Θεωρία Πραγματικοί Αριθμοί Μέρος Β - Κεφάλαιο 1 - Όριο -...

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: -http://sciencephysics4all.weebly.com/ 1

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i) ( ) : ,...3,2,1,0

ii) ( ) : ,...3,2,1,0,1,2,3...,

iii)

.

5

3,

7

2 ,

100

1717,0 ,

100

233,2 ... .

1.

( Q) :

0,|

ZQ .

iv)

. .

...4142135,12 , ...7320508,13 , ...71828,2e

... .

v) ( R)

.

: RQZN .

Venn.

:

i) ,...3,2,10* N

R

QZN
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ii) ,...3,2,1,1,2,3...0*

iii)

** ,|0

QQ

iv) 0* RR

i)=0,333, ii) =2,999, iii) =3,257373 iv)

=4,208313131 .

:

i) =0,333 10=3,333 :

Qa 3

1

9

339(0,333)-(3,333)=-10

ii) =2,999 10=29,999 :

Q 39

27279)(2,999-)(29,999=-10

iii) =3,257373 100=325,7373 10000=32573,7373 :

Q

2475

8062

9900

32248

322489900)(325,7373-)3(32573,737=100-10000

iv) =4,208313131 1000=4208,3131 10000=420831,3131

:

Q

99000

41662341662399000)(4208,3131-)31(420831,31=1000-100000
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R

R

:

i) +=+ = () , R.

ii) (+)+=+(+) ()=() () ,, R.

iii) aaa 00 aaa 11 R

0 1

.

iv)0)()(

aaaa

aaaa

11

R*

a

1 .

:

-=+(-) , R

:

1: a

aa , Ra *R

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Ra *Nv . :

1,

1,

...

a

0a 10 a

1a .

:

aaa , *Ra v,

a

a

a,

*Ra v,

aa )( , *, Ra v

aa )( ,

*, Ra v

aa )( , *Ra v,

:

1)222 2)( aa 222 2)( aa

2) ))((22 a

3) 222)( 2222 aa

4)32233 33)( aa 32233 33)( aa

5) ))(( 2233 aa ))(( 2233 aa

6) )...)(( 12322321 aa

7) ])()())[((2

1

3 222333

aa

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0 3333 a

8)222222 )()())(( xayyaxyxa

Ryxa

,,,,

*

Nv .

) :

)83)(15)(23(3)83()15()23( 333 xxxxxx

) :

0)83()15()23( 333 xxx

:

) 0 a 3333 a .

0831523)83()15()23( xxxxxx :

)83)(15)(23(3)83()15()23( 333 xxxxxx

)

8

3

5

1

3

2

083015023

0)83)(15)(23(30)83()15()23( 333

xxx

xxx

xxxxxx

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R

Ra , : 0 aa .

:

0 aa

0 aa

0 aa

i) a a , ),,( Ra ii) aa , ),,( Ra iii) 0 : aa

0 : aa , ),,( Ra

iv) a a , ),,,( Ra v) 0,,, a a a vi) 0, a *N : vvaa vii) 00

a

a, ),( *Ra

viii) 0a :

11

aa

,, .

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1) Ra , a . :

i)

: xaRxa /),(

ii) : xaRxa /],[

iii) , : xaRxa /],(

iv) , : xaRxa /),[

2) Ra . :

i. axRxa /),( ii. axRxa /),[

iii. axRxa /),( iv. axRxa /],(

3) R . ),( R .

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Ra . ( a )

, 0a 0a . :

0,

0,

a

a

a

aa

:

i. 0 aa ii. aa aa

iii. 22 aa 22 aa iv. 0a : axax v. R : xx

:

i. aa ii.

aa , 0

iii. aaa iv. 0 : ),( xxx .

Rxx o, 0 :

),( oooooo xxxxxxxxxx

: ],[ xxx

],[ oooooo xxxxxxxxxx

v. 0 : ),(),( xxxx Rxx o, 0 :

),(),(

oo

ooooo

xxx

xxxxxxxxxx

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: ),[],( xxxx

),[],(

oo

ooooo

xxx

xxxxxxxxxx

1) : i)

233

2/x

RxA ii)

43

/x

xRxB .

.

:

i) 303 xx . :

5,5

175

5

1723

5

23

235

21

2

3

5

15

3

2123

3

2223

3

2

xxx

xx

xxx

5,

5

17A

ii) 303 xx . :

),4[)3,(

4304030)3)(4(03

4

03

)4(30

3

1230

3

12404

34

3

x

xxxxxxx

x

x

x

x

x

x

xx

x

x

x

x

),4[)3,( B .

2) 0132 zyx ,

.,, zyx

:

02 x , 03y 01z Rzyx ,,

0132 zyx :

02 x 03 y 01z 2x 3y 1z .
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3) Rzyx ,, : zyzxyx .

:

zyzxyzzxyzzxyzzxyx )()(

zyzxyx .

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0a *Nv .

( v a ) x axv .

: axxa vv . v a

axv .

1v aa1 , 2v aa2 .

1) 0a *Nv aaa v vvv 2) 0a , *Nv v , aav v 3) 0, a *,, Nv :

i. vvv aa ii. v

v

v aa

, ( 0 )

iii. vv aa iv. vv aa v. vv aa

vi. vv v aa

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Ra 0a , Z *Nv . a

:

v aa

*, N 00

.

, , xa 0a x .

( vvv

xaa

lim , v x ).

,

(

).

Ra , , 0a , 0 Ryx , :

yxyxaaa

yx

y

x

aa

a

xxx aa )(

x

xx

aa

xyyx

aa

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1) .

i) 3 53 , ii)

232

, iii)123

2

:

i)5

253

5

253

55

53

5

3 3

3 3

3

3 23

3 2

3

ii) )23(2)2()3(

)23(2

)23)(23(

)23(2

23

222

iii)

2

226

2

226

)2(

)123(2

2

123

22

)123(2

)1222(3

)123(2

)12()3(

)123(2

)]12(3)][12(3[

)]12(3[2

)12(3

2

123

2

2

22

2) x ,

2

2

12

xx .

:

xx

xxx

xx

xxxx

xx

2)12(

2144

2

122

2

1

2

122)2(

2

12

22

2

2

2

2)12( x x2 ,

x

x

2

)12( 2

( ).
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,

.

:

2

2 x , *Na 33 x , *N .

2

2x

3

3x

:

3

2

32

32

32

3

2

,*

N 23

2

, *N

2

3 2

.

*

N 2 , *N ( ).

: 2

3 2

2

23 2

2

6

:

3

2

3

626

2

2 22 46 312 (5)

:

22 x 23122 x 61442 x 672x

*

N :

i) 22 x 23122 x

ii) 33 x 3266 62167233 x

,3

12 26 , *N .

*

N x5 5

.

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:

62363257255 x

623

325 5 :

5432532 , *N

:

623 3255 x 6543223 5323255 x 3024181223 5323255 x 56543 5325 x

, 5432 532 , *N :

65432236 5323272 xx 3024181223 53232 x

30242015532 x , *N

, x 1 .

: 242015 532 x .

Θεωρία Πραγματικοί Αριθμοί Μέρος Β - Κεφάλαιο 1 - Όριο -...

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