Zerlegung von Quadraten und ????Zerlegung von Quadraten und ????

1

1 x

x

y

zzy

2x+y=1 y+z=x 3z=y

15/11

11

11 6

4111

4

1 11 111 3

24 4

11

11 4

11

11

11 4

11

3

1 1 1 1111 3 1 1

2 2 24 14 4 13 3

4

4

1 1 1 11 1 1 111 3 1 1

2 2 24 14 4 13 3

11

11 4

3

1

4

4

1

1

15/9

9

9 6

3

Kettenbruchentwicklung und ggT:

Die Länge des kleinsten Quadrats ist der ggT

rn hat fraktionalen

Anteil

an := ganzzahliger Anteil von rn

Ja

Nein

z:=1/(rn – an)

an:= rn Ende

z := x, , n:=0

Euklidischer Algorithmus

rn:= z

n:=n+1

Gegeben x

Kettenbrüche und ähnliche Rechtecke

Kettenbrüche und ähnliche Rechtecke

x

x y

y

x-y

[0;1,1,1,...]x y y

yx x y

[1;1,1,...]

x

x y

y

x-2y

[0;2,1,2,1,...]2

x y yy

x x y

[1;2,1,2,1,2,...]

x

x y

y

x-3y

[0;3,1,3,1,...]3

x y yy

x x y

[1;3,1,3,1,3,...]

x

x y

y

x-ny

[0; ,1, ,1,...]x y y

y n nx x ny

[1; ,1, ,1, ,...]n n n

Was sind die Gleichungen für:

[1;1,2,1,1,2,1,1,2,…]

[1;1,3,1,1,3,1,1,3,…]

[1;2,3,1,2,3,1,2,3,…]

x

x y

y

x

x-2y

2[0;2,2,2,...]

2

x y yy

x x y

[2;2,2,2,...]

x

x y

y

x-3y

3[0;3,3,3,...]

3

x y yy

x x y

[3;3,3,3,...]

nx y y

x x ny

2 1 0y ny

( ) 1y n y

1[0; , , ,...]y y n n n

n y

[ ; , , ,...]n n n n

x rational:

x kann in der Form m/n geschrieben werden; m und n natürliche Zahlenx hat schließlich-periodische Entwicklung bezüglich jeder Basis x hat abbrechende Kettenbruchentwicklung

x irrational:

x nicht als Quotient zweier natürlicher Zahlen als m/n schreibbarx keine Periodizität in der Entwicklung bezüglich jeder Basisx hat Kettenbruchentwicklung, die nicht abbricht

Wenn x algebraisch von der Ordnung 2 (und irrational), dannhat x eine schließlich-periodische Kettenbruchentwicklung. Es gilt auch die Umkehrung!

n [ a; Period ]

√2  1; 2

√3  1; 1,2

√4  2;

√5  2; 4

√6  2; 2,4

√7  2; 1,1,1,4

√8  2; 1,4

√9  3;

√10  3; 6

√11  3; 3,6

√12  3; 2,6

√13  3; 1,1,1,1,6

√14  3; 1,2,1,6

√15  3; 1,6

√16  4;

√17  4; 8

√18  4; 4,8

√19  4; 2,1,3,1,2,8

√20  4; 2,8

√21  4; 1,1,2,1,1,8

√22  4; 1,2,4,2,1,8

√23  4; 1,3,1,8

√24  4; 1,8

√25  5;

√26  5; 10

√27  5; 5,10

√28  5; 3,2,3,10

√29  5; 2,1,1,2,10

√30  5; 2,10

√31  5; 1,1,3,5,3,1,1,10

√32  5; 1,1,1,10

√33  5; 1,2,1,10

√34  5; 1,4,1,10

√35  5; 1,10

√36  6;

√37  6; 12

√38  6; 6,12

√39  6; 4,12

√40  6; 3,12

√41  6; 2,2,12

√42  6; 2,12

√43  6; 1,1,3,1,5,1,3,1,1,12

√44  6; 1,1,1,2,1,1,1,12

√45  6; 1,2,2,2,1,12

√46

 6; 1,3,1,1,2,6,2,1,1,3,1,12

√47  6; 1,5,1,12

√48  6; 1,12

√49  7;

√50  7; 14

n [ a; Period ]

√51  7; 7,14

√52  7; 4,1,2,1,4,14

√53  7; 3,1,1,3,14

√54  7; 2,1,6,1,2,14

√55  7; 2,2,2,14

√56  7; 2,14

√57  7; 1,1,4,1,1,14

√58  7; 1,1,1,1,1,1,14

√59  7; 1,2,7,2,1,14

√60  7; 1,2,1,14

√61  7; 1,4,3,1,2,2,1,3,4,1,14

√62  7; 1,6,1,14

√63  7; 1,14

√64  8;

√65  8; 16

√66  8; 8,16

√67  8; 5,2,1,1,7,1,1,2,5,16

√68  8; 4,16

√69  8; 3,3,1,4,1,3,3,16

√70  8; 2,1,2,1,2,16

√71  8; 2,2,1,7,1,2,2,16

√72  8; 2,16

√73  8; 1,1,5,5,1,1,16

√74  8; 1,1,1,1,16

√75  8; 1,1,1,16

√76  8; 1,2,1,1,5,4,5,1,1,2,1,16

√77  8; 1,3,2,3,1,16

√78  8; 1,4,1,16

√79  8; 1,7,1,16

√80  8; 1,16

√81  9;

√82  9; 18

√83  9; 9,18

√84  9; 6,18

√85  9; 4,1,1,4,18

√86  9; 3,1,1,1,8,1,1,1,3,18

√87  9; 3,18

√88  9; 2,1,1,1,2,18

√89  9; 2,3,3,2,18

√90  9; 2,18

√91  9; 1,1,5,1,5,1,1,18

√92  9; 1,1,2,4,2,1,1,18

√93  9; 1,1,1,4,6,4,1,1,1,18

√94

 9; 1,2,3,1,1,5,1,8,1,5,1,1,3,2,1,18

√95  9; 1,2,1,18

√96  9; 1,3,1,18

√97  9; 1,5,1,1,1,1,1,1,5,1,18

√98  9; 1,8,1,18

√99  9; 1,18

Realisierung des Euklidischen Algorithmusmit Microsoft Excel

Kettenbruchentwicklungen von

2, , , tanh(1), tan(1),...e e

http://en.wikipedia.org/wiki/Continued_fraction

http://wims.unice.fr/wims/wims.cgi?lang=en&module=tool%2Fnumber%2Fcontfrac.en&cmd=new

http://mathworld.wolfram.com/ContinuedFraction.html

Calculator:

Gute Seiten:

http://home.att.net/~numericana/answer/fractions.htm#continued

http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/cfINTRO.html#intro

Contfrac----- Help [Back] -----

Examples of expressions, and how to enter them.

For the expression: You may type: Which gives:

         pi^2-3*e 1.7147589...

           sqrt(2)+5^(1/3) 3.1241895...

46-36-26 4^6-3^6-2^6 3303

222-10! 2^22-10! 565504

(35-1)(25-1)-1 (3^5-1)*(2^5-1)-1 7501

              (15+77-2)/(2^3*3^5-1) 90/1943

More advanced examples .

Contfrac----- Help [Back] -----

Examples of expressions, and how to enter them.

For the expression: You may type: Which gives:

ppcm(15,70)-pgcd(21,33) lcm(15,70)-gcd(21,33) 207

               sum(n=1,10,n^2+n) 440

             prod(n=1,10,n^2/(n^2+1))              

             binomial(30,12) 86493225

integral part of e4 truncate(exp(4)) 54

          2^(2^(2^2))-8! 25216

root of x2+x-1between 0 and 1 (golden ratio)

solve(x=0,1,x^2+x-1) 0.618033988...

Elementary examples . For more information on the functions and their names, please consult the manual

of pari.

= ½ (1+5) [1; 1, 1, 1, 1, 1, 1, 1, 1, ...

½ [k+(k2+4)] [k; k, k, k, k, k, k, k, k, ...

2 [1; 2, 2, 2, 2, 2, 2, 2, 2, ...

3 [1; 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, ...

5 [2; 4, 4, 4, 4, 4, 4, 4, 4, ...

7 [2; 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, ...

41 [6; 2, 2, 12, 2, 2, 12, 2, 2, 12, 2, 2, 12, ...

e = exp(1) [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10, 1, 1, ... 2n+2, 1, 1, ...

e = exp(1/2) [1; 1, 1, 1, 5, 1, 1, 9, 1, 1, 13, 1, 1, 17, 1, 1, ... 4n+1, 1, 1, ...

exp(1/3) [1; 2, 1, 1, 8, 1, 1, 14, 1, 1, 20, 1, 1, 26, 1, 1, ... 6n+2, 1, 1 ...

exp(1/k) [1; k-1, 1, 1, 3k-1, 1, 1, 5k-1, 1, 1, 7k-1, ... (2n+1)k-1, 1, 1 ...

e 2 = exp(2) [7; 2, 1, 1, 3, 18, 5, 1, 1, 6, ... 12n+6, 3n+2, 1, 1, 3n+3 ...

exp(2/3) [1; 1, 18, 7, 1, 1, 10, ... 36n+18, 9n+7, 1, 1, 9n+10 ...

exp(2/5) [1; 2, 30, 12, 1, 1, 17, ... 60n+30, 15n+12, 1, 1, 15n+17 ...

exp(2/7) [1; 3, 42, 17, 1, 1, 24, ... 84n+42, 21n+17, 1, 1, 21n+24 ...

exp(2/(2k+1))[1; k, ... (24k+12)n+12k+6,  (6k+3)n+5k+2,  1,  1, 

(6k+3)n+7k+3 ...

tanh(1) = (e2-1)/(e2+1)

[0; 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, ... (2n+1) ...

tanh(1/k) [0; k, 3k, 5k, 7k, 9k, 11k, 13k, 15k, 17k, 19k, ... (2n+1)k ...

tan(1) [1; 1, 1, 3, 1, 5, 1, 7, 1, 9, 1, 11, 1, 13, 1, 15, 1, ... 2n+1, 1, ...

tan(1/2) [0; 1, 1, 4, 1, 8, 1, 12, 1, 16, 1, 20, 1, 24, 1, 28, 1, ... 4n, 1, ...

tan(1/k) [0; k-1, 1, 3k-2, 1, 5k-2, 1, 7k-2, 1, 9k-2, 1, ... (2n+1)k-2,1, ...