Wednesday, June 28, 2017

On Triple Ratios: 3 of 6



          Reciprocal Additions

         
          Define ‘reciprocal addition’, a.k.a. ‘reduction’ as:

          x [+] y   =  1 / ( (1/x) + (1/y) )   =   xy  /  (y+x)

          Reduction is addition conjugated by reciprocal.
          It is commutative, associative, with identity infinity and attractor zero. Multiplication distributes over it:
          x*(y[+]z)  = (x*y)[+](x*z)
          Reciprocal turn addition and reduction into each other:
          1 / (x+y)       =  1/x  [+]  1/y
          1 / (x[+]y)    =    1/x  +  1/y

          There are three reductions on the triple ratios:

          x [+]1 y   =  1 / ( (1/x) +1 (1/y) )   =   xy  /  (y+1x)
x [+]2 y   =  1 / ( (1/x) +2 (1/y) )   =   xy  /  (y+2x)
x [+]3 y   =  1 / ( (1/x) +3 (1/y) )   =   xy  /  (y+3x)

          Therefore “Complementarity”:
          (x [+]1 y) * (x +1 y)       =       x*y
          (x [+]2 y) * (x +2 y)       =       x*y
          (x [+]3 y) * (x +3 y)       =       x*y

          (x [+]1 y)    =      
( (x1y2+x2y1)(x1y3+x3y1)  ;   x2y2(x1y3+x3y1)   ;   x3y3(x1y2+x2y1)  )
          (x [+]2 y)    =      
(  x1y1(x2y3+x3y2)  ;  (x1y2+x2y1)(x2y3+x3y2)  ; x3y3(x1y2+x2y1) )
          (x [+]3 y)    =      
(  x1y1(x2y3+x3y2)   ;   x2y2(x1y3+x3y1)     ;  (x1y3+x3y1)(x2y3+x3y2) )
          We can interpret reciprocal addition in terms of dual arithmetic:

          x [+] ey      =       xey / (x + ey)  =  (yx/(y2-x2))(y – ex)

          x [+]1 ey     =       xey / (x +1 ey)  =  (yx/(y2-1x2))(y –1 ex)
          x [+]2 ey     =       xey / (x +2 ey)  =  (yx/(y2-2x2))(y –2 ex)
          x [+]3 ey     =       xey / (x +3 ey)  =  (yx/(y2-3x2))(y –3 ex)



Tuesday, June 27, 2017

On Triple Ratios: 2 of 6



Hyperbolic Numbers


          Define the dual, or hyperbolic numbers, as number pairs with pairwise operations:
          (a;b)*(c;d) =       (a*c; b*d)
          1 /(c;d)                 =       (1/c; 1/d)
          (a;b)+(c;d) =       (a+c; b+d)
          -(a;b)                    =       (-a: -b)
          0                           =       (0; 0)
          1                           =       (1; 1)
          r                           =       (r; r)            for any real number r
          e                           =       (1;-1)          this is the dual unit.

          This system is a ring; * and + are commutative, associative, have identities 0 and 1, inverses –x, and sometimes 1/x. Reciprocal fails for the non-zero numbers (a;0) and (0;b).

          The dual numbers can be written in terms of 1 and e:
          (a;b)            =       (a+b)/2   +   e (a-b)/2
         
          Written with e, dual numbers have these laws:
          (x + ey) + (z + ew)       =       (x+z)  + e(y+w)
                   e2                          =       1
          (x + ey) * (z + ew)        =       (xz+yw)  + e(xw+yz)
          1 / (x + ey)                    =       (x/(x2-y2))  - e (y/(x2-y2))    if  x2-y20

          These rules resemble the rules for complex multiplication, with some signs reversed. The analog to cis in complex numbers is hyperbolic cis, or ‘chesh’:
          chesh(t)      =       cosh(t)   +  e sinh(t)

          This rule applies:
          chesh(a) * chesh(b)       =       chesh(a+b)

          And just as cis(a)*z is a rotation of z in the complex plane, so too is chesh(a)*z a Lorentz transformation of z in the hyperbolic plane.

          Each of the three arithmetics on triple ratios is isomorphic to the dual numbers. The e’s correspond to negatives in the other two arithmetics. For instance, in arithmetic 3,
          x +3 (-13)y  is subtraction:  x -3 y;
          but x +3 (-11)y  is a dual number;
as is its conjugate x +3 (-12)y   =   x -3 (-11)y