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