BYPASSING INTEGERS FOR RATIONALS
                                                  STRATEGY TACTICS

       CLOSURE ON INTEGRAL DIVISION
          --------------------->
     SHIFT|                    ^CLOSURE ON
      FROM|                    |DEFINED
  INTEGERS|                    |QUOTIENT
          |                    |DIVISION
          |                    |(RESTRICTED
          V-------------------->SYSTEM)
 DEVELOP DEFINED QUOTIENT ARITHMETIC
 DIVIDEND@D; divisor @ d ¬ = 0;  DEFINED QUOTIENT:  [D d]  s. t.  D is INTEGRAL MULTIPLE of d;
QUOT. CLOSURE: for OPRATN  o  and DFND QUOT:   d1  o  d = d3 (COMBINING DEFND QUOTNTS YIELDS A DEFND QUOT), thus:
 [D1 d1] > [D2 d2] <--> D1 * d2 > D2 * d1;
 [D1 d1] < [D2 d2] <--> D1 * d2 < D2 * d1;
 [D1 d1] = [D2 d2] <--> D1 * d2 = D2 * d1;
 [D1 d1] + [D2 d2]  =  [D1 * d2 + d1 * D2] [d1 * d2];     [D1 d2] - [D2 d2]  =  [D1 * d2 - d1 * D2] [d1 * d2];
 [D1 d1] * [D2 d2]  =  [D1 * D2] [d1 * d2];
 [D1 d1] [D2 d2]  =  [D1 * d2] [D2 * d1]
       CLOSURE ON A TOTAL NUMBER SYSTEM       
          ------------------------->
SHIFT FROM|                        ^ADDITION, SUBTRACTION,
   DEFINED|                        |MULTIPICATION, (NONZERO) DIVISIOM
  QUOTIENT|                        |CLOSED IN TOTAL NUMBER SYSTEM
ARITHMETIC|                        |
          |                        |
          V----------------------- >
     DEVELOP INTEGAL VECTOR ARITHMETIC
         COMBINING
   1st VECTOR COMP. = u   2nd VEC. COMP. = v
VECTOR: [u, v] s.t. the COMPS are INTEGERS;
ARITHMETIC OF VECTORS OF INTEGERS:
 [u1, v1] > [u2], v2] <--> u1 * v2 >u2 * v1;     [u1, v1] < [u2, v2] <--> u1 * v2 < u2 * v1;
 [u1, v1] = [u2, v2] <--> u1 * v2 = u2 * v1;
 [u1, v1] + [u2, v2]  =  [u1 * v1 + u2 * v2, v1 * v2];
[u1, v2] - [ u2, v2]  =  [u1 * v2 - v1 * u2, v1 * v2];
 [u1, v1] * [u2, v2]  =  [u1 * u2, v1 * v2];
 [u1, v1] [u2, v2]  =  [u1 * v2, [u2 * v1]
WHEREAS DEFINED QUOTIENTS HAVE RESTRICTED COMPONENTS, VECTOR COMPONENTS DO NOT, PROVIDED NOT VIOLATING INTEGRAL ARITHMETIC. ALSO, NOTE THAT DIVISION ON VECTORS BECOMES PRODUCT ON COMPONENTS, HENCE ALWAYS WORKS! THUS, CLOSURE GOAL IS ACHIEVED FOR VECTOR DIVISION!
         "AWKWARD" VECTOR NOTATION       
         ------------------------->USE FRACTION
    SHIFT|                        ^SIGNS, RESULTING
     FROM|                        |IN 3 CLASSES
  VECTORS|                        |OF RATIONALS
       OF|                        |CLOSED FOR
 INTEGERS|                        |MULTIPLICATION &
         V----------------------- >DIVISION
   REDUCE: 3 EQUIVALENCE CLASSES OF VECTORS
         COMBINING
 SO "RATIONALS/FRACTIONS" ARE VECTORS OF INTEGERS, BUT THE VECTOR NOTATION IS AWKWARD & CAN BE BYPASSED. HOW? VIA THE VECTOR EQUIVALENCE RULE:  [u1 , v1] = [u2, v2] <--> u1 * v2 = u2 * v1.
GIVEN A VECTOR WHOSE FIRST COMPONENT IS AN INTEGRAL MULTIPLE, k OF THE SECOND COMPONENT -- [u,v] = [kv,v], THEN, BY THE EQUIVALENCE RULE, [u,v] = [kv, v] = [kv/v,v/v] = [k,1], AN EQUIVALENCE CLASS OF INTEGRAL VECTORS. BUT SUPPOSE SECOND COMPONENT IS INTEGRAL COMPONENT OF FIRST COMPONENT -- [u,v] = [v,Kv], SO, BY THE EQUIVALENCE RULE, [u,v] = [v,kv] = [v/v,kv/v] = [1,k], AN EQUIVALENCE CLASS OF "EGYPTIAN FRACTIONS" (PL, as in our coinage). THE THIRD POSSIBILITY IS NEITHER OF THESE CASE, FOR AN EQUIVALENCE CLASS OF "FRACTIONS". SO WE BYPASS THE VECTOR NOTATION BY USING THE SOLIDUS SIGN, "/" BETWEEN INTEGRALS, FOR RATIONALS.
BUT WHAT ABOUT TOTALITY. FROM THE DEFINED QUOTIENT DIVISION RULE -- (a b) (c d) = (a*d) (b * c) , CLOSURE ON DEFINED QUOTIENTS -- THERE FOLLOWS THE DIVISION RULE FOR VECTORS: [a,b] [c,d] = [a*d,b*c], CLOSURE ON VECTORS OF INTEGERS SINCE IT TURNS DIVISION INTO MULTIPLICATION ON INTEGERS, ALWAYS ALLOWED. HENCE (BARRING DIVISIN BY ZERO), WE HAVE TOTALITY FOR DIVISION. BUT NO ONE MADE UP A WEIRD RULE FOR DIVISION OF FRACTIONS -- IT WAS REQUIRED, BACK IN THE INTEGER SYSTEM, SO THAT DIVISION OBEYS CLOSURE, PERHAPS THE MOST "SACRED" RULE IN MATHEMATICS! DIG?