-
GeoGebra 3.2 Helpe in Arabic Markus Hohenwarter, www.geogebra
Translated by Team
From Egypt
Ayman Mohamed
Emad Abd Elkader
Maha Ahmed Ismail
-
arbeGoeG -1
.
"retrawnehoH sukraM"
1-1
weiV cihparG : arbeGoeG
weiV teehsdaerpS weiV arbeglA
.
1-1-1
.
.
.
.
.
-
weiV cirbeglA 2-1-1
raB tupnI
retnE
.
: 2( = )
2^x = )x(f
stcejbO eerF :
stcejbO tnednepeD
:
" seitreporP " " " tcejbO yrailixuA "
." cisaB :
-
"raB tupnI "
" " " dnammoc "
1F
weiV teehsdaerpS 3-1-1
A
1A 1
) arbeGoeG
(
weiV stcejbo yrailixuA
arbeGoeG 2 1
1-2-1
" ". weiV "
"raB tupnI " "weiV ciarbeglA
-
" /"
" "
.
-:
.1
-
( X : Y ) ( ) ( )
.2
( - lrtC) ( + lrtC) .3
" "
"seitreporP "
-
) "sixA "
( ) (
) ( )
(
"dirG "
:
tfihS
) (
( seitreporP )
-
" " slooT "
" rabloot ezimotsuC
stcejbO fo seitreporP eht gnignahC 2-2-1
-:
" seitreporP "
(seitreporP ) ( tidE )
-
" evoM )
....... ( )
" seitreporP"
( )
( )
......... ( )
( )uneM txetnoC 3-2-1
.
: A
( ) ( )
A ( )
-
" seitreporP"
(looT noitatneserP) arbeGoeG 3-1
(raB noitagivaN) 1-3-1
" "weiV " .
" spets noitcurtsnoc rof rab noitagivaN
-:
()
( )
(locotorP noitcurtsnoC) 2-3-1
" locotorP noitcurtsnoC " "weiV "
:
-
-:
( ) ( ) ( emoH) ( dnE) ) ( eteleD)
(
-: " weiV " ( )
-
" tniopkaerB "
weiV " "
: LMTH arbeGoeG
..." "
-:
LMTH
" "
" "
(arbeGoeG fo gnitteS) 3-3-1
" snoitpO "
arbeGoeG
-
" (" ) "
(" )
" "
(looT gnirohtuA) arbeGoeG 4-1
(snoitpO gnitnirP) 1-4-1
(weiV cihparG)
"weiverP tnirP " "eliF "
-
" " " " " "
. " "
(locotorP noitcurtsnoC)
noitcurtsnoC " " weiV " .1
" locotorP
" weiverPtnirP " "eliF " .2
) weiV (
2-4-1
eliF
" " " tropxE "
:
-
" " tamroF "
ipd " noituloseR
:
o
o
draobpilC
" " " trepxE " elif
-
GNP
gnissecorP droW
(segapbeW evitcaretnI) 3-4-1
: arbeGoeG
" " " tropxE " " eliF "
:
" rohtuA " " etaD " " eltiT " o ) " lareneG " o
(
" decnavdA " o
-
)
( .
.
)
( 867 4201
: elcric
( lmth.elcric) lmth .1 ( bgg.elcric) BGG .2 ( ) raj.arbegoeG .3
( ., irafaS ,allizoM ,rerolpxE tenretnI) telppa avaJ
moc.avaj.www//:ptth
( retirW eciffOnepO , egaPtnorF)
-
(stupnI cirtemoeG) . 2
(setoN lareneG) 1-2
) ( weiV cihparG)
(
( weiV ciarbeglA)
( ) -:
( ) ( )
slooT noitcurtsnoC 2-2
.
.
-
(C , B , A) :
( 1)
( 2)
( 3)
LRTC
" "
""
( ) 1-2-2
: . ( )
-
.1
.2
"eteled" ( 1) ( 2)
:
.
(noitaleR ).
:
( 1)
( 2)
( ) ( 3)
( )
.
-
( C , B , A)
( C , A)
2-2-2
) .
......... (
-: ( )
-
. .1
. .2
" "
" " " " "
"
3-2-2
:
4-2-2
:
:
-
5-2-2
:
6-2-2
( )
) :
(
7-2-2
-:
B B ( C , B , A)
-:
A . B A
B
( ) g : g A
A
) B , A :
BA (
-
(roteVralucidnepreP ) BA
A g : A g
(rotceVralucidnepreP ) g
-:
( )
-:
A A g
g
" f" " A"
A
8-2-2
M :
:
P M
PM
( C , B , A)
-
( ) :
( ) :
( ) :
:
9-2-2
( ) :
( B A ) A , B
B A
A M :
( B A ) B
B A :
( C , B , A) :
-
B C A
( C , B , A) :
B C A
01-2-2
-:
) ( )
(
081 :
..."" " " o o
" " " " " o
"
( ) B ( ) A :
-
C B
-:
" " " "" "
" "
naelooB 11-2-2
.
21-2-2
B B A :
A B A
-
A B
) A (
:
2^x=)x(f (1)
A ( 2)
A B ) ) )A(x('f , )A(x ( = B ( 3)
B
)A(x A )x(`f )x( f
f A ) )A(x('f
A B
f A
-
A B B
B A
31-2-2
:
:
:
:
-
:
) :
(
41-2-2
" xeTaL" ( )
( ) (1
(2
" " " " " ... " " " " "
. " txet cimanyD"
A
-
) A (
A " "
A A
arbeGoeG
"+" = " "
"" "+"
=" "
"+"
-
A= " + "
= " "
A
"+ " a= " + "
= " "
""
a
(1
"" B (2
""
xeTaL
xeTaL
}{ xeTaL
-
b/a
2 1
xeTaL
51-2-2
-
:
) (1
(
(2
" "
-:
:
" "
" "
-: " "
: 1
: 2
: 4
( C , B , A)
. B A
B A
C A
3 A
4
A )0,3( + A: )4,0( + A:
A
-
A
" "
" " " "
"" " "
"
% 001 % 0 "
tupnI ciarbeglA .3
setoN lareneG 1-3
( )
raB tupnI
retnE .1
.2 weiV cihparG raB tupnI
stcejbO gnimaN
raB tupnI
sretteL latipaC : stnioP
)4 ,2 ( = C
-
) ) 081 ; 1( = P
( O + tlA
i + 2 = X
sretteL llamS : srotceV
)4 ,2 ( = v
) ) 081 ; 1( = u
( O + tlA
i + 2 = t
: (snoitceS cinoC) ( selcriC) ( seniL) "y" "x"
g 3 + x = y :g
c 4 = 2^)2-y( + 2^)1-x( : c
^
pyh 2 = 2^y 2^x :pyh
: ( ) ( ) : snoitcnuF
4+ 2= f 4 + x 2 = y :f
2( = ) f 2^x = )x(f
= )x(nis = y
.1
raB tupnI .2
-
erocsrednu
1A 1_A BAS }BA{_S
:
raB tupnI .1
retnE
)3 , 2 ( = A
)1 , 2( = A
A retnE
.2
retnE
-
yrotsiH raB tupnI
raB tupnI
raB tupnI weiV ciarbeglA
weiV cihparG
-
tupnI tceriD 2-3
raB tupnI
retnE
1-2-3
retnE
rettel llams
3 = r r
e
a
-
) . ( .1
e .2
arbeGoeG
+ tlA .3 ip P
06 =
3/ip =
". " " "
" "
" "
" " " "
-
2-2-3
latipaC llamS
3-2-3
. " y " x . ( )
" : "
) 3 - , 4 ( t + ) 5 , 5-( = X : g 2 = y 4 + x 3 : g g
1- = d , 2 = k : d + x k = y : g:
(sixAy , sixAx )
sixAy dna sixAx
] sixAx , A [ ralucidnepreP:
A
4-2-3
.
" :"
441 = 2y 61 + 2x 9 : lle llE (espillE)
441 = 2y 61 + 2x : pyh pyH (alobrepyH)
x 4 = 2y : rap raP (alobaraP)
52 = 2y + 2x : 1k 1k (elcriC)
) 0 ; 1 ( = P ) 0 , 1 ( = P P ) 09 ; 5 ( = v ) 5 , 0 ( = v v
-
52 = 2 )2 + y( + 2)5 x( : 2k 2k
b a
4 = a 3 = b
b a ^b 2^a = 2^y 2^a + 2^x + 2^b :lle
5-2-3
(
2^x 3^x3 = )x( f f
) )x(f ( nat = )x( g g
)x( nat + )x3( nis
( , nat , soc , nis )
evitavireD largetnI arbeGoeG
2^x 3^x3 = )x(f
)x(''f )x('f:
2^x 3^x 3 = )x(f
))2+x('f(soc = )x(g
etalsnarT
noitcnuF [ ]
-
6-2-3
+
-
ecaps ro *
/
^
!
) ( ammag
) (
) ( x
) ( y
) ( sba
( 1- 1 ) ) ( ngis
) ( trqs
) ( trbc
) ( pxe
) ( nl ro ) ( gol
) ( dl 2
) ( gl 01
) ( soc
) ( nis
) ( nat
) ( soca
) ( enisa
) ( nata
) ( hsoc
) ( hnis
) ( hnat
-
) ( hnoca
) ( hnisa
) ( hnata
) ( roolf
) ( liec
) ( dnuor
: B A M 2 / ) B + A ( = M
v ) v * v ( trqs = htgneL
snoitarepO dna selbairaV naelooB( ) 7-2-3
( eslaF" )" ( eurT" )"
eslaf = a eurt = a
syeK worrA dna xoB kcehc
snoitarepO naelooB
:
b , a
b a
b == a ==
b a
b =! a =!
< < b < a
> > b > a
=< b a
-
b=< a
b a
b => a =>
b a b && a
&&
b a b || a
||
a
a! !
|| b || a
b a
snoitarepO tsiL stcejbO tsiL 8-2-3
} { stcejbo tsiL
C , B , A } C , B , A { = L
})2,2(, )1,1( , )0,0( { = L
2tsiL == 1 tsiL
eslaF eurT
2tsiL =! 1 tsiL
eslaF eurT
h , g S ]h , g[ tcesretnI = S:
tcesretnI
BAS 1A
}BA{_S 1_A:
-
} {
( )
2tsiL + 1tsiL
( ) rebmuN + tsiL
2tsiL 1tsiL
( ) rebmuN tsiL
2tsiL * 1tsiL
( ) )
(
rebmuN * tsiL
2tsiL / 1tsiL
( ) rebmuN / tsiL
2^tsiL
)tsiL(niS
9-2-3
arbeGoeG
} }9 , 8 , 7 { , } 6 , 5 , 4 { , } 3 , 2 , 1{ {
xirtaM + xirtaM
-
xirtaM - xirtaM
rebmuN * xirtaM
xirtaM * xirtaM
)
(
} }6,5,4{ , }3,2,1{ { * } }6,5{ , }4,3{ , }2,1{ {
} }15,04,92{ , }33,62,91{ , }51,21,9 { { =
)rotceV ro( tnioP * 22xirtaM
( 22)
) 52 , 11 ( = ) 4 , 3 ( * } } 4 , 3 { , } 2 , 1 { {
)rotceV ro( tnioP * 33xirtaM
( 33)
, 1 ( * }} 1 , 0 , 0 { , } 6 , 5 , 4 { , } 3 , 2 , 1 {{
) 02 , 8 ( = ) 2
]xirtaM[ tnanimreteD
]xirtaM[ trevnI
]xirtaM[ esopsnarT
01-2-3
( ) arbeGoeG
i4 + 3 ) 4 , 3 (
i1 + 0 )1 , 0( = i i
+ 3 = q i
i4
( )
-
-:
( seitreporP ) ( arbeglA )
( )
( )
)3 , 1(= )2- , 1 ( ) 1 , 2 ( )1 - , 3 ( = )2 - , 1 ( + ) 1 , 2 (
i3 + 1 = )i2 1( )i1 + 2( i1 3 = )i2 1( + ) i1 + 2 (
/ .
)1 , 0(= )2- , 1 ( / ) 1 , 2 ( )3 - , 4 ( = )2- , 1 ( ) 1 , 2 ( i + 0= )i2 1( / )i1 + 2( i3 4 = )i2 1( * ) i1 + 2 (
B*A B/A B A
i5 + 7= )i5 + 4 ( + 3 ) 5 , 7 ( = )5 , 4 ( + 3
i5 - 1-= )i5 + 4 ( 3 )5- , 1-(= )5 , 4 ( 3
i3 0 = )i1 + 0 ( / 3 ) 3 - , 0 ( = )1 , 0 ( / 3
i6 3 = ) i2 + 1 ( * 3 )6 , 3 ( = )2 , 1( 3
sdnammoC 3 3
" S = "
-
h g S ]h , g[tcesretnI = S
1A 1_A
}BA{_S BAS
retnE
sdnammoC lareneG 1-3-3
eteleD
] tcejbo[ eteleD
noitaleR
b a ]b tcejbo , a tcejbo [ noitaleR
sdnammoC naelooB 2-3-3
fI
noitidnoC ]tcejbo ,noitidnoc[ fI
a noitidnoC ]b tcejbo ,a tcejbo ,noitidnoc[fI
b
denifeDsI
" eslaF " " eurT " ]tcejbo[denifeDsI
petSnoitcurtsnoC
] [ petSnoitcrtsnoC
tcejbo ]tcejbo [ petSnoitcurtsnoC
-
denifeDsI
" eslaF " " eurT " ]tcejbo[denifeDsI
regetnIsI
" eslaF " " eurT " ]rebmuN[regetnIsI
3-3-3
oitaReniffA
]C tnioP ,B tnioP ,A tnioP[oitaReniffA
aerA
.. .. C B A ] ,C tnioP ,B tnioP ,A tnioP[ aerA
largetnI
petSsixA
] [XpetSsixA
] [YpetSsixA
ecneuqeS renroC ) petSsixA
"( "
tneiciffeoClaimoniB
rn ] r rebmuN , n rebmuN[ tneiciffeoClaimoniB
ecnerefmuriC
cinoc ]cinoc[ ecnerefmucriC
oitaRssorC
D C B A ]D tnioP ,C tnioP ,B tnioP ,A tnioP[ oitaRssorC
-
erutavruC
]noitcnuF , tnioP [ erutavruC
] evruC , tnioP [ erutavruC
ecnatsiD
B A ] B tnioP , A tnioP [ ecnatsiD
] eniL , tnioP [ ecnatsiD
h g ] h enil , g eniL [ ecnatsiD
.
htgneLsixAtsriF
]cinoc[ htgneLsixAtsriF
DCG
b a ]b rebmuN , a rebmuN[ DCG
]srebmun fo tsiL[ DCG
noisiviDregetnI
b a ]b rebmuN , a rebmuN[ viD
largetnI
b a noitcnuF ]b , a , noitcnuF [ largetnI
( )x(g )x(f ) ]b , a , g noitcnuF , f noitcnuF [ largetnI
[ a b]
noitaretI
] n rebmuN , x rebmuN , noitcnuF[ noitaretI
n noitcnuF
x
2x = )x(f:
-
( 3=x)
] 2 , 3 , f [ noitaretI
9 = 23 = )3(f -
18 = 29 = )9(f -
3
18
MCL
b a ] b , a [ MCL
] srebmun fo tsil [ MCL
htgneL
]rotcev[ htgneL
A ]A tnioP[ htgneL
b a ] b , a , noitcnuF[ htgneL
B A ]B tnioP , A tnioP ,noitcnuF[ htgneL
2t 1t evruC ]2t , 1t , evruC[ htgneL
B A c ]B tnioP , A tnioP , c evruC [ htgneL
] tsiL [ htgneL
yticirtneccEraeniL
] cinoC [ yticirtneccEraeniL
yticirtneccE raeniL
muSrewoL
]n , b , a ,noitcnuf[ muSrewoL
n ] b , a [ noitcnuf
-
mumixaM dna muminiM
b a ] b , a [ niM
b a ] b , a [ xaM
noitcnuF oludoM
b a ]b rebetni , a regetni[ doM
retemaraP
] alobaraP [ retemaraP
retemireP
] nogyloP [ retemireP
suidaR
] elcriC [ suidaR
sdnammoc modnaR
b a ] b , a [ neewteBmodnaR
n ] p , n [ laimoniBmodnaR
p
]noitaived dradnatS , naeM[ lamroNmodnaR
dradnatS naeM
noitaived
]naeM[ nossioPmodnaR
naeM
htgneLsixAdnoceS
] cinoC [ htgneLsixAdnoceS
epolS
] eniL [ epolS
"" " "
-
muSladiozeparT
n ] n , b , a , noitcnuF [ muSladiozeparT
]b , a[ noitcnuF
muSreppU
]n , b , a , noitcnuF [ muSreppU
n ] b , a [ noitcnuf
n
4-3-3
elgnA
2v 1v ]2v rotcev , 1v rotcev[ elgnA
h g ]h eniL , g eniL[ elgnA
B CB AB ]C tnioP , B tnioP , A tnioP [ elgnA
A B ] elgnA , B tnioP , A tnioP[ elgnA
]B , , A [ tatoR
]cinoC[ elgnA
]rotceV[ elgnA
]tnioP[ elgnA
) ]rebmuN[ elgnA
( 2
]nogyloP[ elgnA
-
5-3-3
retneC
]cinoc[ retneC
:
diortneC
]nogylop[ diortneC
renroC
n ] n [ renroC
n egami ] n , egami [ renroC
n txet ] n , txet [ renroC
}4 , 3 , 2 , 1 { = n
mumretxE
]laimonylop[ mumertxE
sucoF
] cinoc [ sucoF
tnioPnoitcelfnI
] laimonylop [ tnioPnoitcelfnI
tcesretnI
h g ] h eniL , g eniL [ tcesretnI
] cinoC , eniL [ tcesretnI
(2 )
n ] n , cinoC , eniL [ tcesretnI
-
] 2c cinoC , 1c cinoC [ tcesretnI 2c 1c
(4 )
2c 1c n ] n , 2c cinoC , 1c cinoC [ tcesretnI
]2f laimonyloP , 1f laimonyloP [ tcesretnI
2f 1f
] n , 2f laimonyloP , 1f laimonyloP[ tcesretnI n
2f 1f
]eniL , laimonyloP[ tcesretnI
] n , eniL , laimonyloP[ tcesretnI n
] A tnioP , g noitcnuF , f noitcnuF[ tcesretnI g f
A
] A tnioP , eniL , noitcnuF [ tcesretnI
A
tniopdiM
B A ] B tnioP , A tnioP [ tniopdiM
] tnemgeS [ tniopdiM
tnioP
] eniL [ tnioP
] cinoC [ tnioP
] noitcnuF [ tnioP
] nogyloP [ tnioP
] rotceV [ tnioP
] rotcev , tnioP [ tnioP
tooR
] laimonyloP [ tooR
-
a ] a , noitcnuF [ tooR
] b , a [ ] b , a , noitcnuF [ tooR
xetreV
] cinoC [ xetreV
6-3-3
rotceVerutavruC
]noitcnuF , tnioP[ rotceVerutavruC
] evruc , tnioP[ rotceVerutavruC
noitceirD
]eniL[ noitceriD
= y b + x a ( = + ) c
( - ) ) a - , b (
rotceVralucidnepreP
]eniL[ rotceVralucidnepreP
= y b + x a ( = + ) c
( ) ) b , a ( ]v rotceV[ rotceVralucidnepreP
) b , a ( ) a , b - (
rotceVralucidneprePtinU
) ]eniL[ rotceVralucidneprePtinU
(
) ]rotceV[ rotceVralucidneprePtinU
(
rotceVtinU
( ) ]eniL[ rotceVtinU
( ) ]rotceV[ rotceVtinU
-
rotceV
B A ]B tnioP , A tnioP[ rotceV
]tnioP[ rotceV
7-3-3
nemgeS
B A ]B tnioP , A tnioP[ tnemgeS
A a ]a rebmuN , A tnioP[ tnemgeS
8-3-3
yaR
B A ]B tnioP , A tnioP[ yaR
rotceV tnioP ]rotceV , tnioP[ yaR
9-3-3
nogyloP
, C , B , A ]........ , C , B , A [ nogyloP
B , A n ] n rebmuN , B , A [ nogyloP
01-3-3
rotcesiBelgnA
B C , B , A ]C , B , A[ rotcesiBelgnA
CB , BA
h g ]h eniL , g eniL[ rotcesiBelgnA
etotpmysA
]alobrebyH[ etotpmysA
sexA
]cinoC[ sexA
retemaiD
eniL ]cinoC , eniL[ retemaiD
-
cinoC
rotceV ]cinoC , rotceV[ retemaiD
cinoC
xirtceriD
alobaraP ]alobaraP[ xirtceriD
sixAtsriF
]cinoC[ sixAtsriF
eniL
B , A ]B tnioP , A tnioP[ eniL
eniL tnioP ]eniL , tnioP[ eniL
rotceV tnioP ]rotceV , tnioP[ eniL
ralucidnepreP
tnioP ]eniL , tnioP[ ralucidnepreP
eniL
tnioP ]rotceV , tnioP[ ralucidnepreP
rotceV
rotcesiBralucidnepreP
]B tnioP , A tnioP[ rotcesiBralucidnepreP
B , A
]tnemgeS[ rotcesiBralucidnepreP
tnemgeS
raloP
tnioP ]cinoC , tnioP[ raloP
cinoC
sixAdnoceS
]cinoC[ sixAdnoceS
tnegnaT
cinoC ]cinoC , tnioP[ tnegnaT
tnioP
-
cinoC ]cinoC , eniL[ tnegnaT
eniL
nioitcnuF ]noitcnuF , a rebmuN[ tnegnaT
a
nioitcnuF ]noitcnuF , A tnioP[ tnegnaT
A
)A(x A
)A(x = x A
, )A(x [ tnegnaT ] noitcnuf
tnioP evruC ]evruC , tnioP[ tnegnaT
11-3-3
elcriC
r M ]r rebmuN, M tnioP[ elcriC
M ]tnemgeS , M tnioP[ elcriC
tnemgeS
A M ]A tnioP , M tnioP[ elcriC
C , B , A ]C tnioP , B tnioP , A tnioP[ elcriC
cinoC
]E tnioP , D tnioP , C tnioP , B tnioP , A tnioP[ cinoC
E , D , C , B , A
espillE
a G , F ]a rebmuN , G , F[ espillE
-
a 2 GF
G , F ]tnemgeS , G , F [ espillE
tnemgeS
C , B , A ]C , B , A [ espillE
alobrepyH
a G , F ]a rebmuN , G , F [ alobrepyH
a 2 GF
G , F ]tnemgeS , G , F [ alobrepyH
tnemgeS
C , B , A ]C , B , A [ alobrepyH
elcriCgnitalucsO
noitcnuF ]noitcnuF , tnioP[ elcriCgnitalucsO
tnioP
evruC ]evruC , tnioP[ elcriCgnitalucsO
tnioP
alobaraP
g F ]g eniL , F tnioP[ alobaraP
21-3-3
FI :
tcesretni slargetni evitavired
] 2^x , )x( nis , 3 < x [ fi = )x( f )x(nis = )x( f 3 < x
2^x = )x(f
0 b 3 a ( b) ( 3 a )
-
( )
evitavireD
]noitcnuF[ evitavireD
n ]n rebmuN , noitcnuF[ evitavireD
evitavired )x(`f )x(f
)x(``f )x(f
] 2 , f[ evitavired
dnapxE
]noitcnuF[ dnapxE
21- x 2x = )x(f )4-x()3+x( dnapxE
rotcaF
]laimonyloP[ rotcaF
)3+x( )2-x( = )x(f ]6 x + 2^x[ rotcaF
noitcnuF
] b , a [ noitcnuf ]b , a , noitcnuF[ noitcnuF
] 1 , 1- , 2^x[ noitcnuF = )x(f
]1 , 1-[ 2x = )x(f
)x(f 2 = )x(g
]1 , 1-[
largetnI
]noitcnuF[ largetnI
-
laimonyloP
]noitcnuF[ laimonyloP
9 + x6 2x ] 2^)3-x( [ laimonyloP
1-n ]stniop n fo tsiL[ laimonyloP
yfilpmiS
]noitcnuF[ yfilpmiS
x3 = )x(f ] x + x + x [ yfilpmiS )x( nat = )x(f ] )x( soc / )x(nis [ yfilpmiS
)x2-( nis = )x(f ] )x( soc )x( nis 2- [ yfilpmiS
laimonyloProlyaT
n a = x ]n , a , noitcnuF[ laimonyloProlyaT
31-3-3
evruC
, 1e noisserpxE[ evruC
, 2e noisserpxE
]b , a , t retemaraP
1e
2e
]b , a [ t
] ip 2 , 0 , t , )t( nis 2 , )t( soc 2 [ evruc = C
)3(c
3
. tnioP
b , a
evitavireD
-
]evruC[ evitavireD
41-3-3
crA
B , A cinoC ]B tnioP , A tnioP , cinoC[ crA
, 1t rebmuN , cinoc[ crA
]2t rebmun
1t cinoC
2t
:
r ) )t( nis r , )t( soc r [ :
b a ) )t( nis b , )t( soc a (:
crAralucriC
B , A M ]B , A , M [ crAralucriC
B
rotceSralucriC
, A M ]B , A , M [ rotceSralucriC
B
B
crAralucricmucriC
C , B , A ]C , B , A [ crAralucricmucriC
rotceSralucricmucriC
C , B , A ]C , B , A [ rotceSralucricmucriC
rotceS
cinoC ]B tnioP , A tnioP , cinoC[ rotceS
B , A
-
cinoC ]2t rebmuN , 1t rebmuN , cinoC[ rotceS
2t 1t
:
r ) )t( nis r , )t( soc r [ :
b a ) )t( nis b , )t( soc a (:
elcricimeS
B , A ] B , A [ elcricimeS
51-3-3
txeTnoitcarF
re bmun ]rebmuN[ txeTnoitcarF
2 + x 5.1 = y : a
] ]a[ polS [ txeTnoitcarF
xeTaL
tcejbo ]tcejbo[ xeTaL
= )x( f 2 = a
]f[ xeTaL 2x a
2x2
tcejbo ]naelooB , tcejbo[ xetaL
eurT naelooB
eslaF
-
= )x( f 2 = a
2x a
2x 2 ] eurt , f [ xeTaL 2x a ]eslaf , f [ xeTaL
edocinUoTretteL
]retteL[ edocinUoTretteL
79 ]"a"[ edocinUoTretteL
emaN
]tcejbO[ emaN
cimanyd
tcejbO
tcejbO
]txet sa tcejbo fo emaN[ tcejbO
emaN
.. , 2A , 1A n 02A
[ tcejbO ( 2=n) 2 ] n + "A"
2A
txeTelbaT
]... , 3 tsiL , 2 tsiL , 1 tsiL[ txeTelbaT
-
] 4^x , 3^x , 2^x [ txeTelbaT , 2x
4x , 3x
]01 , 1 , i , 2^i[ecneuqeS [ txeTelbaT ]
...... , 3 tsiL , 2 tsiL , 1 tsiL[ txeTelbaT
]rv ,
thgir = r lacitrev = v rv
= "v" = "h" = "l" = "r" = "c"
]"v" , "61,9,4,1{ , }3,2,1{ [ txetelbaT
]"h" , "61,9,4,1{ , }3,2,1{ [ txetelbaT
, 9.32423 , 1.321 , 2.11{ [ txeTelbaT
]"r" , }"0.432"
txeT
tcejbo ]tcejbO[ txeT
-
2a = c 2 = a
4 ]c[ txeT
tcejbo ]naelooB , tcejbo[ txeT
eurT naelooB
eslaF
[ txeT 2a = c 2 = a
eslaf , c [ txeT 4 ] eurt , c
2a ]
tcejbO ] tnioP , tcejbO [ txeT
tnioP
] )3,2( , "olleh" [ txeT
)3,2( olleh
tcejbo ]naelooB , tnioP , tcejbO [ txeT
eurT naelooB
eslaF
tnioP
edocinUoTtxeT
]"txeT"[ edocinUoTtxeT
]"txet emoS"[ edocinUoTtxeT
}611 , 021 , 101 , 611 , 23 , 101 , 901 , 111 , 38 {
"olleh"
} 111 , 801 , 801 , 101 , 401 {
retteLoTedocinU
] regetni [ retteLoTedocinU
-
a ]79[ retteLoTedocinU
fo tsiL[ txeToTeodcinU
]sregetni
] }111 , 801 , 101 , 401{ [ txeToTeodcinU
"olleh"
61-3-3
sucoL
P Q ]P tnioP , Q tnioP[ sucoL
P )
(
71-3-3
dneppA
tsiL tcejbO ]tcejbO , tsiL[ dneppA
)5,5( } )5,5( , }3,2,1{ [ dneppA
} )5,5( ,3,2,1{ }3,2,1{
tcejbO tsiL ]tsiL , tcejbO[ dneppA
] }3,2,1{ , )5,5( [ dneppA
}3,2,1 , )5,5( { )5,5( }3,2,1{
fItnuoC
tsiL ]tsiL , noitidnoc[ fItnuoC
noitidnoC
] } 5, 4 , 3 , 2 , 1 { , 3 < x [ fItnuoC 3 < x
3
] 01A:1A , 3 < x [ fItnuoC
-
3 01A 1A
tnemelE
tsiL n ]n rebmuN , tsiL[ tnemelE
)
(
tsriF
tsiL ]tsiL[ tsriF
tsiL n ]stnemele fo n rebmuN , tsiL[ tsriF
tresnI
noitisoP tsiL tcejbO ]noitisoP , tsiL , tcejbO[ tresnI
] 3 , }5,4,3,2,1{ , 2^x[ tresnI 3 }5,4,3,2,1{ 2x
}5,4,3,2x,2,1{
] 1- , }5,4,3,2,1{ , )2,1( [ tresnI 1 )2,1(
]noitisoP , 2 tsiL , 1tsiL [ tresnI 2 tsiL 1 tsiL
noitisoP
noitcesretnI
1 tsiL ]2 tsiL , 1 tsiL[ noitcesretnI
2 tsiL
tsiLnoitaretI
x 1+n ]n , x , noitcnuF[ tsiLnoitaretI
noitcnuF
2^x = )x(f
] 2 , 3 , f [ tsiLnoitaretI = L
-
} 18 , 9 , 3 { = 2)23( , 23 , 3 { = L
nioJ
1 tsiL ]...... , 2 tsiL , 1 tsiL[ nioJ
...... 2 tsiL
] }3,2,1{ , }3,4,5{ [ nioJ
} 3 , 2 , 1 , 3 , 4 , 5 {
fIpeeK
tsiL ]tsiL , noitidnoC[ fIpeeK
noitidnoC
] } 6 , 5 , 4 , 3 , 2 , 1 { , 3 < x [ fIpeek
} 2 , 1 {
tsaL
tsiL ]tsiL[ tsaL
n ]stnemele fo n rebmuN , tsiL[ tsaL
tsiL
htgneL
tsiL ]tsiL[ htgneL
niM
]tsiL[ niM
xaM
]tsiL[ xaM
tcudorP
tsiL ]srebmun fo tsiL[ tcudorP
denifednUevomeR
-
]tsiL[ denifednUevomeR
, 1- , 3- , i , i^)1-([ ecneuqeS [ denifednUevomeR
] ]5.0
esreveR
]tsiL[ esreveR
ecneuqeS
]b , a , i , noisserpxE[ ecneuqeS
b a i noisserpxE
] 5 , 1 , i , )i,2{ [ ecneuqeS = L
i
5 1
, b , a , i , noisserpxE[ ecneuqeS
]s
s b a i noisserpxE
] 5.0 , 5 , 1 , i , )i,2{ [ ecneuqeS = L
5.0 5 1 i
, )3 , 2( , )5.2 , 2( , )2 , 2( , )5.1 , 2( , )1 , 2( { = L
} )5 , 2( , )5.4,2( , ) 4 , 2( , ) 5.3 , 2 (
b a
troS
]tsiL[ troS
-
}3,2,1{ ] }1,2,3[ [ troS
] } "sgif" , "selppa" , "sraep" [ troS
{ ] } )1,4( , )5,2( , )2,3( { [ troS )1,4( , )2,3( , )5,2(
muS
]tsiL[ muS
6 ] }3,2,1{ [ muS 3x + 2x = )x(f ] } 3^x , 2^x { [ muS 0505 ]] 001 , 1 , i , i[ ecneuqeS [ muS )5 , 3( ]} )3,2( , )2,1( { [ muS )2 , 4 ( ] } 3 , )2,1( { [ muS cbA ] } "c" , "b" , "a" { [ muS
, tsiL[ muS
fo n rebmuN
]stnemele
tsiL n
01 ] 4 , }6,5,4,3,2,1{ [ muS
ekaT
n m ]n dnE , m tratS , tsiL[ ekaT
tsiL
noinU
1tsiL ]2 tsiL , 1 tsiL[ noinU
2tsiL
81-3-3
-
. g A ] g , A [ tcelfer
B g A ] g , A [ tcelfer = B
etaliD
n S A ]S tnioP , n , A tnioP[ etaliD
n S eniL ]S tnioP , n , eniL[ etaliD
S cinoC ]S tnioP , n , cinoC[ etaliD
n
n S nogyloP ]S tnioP , n ,nogyloP[ etaliD
n S egami ]S tnioP , n , egamI[ etaliD
tcelfeR
B A ]B tnioP , A tnioP[ tcelfeR
tnioP eniL ]tnioP , eniL[ tcelfeR
tnioP cinoC ] tnioP , cinoC[ tcelfeR
tnioP nogyloP ]tnioP , nogyloP[ tcelfeR
tnioP egami ]tnioP , egamI[ tcelfeR
eniL tnioP ]eniL , tnioP[ tcelfeR
eniL g ]eniL , g eniL[ tcelfeR
eniL cinoC ]eniL , cinoC[ tcelfeR
eniL nogyloP ]eniL , nogyloP[ tcelfeR
eniL egami ]eniL , egamI[ tcelfeR
elcriC tnioP ]elcriC , tnioP[ tcelfeR
etatoR
elgnA tnioP ]elgnA , tnioP[ etatoR
-
elgnA rotceV ]elgnA , rotceV[ etatoR
elgnA eniL ]elgnA , eniL[ etatoR
elgnA cinoC ]elgnA , cinoC[ etatoR
elgnA nogyloP ]elgnA , nogyloP[ etatoR
elgnA egamI ]elgnA , egamI[ etatoR
B elgnA tnioP ]B tnioP , elgnA , A tnioP[ etatoR
elgnA eniL ]tnioP , elgnA , eniL[ etatoR
tnioP
elgnA cinoC ]tnioP , elgnA , cinoC[ etatoR
tnioP
elgnA nogyloP ]tnioP , elgnA , nogyloP[ etatoR
tnioP
, elgnA egamI ]tnioP , elgnA , egamI[ etatoR
tnioP
etalsnarT
rotceV tnioP ( ) ]rotceV , tnioP[ etalsnarT
eniL ( ) ]rotceV , eniL[ etalsnarT
rotceV
cinoC ( ) ]rotceV , cinoC[ etalsnarT
rotceV
rotceV noitcnuF ( ) ]rotceV , noitcnuF[ etalsnarT
nogyloP ( ) ]rotceV , nogyloP[ etalsnarT
rotceV
rotceV egamI ( ) ]rotceV , egamI[ etalsnarT
tnioP rotceV ( ) ]tnioP , rotceV[ etalsnarT
-
sdnammoC scitsitatS 91-3-3
trahCraB
] b , a [ ]sthgieh fo tsiL , b , a [ trahCraB
tsiL
] } 5 , 4 , 3 , 2 , 1 { , 02 , 01 [ trahCraB
] 02 , 01 [
}5 ,4 , 3 , 2 , 1 {
, noisserpxE , b , a[ trahCraB
]d oT , c morF , k elbairaV
] b , a [
noisserpxE
d c
01 = n 9.0 = q 1.0 = p
[ tneiciffeoClaimoniB , 5.0 + n , 5.0 -[ trahCraB
] n , 0 , k , )k-n(^q*k^p * ] k , n
]5.0 + n , 5.0-[
, noisserpxE , b , a[ trahcraB
] s petS , d oT , c morF , k elbairaV
] b , a [
noisserpxE
s d c
tsiL ]htdiW ,atad war fo tsiL[ trahCraB
htdiw
, 5 , 5 , 3 , 3 , 3 , 2 , 2 , 2 ,1,1,1{ [ trahCraB
] 1 , }5
fo tsiL , atad fo tsiL[ trahcraB
]seicneuqerf
tsiL
fo tsiL atad fo
ycneuqerf
fo tsiL atad
}1 , 0 , 21 , 8 , 5{ , }41,31,21,11,01{ [ trahCraB
-
]
] }3 , 34 , 21 , 0 , 1{ , }9,8,7,6,5{ [ trahCraB
, 31 , 33 , 21{ , }6.0 , 5.0 , 4.0 , 3.0{ [ trahCraB
] }4
fo tsiL , atad fo tsiL[ trahcraB
]w srab fo htdiW ,seicneuqerf
tsiL
fo tsiL atad fo
w ycneuqerf
fo tsiL atad
}1 , 0 , 21 , 8 , 5{ , }41,31,21,11,01{ [ trahCraB
] 5.0 ,
5.0
}1 , 0 , 21 , 8 , 5{ , }41,31,21,11,01{ [ trahCraB
]0 ,
tolPxoB
fo tsiL , elacSy , tesffOy[ tolPxoB
]atad war
elacSy tesffOy
atad war fo tsiL
}9 ,8,8,8,7,7,6,5,5,4,3,2,2{ , 1 , 0 [ tolPxoB
1Q ,tratS , elacSy , tesffOy[ tolPxoB
]dnE , 3Q , naideM ,
]dne , trats [
tneiciffeoCnoitalerroC
-x fo tsiL[ tneiciffeoCnoitalerroC
]setanidrooc-y fo tsiL , setanidrooc
x
y
fo tsiL[ tneiciffeoCnoitalerroC
]stniop
ecnairavoC
-
]2 tsiL ,1 tsiL[ ecnairavoC
y x ]stniop fo tsiL[ ecnairavoC
eniLtiF
x y ]stniop fo tsiL[ eniLtiF
sdnammoC tif rehtO
]stniop fo tsiL[ pxEtiF
y x ]stniop fo tsiL[ XeniLtiF
]stniop fo tsiL[ goLtiF
))xk-( ^x b+1( / a ]stniop fo tsiL[ citsigoLtiF
.
fo n eergeD , stniop fo tsiL[ yloPtiF
]laimonylop
n
)d + xc( nis b + a ]stniop fo tsiL[ woPtiF
.
margotsiH
seiradnuob ssalc fo tsiL[ margotsiH
]sthgieh fo tsiL ,
seiradnuob sslc fo tsil
sthgieh fo tsil
] }1,3,8,6,2{ , }5,4,3,2,1,0{ [ margotsiH
}1,3,8,6,2{ 5
]1,0[
]2,1[
seiradnuob ssalc fo tsiL[ margotsiH
-
seiradnuob sslc fo tsil ]atad war fo tsiL ,
, }5,4,3,2,1,0{ [ margotsiH
]}0.4,5.2,2.2,7.1,3.1,2.1,1.1,0.1{
5
1 ( ) 2 ( )
( )
lamroNesrevnI
dradnatS , naeM[ lamroNesrevnI
]ytilibaborp , noitaived
()1-
( ) ( )1-
ytilibaborp
naeM
]srebmun fo tsiL[ naeM
]stniop fo tsiL[ XnaeM
]stniop fo tsiL[ YnaeM
naideM
]srebmun fo tsiL[ naideM
edoM
]srebmun fo tsiL[ edoM
} { } 4 , ,3 , 2 , 1{ [ edoM }1{ ] }4 ,3 ,2 ,1 , 1 , 1 {[ edoM
}3, 2 , 1{ ] }4,3,3 ,2,2,1,1{ [ edoM
lamroN
, noitaived dradnatS , naeM[ lamroN
]eulav elbairaV
[/ ( )]
( )
-
)
(
sdnammoC elitrauQ
]srebmun fo tsiL[ 1Q
]srebmun fo tsiL[ 3Q
DS
]srebmun fo tsiL[ DS
sdnammoC amgiS
]srebmun fo tsiL[ XXamgiS
2^]tsil[naeM ]tsil[htgnel / ]tsil[XXamgiS
]stniop fo tsiL[ XXamgiS
, etanidrooc-x fo tsiL[ YXamgiS
]etanidrooc-y fo tsiL
]stniop fo tsiL[ YXamgiS
]tsil[htgnel / ]tsil[YXamgiS
]tsil[YnaeM*]tsil[XnaeM
]stniop fo tsiL[ YYamgiS
seititnauq scitsitats rof sdnammoC
(/) ( ) ( 2) ]srebmun fo tsiL , srebmun fo tsiL[ xxS
(/) ( ) ( 2) ]stniop fo tsiL[ xxS
(/) ( ) ( ) ]srebmun fo tsiL , srebmun fo tsiL[ yxS
(/) ( ) ( ) ]stniop fo tsiL[ yxS
(/) ( ) ( 2) ]srebmun fo tsiL , srebmun fo tsiL[ yyS
-
(/) ( ) ( 2) ]stniop fo tsiL[yyS
ecnairaV
]srebmun fo tsiL[ ecnairaV
02-3-3
egnaRleC
dnE tratS ]llec dnE , llec tratS[ egnaRleC
}3A , 2A , 1A{ ]3A , 1A[ egnaRlleC
nmuloC
) ]llec teehsdaerpS[ nmuloC
(1
nmuloC 3B
B 2 ]3B[
emaNnmuloC
teehsdaerpS[ emaNnmuloC
]llec
1A
A ]1A[ emaNnmuloC
woR
) ]llec teehsdaerpS[ woR
(1
woR 3B
3 ]3B[
sdnammoC xirtaM 12-3-3
tnanimreteD
]xirtaM[ tnanimreteD
2- ] } }4,3{ , }2,1{ { [ tnanimreteD
-
trevnI
]xirtaM[ trevnI
} }5.0- , 5.1{ , }1,2-{ { ] } }4,3{ , }2,1{ [ trevnI
esopsnarT
]xirtaM[ esopsnarT
} }4,2{ , }3,1{ { ] } }4,3{ , }2,1{ { [ esopsnarT
-4
uneM eliF" " 1-4
.
-
.
arbeGoeG
bgg
arbeGoeG
bgg arbeGoeG
bgg arbeGoeG
(lmth)
.
:
cihparG krowteN elbatroP GNP
( ipd) pid 003)
( GNP
.
-
% 001 drow
tpircstsoP detaluspacnE -SPE
warD leroC .
ipd 27
tamroF tnemucoD elbatroP -FDP cihparG rotceV elbaelacS GVS
GVS FDP
( ) )
( tamroF ateM decnahnE FME
skcirTSP
xeTaL
xeTaL ZkirT/FGP
" "
" " " " " " " "
-
retnE
. arbeGoeG
uneM tidE" " 2-4
" "
" "
-
"" 3-4
" "" "
" "" "
A + tfihS + lrtC
S + tfihS + lrtC
-
" "
" " " " " "
" "
"" 4-4
" "
" " (" ) " " "
( )
" "
-
()
) .
(
"" " "
( y , x)
(y | x)
" "
-
arbeGoeG
" " " "
)
(
"" 5-4
.
-:
.1
.2
.3
.
"BGG"
( TGG) .
-
""
( TGG)
( BGG)
(BGG)
"" 6-4
""
"" 7-4
.
-:
tratsbew
-
pleh/gro.arbegoeg.www//:ptth
ikiw
: moc.secapsikiw.tpyge-arbegoeg//:sptth
arbeGoeG -5
noitaminA( ) 1 -5
( ) arbeGoeG
.
" "
-
" "
01 1 (. ) :
: < =>
: = >
: < =
-
noitaminA launaM
+
) k , k 2 ( = P k P
k
( )
1.0 ( yek worra + tfihS) 01 ( yek worra + lrtC) 001 ( yek worra + tlA)
ytilibisiV lanoitidnoC 2-5
.
.
( eurt = b )
" . "
" "
" " " "
2 < a a 2
-
( eslaf eurt) b eslaf = b eurt = b
h g g || h
slooT denifeD resU 3-5
. arbeGoeG
.
"
" " " " "
" " " "
-:
( )
" " " "
" "
" "
-
" "
erauqs " "
"
"
" " " "
-
.... " " .
( tgg) (bgg)
" " " "
" "
-:
" " " " .1
" " " "
( tgg) .2
" " " "
( tgg)
sroloC cimanyD 4-5
" " arbeGoeG
-
: ". "
" "
( ) " "
" "
[ 1 0]
[ 1 0] c , b , a " " " "
( )
tpircSavaJ 5-5
arbeGoeG pircS avaJ LMTH
pircSavaJ dna stelppA arbeGoeG
-
6-5
tlA tfihS + lrtC lrtC
A /
ahpla
ateb B
C
atled D
E
"" reluE
F
ihp
ammag G
H
I
J
K
L
adbmal
um M
N
O eerged
lobmys
P
ip
-
(spe , gnp)
Q
R
S /
amgis
ateht skcirTSP T
U
V )
(
W
agemo
X
Y
Z
0
0
1
1
2
2
3
3
4
4
5
-
5
6
6
7
7
8
8
9
9
-
+
=
.
-
*
1F
2F
3F
4F
9F
retnE
kcilc-tfeL
kcilc-thgiR
kcilC
dna kcilC
gard
llorcS
leehW /
-
eteleD
ecapskcaB
01
1.0
001
01
1.0
001
01
1.0
001
-
01
1.0
001
pUgP/emoH
nDgP/dnE
tfihS + tlA ( ) c + tlA + lortC( : )
( P + tlA) ( O + tlA)
snoitpaC dna slebaL 7-5
:
( / ) /
" "
" " " " " "
-
.
" " " " " "
" " " " " " " "
" " " "
A
" "
-
"" " " " "
sreyaL 8-5
( )
( 0)
( 01) .
" "
(9 0 ) " "
"" " "
GVS pircS avaJ
enifedeR( ) 9-5
. ( )
:
retnE
retnE
" "
" "
" " " "
h A (1
-
" A
. h A ]h[tniop "
( 2 ,1)
h B A (2
, A [ enil " " B , A
]B
sucoL dna ecarT 01-5
"
" " "
arbeGoeG
sucoL
)
(
B , A B , A a C ) a C
(a
C P ) 3 , )C( x ( = P
C P
sucoL P C P
P
/ColorImageDict > /JPEG2000ColorACSImageDict > /JPEG2000ColorImageDict > /AntiAliasGrayImages false /CropGrayImages true /GrayImageMinResolution 300 /GrayImageMinResolutionPolicy /OK /DownsampleGrayImages true /GrayImageDownsampleType /Bicubic /GrayImageResolution 300 /GrayImageDepth 8 /GrayImageMinDownsampleDepth 2 /GrayImageDownsampleThreshold 1.50000 /EncodeGrayImages true /GrayImageFilter /FlateEncode /AutoFilterGrayImages false /GrayImageAutoFilterStrategy /JPEG /GrayACSImageDict > /GrayImageDict > /JPEG2000GrayACSImageDict > /JPEG2000GrayImageDict > /AntiAliasMonoImages false /CropMonoImages true /MonoImageMinResolution 1200 /MonoImageMinResolutionPolicy /OK /DownsampleMonoImages true /MonoImageDownsampleType /Bicubic /MonoImageResolution 1200 /MonoImageDepth -1 /MonoImageDownsampleThreshold 1.50000 /EncodeMonoImages true /MonoImageFilter /FlateEncode /MonoImageDict > /AllowPSXObjects false /CheckCompliance [ /None ] /PDFX1aCheck false /PDFX3Check false /PDFXCompliantPDFOnly false /PDFXNoTrimBoxError true /PDFXTrimBoxToMediaBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXSetBleedBoxToMediaBox true /PDFXBleedBoxToTrimBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXOutputIntentProfile () /PDFXOutputConditionIdentifier () /PDFXOutputCondition () /PDFXRegistryName () /PDFXTrapped /False
/Description > /Namespace [ (Adobe) (Common) (1.0) ] /OtherNamespaces [ > /FormElements false /GenerateStructure true /IncludeBookmarks false /IncludeHyperlinks false /IncludeInteractive false /IncludeLayers false /IncludeProfiles true /MultimediaHandling /UseObjectSettings /Namespace [ (Adobe) (CreativeSuite) (2.0) ] /PDFXOutputIntentProfileSelector /NA /PreserveEditing true /UntaggedCMYKHandling /LeaveUntagged /UntaggedRGBHandling /LeaveUntagged /UseDocumentBleed false >> ]>> setdistillerparams> setpagedevice
