Πλατάος Γιάννης Συμπληρωματική συλλογή μαθηματικών...
136
Μεσσήνη 2015 Συμπληρωματικός τόμος 7ος Γιάννης Π. Πλατάρος Εκπαιδευτικές Εργασίες (Συλλογή)
Transcript of Πλατάος Γιάννης Συμπληρωματική συλλογή μαθηματικών...
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Volume 1, Number 6, November 1979
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answers of philosophical thought. Some mathematical models, expand the
boundaries between possible, improbable, possible and impossible.
Mathematics, have limits on their answers. But still a super tool for
discussion on all questions of science and particularly the philosophy. The
questions concerning the infinity are of particular interest because, whatever
the findings set out in it, is not easily accepted by the finite human nature.
Here comes the mathematical truth to venture some answers more or less
acceptable.
: [1] Wigner, Eugene The Unreasonable Effectiveness of Mathematics in the
Natural Sciences.
https://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html
[2]
.
: http://thalesandfriends.org/wp-
content/uploads/2012/03/efficiency.pdf
[3] , :
http://panayiot.simor.ntua.gr/attachments/039_06MBAOR.pdf
[4] Pascal Blaise ( 233), ,
:
https://onthewaytoithaca.wordpress.com/2010/08/23/pascals-wager-the-
whole-thing/
[5] https://el.wikipedia.org/wiki/__ [6] http://westcult.gr/index.php/arthrografia/philosophizing/posoi-kokkoi-ammou-
apaitoyntai-gia-na-katalavoun-ton-synoliko-ogko-tou-sympantos
[7] ( )
2012 : http://www.math.ntua.gr/~sofia/dissertations/Larentzaki.pdf
[8] Gottfried Wilhelm Leibnitz http://www.biblical-studies.gr/kbma/Portals/0/PDF/Tehnes/Laibnitz.pdf
[9] (
) ,
http://manosdanezis.gr/index.php/blog/307-2015-02-07-18-53-16
[10] 0,99999;
. 115 . :
http://forum.math.uoa.gr/viewtopic.php?f=15&t=10116&start=0
http://thalesandfriends.org/wp-content/uploads/2012/03/efficiency.pdfhttp://thalesandfriends.org/wp-content/uploads/2012/03/efficiency.pdf
-
[11] Does 0.9999999... truly equal 1?
, 76 . :
https://www.linkedin.com/grp/post/1872005-6044962556768956419
[12] Wikipedia. :
https://el.wikipedia.org/wiki/_
[13]
https://onthewaytoithaca.wordpress.com/2010/08/23/pascals-wager-the-whole-
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[1] Nicolas Bourbaki, Theorie des ensembles, Hermann, Paris, 1970.
[2] Alan Sokal, Trangressing the boundaries: Toward a transformative hermeneutics of quantum gravity, Social Text 46/47 (1996), 217-252.
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Cauchy
() Cauchy , >0 0 , d(-)
-
3 14/4/2004
Banach V C ,
Banach : i) V , norm . ,
(V , . ) norm ii) + V, d(,)= norm . , (V , . ) , norm iii) norm (V , . ) Banach iv) , V,
, nxn(C) nxn , C :
+ : nxn(C) x nxn(C) nxn(C) .
. : nxn(C) x nxn(C) nxn(C) .
: nxn(C) x nxn(C) nxn(C) .
nxn(C) C , , , , , nxn(C)
C : i)() =( )= () ii)(A+B) = + iii)A (B+)= +
, nxn(C) , C
)C(:. nxn nxn(C) :
.n,...4,3,2,1j,i C,][ |}|{maxA j,in
1jj,ini1
=== =
norm : ,,, nxn(C) C
-
4 14/4/2004
1. 0|}n
1jj,i
nj1|{max
==
2.
OAn....4,3,2,1j,i0
n,...4,3,2,1i0||0|}n
1jj,i
nj10
j,i
n
1jj,i|{max
===
====
==
3.
=
=
==
==
=
||
|}]n
1jj,i
nj1|}
n
1jj,i
nj1|}
n
1jj,i
nj10 |{[max||||{|max|{max
4.
=
=
=
=
=
+=++
=++=+
BA|}|{max|}|{max
|}|max||max{||max
j,i
n
1jj,inji
n
1jj,inji
j,i
n
1jnji
n
1jj,injij,i
n
1jj,inji
. norm (nxn(C) , . )
norm , (,
. ) norm , :
+ : x . : Rx , , norm
. .
, + (nxn(C) , . )
0|}n
1jj,i
nj1|{max
==
(nxn(C) , . ) Banach.
(nxn(C) =[i,j] ,
C
-
5 14/4/2004
= }...|{max |njnj,2,2j,1
n
1j,1ni1
+++=
|}|||...|||||||{|max |njn
1jnj,2
n
1j,2j,1
n
1j,1ni1
===
+++
|}}|{max||...|}|{max|||}|{max|{|max |njn
1jni1nj,2
n
1jni1,2j,1
n
1jni1,1ni1
=
=
=
+++
|)}|...||||}(||{max{max n,,2,1j,1n
1jni1ni1++
=
|}||}]|{max{[max ,1n
1jj,1
n
1jni1ni1
==
=
|}]|max|}][|{[max ,1n
1jni1j,1
n
1jni1
=
==
= = BAAB
= BABA norm (nxn(C) , . )
Banach n
=
9.01.0
01
12.0|1.0||1.0|9.01.0
012
-
6 14/4/2004
:
=
01.001.0
001.01.0
001.01.0
00
1.01.0
00
01.001.0
00 =
001.0001.0
00
=
nn
n
101
101
00
01.001.000 n (1)
, (1) n=1, 2 3 , . (1) n=k n=k+1 .:
=
101
101
00
01.001.000 (2)
:
=
++
+
11
1
101
101
00
01.001.000 (3)
, (1) , :
(3) 10
110
100
01.001.000
)2(10
110
100
101
101
00
01.001.000
)2(10
110
100
01.001.000
01.001.000
01.001.000
11
1k
11
=
=
=
++
+
++
(1) . :
-1= 2+
=
+
=
+=
=
=
=
=
910
91
01
91
91
00
1001
0
91
91
00I
101
101
00
1.01.000
2
1nn
1nn
n
1n
-
7 14/4/2004
1/10 1/10 ,
91
109
101
1011
101
101
1nn ==
=
=
-1=
910
91
01
-
0.9999..=1
.
.
: 0,999=1
, .
. , .
, ,
,
.
: 0.9999.=1, [1],[2],[3]
, -
,
.
, ,
!
1: , 1
0.111111111...9
1
1 9 9 0,111111.... 0,99999999......9
2 : 1
0,33333.....3
11 3 3 0,33333..... 0,99999.....
3
-
1/3 = 0.333333... 2/3 = 0.666666... 1/3 + 2/3 = .999999... = 1.
3: =0,999999.. (1)
10=9,999999999. (2)
(2)-(1) :
10- =9,9999999999999..-0,9999999999. .
9=9,0000000000 9=9 =9
19 .
10=9,999=9+0,999=9+, =1.
4 : 0,999999999999999999999..=
9 9 9 9 9 9...
10 100 1.000 10.000 100.000 1.000.000
1 2 3 4 5 6
9 9 9 9 9 9...
10 10 10 10 10 10 (
1
10 ,
1
9 9
10 10 11 9
110 10
5.
1
9 1 10,999... lim0, 99...9 lim lim 1 1 lim 1 0 1
10 10 10
( 4, )
-
6.
0, 9 0, 9 9 0, 9 9 0, 9 9 0, 9 (10 1) 0, 9
9 0, 9 9, 9 0, 9 9 0, 9 9 0, 9 1
.
7
( 3)
( 10)
( 10) ( 3) ( 3)
( 10)
( 10) ( 10) ( 3)
( 10) ( 3)
( 10) ( 10)
10,1
3
13 10 0,1
3
3 0,33333.... 1
0,999999....... 1
0,999999....... 1 . . .
(
.)
8: [0. 9....9 ,1]
(i)1 2 3 4 ..... (ii)
1
(.. 2
)
(iii) lim | 0. 9....9 1| 0
,
0, 0.....0 1
0, >0() ,
0()
,
, 1, .
.
1. 0,9999..(
-
) . 1=0,99999999999999
9. =1 =0,99999 ,
1,999999....0,99999.....
2 2
2 2 2
10. ( )
: |-|0, =.
: . *| | 0
.. *
2
*
*, .
2
= .
1-0.9999
.( 8.)
11: (0,999) (0,999)=1 ,
0,999=1 , (0,999) (0,999)=
1 2 3 4 5 6 1 2 3 4 5 6
1 1 2 3 2 1 2 3 3 1 2 3
9 9 9 9 9 9 9 9 9 9 9 9... ...
10 10 10 10 10 10 10 10 10 10 10 10
9 9 9 9 9 9 9 9 9 9 9 9... ... ... ...
10 10 10 10 10 10 10 10 10 10 10 10
2 3 4 3 4 5 4 5 6
81 81 81 81 81 81 81 81 81... ... ... ...
10 10 10 10 10 10 10 10 10
2 1 2 3 1 2 4 1 2
81 1 1 81 1 1 81 1 11 ... 1 ... 1 ...
10 10 10 10 10 10 10 10 10
2 3 4
0 0 0
81 1 81 1 81 1...
10 10 10 10 10 10
2
2
0 0 0
1 81 1 1 9
10 10 10 10 10
-
2 2 21 9 10 9
1 . . .1 10 9 10
110
12: 0,9990: 0,999+=1.
* , ,
1
0.
*
( 1)
0,999 0,999 0,000...0001000.... 1,000...000999.... 1 .
0,999 *
1 1.
13. :
0,999 . 0,999 . (0,9 0,09 0,009 ...) (0,9 0,09 0,009 ...)
(0,9 0,9) (0,09 0,09) (0,009 0,009) ...
1,8 0,18 0,018 ... (1 0,8) (0,1 0,08) (0,01 0,008) ...
1 (0,8 0,1) (0,08) 0,01) (0,008 0,001) ...
1 0.9 0.09 0.009 ... 1 0,999....
0,999
, 1=0,999
14: 0,999...
0,999... 0,9 0,0999.... 0,910
90,9 0,9 1
10 10
15 :
13 ,
, ,
, ,
... , .
. 1-0,9=0,1 0,1-0,09=0,010,01-0,009=0,001 ...
-
( 10)
. . 1=0,999
:
,
,
,
. ,
,
: .
. Google
0.999 , 1=0.999.., equal 1=0.999
9.390.000 34.000
8.540 proof 1=0.999 6.000 ( 9/9/2015)
, , , ,
:
1 .
. .
.
, ,
,
.
1, 2, 3, ..., , ...
,
.
, ,
.
.
0,999=1 ( ,
-
-
)
1 , .
- .. 1
0
. 1
lim 0
, ,
( 1)
0
(
0, )
{} : =
12 1
0 2
12 2
.
, ,
0,999=1 ,,
-
1
11
2
. ,
,
0,999 1, 1
.
!
-
2) 1=0,999.
.
(
, , /(25)
, ) .
, .. 1,2=1,1999 ,
!
3) ,
, . ,
- ( )
, .
Grandi 1
( 1)
, [4] ,
0, 1 , .
,
, 0,999
, ,
,
, ,
.
: ,
,
.
.
,
,
, . ,
,
, !
-
Summary : The 0,999...=1 equivalence is an issue of student internet
discussions and far beyond. The mentioned parity (equivalence) is
stubbornly doubted. Even specific mathematical proofs are not persuasive. It
seems that the difficulty in the intuitive comprehension of the infinite,
indicates the limits of the finite nature of human beings while at the same
time the power as well as the practical value of mathematical proofs are
highlighted
:
[1] Wikipedia . : 0.999 :
https://el.wikipedia.org/wiki/0,999...
[2] Bogomonly Alexander : : .999=1? :
http://www.cut-the-knot.org/arithmetic/999999.shtml
[3] Kalid Azad . : A Friendly Chat About Whether
0.999.. = 1: http://betterexplained.com/articles/a-friendly-chat-
about-whether-0-999-1
[4] Wikipedia. : Grandi's series :
https://en.wikipedia.org/wiki/Grandi's_series
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