Tuesday, June 27, 2017

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






Monday, June 26, 2017

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
(0n)2   =   0n
0a0b = 0b0c =  0c0a =  (0;0;0)
1/0n  =  (0;0;0)
(∞n)2   =   n
ab = 0c 
bc = 0a 
ca = 0b 
abc  =  (0;0;0)
1/∞n  =  0n
(-1n)2     =    1
-1a*-1b  =  -1c   
-1b*-1c  =  -1a
-1c*-1a  =  -1b 
-1a*-1b*-1c   =   1
1/-1n     =    -1n

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 -1n, 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 0n and negative (-1n)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.