Www.science-ki.blogspot.com MATH SM a (3)

7/23/2019 Www.science-ki.blogspot.com MATH SM a (3)

1/16

http://arabmaths.site.voila.fr Moustaouli Mohamed

I--1

ba=a kbkb aab

( ) 2;a b b a k a kb = -2

-* baabab

b-* b { }/b k b k =

*b a -* :b a b a

""b a-3

*-a a a "b a"

*-3( ; ; )b a

a b c b ca c

"b a"

*-3( ; ; )b a

a b c a ba b

=

3( ; ; )

b aa b c a b

a b

=

"b a"

( )-1 2;a b b a a b

2-

( ) ( ) ( ) ( )51 2 1 2 1 1 2 2 1 2 1 2; ; ; ;a x x y y a x y a x y a x x y y II-

-1a bba

a2( );q r bq r 0+ = r b

a( );q r bq r 0+ = r b a

b

.rqba

-2

a b*baa ( );q r bq r 0+ = r b

a ( );q r bq r 0+ = r b ba

a

b

q

r


2/16


2qq7x

qb'aba

'a x a qxb

III-n-1

nba

a

b

n

[ ]a b n a bn

( ) [ ]2;a b a b n n a b k a b kn = "n"-2

-[ ]a a a n "n"

( )- [ ] [ ];a b a b n b a n "n"

( )- [ ] [ ] [ ]; ( ) ( )a b a b n et b c n a c n "n"

"n"

-nba

[ ]a b nabn

1 1anba nq r 2 2b+ = nq r 10+ = r n 20 r n

( )1 2a=1 2r rnba b n q q =

[ ]a b n

[ ]a b n

k

a b nk =

( )1 2 1 2r r k q q n = 1 2r rn

10 r n 20 r n 1 2r r n

1 2 0r r =1 2r r=

-3n

*-( ) ( ); ; 0a n q r a nq r et r n = +

-( ) [ ] { }; 0;1;.....; 1a n r a r n et r n

[ ]{ }/x x r n - rrn

"n"rr

-[ ]x r x r n

*-{ } [ ]0;1;....; 1 /a r n a r n { }0;1;....; 1 /a r n a r -* 0='r r r n 0 'r n ='r r*-( ) { }; 0;1;..; 1 /x n r n x r )rn(

( )0 1 2 .... 1n=

{ }0;1;.....; 1n

n

n


3/16


*{ }0; 12 =

=0 2 ( ){ }1 / 2 1x x k k= = +

*{ }0;1;2;3;4;5;67 =

=0 7 ( ){ }1 / 7 1x x k k= = +

( ){ }2 / 7 2x x k k= = + ( ){ }3 / 7 3x x k k= = + .................( ){ }6 / 7 6x x k k= = +

7

532 [ ]532=4 4 7

[ ]36= 6 36 6 7 "n"-4-

nxyzt

+ +[ ]x z y t n[ ]z t n[ ]y n

[ ]x z y t n[ ]z t n[ ]y n"n"

-

*-

x r' 'x r' 'x x r r+ +' 'x x r r ' 'r r r r + = +' 'r r r r =

*-( ) ( ) [ ] [ ]2 *; ; p pa b p n a b n a b n

*5

3 4 7 2+ = = ,0 1 2 3 4 10 0+ + + + = = ,3 4 12 2 = =

x7

5 2x+ =

-14

70-2 13+3 702

( )-1 [ ]4 1 0n n n n 217-2 1 3 23 5 2n n +n*

1n-3 2 3 4n n n n+ + +4IV-

aDa

1-

*ba ba ab

a b

a b

a b

D Da b

x D D x

=


4/16


2-

*cba

( ) ( )

a b b a

a b c a b c

a a a

=

=

=

48 60 12 =""-3

-

** a aa D D =

*( ) 2*

;a b a b a b =

.

*ba-- = a b bb a

- a* ( );q rab bq r 0+ = r b r a bq= rab

ab

b

r

a b r bD D D D a(arb bq r = +(

br abr b a bD D D D

a b r bD D D D = a b r b =

barab*baa b r b =

b a*ba- ba

1 1a bq r + =

10 r b

1 0r = a b bb a=

1 0r b1r1 2 2b r q r 1 20+ = r r

2 0r 1 1a= 1 1b r r= b b r r = =

2 0r 1r2r1 2 3 3r r q r 3 20+ = r r

.................

n

1 1a bq r = +10 r b 1a b b r =

1 2 2b r q r = +2 10 r r 1 1 2b r r r =

1 2 3 3r r q r = +3 20 r r 1 2 2 3r r r r = . . .

. ... . .. . .

2 1n n n nr r q r = +10 n nr r 2 1 1n n n nr r r r =

1 1 2 2 3 2 1 1............. n n n na b b r r r r r r r r r = = = = = =

1 3 2 10 .......n nr r r r r b

{ }1 2 3; ; ........ ;.....nA r r r r=

b

/1 0 ; 0p pp r r+ =


5/16


1 0pr + 1p p br r r=+ 1 1p p pr r q= =

= pa b r

*ba

ab ab

1561640

1640 156 10 80156 80 1 76

80 76 1 4

76 4 19 0

= += +

= +

= +

1640 156 4 =-3-

=a b*ba

uvau bv= +

=a b*ba

=( ){ }* 2/ ; ;A n n au bv u v = + +2 2a b A/

p A x A x p

+ =0 0p au bv p=

abpp

( ); ; 0qpa r a pq r r p = +

( ) ( )0 0 0 01 ( )r a q au bv a qu b qv= + = + r pr A0r r p

0r=p a p b

pbap

=ppp

-* =a b

( ){ }* 2/ ; ;B n n au bv u v= = +

*

a

b

a

b ( )bac 21 2 1 2; ;k k a k c b k c = =

( ) =a b 2; /u v au bv = +

( )1 2k u k v c= +c

=a b*ba ab( )a bD D D =

a

bc

a b ca cb c = =


6/16


4-

1a2a3a.....ka*

1a2a3a.....ka 1a

ka.....2a3a

12 18 15 3 =

1a

2a3a.....

ka

1

23.....

k1

i k

i ii

a=

=

V--1

*ba= 1a bba

-2 Bezout

*ba

1a b =( ) 2; / 1u v au bv + =

:( )*ba 2; / 1u v au bv + =

= 1a b = { }1;1a bD D1ba Bezout

*ba

( ) 21 ; / 1a b u v au bv = + =

-3

bad*ba

1a ba b dd d

= =

a b*ba =q*

1 ; ;p q a p b q = = =

-4 GAUSSdeThorme-

*bca abc bc= 1a c

= 1a cc ab*bca

;( ) ; / 1 ;u v k au cv ab kc + = =

) ) ( )1b b b au cv bau bcv kcu bcv c ku bv= = + = + = + = +bc

-a-

*bca

1a b et a c et b c ab c =


7/16


x6 32

= 1a b664236 4 24 =

b-

*n*bca

[ ][ ]

1

ab ac nb c n

a n

=

[ ]

( )

ab ac n k ab ac kn

n a b c

=

[ ]b c n( )n b c= 1a n

-5

*bca-*1

11

a ba bc

a c

= =

=

*1 1nan*ba-* b a b = =

*1 1m namn*ba-* b a b = =

ab*

( )= 1a b 2;u v x au bv = +

17 3 94y+ =

117 2 3 20 94 + 17= 3 94x y+ =( ) ( )17 2 3 20 0x y + =

) ) ( )17 2 3 20x y =

( )3 17 2x 17 3 ( )3 2x= 1 2 3k x k =3 2k x k = +

( )17 3 2 3 94k k y + + =17 20k y k = +

( ){ }3 2; 17 20 /S k k k = + +

2

[ ]

[ ]

[ ]

( ) [ ]

17 3 94 17 94 3

17 94 3

2 1 3

1 3

1 3

x y x y

x

x

x

x

+ = =

3 1k x k =[ ]1 3x= 1 3 1 =

17 37k y k = +

( ){ }3 1;17 37 /S k k k = +


8/16


0uq*

0 1u q =0u1u2u3u

= +21 3 02 44u u uq

1a b =( ) 1a b b+ =( ) 1a b ab+ =

2

2 3

3 2

n

n n

+

+ +

2

3

-6

3a......na2a1a*

1

3a......na2a1a*

3a......na2a1a*

1u2u3u......nu*

11

1n

ii

u a=

=

( ) 2;x y ax by c + =-

21075 64 9y+ =

1075 641075 64 16 51

64 51 1 13

51 13 3 12

13 12 1 1

12 12 1 0

= +

= +

= +

= +

= +

1075 64 1 =

10752( );y 64 9y+ =

64 1075b a= =

( )

( )( )

( ) ( )( )( )

51 16

13 16

12 16 3 16

1 16 16 3 16

1 5 84

a b

b a b

a b b a b

b a b a b b a b

a b

=

=

=

=

= +

9 45 756a b= +9 45 1075 756 64= +

1075756;45( ) 64 9x y+ =1075( 45) 64( 756) 0x y+ + =( )64 1075 45x 1075+ 64 ( )64= 1 45x +


9/16


45 64k x k + =64 45k x k = 1075 756k y k = +

1075 756 ; 64 45y k x k= + = 1075 64 9x y+ =

( ){ }64 45;1075 756 /S k k k = +

) ) ( ) 2;E x y ax by c + =0a0b

a b= ( ) *2'; ' / ' ' ' ' 1a b a a et b b a b = = =

*

c

' ' 'a x b y c+ =

'c c=

( )0 0;u v =1 ' 'a b

*20 0' ' 'a u b v c+ =' ' 'a x b y c+ =

* ax by c+ =2( )0 0;y0 0ax by c+ =)0 0 ' ' )a x b y c c= +

( );a b2a b=

ax by c+ =2 c

ax by c+ =' ' 'ac x b y c+ =

' ' 1a( )0 0;u v =1 ' 'a b x b y+ =0 0' ' ' ' 'a c u b c v c+ =)'0 0 ' ) '( ' ) 0a x c u b y c v + )'0 0= ' ) '( ' )a x c u b y c v =

( )0' ' 'a b c v y1 ' 'a b= ( )0' 'a c v yk+ =0' 'y a k c v 0' 'kb c u= +

'( )0 0 ' ; ' 'kb c u a k c v+ + ' ' 'a x b y c+ =

'( ){ }0 0 ' ; ' ' /kb c u a k c v k + +

= 7 3 1x y2

2137aaa35

VI--1

( ) *2;a b ab aba b

-2

-*-

a

b

c

*

( )

a b b a

a b c ac bc

a a a

b a a b a

=

=

=

=

= *a b mba-*- ba m

-

a

b

*a b m =a b =m ab=


10/16


*ba

1a b a b ab = =

-3

1a2a3a.....ka*

1aka.....2a3a1a

2a3a.....

kaVII-

-1-

a

ad { }1;1; ;d a aad

6-* -33-22

-*{ }7

1; 1;7; 7D 7 =

-

a a -11a

a{ }1; 1; ;aD a a= 1a

P-2

- q pq )(

- apa

a- .-11*a

-

+P

P+ 2 P+

+P P+.! 1m p= +m p

m P+

m q Pqm+

q pq pq!p)q!p(

1q( !)q m p!q pq m q

+P P-3

n 2n

n 2p nnp

n 2nn pnk*n pk=


11/16


1 n

1 k kp n= knp k

= 2p pk n

n 2n

n. 2p n

n n

)

2

p n( 13115732179

2 217 289 ; 13 169= =

-4

-*

1 2 ...... na a a a= n

;{ }1p a 2;.....; / ii n p a

2n=1 2p a a. 1p a

11a 1a p =1 2p a aGAUSS2p a

n1n +......1 2 1 n nb a a a a += p b

1np a +

1na 1+ 1na p+ GAUSS1 2p b= ... np a a a

{ }1;2;.....; / ii n p a

1p

2p......np

{ }1 2 ...... 1; 2;....;n ip p p p i n p p =

-5 -1

n11- 1 2

1 2 ...... k

kn p p p

= 11p2p......np

n......2 =1

1 21 2n ...... k

kn p p p

= n

-1752

-2)A(

1

1 21 2 ...... k

kn p p p

= 2p......np1p

nd d 1 2

1 2 ...... k

kd p p p

=

0 i i i{ }1;2;....;k

2

1 21 2 ...... kkn p p p = 2p......np1p nm m


12/16


1 21 2 ......

kkd p p p

=

0 i i i{ }1;2;....;k

)B( +

1 21 2*ba ...... k

ka p p p

= 1 21 2 ...... k

kb p p p

=

2p......np1p

ba 1 21 2 ...... k

kp p p

=

( )inf ;i i i =i{ }1;2;....;k

+

1 21 2*ba ...... k

ka p p p

= 1 21 2 ...... k

kb p p p

= 1p

2p......np

ba 1 21 2 ...... k

km p p p =

( )sup ;i i i =i{ }1;2;....;k

180 1170 180 1170n p

1- ' ' ' 1md p d m d d p d p d = =

2- { } { }* 1 1; 2;....;m kq q p k n q p = n

p

1 21 2-3 ...... n

na p p p = a

( )a

a) ) ( )( ) ( )1 21 1 ..... 1 na = + + +

VIII--1

)-1 ) ( )2; 1 1 1nm n m m n m +

1-2 nm m p n m p

! 11-3 k kb b n k b n b +

)-1 ) ( )2; 1 1 1nm n m m n m +

( ) 2;m n ( )( ) ( ) ( ) ( )0 2

1 1 1 1 1 1i n i nn i in i i

n ni i

m m C m n m C m= =

= =

= + = = + +

1 0m + ( )1 1nm n m

1-2 nm m p n m p

m 1m

+ ( )1 1nm n mnp

n

( )1 1n m p

( )1 1n m p

+

n

m p

! 11-3 k kb b n k b n b +


13/16


{ }1/ knA k n b += )2(11 n nk knb b n k b b n

+ nA

nA n0nk0 0

1n nk kb n b

+

0nk

b n2n0 2 1nk

b 0

1nk 0 1 0nk

0( 1) 1nk

b n +

0

1n nk A 0nkn

1n=0 0nk =( )0 0 11b b + -2

.10 8976542310 .2 10

.12 897654231011101110

.8 76542310

3-

b.( )1b 1-

( )1bb

kqkrknk

k kn b q r 0+ = k

kr b 0 kq b

( ) 2;b n ( )1b =0n

*n 1! k kk b n b +

n

kb

k k kn b q r 0+ = kkr b kq 0 kq b

kq b1k k

kq b b 1k kk kn+ b q r b

+ =+ 1kn b +

0 kq b

k1- k kn b q r 0+ = k

kr b 0 kq b

kr1

1 1k

k k kr b q r

+ = 1

10 k

kr b

10 kq b (k

kr b(

1k

r 1

r2

1 2 2k

k k kr b q r

+ = 2

20 k

kr b

20 kq b

. . .

. . .

. . .

2 1 1r bq r = + 10 r b 10 q b

1 01r q= 0 1q r=

0

i ki

ii

n q b=

=

=

0 iq b { }1;2;....;ki


14/16


2

( )1bb

kn0

i ki

ii

n q b=

=

= 0 iq b

i{ }1;2;....;k0kq 0n

0

i ki

iin q b

=

== 0iq b

i

{ }1;2;....;k

n

b

( )1 0...........k kbnb bn q q q=

* 2703 3 22703 2 10 7 10 0 10 3= + + +* 158

3 28 1 2 0 2 0 2 0= + + 10008+ 3 215 1 2 1 2 1 2 1= + + 111115+

*

15 1 8 ( )815+ =7 17=

2131 2 8 0 8 3= + =( )8131 203+

-1 ;n b b

0 1 0

1 1 2 1

1

1 2

0 ;

0 ;

. .0 ;

.....

k k k k

q

r b n bq r

r b q bq r

r b q bq r

n q q q

+

= +

= +

= +

;{ }1 2 ....A q q=kk kq r=

0 1 0

1 1 2 1

1 1 1

0 ;

0 ;

. .

0 ;k k k k

k k

r b n bq r

r b q bq r

r b q bq r q r

= +

= +

= +=

ibi

1 0

21 2 1

11

. .

i i ii i i

n bq r

bq b q br

b q b q b r +

+

= +

= +

= +


15/16


1 11 1

. .

k k kk k k

k kk k

b q b q b r

b q b r

= +

=

0

0i k

ii i

i

r b n b r =

=

= { }1;2;....;i k

0kr 1 1 0......k kn r r r r =

bn( (0 ii k r

( )1 1 0......k k bn r r r r =

3254

( )83254 6266=( )123254 1 72=

-4

y

( )

1 0.....n n b

x a a a=( )

1 0......m m b

y c c c=

y xm n

( )1 0.....n ny bx a a a=( )1 0......n n by c c c=

1 1 1 1.......i i i i n n n na c et c a c a c a+ + = = yx= iaic

-5

bxb

b = xxx xxx yyyyyy


16/16

http://arabmaths site voila fr Moustaouli Mohamed

6-

x 1 1 0.....n na a a a=

[ ]

[ ]

[ ] { }

1 0

0 1

1 0

0 4 4

0 5 0 5

0 5 00; 25;50;75

x a a

x a ou a

x a a

= =

[ ] [ ]

[ ] [ ]

[ ] ( ) [ ]

0

0

0

0 3 0 3

0 9 0 9

0 11 1 0 11

i n

ii

i n

ii

i ni

ii

x a

x a

x a

=

=

=

=

=

=

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