jueves, 6 de abril de 2017

1429. Over and under Euclid's algorithm...

     Pepe Chapuzas went to the blackboard to correct a gcd exercise, but didn't factorize the numbers but used Euclid's algorithm... Then... something else.



    Dear Teacher:
    The  gcd (203, 161) = 7 . 

    As well, Euclid's algorith allows me to write  203:161  as a continued fraction:


203:161 =
= 1 + 42:161 =
= 1 + 1:(161:42) =
= 1 + 1:(3+35:42) =
= 1 + 1:(3+1:(42:35)) = 
= 1 + 1:(3+1:(1+7:35))) =
= 1 + 1:(3+1:(1+1:5))) =
= [ 1; 3, 1, 5 ]

    As well, Euclid's algorith allows me to divide a  203 x 161  rectangle into squares:

One  161 x 161  square
Three  42 x 42  squares
One  35 x 35  square
Five  7 x 7  squares


    In total there are  1+3+1+5 = 10  squares...

    As well, Euclid's algorithm allows me to calculate the gcd of two polynomials (even if Ruffini's rule isn't viable)...


    So, gcd ( x5 + 2x4 + 6x3 + 3x2 + 8x – 2 ,  x4 + x3 + 5x2 + 2x + 6 )  =   x2 + 2.

    Pepe continued talking about continued fractions...

    Dear Teacher:
    Look at these periodic continued fractions:


The golden ratio  [ 1; 1, 1, 1, 1 ... ] = (1+5)/2
The silver ratio  [ 2; 2, 2, 2, 2 ... ] = 1+2
The bronze ratio  [ 3; 3, 3, 3, 3 ... ] = (3+13)/2
    
    These metallic ratios are solutions to these equations:


Golden ratio  1+1/x = x.
Silver ratio  2+1/x = x.
Bronze ratio  3+1/x = x.
    Because if we rationalize:


1+2/(1+5) = 1+(5–1)/2 = (1+5)/2.
2+1/(1+2) = 2+(2–1)/1 = 1+2.
3+2/(3+13) = 3+(13–3)/2 = (3+13)/2.

    The golden ratio is found in many canvases of great painters...
    The silver ratio is found in the rectangle we discard in a paper sheet (similar to DIN A4) to get a square in origami works.
    But my favorite infinite continued fraction is not periodic

e = [ 2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10, 1, 1, 12, 1 ... ]