**On Triple Ratios**

**Triple Ratios and their Arithmetics**

A triple ratio is (a;b;c), with this
equality rule:

(a;b;c) = (A;B;C) if
and only if

aB=Ab and bC=Bc and cA=Ca.

So if A, B and C are not zero, then

a/A =
b/B = c/C

This is triple equal proportion. It
applies, for instance, to the sides of similar triangles; and also the Sine
Law:

(a;b;c) =
(sin(α), sin (β), sin(γ))

The
definition of equality implies the Cancellation Law:

(ka;kb;kc) =
(a;b;c)

This
in turn implies, if a, b, and c are all nonzero:

(a;b;c) = (1 ;
b/a ; c/a) = (a/b ; 1 ; c/b) = (a/c
; b/c ; 1)

We
define multiplication, unity and inverses this way:

(a;b;c)*(x;y;z) = (ax;
by; cz)

(1;1;1) = 1

1/(a;b;c) = (bc;
ca; ab)

So if a, b and c are all
nonzero:

1/(a;b;c) = (1/a;
1/b; 1/c)

Define these trios of
zeros, infinities and negatives:

0

_{1}= (1;0;0)
0

_{2}= (0;1;0)
0

_{3}= (0;0;1)
∞

_{1}= (0;1;1)
∞

_{2}= (1;0;1)
∞

_{3}= (1;1;0)
-1

_{1}= (-1;1;1) = (1;-1;-1)
-1

_{2}= (1;-1;1) = (-1;1;-1)
-1

_{3}= (1;1;-1) = (-1;-1;1)
(0;0;0) is the indefinite
triple ratio. It equals all ratios, and it is the only one that does so.

For any n = 1, 2, or 3,
and if abca,
then

(0

_{n})^{2}= 0_{n}
0

_{a}0_{b}= 0_{b}0_{c}= 0_{c}0_{a}= (0;0;0)
1/0

_{n}= (0;0;0)
(∞

_{n})^{2}= ∞_{n}
∞

_{a}∞_{b}= 0_{c}
∞

_{b}∞_{c}= 0_{a}
∞

_{c}∞_{a}= 0_{b}
∞

_{a}∞_{b}∞_{c}= (0;0;0)
1/∞

_{n}= 0_{n}
(-1

_{n})^{2}= 1
-1

_{a}*-1_{b}= -1_{c}
-1

_{b}*-1_{c}= -1_{a}
-1

_{c}*-1_{a}= -1_{b}
-1

_{a}*-1_{b}*-1_{c}= 1
1/-1

_{n}= -1_{n}
For any real number R,
define these triple ratios:

R

_{1}= (1; R; R)
R

_{2}= (R; 1; R)
R

_{3}= (R; R; 1)
Then:

R

_{1}R_{2}R_{3}= 1
R

_{1}R_{2}= 1/R_{3}
R

_{2}R_{3}= 1/R_{1}
R

_{3}R_{1}= 1/R_{2}
Define these three
additions:

(a;b;c) +

_{1}(x;y;z) = (ax; bx+ya ; cx+za)
(a;b;c) +

_{2}(x;y;z) = (ay+xb; by; cy+zb)
(a;b;c) +

_{3}(x;y;z) = (az+xc; bz+yc ; cz)
“Two-Denominators Rule”

For each +

_{n}, the nth term is the denominator, and the other two terms are independent numerators. These rules follow:
(a;b;c) +

_{1}(a;y;z) = (a; b+y ; c+z)
(a;b;c) +

_{2}(x;b;z) = (a+x; b; c+z)
(a;b;c) +

_{3}(x;y;c) = (a+x; b+y ; c)
“Common Denominators
Rule”

(1;b;c) +

_{1}(1;y;z) = (1; b+y ; c+z)
(a;1;c) +

_{2}(x;1;z) = (a+x; 1; c+z)
(a;b;1) +

_{3}(x;y;1) = (a+x; b+y ; 1)
“Unit Denominators Rule”

From +

_{n}and -1_{n}, define –_{n}:
x –

_{1}y = x +_{1}(-1_{1})y
x –

_{2}y = x +_{2}(-1_{2})y
x –

_{3}y = x +_{3}(-1_{3})y
Each of
the three additions, subtractions, units and zeros form a ring with * and
reciprocal:

+

_{n}is commutative, associative, has identity 0_{n}and negative (-1_{n})x
Distribution
works: a*(b+

_{n}c) = (a*b)+_{n}(a*c)
Multiplication
is commutative, associative, has identity 1.

However
reciprocal is problematic with the zeros and the infinities. The reciprocal of
an infinity is a zero, the reciprocal of a zero is the indefinite ratio, and an
infinity times its zero is indefinite.

In the
3 arithmetic, any triple ratio (a;b;c) is either an infinity (a;b;0) or it
equals (a/c; b/c; 1). In the unit-denominator ratios, all operators work
independently on the first two terms, and leave the last term equal to one:

(a;b;1)*(A;B;1) =
(aA; bB; 1)

1 /
(a;b;1) = (1/a; 1/b; 1)

(a;b;1)+

_{3}(A;B;1) = (a+A; b+B; 1)
(a;b;1)-

_{3}(A;B;1) = (a-A; b-B; 1)
So
unit-third-term ratios under the third arithmetic are isomorphic to pairs of
numbers operating in parallel; the

*dual*, or*hyperbolic*numbers. Similarly with unit-first-term ratios under the first arithmetic, and unit-second-term ratios under the second arithmetic.
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