Lattice Rationals, 9 of 10

The Wheel Numbers

               TheWheel Numbers arise from the lattice rationals if you add the axiom:

                              0/n     =    0/1        for     n>0

               This is equivalent to “Positive Cancellation”:

                              (ac)/(bc)        =       a/b       if   c>0

               Note that the alternator is still unequal to zero, and negatives do not cancel.

               The wheel numbers can be divided in mainstream, alternates, and null quotients.

               The mainstream numbers are of the form a/b , in lowest terms, with b>0.

               The alternate numbers are of the form a/b , in lowest terms, with b<0.

               The null quotients are 1/0, infinity; -1/0, negative infinity; and 0/0, indefinity.

               The wheel numbers correspond to a circle surrounding a point. Each wheel number corresponds to the slope from the center to the point. The point at the center corresponds to 0/0; the points directly above and below the center correspond to   +1/0   and  - 1/0;  the points directly right and left of the center  correspond to 0 and @; the right half of the circle corresponds  to the mainstream numbers, the left half of the circle corresponds to the alternate numbers.

                                                            +1/0  

                                               -1+@              1

                                             @           0/0            0                        

                                                1+@              -1

                                                             - 1/0

     Note that            - 1/0         =             +1/0  + @

               and                  +1/0         =             -1/0  + @

Reciprocal flips at the infinities between both sign and alternativity.

               The wheel numbers have these laws:

               Addition, reduction and multiplication are commutative and associative.

               Identities:

                              x + 0/1                  =             x  1/+  1/0                       =             x*1/1     =             x

               Attractors:

                              x + 1/0  =  1/0   if x is finite

                              x  1/+  0/1   =   0/1 if x is nonzeroid.

               Indefinities:

                              0  1/+ @           =             1/0 +(-1/0)    =    0 * 1/0  =   0/0

               Alternator:

                              X + @                    =             x  1/+ -1/0         =    -x/-1

                              X + @ + @           =             x  1/+ -1/0  1/+ -1/0 =     x

               Triple Distribution:

                              X*(A+B+C)           =             X*A   +   X*B   +   X*C                       if X is finite

                              X*(A 1/+ B 1/+ C)       =             X*A   1/+   X*B   1/+   X*C                          if X is nonzeroid

               Since @+@ = 0,  it’s consistent with exponential arithmetic to posit, for A not equal to 0:

                                             A^@      =    -1

                              and        A^(x+@)   =   -(A^x)

               So in general we can posit:

                                             log_A(-1)     =    @

                              and        log_A(-x)     =    log_A(x) + @

               This is a theory of logarithmic negation without reference to the complex numbers.