kn os < jrkups kn 4 kr < Ú 4 kr 4 + 4. ○
○
○
○
○
○
○
○
Tbm prkduft 4 Ú 4 agvkivms 4 jrkups kn 4, wbafb hmogs tbm dkts nkrh o squorm. 4 Ú 4 fog lm rmod os ‒4 squormd‑ ogd wrattmg os 4<. Tbm guhlmr ?> as o squorm guhlmr. ○○○○
<Ú4
○○○○ ○○○○
4Ú4
○○○○
Imt‑s stort3 Zbafb as tbm fkrrmft hmgtoi strotmjy; Tbrmm studmgts mxpioag tbmar hmtbkd nkr gdagj tbm ogswmr tk ?<4 Ú 2. • Laiiy soys tbot yku fog dk ?<4 Ú ?5 tk jmt ?<45, tbmg sultroft < tk jmt ?<62. • Imo soys tbot yku boivm ?<4 ogd 2 twafm mofb tk javm 6?Ú < 8 ><. • Vuroa soys tbot yku huitapiy 2 ly 4 tk javm 6<, 2 ly < tk javm ?> ogd 2 ly ? tk javm 2. Vbm soys tbm tktoi as tbmrmnkrm 6< + ?> + 2 8 =>. Orm ogy kn tbm studmgts fkrrmft ogd fog yku mxpioag ogy mrrkrs ag tbmar tbag`agj;
▥ ▥
▥
▥
▥
Nagdagj tbm prkduft kn twk guhlmrs agvkivms huitapiafotakg. Zm soy ‒tbm prkduft kn < ogd 6 as >‑. Ú686Ú< oÚl8lÚo m.j. < ‖ Tbas as tbm fkhhutotavm iow nkr huitapiafotakg, hmogagj tbot tbm krdmr dkms gkt hottmr. (o Ú l) Ú f 8 o Ú (l Ú f) m.j. (6 Ú =) Ú 4 8 6 Ú (= Ú 4) ‖ Tbas as tbm osskfaotavm iow nkr huitapiafotakg, hmogagj at dkms gkt hottmr wbafb poar as huitapiamd rst. Tbm huitapiafotakg oijkratbh nkr huitapiyagj ly o sagjim dajat agvkivms3 ‖ Huitapiyagj tbm sagjim dajat ly mofb dajat ag tbm ktbmr guhlmr, ? stortagj nrkh tbm rajbt. <6 Ú 4 ‖ Forry ogd odd ogy dajats watb o bajbmr piofm voium tk tbm tktoi ag 1< tbm gmxt fkiuhg. Hmgtoi strotmjams nkr huitapiafotakg agfiudm3 ‖ Hmhkrasagj tbm huitapiafotakg tolims. ‖ ]sagj tbm fkhhutotavm iow ly fbogjagj tbm krdmr. Nkr
1 Ú : 8>
mxohpim, 46 Ú < hajbt lm tbkujbt kn hkrm mosaiy os < jrkups kn 46 kr < Ú 46. ‖ ]sagj tbm fkhhutotavm ogd osskfaotavm iow ly oitmragj tbm krdmr an hkrm tbog kgm guhlmr as lmagj huitapiamd. ▥
6
4 Ú 6 8 ?< 4Ú<+?81
?<Ú 6 8 6>
== Ú <5 8 = Ú ?? Ú < Ú ?5 8 = Ú < Ú ?? Ú ?5 8 ?5 Ú ?? Ú ?5 8 ??55
Zbmg o guhlmr as huitapiamd ly atsmin at as soad tk lm ‒squormd‑. Ú o 8 o< m.j. = Ú = 8 =< 8 <= Zbmg tbm guhlmr = as squormd, tbm rmsuit as <=. Vk tbm guhlmr <= as o squorm guhlmr.
o
© Dovad Jrmmgwkkd mt oi. <5?6 AVLG3 1:2??5:><>1:6 Ubktkfkpyagj as rmstraftmd ugdmr iow ogd tbas hotmraoi hust gkt lm trogsnmrrmd tk ogktbmr porty
Fohlradjm ]gavmrsaty Urmss
s o m a d y m @
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Fboptmr ? Fkhputotakg watb pksatavm agtmjmrs
‖ ]sagj tbm dastralutavm iow ly ho`agj o ?5, ?55 mtf. ogd tbmg odcustagj ly oddagj kr sultroftagj. Tbm dastralutavm iow aso Ú (l + f) 8 (o Ú l) + (o Ú f) kr o Ú (l ∜ f) 8 (o Ú l) ∜ (l Ú f). Tbas waii lm usmd hkrm mxtmgsavmiy ag tbm oijmlro fboptmrs.
s o m d a y m @
> Ú Ú <5)?)+ (> Ú 8 ?<5 + > 8 ?<>
:
Ú ?2 8 (: Ú <5) ∜ (: Ú <) 8 ?45 ∜ ?4 8 ?<>
‖ ]sagj tbm dkuliagj ogd boivagj strotmjy ly dkuliagj kgm
= Ú : Ú 4 8 ?5 Ú : Ú <
8 :5 Ú < 8 ?45 ?? Ú ?< 8 ?? Ú > Ú < 8 >> Ú < 8 ?6<
guhlmr ogd boivagj tbm ktbmr.
‖ ]sagj noftkrs tk spiat o guhlmr.
Mxohpim : ]sagj hmgtoi strotmjams nkr huitapiafotakg ]sm o hmgtoi strotmjy tk gd tbm ogswmr tk mofb kn tbmsm prkdufts. o :Ú> l 6 Ú ?6 f 4 Ú <1 d = Ú <4 m : Ú ?4 n ?<< VKI]TAKG
M R U I O G OT A K G
o4 : Ú > 8
Ú > kr > Ú: : sbkuid lm hmhkrasmd (nrkh huitapiafotakg tolims).
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l 66 Ú ?6 8
Ú ?6 8 (66 Ú ?5) + (6 Ú 6) 8 65 + 1 8 61 (Tbm dastralutavm iow as lmagj usmd.)
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Ú <1 84 (4 Ú 65) ∜ (4 Ú ?) 8 ?<5 ∜ 4 8 ??> (Tbm dastralutavm iow as lmagj usmd.)
d = ?Ú <4 8
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Ú <4 8= ?5 Ú ?< 8 ?<5 (Tbm dkuliagj ogd boivagj strotmjy as lmagj usmd.)
f
m 1: Ú ?4 8
n
?<< 8 ?< Ú ?< 8?44
Ú ?4 8 ::Ú : Ú < 8 41 Ú < 8 12 (Noftkrs kn ?4 orm usmd.)
2
Tbas as rmod os ‒?< squormd‑, wbafb hmogs tbot ?< as huitapiamd ly atsmin. Tbm guhlmr ?44 as o squorm guhlmr.
Mxohpim 2 ]sagj tbm huitapiafotakg oijkratbh Javm tbm rmsuit nkr mofb kn tbmsm prkdufts. o 6? Ú 4 l
?1: Ú :
© Dovad Jrmmgwkkd mt oi. <5?6 AVLG3 1:2??5:><>1:6 Ubktkfkpyagj as rmstraftmd ugdmr iow ogd tbas hotmraoi hust gkt lm trogsnmrrmd tk ogktbmr porty
Fohlradjm ]gavmrsaty Urmss
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4Ú?84 4 Ú 6 8 ?<
6? Ú
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Zratm tbm gmxt tbrmm guhlmrs ag tbmsm pottmrgs. o 4, 2, ?<, ?>, [[ l ??, <<, 66, [[
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Orm tbmsm stotmhmgts trum kr noism; o 4Ú686Ú4 l <Ú=Ú>8>Ú=Ú< d 6 Ú 6< 8 6 Ú 65 + 6 Ú < m = Ú ?2 8 ?5 Ú 1 j ?1 Ú : 8 <5 Ú : ∜ ?1 b 61 Ú 4 8 45 Ú 4 ∜ ? Ú 4
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Mxohpim :o,n
4
]sagj ykur `gkwimdjm kn huitapiafotakg tolims, javm tbm ogswmr tk tbmsm prkdufts. o 2Ú: l >Ú1 f ?< Ú 4 d ??< m > Ú ?< n :Ú= j ?< Ú 1 b ?6 Ú 6
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Nagd tbm rmsuits tk tbmsm prkdufts hmgtoiiy. Bagt3 ]sm tbm dastralutavm iow strotmjy ‖ sultroftakg nkr o tk d ogd oddatakg nkr m tk b. o 6 Ú ?1 l > Ú <1 f 4 Ú <2 d 62 Ú : m = Ú Ú 4< b =6 Ú 6
Mxohpim :d,m
>
Nagd tbm ogswmr tk tbmsm prkdufts hmgtoiiy. Bagt3 ]sm tbm dkulim ogd boivm strotmjy kr spiat o guhlmr usagj ats noftkrs. o 4 Ú <4 l 6 Ú ?2 f > Ú ?> d <4 Ú 6
Mxohpim 2
:
Javm tbm rmsuit kn mofb kn tbmsm prkdufts, usagj tbm huitapiafotakg oijkratbh. Fbmf` ykur rmsuits watb o foifuiotkr. 66 o l f d 46 :< ==
m
<
Ú
?<1 Ú
<
n
6
Ú
45: Ú
:
j
>
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=<> Ú
=
b
6
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© Dovad Jrmmgwkkd mt oi. <5?6 AVLG3 1:2??5:><>1:6 Ubktkfkpyagj as rmstraftmd ugdmr iow ogd tbas hotmraoi hust gkt lm trogsnmrrmd tk ogktbmr porty
1
Fohlradjm ]gavmrsaty Urmss
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H O T AF
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F O
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2
Nagd tbm ogswmr tk tbmsm prkdufts, sbkwagj ykur wkr`agj. o 4: Ú = l ?61? Ú 6 f 1 Ú 4<=
\@ A GJ
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d : Ú 4?:5
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1
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O farfuior rofm trof` as <45 hmtrms ikgj ogd \kry rugs smvmg iops. Bkw nor dkms \kry rug ag tktoi;
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?5 Majbt taf`mts fkstagj $66 mofb orm purfbosmd nkr o fkgfmrt. Zbot
as tbm tktoi fkst kn tbm taf`mts; ?? \mjjam ogd Ogjmik fkhlagm tbmar pof`s kn fords. \mjjam bos vm
smts kn ?6 fords ogd Ogjmik bos tbrmm smts kn ?: fords. Bkw hogy fords orm tbmrm ag tktoi; ?< Voio purfbosms skhm jkkds nkr o porty ot og kutimt stkrm ogd bos $?55 tk spmgd. Vbm smimfts majbt
lkttims kn drag` nkr $< mofb, ?6 nkkd pof`s ot $> mofb ogd ?2 porty bots ot =5 fmgts mofb. Dkms sbm bovm mgkujb hkgmy tk poy nkr oii tbm atmhs; ?6 Nagd tbm hassagj dajats ag tbmsm prkdufts. 6
o
m
1
Ú
:
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6
< 2
f
Ú
1
n
d
: 1 Ú
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:
Ú
< =
l
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6 :
< 6
j
1
Ú
=
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6 = ?
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5
>
b
4 Ú
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1
2 \@ A GJ
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?4 Tbm fkhhutotavm ogd osskfaotavm iows nkr huitapiafotakg hmog tbot guhlmrs fog lm huitapiamd H
ag ogy krdmr. Vk (o Ú l) Ú f 8 (l Ú o) Ú f 8 l Ú (o Ú f) 8 [[[, wbmrm tbm lrof`mts sbkw wbafb guhlmrs orm huitapiamd rst. Ag bkw hogy woys fog <Ú 6 Ú = lm foifuiotmd;
]
T
\
BM
?= Tbm dastralutavm iow fog bmip tk wkr` kut prkdufts hmgtoiiy.
m.j. : Ú 6? 8 (: Ú 65) + (: Ú ?) 8
℮Ϗατν νοϊκ λώ τκν ώλΰΰλραθζ οφ φαθζΰν υϏλδύϊτφ. ∎λ θλτ ώαθδ τκν οθφρνϏ. o 6 Ú <5 + 6 Ú ? l 1 Ú =5 + 1 Ú < f : Ú 65 + : Ú < < d = Ú ?55 ∜ = Ú 6 m o Ú 45 ∜ o Ú < n o Ú <55 + o Ú 6 Ú
?> Bkw hogy dannmrmgt woys fog tbm twk spofms lm iimd ag tbas prklimh; Mxpioag wby.
2
Mgrafbhmgt3 Hassagj dajats ?: Nagd oii tbm hassagj dajats ag tbmsm prkdufts. o
l
? :
Ú
= ?
< 1 6
Ú
2
?2 Tbm prkduft kn twk guhlmrs as ?6< ogd tbmar suh as <2. Zbot orm tbm twk guhlmrs; © Dovad Jrmmgwkkd mt oi. <5?6 AVLG3 1:2??5:><>1:6 Ubktkfkpyagj as rmstraftmd ugdmr iow ogd tbas hotmraoi hust gkt lm trogsnmrrmd tk ogktbmr porty
Fohlradjm ]gavmrsaty Urmss
N
F O
6 4 <
UV
H O T AF
I
Q
<:
Guhlmr ogd Oijmlro
?N
Huitapiyagj iorjm pksatavm agtmjmrs Tbmrm orm hogy satuotakgs tbot rmquarm tbm huitapiafotakg kn iorjm guhlmrs0 nkr mxohpim, tbm tktoi rmvmgum nrkh smiiagj 45 555 taf`mts ot $<6 mofb, kr tbm ormo kn o rmftogjuior por` watb imgjtb ogd wadtb dahmgsakgs kn >= hmtrms ly ?<< hmtrms. Tk fkhpimtm sufb foifuiotakgs ly bogd rmquarms tbm usm kn o suatolim oijkratbh.
Bkw hufb rmvmgum fohm nrkh smiiagj taf`mts tk tbas johm;
Imt‑s stort3 Vpkt tbm mrrkrs
<:?
Tbmrm orm tbrmm typms kn mrrkrs ag tbm wkr`agj sbkwg nkr tbas prklimh. Nagd tbm mrrkrs ogd dmsfralm tbmh.
Ú
?6
>?6 <:? ?<:6
▥
▥
▥
Zbmg huitapiyagj ly ?5, ?55, ?555, ?5 555 mtf. mofb dajat hkvms tk tbm imnt ly tbm guhlmr kn zmrks0 m.j. 4= Ú ?555 8 4= 555. O strotmjy nkr huitapiyagj ly huitapims kn ?5, ?55 mtf. as tk rst huitapiy ly tbm guhlmr watbkut tbm zmrks tbmg odd tbm zmrks tk tbm ogswmr iotmr. Nkr mxohpim, 6 Ú ?555 8 >6 555 Tbm oijkratbh nkr huitapiyagj iorjm guhlmrs agvkivms davadagj tbm prklimh agtk shoiimr prkdufts ogd tbmg oddagj tbm tktois0 m.j.
?46 Ú
?4
=:< ?465 <55<
© Dovad Jrmmgwkkd mt oi. <5?6 AVLG3 1:2??5:><>1:6 Ubktkfkpyagj as rmstraftmd ugdmr iow ogd tbas hotmraoi hust gkt lm trogsnmrrmd tk ogktbmr porty
↕ ?46 Ú 4 ↕ ?46 Ú ?5 ↕ ?465 + =:<
Fohlradjm ]gavmrsaty Urmss
s o m d a y m @
<2
Fboptmr ? Fkhputotakg watb pksatavm agtmjmrs
Mxohpim 1 Huitapiyagj iorjm guhlmrs Javm tbm rmsuit nkr mofb kn tbmsm prkdufts. o 6: Ú ?55 l 4= Ú :5 VKI]TAKG
d <=<
>?4 Ú ?4
M R U I O G OT A K G
6: Ú ?55 8 6 :55
o
f
Hkvmtbm6ogdtbm:twkpiofmstktbmimnt ogdodd twk zmrks.
l 4= Ú :5 8 4= Ú : Ú ?5 8 6?= Ú ?5
Narst huitapiy ly :, tbmg huitapiy ly ?5 iotmr. 4= :
Ú
8 6?=5
6?=
f
>?4 Ú
Narst huitapiy >?4 Ú 4. Tbmg huitapiy >?4 Ú ?5. Odd tbm tktois tk javm tbm ogswmr.
?4
<4=> >?45 2=1>
Narst huitapiy <= Ú =. Tbmg huitapiy <= Ú <5.
d <= Ú
<=
?<=
Odd tbm tktois tk javm tbm ogswmr.
=55
<
Tbas fog lm wrattmg os <= 8 ><=.
><=
Mxmrfasm ?N
\@ AGJ
ZK
?
<
6
Zbot as tbm hassagj dajat ag tbmsm prkdufts; o :< Ú ?5 8 :;5 l ?6 Ú ?55 8 ?65; f
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41 Ú ?55 8 41;5
Zbot as tbm hassagj guhlmr ag tbmsm prkdufts; o ?= Ú [[ 8 ?=55 l 6<1 Ú [[ 8 6<15
1<4 Ú ?5 8 1<;5
d
N
F O
T
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H O T AF
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f 1< Ú [[ 8 1<5 555
Vtotm an tbm nkiikwagj foifuiotakgs orm fkrrmft. An tbmy orm agfkrrmft, gd tbm fkrrmft ogswmr. <>
o Ú
4
24
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l Ú
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1<
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<4
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d
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452
61 ?1=
?245 <<52
<45 >42 \@ AG J
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Mxohpim 1o
4
Javm tbm rmsuit kn mofb kn tbmsm prkdufts. o 4 Ú ?55 l <1 Ú ?5 f m =5 Ú ?555 n >65 Ú ?55 j
H
?26 Ú ?5 ?44? Ú ?5
d 4> Ú ?55 b <1?5 Ú ?5 555
© Dovad Jrmmgwkkd mt oi. <5?6 AVLG3 1:2??5:><>1:6 Ubktkfkpyagj as rmstraftmd ugdmr iow ogd tbas hotmraoi hust gkt lm trogsnmrrmd tk ogktbmr porty
Fohlradjm ]gavmrsaty Urmss
N
F O
T
\
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UV
H O T AF
O
I
I
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Bkw fkuid yku foifuiotm tbm guhlmr kn pamfms kn nruat kg tbas stoii watbkut fkugtagj tbmh oii; \@ AGJ
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Mstahotm tbm ogswmrs tk tbmsm prkdufts, tbmg usm o foifuiotkr tk fbmf`. o ?1 Ú ?? l <> Ú
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O pkki ormo agfiudms >2 squorm hmtrms kn povagj ot $6< pmr squorm hmtrm. Zbot as tbm tktoi fkst kn povagj;
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?< Tbmrm orm 6>5 dmjrmms ag o nuii turg. Bkw hogy dmjrmms dkms tbm hagutm bogd kg o fikf` turg ag kgm
wmm`;
© Dovad Jrmmgwkkd mt oi. <5?6 AVLG3 1:2??5:><>1:6 Ubktkfkpyagj as rmstraftmd ugdmr iow ogd tbas hotmraoi hust gkt lm trogsnmrrmd tk ogktbmr porty
Fohlradjm ]gavmrsaty Urmss
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Fboptmr ? Fkhputotakg watb pksatavm agtmjmrs
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?6 Tbm prkduft kn twk wbkim guhlmrs as imss tbog tbmar suh. Gmatbmr guhlmr as zmrk kr imss.
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Zbot hust lm trum olkut kgm kn tbm guhlmrs;
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12 Ú ?> + < Ú ?> 66 Ú <> ∜ 6 Ú <> f ?1 Ú ?= + 64 Ú ?: ∜ 4 Ú ?: + ? Ú ?= d << Ú ?1 ∜ 6 Ú ?: + =? Ú 1 ∜ ? Ú 1 + ?6 Ú ?: ∜ < Ú ?1 o
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Mgrafbhmgt3 Huitapiafotakg puzzim Zbot as tbm iorjmst guhlmr yku fog ho`m ly fbkksagj vm dajats nrkh tbm iast ?, <, 6, 4, =, >, :, 2, 1 ogd piofagj tbmh agtk tbm prkduft sbkwg ot rajbt; l Zbot as tbm shoiimst guhlmr yku fog ho`m ly fbkksagj vm dajats nrkh tbm iast ?, <, 6, 4, =, >, :, 2, 1 ogd piofagj tbmh agtk tbm prkduft sbkwg ot rajbt;
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?: Tbm prkduft kn twk wbkim guhlmrs as ?4 61? ogd tbmar dannmrmgfm as >. Zbot orm tbm twk guhlmrs;
2< 8 2 Ú 2 8 >4 >4 as o squorm guhlmr. Nagd oii tbm tbrmm-dajat squorm guhlmrs ag wbafb tbm rst dajat as ? kr <. l Zbot as tbm iorjmst tbrmm-dajat squorm guhlmr; f Zbot as tbm iorjmst nkur-dajat squorm guhlmr;
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© Dovad Jrmmgwkkd mt oi. <5?6 AVLG3 1:2??5:><>1:6 Ubktkfkpyagj as rmstraftmd ugdmr iow ogd tbas hotmraoi hust gkt lm trogsnmrrmd tk ogktbmr porty
Fohlradjm ]gavmrsaty Urmss
UV
H O T AF
?4 An lktb guhlmrs ag o huitapiafotakg fkhputotakg bovm ot imost tbrmm dajats, tbmg tbm oijkratbh gmmds
tk lm mxpogdmd. ]sm tbm oijkratbh tk gd tbmsm prkdufts. ?5?6 614: o l f <14
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Davadagj pksatavm agtmjmrs ogd dmoiagj watb rmhoagdmrs Davasakg agvkivms gdagj tbm guhlmr kn mquoi jrkups agtk wbafb o portafuior guhlmr fog lm davadmd. Tbas fog lm ofbamvmd lktb watb ogd watbkut o rmhoagdmr kr ‒imnt kvmr‑. Davadagj <5 oppims ohkgjst vm pmkpim kr $?5 555 lmtwmmg tbrmm log` offkugts orm mxohpims kn wbmg davasakg fog lm usmd. Huitapiafotakg ogd davasakg orm rmvmrsm kpmrotakgs, ogd tbas as sbkwg ag tbas sahpim mxohpim3 : Ú
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ú 6:8
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Imt‑s stort3 Orrogjagj fkugtmrs O tktoi kn <4 fkugtmrs sat kg o tolim. ]sagj wbkim guhlmrs, ag bkw hogy woys fog tbm fkugtmrs lm davadmd agtk mquoi-sazmd jrkupswatb gk fkugtmrs rmhoagagj; • As at oisk pkssalim tk davadm tbm fkugtmrs agtk mquoi-sazmd jrkups lut watb twk fkugtmrs rmhoagagj; • An vm fkugtmrs orm tk rmhoag, bkw hogy mquoi-sazmd jrkups fog lm nkrhmd ogd wby;
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••••• ••••• ••••• ••••• ••••
Tbm guhlmr kn mquoi-sazmd jrkups nkrhmd nrkh tbm davasakg kpmrotakg as foiimd tbm quktamgt. Tbm tktoi lmagj davadmd as foiimd tbmdavadmgd ogd tbm sazm kn tbm mquoi jrkups as foiimd tbm davaskr. :1 ú : 8 ?? ogd < rmhoagdmr 8 ?? < Ogy ohkugt rmhoagagj ontmr davasakg agtk :
mquoi-sazmd jrkups as foiimd tbm rmhoagdmr. quktamgt tktoi lmagj davadmd sazm kn mquoi ‖ :1 ú : 8 ?? ogd < rmhoagdmr hmogs (davadmgd) jrkups (davaskr) :1 8 ?? Ú : + <. o ú l ≥ l ú o (jmgmroiiy) ‖ Tbm fkhhutotavm iow dkms gkt bkid nkr davasakg. 2 <ú < 28/ ú (o ú l) ú f ≥ o ú (l ú f) (jmgmroiiy) ‖ Tbm osskfaotavm iow dkms gkt bkid nkr davasakg. (2 ú )4 <ú 2(8/ 4) ú< ú Tbm sbkrt davasakg oijkratbh agvkivms rst davadagj ?? ú 6 8 6 ogd agtk tbm dajat watb tbm bajbmst piofm voium ogd tbmg 4 ú 6 8 ? ogd < rmh. forryagj ogy rmhoagdmr tk tbm gmxt dajat, wkr`agj ? rmh. <6 ú 6 8 : ogd nrkh imnt tk rajbt. 4?6 ú 6 8 ?6: ogd < rmhoagdmr < rmh. ?6
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Hmgtoi davasakg fog lm dkgm usagj dannmrmgt strotmjams. ‖ @gkwagj ykur huitapiafotakg tolims knn ly bmort. ‖ >6 ú 1 8 ; as tbm sohm os os`agj 1Ú ; 8 >6. ‖ Ho`agj o fkgvmgamgt huitapim kn tbm davaskr ogd tbmg odcustagj ly oddagj kr sultroftagj.
© Dovad Jrmmgwkkd mt oi. <5?6 AVLG3 1:2??5:><>1:6 Ubktkfkpyagj as rmstraftmd ugdmr iow ogd tbas hotmraoi hust gkt lm trogsnmrrmd tk ogktbmr porty
Fohlradjm ]gavmrsaty Urmss
s o m ad y m @
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Fboptmr ? Fkhputotakg watb pksatavm agtmjmrs
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Lmikw as og oppiafotakg kn tbm dastralutavm iow. ú 6 8 (15 ∜ >) ú 6 24 ú 6 8 (>5 6 + <4) ú 24 8 (>5 ú 6) + (<4 ú 6) 8 (15 ú 6) ∜ (> ú 6) 8 <5 + 2 8 65 ∜ < 8 <2 8 <2 ‖ Boivagj lktb guhlmrs. An lktb guhlmrs ag tbm davasakg orm mvmg, tbmg boivm lktb guhlmrs.
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Mxohpim ?? ]sagj tbm sbkrt davasakg oijkratbh ]sm tbm sbkrt davasakg oijkratbh tk gd tbm quktamgt ogd rmhoagdmr. o 6) 6: l :)?1= VKI]TAKG o
M R U I O G OT A K G
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© Dovad Jrmmgwkkd mt oi. <5?6 AVLG3 1:2??5:><>1:6 Ubktkfkpyagj as rmstraftmd ugdmr iow ogd tbas hotmraoi hust gkt lm trogsnmrrmd tk ogktbmr porty
Fohlradjm ]gavmrsaty Urmss
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Zratm tbm guhlmr tbot as hassagj ag tbmsm stotmhmgts. o 2 ú < 8 4 as tbm sohm os 4 Ú ; 8 2. l 6> ú ?< 8 6 as tbm sohm os ; Ú ?< 8 6>. f 4< ú ; 8 > as tbm sohm os > Ú : 8 4<. d :< ú > 8 ; as tbm sohm os ?< Ú > 8 :<. Zbot as tbm rmhoagdmr wbmg3 o < as davadmd agtk :; f 4< as davadmd ly 2;
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Nagd tbm ogswmr tk tbmsm usagj o hmgtoi strotmjy. Bagt3 ]sm tbm dastralutavm iow strotmjy. o >6 ú 6 l :> ú 4 f =: ú 6 d <5= ú = m <56 ú : n ?21 ú 1 j 15> ú 6 b 415 ú =
Mxohpim ?5f
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Nagd tbm ogswmrs tk tbmsm usagj o hmgtoi strotmjy. Bagt3 ]sm tbm boivagj strotmjy ly boivagj lktb guhlmrs. o 22 ú 4 l ?<4 ú 4 f ?6> ú 2 d ??< ú ?>
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]sm tbm sbkrt davasakg oijkratbh tk gd tbm quktamgt ogd rmhoagdmr. Fbmf` ykur ogswmrs usagj o foifuiotkr. o 6) :? l :) 1< f <)?61 d >)<4:
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]sm tbm sbkrt davasakg oijkratbh ogd mxprmss tbm rmhoagdmr os o nroftakg. o =<> ú 4 l ?>1? ú : f <64= ú > d 1< 66: ú 2
© Dovad Jrmmgwkkd mt oi. <5?6 AVLG3 1:2??5:><>1:6 Ubktkfkpyagj as rmstraftmd ugdmr iow ogd tbas hotmraoi hust gkt lm trogsnmrrmd tk ogktbmr porty
Fohlradjm ]gavmrsaty Urmss
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H O T AF
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Fboptmr ? Fkhputotakg watb pksatavm agtmjmrs
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?5 An ??: nkkd pof`s orm davadmd mquoiiy ohkgj gagm nohaiams, bkw hogy pof`s dkms mofb nohaiy H
rmfmavm;
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?? Vpragj Nrmsb Fkhpogy smiis hagmroi wotmr ag
pof`s kn sax lkttims. Bkw hogy pof`s orm tbmrm ag o truf` fkgtoagagj :44 lkttims;
?< O lraf`ioymr morgs $?
Bkw hufb dkms bm morg pmr doy an bm wkr`s Hkgdoy tk Nradoy; l Bkw hufb dkms bm morg pmr bkur an bm wkr`s 1 bkurs pmr doy Hkgdoy tk Nradoy;
bm
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ktbmr pksts tbot orm davadmd mvmgiy oikgj tbm nmgfm 4 hmtrms oport. An tbm nmgfm as tk lm <>4 hmtrms ikgj, bkw hogy pksts orm gmmdmd, agfiudagj tbm mgd pksts; ?4 Nramgdiy Toxas fog to`m up tk nkur possmgjmrs
mofb. Bkw hogy toxas orm rmquarmd tk trogspkrt =1 pmkpim;
?= O truf` fog forry up tk : tkggms kn rkf`. Zbot as tbm hagahuh guhlmr kn traps gmmdmd tk trogspkrt
?65 tkggms kn rkf`; ?> Oii tbm rkws, fkiuhgs ogd hoag daojkgois ag tbm hojaf squorm huitapiy tk
javm . Fog yku gd tbm hassagj guhlmrs;
1
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© Dovad Jrmmgwkkd mt oi. <5?6 AVLG3 1:2??5:><>1:6 Ubktkfkpyagj as rmstraftmd ugdmr iow ogd tbas hotmraoi hust gkt lm trogsnmrrmd tk ogktbmr porty
Fohlradjm ]gavmrsaty Urmss
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H O T AF
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ogd = rmhoagdmr hmogs 1> 8 ?6 Ú rmhoagdmr hmogs ?54 8
+ =. Ú <5 + 4.
?2 Uams orm purfbosmd wbkimsoim ot
gagm nkr $4. Bkw hufb waii at fkst tk purfbosm ?=6 pams; ?1 Javm tbm rmsuits tk tbmsm prklimhs,
an o rmprmsmgts ogy guhlmr ktbmr tbog 5. o oú? l 5úo f oúo <5 O guhlmr imss tbog 65 imovms o
rmhoagdmr kn 6 wbmg davadmd ly = ogd o rmhoagdmr kn < wbmg davadmd ly 6. Zbot twk guhlmrs hmmt tbm javmg fkgdatakgs;
ú l 8 l ú o;
<< Tbm sbkrt davasakg oijkratbh fog oisk lm usmd tk davadm ly guhlmrs watb hkrm tbog kgm dajat.
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]sm tbm sbkrt davasakg oijkratbh ogd mxprmss tbm rmhoagdmr os o nroftakg. o 6:? ú ?? l 1<> ú ?: f 454 ú ?6 d ?>1?4 ú =>
Mgrafbhmgt3 Ikgj, sbkrt davasakg <6 ]sm tbm sbkrt davasakg oijkratbh tk gd tbm quktamgt ogd rmhoagdmr. o
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<4 Tbm hojaf prkduft nkr tbas squorm as >:<5. Nagd tbm hassagj guhlmrs.
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Fohlradjm ]gavmrsaty Urmss
N
F O
T
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UV
H O T AF
O
I
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Q
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Fboptmr ? Fkhputotakg watb pksatavm agtmjmrs
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Mstahotagj ogd rkugdagj pksatavm agtmjmrs Hogy tbmkrmtafoi ogd proftafoi prklimhs dk gkt gmmd prmfasm kr mxoft ogswmrs. Ag sufb satuotakgs rmoskgolim mstahotakgs fog prkvadm mgkujb agnkrhotkg tk skivm tbm prklimh. Tbm tktoi rmvmgum nrkh tbm Oustroiaog Kpmg tmggas tkurgohmgt dmpmgds kg frkwd guhlmrs. Mstahotms wkuid lm usmd lmnkrm tbm tkurgohmgt lmjags tk prmdaft tbmsm guhlmrs. Og mstahotm nkr tbm tktoi rmvmgum hajbt lm $2 haiiakg.
Imt‑s stort3 Tbm tmggas frkwd Bmrm as o pbktk kn o frkwd ot o tmggas hotfb. Dmsfralm bkw yku hajbt mstahotm tbm guhlmr kn pmkpim ag tbm pbktk. Zbot as ykur ogswmr; Bkw dannmrmgt as ykur ogswmr nrkh tbksm kn ktbmrs ag ykur fioss;
Bkw fog yku mstahotm tbm guhlmr kn spmftotkrs;
s o m d a y m @
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Mstahotms kr opprkxahotakgs tk tbm ogswmrs kn prklimhs fog lm nkugd ly rkugdagj guhlmrs tk tbm gmormst ?5, ?55, ?555 mtf. An tbm gmxt dajat as 5, ?, <, 6 kr 4, tbmg rkugd dkwg. An tbm gmxt dajat as =, >, :, 2 kr 1, tbmg rkugd up. Imodagj dajat opprkxahotakg rkugds tbm rst dajat tk tbm gmormst ?5 kr ?55 kr ?555 mtf. m.j. Nkr 16< usm 155 Nkr 1>2 usm ?555 . Tbm syhlkis ≂ kr 8. hmog ‒as opprkxahotmiy mquoi tk‑.
Mxohpim ?< \kugdagj \kugd tbmsm guhlmrs os agdafotmd. o
2> (tk tbm gmormst ?5)
VKI]TAKG o
. 2> ≂ 15 kr 2> 8. 15
. l 4?4< ≂ 4?55 kr 4?4< 8. 4?55
l
4?4< (tk tbm gmormst ?55)
M R U I O G OT A K G
Tbm dajat ontmr tbm 2 as jrmotmr tbog kr mquoi tk =, sk rkugd up. Tbm dajat ontmr tbm ? as imss tbog kr mquoi tk 4, sk rkugd dkwg.
© Dovad Jrmmgwkkd mt oi. <5?6 AVLG3 1:2??5:><>1:6 Ubktkfkpyagj as rmstraftmd ugdmr iow ogd tbas hotmraoi hust gkt lm trogsnmrrmd tk ogktbmr porty
Fohlradjm ]gavmrsaty Urmss
6:
Guhlmr ogd Oijmlro
Mxohpim ?6 ]sagj imodagj dajat opprkxahotakg Mstahotm tbm ogswmrs tk tbmsm prklimhs ly rkugdagj mofb guhlmr tk tbm imodagj dajat. o 4< Ú : l 1= Ú 6<> VKI]TAKG o
M R U I O G OT A K G
4< Ú : ≂ 45 Ú : 8 <25
. l 1= Ú 6<> 8. ?55 Ú 655 8
65 555
Tbm imodagj dajat ag 4< as tbm 4 ag tbm ‒tmgs‑ fkiuhg.
Tbm gmormst ‒tmg‑ tk 1= as ?55, ogd tbm imodagj dajat ag 6<> as ag tbm ‒bugdrmds‑ fkiuhg.
Mxohpim ?4 Mstahotagj watb kpmrotakgs Mstahotm tbm ogswmrs tk tbmsm prklimhs ly rkugdagj lktb guhlmrs os agdafotmd. o ??= Ú 1< (tk tbm gmormst ?55) l <<>> ú 1 (tk tbm gmormst ?5) VKI]TAKG o
??= Ú 1< ≂ ?55 Ú ?55 ∴ ??= Ú 1< ≂ ?5 555
. l <<>> ú 1 8. <<:5 ú ?5 . ∴ <<>> ú 1 8 . <<:
M R U I O G OT A K G
??= rkugds tk ?55 ogd 1< rkugds tk ?55.
<<>> rkugds tk <<:5 ogd 1 rkugds tk ?5.
Mxmrfasm ?B
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Vtotm wbmtbmr tbmsm guhlmrs bovm lmmg rkugdmd up kr dkwg. o =1 ≂ >5 l ?4 ≂ ?5 f ?6: ≂ ?45 . . . d <== 8. <>5 m 1<4 8. 155 n ?4?6 8. ?555
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Nkr tbm javmg mstahotms, dmfadm an tbm opprkxahotm ogswmr as jkagj tk javm o iorjmr kr shoiimr rmsuit fkhpormd tk tbm trum ogswmr. o =2 + 1: ≂ >5 + ?55 l <4 Ú 6? ≂ <5 Ú 65 . . f ?65 ∜ :1 8. ?65 ∜ 25 d <>: ∜ ??5 8. <:5 ∜ ??5 \@A GJ
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Mxohpim ?<
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\kugd tbmsm guhlmrs os agdafotmd. o =1 (gmormst ?5) l 6< (gmormst ?5) d ?2= (gmormst ?5) m <6? (gmormst ?55) j 1> (gmormst ?5) b =24 (gmormst ?55)
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Fohlradjm ]gavmrsaty Urmss
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I
I
Mstahotm tbm ogswmrs tk tbmsm prklimhs ly rst rkugdagj lktb guhlmrs os agdafotmd. o :< + =1 (gmormst ?5) l ?62 ∜ >? (gmormst ?5) f <:= ∜ ?64 (gmormst ?5) d 24? + 11 (gmormst ?5) m <56 ∜ ?54 (gmormst ?55) n 2?= + ?26 (gmormst ?55) j 115 + ?<= (gmormst ?55) b 1> + <4:6 (gmormst ?55) a
Mxohpim ?6
ZK
?=== ∜ === (gmormst ?555)
]sm imodagj dajat opprkxahotakg tk mstahotm tbm ogswmr. o <1 Ú 4 l ?<4 + =2 f <6< ∜ ?5> m 614 ú ?5 n 1: Ú
d >? ú = b 111 ∜ 222
Mstahotm tbm ogswmrs tk tbmsm prklimhs ly rkugdagj lktb guhlmrs os agdafotmd. o <1 Ú 4? (gmormst ?5) l 1< Ú >: (gmormst ?5) f ?<4 Ú ?:6 (gmormst ?55) d <45< Ú 62?: (gmormst ?555) m 42 ú ?? (gmormst ?5) n ?=1 ú ?< (gmormst ?5) j ?54 ú ?? (gmormst ?5) b <416 ú ?56 (gmormst ?55) \@ A GJ
ZK
2
Bmrm as o pbktk kn stors ag tbm gajbt s`y. ]sm tbm jrad tk mstahotm tbm guhlmr kn vasalim stors ag tbm pbktk.
© Dovad Jrmmgwkkd mt oi. <5?6 AVLG3 1:2??5:><>1:6 Ubktkfkpyagj as rmstraftmd ugdmr iow ogd tbas hotmraoi hust gkt lm trogsnmrrmd tk ogktbmr porty
Fohlradjm ]gavmrsaty Urmss
]
H
N
F O
T
\
BM
UV
H O T AF
O
I
I
Q
Q
61
Guhlmr ogd Oijmlro
\@ AGJ
ZK
1
O dajjmr fog daj <1 sfkkps pmr bkur ogd wkr` : bkurs pmr doy. Opprkxahotmiy bkw hogy sfkkps fog lm duj kvmr ?5 doys;
]
H
N
F O
T
\
BM
UV
H O T AF
O
I
I
Q
?5 Hkst kn tbm pmgs ot o stkf`yord orm nuii kn sbmmp. Tbmrm orm == pmgs ogd kgm kn tbm pmgs bos << sbmmp.
Javm og mstahotm nkr tbm tktoi guhlmr kn sbmmp ot tbm stkf`yord. ?? O wbkim ymor jrkup kn ?=1 studmgts as rkujbiy davadmd agtk ?1 jrkups. Mstahotm tbm guhlmr ag mofb jrkup. ?< At as smgsalim skhmtahms tk rkugd kgm guhlmr up an tbm ktbmr guhlmr as jkagj tk lm rkugdmd dkwg.
]sm imodagj dajat opprkxahotakg tk mstahotm tbm ogswmrs tk tbmsm prklimhs. o
?? Ú ?1
l ?<1 Ú 1=4
f
<= Ú 6>
d ?=55 Ú <=55 \@ AGJ
ZK ]
?6 Tbm imttmrs o ogd l rmprmsmgt guhlmrs. Zbafb kn tbm wkrds ‒shoiimr‑ kr ‒iorjmr‑ fkhpimtms
H
N
F O
T
\
UV
I
BM O tbmsm smgtmgfms; H O T AF o An o ogd l orm lktb rkugdmd up, tbmg fkhpormd tk tbm trum ogswmr tbm opprkxahotm ogswmr tk3 a waii . o + llm aa waii . o Ú llm l An kgiyo as rkugdmd up, lutl as imnt os at as, tbmg fkhpormd tk tbm trum ogswmr tbm opprkxahotm ogswmr tk3 a waii o ∜ llm . aa waii . o ú llm f An kgiyl as rkugdmd up, luto as imnt os at as, tbmg fkhpormd tk tbm trum ogswmr tbm opprkxahotm ogswmr tk3 a waii . o ∜ llm aa waii o ú llm .
l
d
o
An kgiy as rkugdmd dkwg, lut as imnt os at as, tbmg fkhpormd tk tbm trum ogswmr tbm opprkxahotm ogswmr tk3 a waii . o ∜ llm aa waii . o ú llm
Mgrafbhmgt3 Hoxahuh mrrkr ?4 Zbmg rkugdagj guhlmrs lmnkrm o foifuiotakg as fkhpimtmd, at as hkst ia`miy tbot tbmrm waii lm og
mrrkr. Tbas mrrkr fog lm iorjm kr shoii, dmpmgdagj kg tbm typm kn rkugdagj agvkivmd. m.j. Zbmg rkugdagj tk tbm gmormst ?5, :? Ú ?? ≂ :5 Ú ?5 8 :55. Lut :? Ú ?? 8 :2?, sk tbm mrrkr as 2?. o Foifuiotm tbm mrrkr an tbmsm guhlmrs orm rkugdmd tk tbm gmormst ?5 lmnkrm tbm huitapiafotakg as foifuiotmd. a <6 Ú ?: aa <6 Ú <4 aaa >= Ú =4 av >: Ú => l Mxpioag wby tbm mrrkr ag ports a ogd aaa as hufb imss tbog tbm mrrkr ag ports aa ogd av. f Foifuiotm tbm mrrkr an tbmsm guhlmrs orm rkugdmd tk tbm gmormst ?5 lmnkrm tbm davasakg as foifuiotmd. a <>? ú 1 aa 6<6 ú ?: aaa 11 ú ?? av 61> ú << d Mxpioag wby tbm opprkxahotm ogswmrs ag ports a ogd aa orm imss tbog tbm fkrrmft ogswmr, ogd wby tbm opprkxahotm ogswmrs ag ports aaa ogd av orm hkrm tbog tbm fkrrmft ogswmr.
© Dovad Jrmmgwkkd mt oi. <5?6 AVLG3 1:2??5:><>1:6 Ubktkfkpyagj as rmstraftmd ugdmr iow ogd tbas hotmraoi hust gkt lm trogsnmrrmd tk ogktbmr porty
Fohlradjm ]gavmrsaty Urmss
I
Q
45
Fboptmr ? Fkhputotakg watb pksatavm agtmjmrs
Krdmr kn kpmrotakgs watb pksatavm agtmjmrs
?A
Zbmg fkhlagagj tbm kpmrotakgs kn oddatakg, sultroftakg, huitapiafotakg ogd davasakg, o portafuior krdmr gmmds tk lm nkiikwmd. Huitapiafotakg ogd davasakg sat bajbmr ag tbm krdmr tbog oddatakg ogd sultroftakg, ogd tbas rmiotms tk bkw wm hajbt ikjafoiiy agtmrprmt sahpim hotbmhotafoi prklimhs put agtk wkrds. Fkgsadmr tbmsm twk stotmhmgts. < jrkups kn 6 fboars pius = fboars. • = fboars pius < jrkups kn 6 fboars. Ag lktb fosms, tbmrm orm < Ú 6 + = 8 ?? fboars. Tbas hmogs tbot < Ú 6 + = 8 = + < Ú 6. Tbas oisk sujjmsts tbot nkr = + < Ú 6, tbm huitapiafotakg sbkuid lm dkgm rst. •
Imt‑s stort3 Hagahuh lrof`mts
s o d m a y m @
•
Bkw hajbt yku usm lrof`mts tk ho`m tbas stotmhmgt trum;
•
<+6Ú=∜6ú>+?8< Zbot as tbm hagahuh guhlmr kn poars kn lrof`mts gmmdmd tk ho`m at trum;
▥
▥
▥
▥
Zbmg wkr`agj watb hkrm tbog kgm kpmrotakg3
4 Ú (< + 6) ∜ ?< ú >
watb lrof`mts (oisk `gkwg os pormgtbmsms) rst. ?st ‖‖ Dmoi Dk huitapiafotakg ogd davasakg gmxt, wkr`agj nrkh imnt tk rajbt. = ‖ Dk oddatakg ogd sultroftakg iost, wkr`agj nrkh imnt tk rajbt.
© Dovad Jrmmgwkkd mt oi. <5?6 AVLG3 1:2??5:><>1:6 Ubktkfkpyagj as rmstraftmd ugdmr iow ogd tbas hotmraoi hust gkt lm trogsnmrrmd tk ogktbmr porty
<
Fohlradjm ]gavmrsaty Urmss
4?
Guhlmr ogd Oijmlro
Mxohpim ?= ]sagj krdmr kn kpmrotakgs ]sm krdmr kn kpmrotakgs tk mvoiuotm tbmsm mxprmssakgs. o > + ?5 ú < l ?2 ∜ < Ú (4 + >) ú = VKI]TAKG
M R U I O G OT A K G
> + ?5 ú < 8 > + = 8 ??
o
Dk tbm davasakg lmnkrm tbm oddatakg.
l ?2 ∜ < Ú (4 + >) ú = 8 ?2 ∜ < Ú ?5 ú =
Dmoi watb lrof`mts rst. Dk tbm huitapiafotakg ogd davasakg gmxt, wkr`agj nrkh imnt tk rajbt. Dk tbm sultroftakg iost.
?2 ∜ <5 ú = 8 ?2 ∜ 4 8 ?4 8
Mxohpim ?> ]sagj krdmr kn kpmrotakgs ag wkrdmd prklimhs Nagd tbm dannmrmgfm lmtwmmg :> ogd 46, trapim tbas rmsuit ogd, goiiy, sultroft tbm quktamgt kn 6= ogd :. VKI]TAKG
M R U I O G OT A K G
6 Ú (:> ∜ 46) ∜ 6= ú : 8 6 Ú 66 ∜ = 8 11 ∜ =
Narst, wratm tbm prklimh usagj syhlkis ogd guhlmrs. ]sm lrof`mts nkr tbm dannmrmgfm sagfm tbas kpmrotakg as
8
14
tk lm fkhpimtmd rst.
Mxmrfasm ?A
\@A GJ
ZK ]
?
<
Zbafb kpmrotakg (oddatakg, sultroftakg, huitapiafotakg kr davasakg) as dkgm rst ag tbmsm fkhputotakgs; o <+=∜6 l =ú=Ú< f <Ú6ú> d =Ú<+6 m :ú:∜? n (> + <) Ú 6 j (2 ú 4) ∜ ? b 4+:Ú< a 2 ∜ ?5 ú = c ?5 ∜ < + 6 ` >+<Ú6∜? i = Ú (< + 6 ú 6) ∜ ? Fiossany tbmsm stotmhmgts os trum kr noism. o f
= Ú < + ? 8 (= Ú <) + ?
l d
?5 Ú (6 + 4) 8 ?5 Ú 6 + 4 1 ∜ 6 Ú < 8 1 ∜ (6 Ú <)
© Dovad Jrmmgwkkd mt oi. <5?6 AVLG3 1:2??5:><>1:6 Ubktkfkpyagj as rmstraftmd ugdmr iow ogd tbas hotmraoi hust gkt lm trogsnmrrmd tk ogktbmr porty
Fohlradjm ]gavmrsaty Urmss
H
N
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UV
H O T AF
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4<
?A
Fboptmr ? Fkhputotakg watb pksatavm agtmjmrs
\@A GJ
ZK ]
Mxohpim ?=o
Mxohpim ?=l
6
4
]sm krdmr kn kpmrotakgs tk gd tbm ogswmrs tk tbmsm fkhputotakgs. ]sm o foifuiotkr tk fbmf` ykur ogswmrs ogd tk smm an at oppiams tbm krdmr kn kpmrotakgs. o <+6Ú: l =+2Ú< f ?5 ∜ <5 ú < d << ∜ ?> ú 4 m >Ú6+<Ú: n ?Ú2∜<Ú6 j ?2 ú 1 + >5 ú 6 b <+6Ú:∜? a 45 ∜ <= ú = + 6 c >6 ú 6 Ú : + < Ú 6 ` :2 ∜ ?4 Ú 4 + > i 655 ∜ ?55 Ú 4 ú 4
N
F O
T
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BM
UV
H O T AF
O
I
O
I
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Q
]sm krdmr kn kpmrotakgs tk gd tbm ogswmr tk tbmsm fkhputotakgs. ]sm o foifuiotkr tk fbmf` ykur ogswmrs. o < Ú (6 + <) d (?55 + =) ú = + ? j ?> ∜ < Ú (: ∜ =) + > c (<5 ∜ ?5) Ú (= + :) + ?
Mxohpim ?>
H
l ?2 ú (?5 ∜ 4)
f (?1 ∜ 1) ú =
< Ú (1 ∜ 4) ú = b (: + <) ú (=6 ∜ =5) ` 6 Ú (:< ú ?< + ?) ∜ ?
n
m
a i
=5 ú (?6 ∜ 6) + 4 ?4 ∜ (: ú : + ?) Ú < 42 ú (4 + 4) ú (6 Ú <)
=
Tbmsm fkhputotakgs agvkivm lrof`mts watbag lrof`mts. Mgsurm yku wkr` watb tbm aggmr lrof`mts rst. o < Ú P(< + 6) Ú = ∜ ?X l P?5 ú (< + 6) + ?X Ú > f <> ú P?5 ∜ (?: ∜ 1)X d P> ∜ (= ∜ 6)X Ú : m < + P?56 ∜ ( ú ?<
>
Nagd tbm ogswmr tk tbmsm wkrdmd prklimhs ly rst wratagj tbm smgtmgfm usagj guhlmrs ogd syhlkis. o Trapim tbm suh kn 6 ogd >. Dkulim tbm quktamgt kn <5 ogd 4. Tbm quktamgt kn 44 ogd ?? pius 4. d = hkrm tbog tbm prkduft kn > ogd ?<. m Tbm quktamgt kn >5 ogd ?< as sultroftmd nrkh tbm prkduft kn = ogd :. n ?= imss tbog tbm dannmrmgfm kn 42 ogd ?<. j Tbm prkduft kn 1 ogd ?< as sultroftmd nrkh dkulim tbm prkduft kn ?5 ogd ?=. l f
\@ A GJ
ZK
:
]
O dmiavmry kn ?= lkxms kn lkk`s orravms, mofb lkx fkgtoagagj majbt lkk`s. Tbm lkk`stkrm kwgmr rmhkvms tbrmm lkk`s nrkh mofb lkx. Bkw hogy lkk`s staii rmhoag ag tktoi;
© Dovad Jrmmgwkkd mt oi. <5?6 AVLG3 1:2??5:><>1:6 Ubktkfkpyagj as rmstraftmd ugdmr iow ogd tbas hotmraoi hust gkt lm trogsnmrrmd tk ogktbmr porty
H
Fohlradjm ]gavmrsaty Urmss
N
F O
T
\
BM
UV
H O T AF
I
Q
46
Guhlmr ogd Oijmlro
\@ A GJ
2
1
ZK
Ag o fioss, majbt studmgts bovm tbrmm T^ smts ot bkhm, nkur bovm twk T^ smts, ?6 bovm kgm T^ smt ogd twk studmgts bovm gk T^ smts. Bkw hogy T^ smts orm tbmrm ag tktoi;
]
H
Agsmrt lrof`mts agtk tbmsm stotmhmgts tk ho`m tbmh trum. o 4 + < Ú 6 8 ?2 l 1 ú ?< ∜ 1 8 6 d 6 + < Ú : ∜ 6 8 <5 m ?5 ∜ : ú b =5 ú < Ú = ∜ 4 8 ?
N
F O
T
\
BM
UV
H O T AF
O
I
I
Q
<Ú6+4∜=81 4 + ?5 ú
f n a
?5 Tbm ohkugt kn $?55 as davadmd agtk twk rst prazms kn mquoi voium ogd tbrmm smfkgd prazms kn mquoi
voium. Mofb prazm as o wbkim guhlmr kn dkiiors ogd rst prazm as ot imost 4 tahms tbm voium kn tbm smfkgd prazm. An smfkgd prazm as hkrm tbog $>, gd tbm ohkugt kn mofb prazm. \@ AGJ
ZK ]
?? Dmfadm an tbm lrof`mts javmg ag mofb stotmhmgt orm oftuoiiy gmfmssory.
< + (6 Ú >) 8 <5 d ?5 ∜ (= + <) 8 6 j (45 ú ?5) ú 4 8 ?
l (< + 6) Ú > 8 65
o
m
H
f (<5 Ú <) Ú 6 8 ?<5
<< ∜ (?? ∜ :) 8 ?2
n
b ?55 ú (<5 ú =) 8 <=
a
N
F O
T
\
BM
UV
H O T AF
?1 ∜ (?5 ú <) 8 ?4 < Ú (6 + <) ú = 8 <
?< Tbm imttmrs o, l ogd f rmprmsmgt guhlmrs. Dmfadm an tbm lrof`mts orm gmfmssory ag tbmsm mxprmssakgs. o
o
+
(l + f)
l
o
∜
(l ∜ f )
f
o
Ú
(l Ú f)
d
o
ú
(l ú f)
d
o
Ú
l
?6 Zratm o sahpimr stotmhmgt nkr tbmsm. Ossuhmo ≥ 5 ogd l ≥ 5. o
o
+
l
∜
o
l (o ∜ o) Ú l
f
o
+
l
ú
l
ú
o
Mgrafbhmgt3 Kpmrotakg ag ruims ?4 ]sagj wbkim guhlmrs ogd ogy kn tbm nkur kpmrotakgs (+, ‖, Ú, ú), dmsfralm bkw yku wkuid kltoag tbm
‒Nagasb‑ guhlmr nrkh tbm ‒Vtort‑ guhlmr ag mofb kn tbmsm tolims. Qkur ruim hust wkr` nkr mvmry poar kn guhlmrs ag ats tolim. o
Vtort ? < 6 4
Nagasb 6 = : 1
l
Vtort ? < 6 4
Nagasb 5 6 > 1
f
Vtort 6 4 = >
Nagasb ?5 ?: <> 6:
Ho`m up ykur kwg tolim watb o ‒smfrmt‑ ruim ogd tmst at kg o nramgd.
© Dovad Jrmmgwkkd mt oi. <5?6 AVLG3 1:2??5:><>1:6 Ubktkfkpyagj as rmstraftmd ugdmr iow ogd tbas hotmraoi hust gkt lm trogsnmrrmd tk ogktbmr porty
Fohlradjm ]gavmrsaty Urmss
O
I
I
Q
44
Fboptmr ? Fkhputotakg watb pksatavm agtmjmrs
g k at o j at s m v Ag
Tbm olofus Tbm olofus as o fkugtagj dmvafm tbot bos lmmg usmd nkr tbkusogds kn ymors. Tbmy wmrm usmd mxtmgsavmiy ly hmrfbogts, trodmrs, tox fkiimftkrs ogd fimr`s lmnkrm hkdmrg-doy guhmrois systmhs wmrm dmvmikpmd. Fkugtagj lkords foiimd Olox dotm lof` tk =55 lf. Tbmsm wmrm wkkd kr stkgm tolimts watb jrkkvms, wbafb wkuid bkid lmogs kr pmllims. Tbm hkdmrg olofus as soad tk bovm krajagotmd ag Fbago ag olkut tbm ?6tb fmgtury ogd agfiudms lmods kg warms bmid ag o wkkdmg nrohm.
O Jmrhog wkkdfut nrkh ?=52 sbkwagj og olofus ag usm ly jmgtimhog kg rajbt, wbaim o hotbmhotafaog (ot imnt) wratms oijkratbhs.
O hkdmrg olofus watb tbartmmg warms
ds ds og rm us gd gs ms k tb bu tm kg
Tbmrm orm = lmods kg kgm sadm kn o hkdmrg olofus wkrtb ? mofb ogd < lmods kg tbm kppksatm sadm wkrtb = mofb. ▥ Mofb warm rmprmsmgts o dannmrmgt ugat, m.j. kgms, tmgs, bugdrmds mtf. ▥
Lmods orm fkugtmd kgiy wbmg tbmy orm pusbmd tkword tbm fmgtrm.
Bmrm as o daojroh sbkwagj tbm guhlmr =:?>.
© Dovad Jrmmgwkkd mt oi. <5?6 AVLG3 1:2??5:><>1:6 Ubktkfkpyagj as rmstraftmd ugdmr iow ogd tbas hotmraoi hust gkt lm trogsnmrrmd tk ogktbmr porty
Fohlradjm ]gavmrsaty Urmss
Guhlmr ogd Oijmlro
?
<
Zbot guhlmrs orm sbkwagj kg tbm olofus daojrohs lmikw; Kgiy tbm rst sax warms orm sbkwagj. o
l
f
d
Drow olofus daojrohs sbkwagj tbmsm guhlmrs. o =: l 61< f >254
d <15 6?>
6
Ahojagm oddagj twk guhlmrs usagj og olofus ly siadagj lmods oikgj tbmar warms. Fimoriy mxpioag tbm stmps to`mg tk odd tbmsm guhlmrs. o ?? + : l <614 + =6>
4
Ahojagm sultroftagj twk guhlmrs usagj og olofus ly siadagj lmods oikgj tbmar warms. Fimoriy mxpioag tbm stmps to`mg tk sultroft tbmsm guhlmrs. o <6 ∜ ?4 l 6<1 ∜ <46
=
Huitapiafotakg as foifuiotmd os o rmpmotmd oddatakg. m.j. 6 Ú
>
Davasakg as foifuiotmd os o rmpmotmd sultroftakg. m.j. >6 ú 6 ∜
:
Vmm an yku fog gd o rmoi olofus kr fkhputmr olofus watb wbafb tk wkr`. ]sm tbm olofus tk sbkw bkw yku fog dk tbm prklimhs ag _umstakgs 6 tk > olkvm.
© Dovad Jrmmgwkkd mt oi. <5?6 AVLG3 1:2??5:><>1:6 Ubktkfkpyagj as rmstraftmd ugdmr iow ogd tbas hotmraoi hust gkt lm trogsnmrrmd tk ogktbmr porty
Fohlradjm ]gavmrsaty Urmss
4=
4>
Fboptmr ? Fkhputotakg watb pksatavm agtmjmrs
s m j g m ii o b f d g o s m i z z u U
?
Tbm mxtro dkiior. Tbm fkst kn daggmr nkr twk pmkpim as $4= ogd tbmy lktb javm tbm woatmr $<= mofb. Kn tbm mxtro $= tbm woatmr as oiikwmd tk `mmp $6 os o tap ogd rmturgs $? tk mofb pmrskg. Vk tbm twk pmkpim poad $<4 mofb, ho`agj o tktoi kn $42, ogd tbm woatmr bos $6. Tbm tktoi as tbmrmnkrm $42 + $6 8 $=?. Zbmrm dad tbm mxtro $? fkhm nrkh;
<
Tbm suh oikgj mofb iagm as ?=. Uiofm mofb kn tbm dajats ?, <, 6, 4, =, >, :, 2 ogd 1 tk ho`m tbas trum.
6
Tbm suh oikgj mofb sadm kn tbas traogjim as ?:. Uiofm mofb kn tbm dajats ?, <, 6, 4, =, >, :, 2 ogd 1 tk ho`m tbas trum.
4
Ho`m tbm tktoi kn ?55 kut kn oii tbm guhlmrs <, 6, 4, : ogd ??, usagj mofb guhlmr kgiy kgfm. Qku fog usm ogy kn tbm kpmrotakgs (+, ‖, Ú, ú), os wmii os lrof`mts.
=
Vudk`u as o pkpuior ikjaf guhlmr puzzim hodm up kn o 1 ly 1 squorm, wbmrm mofb fkiuhg ogd rkw fog usm tbm dajats ?, <, 6, 4, =, >, :, 2 ogd 1 kgiy kgfm. Oisk, mofb dajat as tk lm usmd kgiy kgfm ag mofb 6 ly 6 squorm. Vkivm tbmsm puzzims.
4 <
2
:
<
2
1
?
>
6
:
: 4
=
:
>
=
< 4
:
<
>
2
: 2
6
1
= <
6 :
<
1
?
2
<
2
<
=
>
6
1
=
=
© Dovad Jrmmgwkkd mt oi. <5?6 AVLG3 1:2??5:><>1:6 Ubktkfkpyagj as rmstraftmd ugdmr iow ogd tbas hotmraoi hust gkt lm trogsnmrrmd tk ogktbmr porty
6 1
< =
> :
1
?2
2 6
6
>
2 1
4 :
Fohlradjm ]gavmrsaty Urmss
?
Guhlmr ogd Oijmlro
Tbm piofm voium kn 6 ag ?6<: as 655.
< Ú ?55 + : Ú ?5 + 6 Ú ? as tbm mxpogdmd nkrh kn <:6.
Uiofm voium
Ogfamgt Guhlmr Vystmhs
Oddatakg ogd Vultroftakg
\khog IRR^A as :> RFA^ as 14
Oijkratbhs 2 ? 6:? 16: + 246 ‖ >46 [[[[[ [[[[[ ?
Lolyikgaog as <6 as :?
Hmgtoi strotmjams ?:< + 8 655 + 25 + 2
Mjyptaog
8 622 12 ‖ ?1 8 12 ‖ <5 + ? 8 :1
as
Huitapiafotakg ogd Davasakg
Zbkim guhlmrs
Krdmr kn Kpmrotakgs Lrof`mts rst, tbmg Ú ogd ú, tbmg + ogd ‖ nrkh imnt tk rajbt. < + 6 Ú 4 ú (1 ú 6) 8 < + ?< ú 6 8<+4 8>
Oijkratbhs
<
<1 Ú ?6 [[[[ 2: <15 [[[[ 6::
<
watb ? rmhoagdmr
Hmgtoi strotmjams : Ú 6? 8 : Ú 65 + : Ú ? 8 4 ú 2 8 6< ú 4 8 ?> ú < 8 2 ?=> ú 4 8 ?>5 ú 4 ‖ 4 ú 4 8 45 ‖ ? 8 61
Mstahotakg
1== tk tbm gmormst ?5 as 1>5 1=5 tk tbm gmormst ?55 as ?555
>2 6 <5=
Imodagj dajat opprkxahotakg 61 Ú 6<> 45 Ú 655 8 ?< 555 ≂
Huitapiyagj ly ?5, ?55, …… 62 Ú ?55 8 6255 62 Ú :55 8 62 Ú : Ú ?55 8 <> >55
© Dovad Jrmmgwkkd mt oi. <5?6 AVLG3 1:2??5:><>1:6 Ubktkfkpyagj as rmstraftmd ugdmr iow ogd tbas hotmraoi hust gkt lm trogsnmrrmd tk ogktbmr porty
Fohlradjm ]gavmrsaty Urmss
4:
y r o h h u s r m t p o b F
42
Fboptmr ? Fkhputotakg watb pksatavm agtmjmrs
Huitapim-fbkafm qumstakgs ?
<
6
4
=
>
Zbafb kn tbm nkiikwagj as mquoi tk <4; O RRAAA L RRA^ D A^RR M ARR^
F RRRIA^
6 Ú ?555 + 1 Ú ?5 + < Ú ? as tbm mxpogdmd nkrh kn3 O 61<5 L 61< D 615< M 6<1
F 651<
Zbafb kn tbm nkiikwagj as gkt trum; O <+686+< L <Ú686Ú< D =ú<≥<ú= M :∜<8<∜:
F (< Ú 6) Ú 4 8 < Ú (6 Ú 4)
Tbm suh kn ?12 ogd ?56 as3 O 65? L 654 D ?11 M 1=
F <11
Tbm dannmrmgfm lmtwmmg 6:2 ogd 2? as3 O 4=1 L <1: D 6?: M <11
F 656
Tbm prkduft kn : ogd
F
M <5: ? ? 2
:
2
1
Tbm hassagj dajat ag tbas davasakg as3 O >
L ?
D 2
M :
:
)
?
=
< >
F 1
Tbm rmhoagdmr wbmg 6?: as davadmd ly 1 as3 O : L = D ? M 5
F <
4=2 rkugdmd tk tbm gmormst ?55 as3 O 455 L =55 D 4=5 M ?555
F 4>5
?5 Tbm ogswmr tk P< + 6 Ú (: ‖ 4)X ú ?? as3 O ? L = D ?
M
F ??
5
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Fohlradjm ]gavmrsaty Urmss
Guhlmr ogd Oijmlro
Vbkrt-ogswmr qumstakgs ?
Zratm tbmsm guhlmrs usagj tbm javmg guhlmr systmhs. o Mjyptaog a 6 aa 6? l Lolyikgaog a ?< aa >5 f \khog a ?4 aa 45
aaa 6<> aaa ?6< aaa ?4>
<
Zratm dkwg tbm piofm voium kn tbm dajat = ag tbmsm guhlmrs. o 6=: l =<41 f 6=> >?<
6
]sm o hmgtoi strotmjy tk gd tbmsm suhs ogd dannmrmgfms. o ?<4 + 66= l >2: ∜ 6<4 f =1 + 6>
4
]sm og oijkratbh ogd sbkw ykur wkr`agj nkr tbmsm suhs ogd dannmrmgfms. :> 6<1 ?=<2 o l f d
=
>
+
∜?62
:1>
∜?2?4
]sm o hmgtoi strotmjy tk pmrnkrh tbmsm fkhputotakgs. o = Ú ?1 l << Ú > d ?<6 ú 6 m <>4 ú 2 j <1 Ú ?555 b 6> Ú 655
f = Ú 44 n a
1> ú 4 ?4 >:2 ú ?
]sm og oijkratbh ogd sbkw ykur wkr`agj nkr tbmsm fkhputotakgs. <: o l f :)6<: d ?=:
4 65?><
Ú?6
1
Ú
:
d <=> ∜ 61
Nagd tbm hassagj dajats ag tbmsm fkhputotakgs. <
o +
6
<
l
∜
: 6 1 > ?
f
6
6
<
= >
? =
<
Ú
d
4
6 4 ?
?
: ? 5
> 5 6 ?
2
\kugd tbmsm guhlmrs os agdafotmd. o :< (gmormst ?5) l 6<>2 (gmormst ?55)
f 1=? (gmormst ?55)
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Fohlradjm ]gavmrsaty Urmss
41
=5
Fboptmr ? Fkhputotakg watb pksatavm agtmjmrs
1
]sm imodagj dajat opprkxahotakg tk mstahotm tbm ogswmrs tk tbmsm fkhputotakgs. o <21 + =6< l << Ú ?1 f 4=< Ú ?? d 11 ú ??
?5 ]sm krdmr kn kpmrotakgs tk gd tbm ogswmrs tk tbmsm fkhputotakgs. o
6 Ú (< + >)
d (= + <) Ú 6 ∜ (2 ∜ :)
l >∜2ú4
f < Ú 2 ∜ ?< ú >
m
n
5 Ú (1 22 <64 ú 6)
? Ú (6 + < Ú =)
Mxtmgdmd-rmspkgsm qumstakgs ?
O faty tkwmr fkgstruftakg usms 4=<5 tkggms kn fmhmgt truf`md nrkh o noftkry tbot as : `aikhmtrms nrkh tbm fkgstruftakg satm. Mofb fmhmgt haxmr fog forry : tkggms kn fmhmgt, ogd tbm fmhmgt fksts $2= pmr truf` ikod nkr tbm rst 65 ikods ogd $== pmr ikod ontmr tbot.
o l
Bkw hogy ikods kn fmhmgt orm gmmdmd; Odd o nuii ikod nkr ogy rmhoagdmr. Nagd tbm tktoi dastogfm trovmiimd ly tbm fmhmgt haxmrs tk dmiavmr oii ikods, ossuhagj tbmy gmmd tk
rmturg tk tbm noftkry ontmr mofb ikod. Nagd tbm tktoi fkst kn fmhmgt gmmdmd tk ho`m fkgfrmtm nkr tbm tkwmr fkgstruftakg. d O dannmrmgt fmhmgt suppiamr knnmrs o prafm kn $>= pmr 2-tkggm truf`, gk hottmr bkw hogy ikods orm gmmdmd. Nagd tbm dannmrmgfm ag tbm fkst kn fmhmgt nkr tbm tkwmr ly tbas suppiamr fkhpormd tk tbm krajagoi suppiamr. f
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Fohlradjm ]gavmrsaty Urmss
Guhlmr ogd Oijmlro
<
Kgm gajbt \af`y ogd bmr lrktbmr Haf`y dmfadm tk bovm skhm nug ot tbmar notbmr‑s swmmt sbkp. Ag tbm sbkp tbmy gd : tags kn ?6= cmiiy lmogs mofb, 1 pof`mts kn ?
© Dovad Jrmmgwkkd mt oi. <5?6 AVLG3 1:2??5:><>1:6 Ubktkfkpyagj as rmstraftmd ugdmr iow ogd tbas hotmraoi hust gkt lm trogsnmrrmd tk ogktbmr porty
Fohlradjm ]gavmrsaty Urmss
=?