Zerlegung von Quadraten und ????

15/11 2x+y=1 y+z=x 3z=y

4

4 4

Kettenbruchentwicklung und ggT: Die Länge des kleinsten Quadrats ist der ggT

Euklidischer Algorithmus Gegeben x z := x, , n:=0 n:=n+1 z:=1/(rn – an) rn hat fraktionalen Anteil rn:= z Ja an := ganzzahliger Anteil von rn Nein an:= rn Ende

Kettenbrüche und ähnliche Rechtecke

Kettenbrüche und ähnliche Rechtecke

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 rational: x kann in der Form m/n geschrieben werden; m und n natürliche Zahlen x hat schließlich-periodische Entwicklung bezüglich jeder Basis x hat abbrechende Kettenbruchentwicklung x irrational: x nicht als Quotient zweier natürlicher Zahlen als m/n schreibbar x keine Periodizität in der Entwicklung bezüglich jeder Basis x hat Kettenbruchentwicklung, die nicht abbricht Wenn x algebraisch von der Ordnung 2 (und irrational), dann hat x eine schließlich-periodische Kettenbruchentwicklung. Es gilt auch die Umkehrung!

n [ a; Period ] √2 √3 √4 √5 √6 √7 √8 √9 √10 √11 √12 √13 √14 √15 √16  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 √52 √53 √54 √55 √56 √57 √58 √59 √60 √61 √62 √63  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 √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 √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

Realisierung des Euklidischen Algorithmus mit Microsoft Excel

Kettenbruchentwicklungen von Gute Seiten: http://mathworld.wolfram.com/ContinuedFraction.html http://home.att.net/~numericana/answer/fractions.htm#continued http://en.wikipedia.org/wiki/Continued_fraction http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/cfINTRO.html#intro Calculator: http://wims.unice.fr/wims/wims.cgi?lang=en&module=tool%2Fnumber%2Fcontfrac.en&cmd=new

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-1 between 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.

f = ½ (1+Ö5) [1; 1, 1, 1, 1, 1, 1, 1, 1, ... ½ [k+Ö(k2+4)] [k; k, k, k, k, k, k, k, k, ... Ö2 [1; 2, 2, 2, 2, 2, 2, 2, 2, ... Ö3 [1; 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, ... Ö5 [2; 4, 4, 4, 4, 4, 4, 4, 4, ... Ö7 [2; 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, ... Ö41 [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, ... Öe = 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, ...