On Triple Ratios: 1 of 6

          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:

0₁  =  (1;0;0)                

0₂  =  (0;1;0)                

0₃  =  (0;0;1)      

∞₁  =  (0;1;1)               

∞₂  =  (1;0;1)      

∞₃  =  (1;1;0)      

-1₁  =  (-1;1;1)     =  (1;-1;-1)

-1₂  =  (1;-1;1)     =  (-1;1;-1)

-1₃  =  (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_a0_b = 0_b0_c =  0_c0_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:

R₁     =       (1; R; R)

R₂     =       (R; 1; R)

R₃     =       (R; R; 1)

Then:

R₁ R₂ R₃    =       1

R₁ R₂          =       1/R₃ 

R₂ R₃          =       1/R₁ 

R₃ R₁          =       1/R₂ 

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 (-1₁)y

x –₂ y          =       x  +₂ (-1₂)y

x –₃ y          =       x  +₃ (-1₃)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+_nc)  = (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.