**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.

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