pbïšbî‹Üa@ò†b¾@ðjî‹vnÜa@çbznàfia@|ïz—m pbïšbî‹Üa@ò†b¾@ðjî‹vnÜa@çbznàfia@|ïz—m pbïšbî‹Üa@ò†b¾@ðjî‹vnÜa@çbznàfia@|ïz—m pbïšbî‹Üa@ò†b¾@ðjî‹vnÜa@çbznàfia@|ïz—m
@ @ @ @ @ @
@ @
@ñbà@òŠì†
@ñbà@òŠì†
@ñbà@òŠì†
@ñbà@òŠì†
@@@@ 2010
@ @
@ @
@ @
@ @
••••
@âÔŠ@æî‹ánÜa
@âÔŠ@æî‹ánÜa @âÔŠ@æî‹ánÜa
@âÔŠ@æî‹ánÜa ZZZZ 01
óÈíáa@†ìä ℝ2
ðÝ‚a‡Üa@kïØÜa@çíäbÕi@
@ðÝî@b·@Ӌɾa@Τ Z
( ) ( ) ( x y
@ @, Τ a b , = xa − yb xb , + ya + 2 yb )
@ @
@óï÷§a@óÈíáa@nÉä@ì
( )
@Z{ ,
2/ 0 }
G = x y ∈
ℝx + ≠ y
N@@ @
(1
− çc@µjåÜ æà@‹Õnà@öu@
G
(
ℝ2, Τ )
N@
@ @
ÞÙÜ
( ) x y , ∈ G
ÞÙÜ@ì@
( ) a b , ∈ G
@båî‡Ü@L@
Z
( ) ( ) ( x y , Τ a b , = xa − yb xb , + ya + 2 yb )
@ @
( xa − yb ) ( + xb + ya + 2 yb ) ( = x + y )( a + b )
@@ì@ @
@çc@b·@ì
( ) x y , ∈ G
@Z( ) a b , ∈ G
ì@@çhÐ@L@
@Z
0 x + ≠ y
ì@0 a + ≠ b
@ @
@çˆg
( xa − yb xb , + ya + 2 yb ) ∈ G
@Z@ @
@çc@ôåÉ·
( ) ( ) x y , Τ a b , ∈ G
@Z N@@ @
çhÐ@êåà@ì æà@‹Õnà@öu@
G
(
ℝ2,Τ )
N@
@ @
i
çc@µj
( G , Τ )
@óï܆bjm@ò‹àŒ@
N
ÞÙÜ
( ) x y , ( ) a b ,
ì@æà@
@båî‡Ü@L@
G
Z( ) ( ) ( )
( )
( ) ( )
, , , 2
, 2
, ,
x y a b xa yb xb ya yb ax by bx ay by a b x y
Τ = − + +
= − + +
= Τ
@ @
çíäbÕÜa@çˆg À@ð܆bjm@
Τ
N@
G
@ @
Ü
( ) x y ,
ÞÙ( ) a b ,
ì@( α β , )
@ì@æà@
@båî‡Ü@L@
G
Z( ) ( ) ( x y , Τ a b , Τ α β , ) ( = xa − by bx , + ya + 2 yb ) ( Τ α β , )
@ @
[ ] [ ]
[ ] [ ] [ ]
2 ,
2 2 2
xa yb xb ya yb
xa yb xb ya yb xb ya yb
α β
β α β
− − + +
= − + + + + + +
@ @
[ ] [ ]
[ ] [ ] [ ]
2 ,
2 2 2
x a b y b a b
x a b b y a b y b a b
α β α β β
β α β α β α β β
− − + +
= + + + − + + +
@ @
@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@
( ) ( )
( ) ( ) ( )
, , 2
, , ,
x y a b b a b
x y a b
α β α β β
α β
= Τ − + +
= Τ Τ
@ @
çíäbÕÜa@çˆg À@ðÉïávm@
Τ
N@
G
@ @
‹—åÈ@æÈ@szjä
( ) a b ,
æà@
@sï¢@
G
( ) x y , G x y
Z; ( ) ( ) ( ) , a b , x y ,
∀ ∈ Τ =
@ @
ñc çc@
@Z@
( ) , ;
@2 xa yb x x y G
xb ya yb y
− =
∀ ∈
+ + =
N@
@ @
@‰‚dä
( ) ( ) x y , = 1, 0
@Z@ôÝÈ@Þ—zåÐ@L@
1
@Z0 a b
=
=
N@@@ @
ÞÙÜ@L@bïÙÈ
( ) x y ,
æà@
@båî‡Ü@L@
G 1 0
@Z0 1 2 0
x y x
x y y y
× − × =
× + × + × =
N@
@ @
çíäbÕÜa@L@çˆg À@a‡îb«@a‹—åÈ@ÞjÕî@
Τ
@xìÜa@íè@
G
( ) 1, 0
N@
@ @
æÙïÜ
( ) a b ,
@a‹—åÈ@
æà N@
G
À@ÞzåÜ
@ó܆bɾa@
G
( ) ( ) ( ) ( ) Ε : x y , Τ a b , = 1, 0
@Z@båî‡Ü@L@
Z
@ @
( )
( )
1
2 0
1
2 0
xa yb xb ya yb ax by
bx a b y
− = Ε ⇔
+ + =
− =
⇔
+ + =
@ @
@ðè@óáÅåÜa@ë‰è@ò†‡zà
( )
2@Z2
a b
a b
b a b
∆ = − = +
N@+
@ @
@çc@b·@ì
( ) a b , ∈ G
çhÐ@
0 a + ≠ b
@çˆg@L@
@Z
∆ ≠ 0
N@@a‡ïyì@þy@ÞjÕm@óÕibÜa@óáÅåÜa@çhÐ@L@ðÜbnÜbi@ì
( ) x y ,
@sï¢@
Z
@ @
( )
2( )
21
0 2 2
x
b
a b a b
x a b a b
−
∆ + +
= = =
∆ + +
( ) (
2)
2 @ì@0
y
1 a
b b
y a b a b
∆ −
= = =
∆ + +
@ @
@ça@b·@ì
1
@Z0 x y
a b
+ = ≠
@çhÐ@L@
+
( ) x y , ∈ G
@Z@ @
‹—åÈ@ÞØ@L@çˆg
( ) a b ,
æà@
çíäbÕÝÜ@ójåÜbi@þqb¿@ÞjÕî@
G
@íè@Þqbá¾a@a‰è@ì
Τ
Z@ @
( ) ( ) ( )
'
2 2
, a 2 b , b
a b a b a b
+ −
=
+ +
N@
@ @
@ó–þ‚
Z
çíäbÕÜa@çc@b·
æà@‹—åÈ@ÞÙÜ@ì@a‡îb«@a‹—åÈ@ÞjÕî@ì@ðÉïá¤@ì@ð܆bjm@
Τ
Þqb¿@G
@ @
( G , Τ )
@çhÐ@óï܆bjm@ò‹àŒ@
N
@ @
(2
− íáa@æÙnÜ
@óÈ
( ) , / ( ) ,
@Z0
x y y
M x y x y G
x y +
Ε = = + ∈
N@@ @
çc@µjåÜ@− æà@‹Õnà@öu@
Ε
( Μ
2( )
ℝ, × )
N@
@ @
ÞÙÜ
( ) x y , ( ) a b ,
@ì@æà
@båî‡Ü@L@
G
Z( ) , ( ) ,
0 0
x y y a b b
M x y M a b
x y a b
+ +
× = + × +
@ @
( )( ) ( ) ( )
( )( )
( )
( ) ( )
( )
0
2 2
0 2
, 2
, ,
x y a b x y b y a b x y a b
xa yb xb ya yb xb ya yb
xa yb xb ya yb M xa yb xb ya by
M x y a b
+ + + + +
=
+ +
− + + + + +
= − + + +
= − + +
= Τ
@ @
@çc@b·@ì
( ) x y , ( ) a b ,
@ì@¶g@çbïánåî
@çhÐ@L@
G
( ) ( ) x y , Τ a b , ∈ G
@Z@ @
@çˆg
Ε
@Zæà@‹Õnà@öu@
( Μ
2( )
ℝ, × )
N@
@ @
−
@ÖïjXnÜa@çc@µjåÜ@
( ) ( )
Z:
, ,
f G
x y M x y
→ Ε
æà@ÞØb“m@L@ ֏( G , Τ )
@
( ) Ε × ,
í®N@
ÞÙÜ
( ) x y , ( ) a b ,
@ì@æà L@
G
@båî‡Ü@
Z
( ) ( )
( ) ( ( ) ( ) )
( ) ( ) ( ) ( )
, , , ,
, , , ,
f x y a b M x y a b M x y M a b
f x y f a b
Τ = Τ
= ×
= ×
@ @
çˆg
æà@ÞØb“m@
f
( G , Τ ) ( ) Ε × ,
í®@N@
@ @
çc@µjä
@ðÝibÕm@ f N
( ) x y ,
ÞÙÜ( ) a b ,
@ì@æà
@båî‡Ü@L@
G
Z( ) ( ) ( ) ( )
( ) ( )
, , , ,
0 0
, ,
f x y f a b M x y M a b
x y y a b b
x y a b
x y a b y b
x a y b x y a b
= ⇔ =
+ +
⇔ + = + + = +
⇔
=
=
⇔
=
⇔ =
@ @
ÖïjXnÜa@çˆg
@ðåîbjm@ f N
@båî‡Ü
( ∀ ∈Ε ∃ M ) ( ) , ( x y , ∈ G ) / M = x + 0 y x + y y
Z@
@çˆg
( ) , ( ) ,
@ZM = M x y = f x y
@ @
ÖïjXnÜa@çˆg
@ðÜí@
f
N@ @
çhÐ@L@ðÜbnÜbi@ì æà@ðÝibÕm@ÞØb“m@
f
( G , Τ ) ( ) Ε × ,
í®@N@
@ @
çc@b·@ì
( G , Τ )
çhÐ@L@óï܆bjm@ò‹àŒ@
( ) Ε × ,
@óï܆bjm@ò‹àŒ@b›îc@
N
@ @
− ÞÙÜ
( ) x y ,
æà@
@båî‡Ü@L@
G
Z@ @
( ) ( )
( )
( ) ( ) ( ) ( )
( )
1 1
'
2 2
2 2
2
, ,
,
2 ,
2 ,
1 0 1 M x y f x y f x y x y y f x y x y x y y M x y x y y x y x y x y
− −= = + − = + + + − = + + − + + = +
(3 − À@ÞzåÜ @ó܆bɾa@Ε
@Z 2X = Ι
2 N@ @ @ @Ê›ä( ) ,
@ZX = M x y
sïy@L@( ) x y , ∈ G
@çˆg@L@ Z @ @( ) ( ) ( ) ( )
2 1 2 2 2 2, , , 1 1 0 0 1 X M x y M x y M x y y x y x y x y y x y x y x y x y y = Ι ⇔ = Ι ⇔
−= − + + + ⇔ = + + + = + ⇔ − ( ) ( ) ( ) ( ) ( { ) ( ) }
2 2 21 1 0 1 1 , 1, 0 ; 1, 0 0 0 x y y x y x y y x y x y y x y + = ⇔ + + = = + + = = ⇔ ⇔ ⇔ ∈ − = =
@ @ @ó܆bɾa@çhÐ@êåà@ì @Z 2X = Ι
2 À@µÝy@ÞjÕm@L@ @báè@Ε ( ) 1, 0
2@ZM = Ι ( 1, 0 )
2ì@M − = −Ι
N@ @ @•••• @âÔŠ@æî‹ánÜa @âÔŠ@æî‹ánÜa @âÔŠ@æî‹ánÜa @âÔŠ@æî‹ánÜa ZZZZ 02
(1 − æÙïÜ @ÞØ@Âi‹î@ñ‰Üa@ÞîíznÜaR ( )
óXÕä óXÕåÜbi@M z ( )
' 'M z
@sï¢@ Z @ @( )
'1 z = + + iz i
N@ @ @ − @Ê›ä @Za = i
@ì@1 b = + i
N@ @ @ çc@b·{ } 1
@Z@a ∈ −
ℂ @ì@1 a =
çhÐ@L@ ëØ‹à@Ö¨@çaŠì†@R
@íè@Ω
Z @ @1 1 1 b i a i i ω = = + = − −
@ @ @íè@ênîìaŒ@‘bïÔ@ì( ) ( )
@Zarg arg a = i = π 2
N@ @ @ − ÞÙÜ æà@z
@båî‡Ü@L@ℂ Z @ @( ) ( ) ( ) ( )
1 ' ' ' ' ' '1
1
1
1
R M M R M M
z iz i
iz z i i iz z i
z iz i
−
= ⇔ =
⇔ = + +
⇔ − = − +
⇔ − = + −
⇔ = − − +
@ @
çaŠì‡Üa@çˆg
R
−1óXÕåÜa@Âi‹î@
( )
' '
M z
óXÕåÜbi@( )
@sï¢@
M z
Z@ @
'
1
z = − − + iz i
N@@ @
(2
− À@nÉä Z@ℂ
( ) Ε − − : iz
33 1 ( ) + i z
2− 6 z − + = 10 2 i 0
N@
@ @
ÞÙÜ@− æà@
z
@båî‡Ü@L@ℂ Z
@ @
( ) ( )
( ) ( ) ( ) ( ) ( )
( ) ( )
( )
3 3
'
3 2 2 3
3 2
3 2
1
3 1 3 1 1
3 1 3 2 2 1
3 1 6 2 2
z iz i
iz iz i iz i i
iz i z iz i i i
iz i z z i
= + +
= + + + + + +
= − − + + × + +
= − − + − − +
@ @
@çˆg
( ) z
' 3− = − − 8 iz
33 1 ( ) + i z
2− 6 z − + 10 2 i
@Z@ @
çíÙî@L@ðÜbnÜbi@ì ó܆bÉáÝÜ@þy@
z
( ) Ε
@ÂÕÐ@ì@aˆg@
@çbØ@aˆg Z
@ @
( )
3 2
3 1 6 10 2 0
iz i z z i
− − + − − + =
@ @
@ñc
( ) z
' 3− = 8 0
@Z@ @
@ñc
( ) z
' 3= 8
@Z@ @
çc@ôåÉ·
z
'@†‡ÉÝÜ@kÉÙà@Š‰u@
N@
8
@ @
− çc@b·
2
3= 8
çhÐ@Š‰u@
2
@†‡ÉÝÜ@kÉÙà@
N@
8
@ @
†‡ÉÝÜ@ójÉÙ¾a@Šì‰§a@ì
@ðè@
8
@Z ì@
2
ì@2 j
@sïy@
2 j 1 3
@Z2 2
j = − + i
@N@
@êåà@ì Z
@ @
( ) { }
{ }
( ) ( )
{ }
'
2; 2 ; 2
2 1 ; 2 1 ; 2 1
1 ; 3 1 2 ; 3 1 2
z j j
z i i ij i i j i
z i i i
Ε ⇔ ∈
⇔ ∈ − − + − − + − − +
⇔ ∈ − − − + − + +
@ @
@
@çhÐ@ðÜbnÜbi@ì
( ) ( ) ( )
@Z{ 1 ; 3 1 2 ; 3 1 2 }
S = − + i − + i − + + i
N@@ @
ج
− æÙnÜ ì@
A
ì@
B
@ðÜaínÜa@ôÝÈ@béÔb¨c@ðnÜa@ÂÕåÜa@
C
Z@ @
( ) 1
a = − + i
( 3 1 ) 2
@ì@b = − + i
( 3 1 ) 2
@ì@c = − + + i
@ @
@Ê›ä@ì
( )
@ZA
'= R A
( )
@ì@B
'= R B
( )
@ì@C
'= R C
@ @
@ÂÕåÜa@×bzÜc Z
A
'ì@
B
'ì@
C
'@ðÜaínÜa@ôÝÈ@ðè@
@Z ì@
2 2 j
ì@2 j
sÝr¾a@ì@' ' '
A B C
@Ãb«@
ò‹÷a‡Übi
( ) C
'@bèØ‹à@ðnÜa@
béÈbÉ’@ì@
O 2
r =
N@@ @
ÞÙÜ æà@
z
@båî‡Ü@L@ℂ Z
( )
( ) ( )
( ) ( )
'
' ' '
1 '
1 2 2
iz i z
M z C
M z R
−C + + = ⇔ =
⇔ ∈
⇔ ∈
@ @
ÂÕåÜa@óÈíáª@L@çˆg
( )
@sï¢@
M z
( ) 1 2
@Ziz + + = i
ò‹÷a‡Üa@òŠí–@ðè@( ) C
'@I@
@óXïa
@sÝr¾bi
' ' '
A B C
@H@
çaŠì‡Übi
R
−1c@
ò‹÷a‡Üa@ñ
( ) C
sÝr¾bi@óXïa@
N@
ABC
@ @
@çÿ
( )
@Z1 '
A = R
−A
( )
@ì@1 '
B = R
−B
( )
@ì@1 '
C = R
−C
N@@ @
ò‹÷a‡Üa@Ø‹à
( ) C ( )
@íè@' 1
O = R
−O
@íè@êÕ¨@ì@
@Z
'
1 i
ω = − +
N@@âÔŠ@æî‹ánÜa ••••
@âÔŠ@æî‹ánÜa @âÔŠ@æî‹ánÜa
@âÔŠ@æî‹ánÜa ZZZZ 03
À@nÉä
×
ℤ ℤ¾a@
@ó܆bÉ
( ) Ε : 28 x − 15 y = − 6
@Z N@(1
−
−
@båî‡Ü Z
@ @
28 3 15 6 × − × = 84 90 − = − 6
@ @@ó܆bÉáÝÜ@™b‚@Þy@
( ) 3, 6
@xìÜa@L@çˆg( ) Ε
N@
ÞÙÜ@−
( ) x y ,
×
æà@ℤ ℤ
@båî‡Ü@L@
Z
( )
@ @( ) ( )
28 15 28 3 15 6
28 3 15 6
x y
x y
Ε ⇔ − = × − ×
⇔ − = −
@çˆg
( )
Z@15 / 28 x − 3
@
@ @
@çc@b·@ì
@Z
15 ∧ 28 1 =
@™íØ@óåèà@ky@çhÐ@
( )
@Z15 / x − 3
N@@ @
@çhÐ@êåà@ì Z
@ @
@
( ) ( )
( ) ( )
( ) ( )
( )
15 / 3
28 3 15 6
/ 3 15
28 15 15 6
/ 3 15
6 28
x
x y
x k
k k y
x k
k y k
− Ε ⇔
− = −
− =
⇔ ∃ ∈
× = −
= +
⇔ ∃ ∈
= +
ℤℤ
@ @
@ðÜbnÜbi@ì
( )
@Z{ 3 15 , 6 28 / }
S = + k + k k ∈
ℤ N@@ @
(2
− À@Þzä
@óáÅåÜa@ℤ
( ) [ ]
@Z[ ]
: 8 28 2 15 S z
z
≡
≡
N@@ @
ÞÙÜ æà@
z
@båî‡Ü@L@ℤ Z
( ) ( )
( ) ( )
228 / 8 8 28
, /
2 15
15 / 2
z z
S z z
α β α
β
−
− =
⇔ ⇔ ∃ ∈
− − =
@ @ ℤ
@çˆg
( z − − − = 8 ) ( z 2 ) 28 α − 15 β
@Z@
@ @
@çc@ôåÉái
@Z
28 α − 15 β = − 6
@ @
@çˆg
( α β , ) ( = + 3 15 , 6 k + 28 k ) / k ∈
ℤ@Z@ @
@êåà@ì
( )
@Z8 28 3 15 /
z − = + k k ∈
ℤ@ @
@ñc
@Z
92 420 /
z = + k k ∈
ℤ@ @
@çbØ@aˆg@L@bïÙÈ
@Z
92 420
z = + k
@sïy@
k ∈
ℤ N@@çhÐ
( )
@Z8 28 3 15
z − = − k
( )
@ì@2 15 6 28
z − = − k
@ @
@çˆg
[ ]
@Z[ ]
8 28 2 15 z
z
≡
≡
N@@ @
óáÅåÜa@ßíÝy@óÈíáváÐ@L@ðÜbnÜbi@ì
( ) S
À@
@ðè@ℤ
{ 92 420 / }
@ZS = + k k ∈
ℤ N@@ @
(3
−
@óïåàÜa@ò‡¾a@ßþ‚@çíÙî@öa‹Ñ—Üa@òŠb’fia@tbÉjäg
@I
@óÕïÔ‡Übi
@H
@ÞïÝÜa@Ò—nåà@æà@bÔþXäg I
@@H
0h
@
2 28 d
J= + α
@sïy@L@
α ∈
ℕ N@@ @
@óïåàÜa@ò‡¾a@ßþ‚@çíÙî@öa‹á¨a@òŠb’fia@tbÉjäg@ì
@Z
R
8 15 d = + β
@sïy@L@
β ∈
ℕ N@@ @
m
@óïåàÜa@ò‡¾a@ßþ‚@bÉà@µmŠb’fia@tbÉjäg@ÖibXn
@Z
J R
d = d = d
N@@çc@ñc
@Z
@óáÅåÜa@Þy@íè@
d
[ ]
@Z[ ]
8 28 2 15 d
d
≡
≡
@óÈíáa@À@
N@ℕ
@ @
@ßaüÜa@óvïnä@ßbáÉnbi (2
−
@çc@wnånä@L Z@
92 420 / d = + k k ∈
ℕ N@@ @
@ðè@bÉà@µmŠb’fia@béïÐ@ÖibXnm@óïåàŒ@ò‡à@‹Í–c
@Z
92 min d =
@ @
@„îŠbnÜa@paˆ@óÅzÝÜa@À@Ú܈@ì
1
1 32 min
@Zt = h
N@
@ @
_@ò‹à@ðäbrÜ@µnï÷í›Üa@µmŠb’fia@tbÉjäg@béïÐ@ÖibXnn@ðnÜa@óÅzÝÜa@„îŠbm@íè@bà
@ @
@ @
@ @