BYPASSING RATIONALS FOR REALS
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   CLOSURE ON RATIONAL NUMBER LOGARITHM
          --------------------->
     SHIFT|                    ^CLOSURE ON
      FROM|                    |DEFINED
  RATIONAL|                    |RATIONAL
   NUMBERS|                    |LOGARITHM
          |                    |(RESTRICTED
          V-------------------->SYSTEM)
 DEVELOP DEFINED RATIONAL ARITHMETIC
 In a DECIMAL NUMBER, the digits prior to the DECIMAL POINT form the CHARACTERISTIC (denote as "C"); posterior digits form the MANTISSA (denote "M"); many decimals are formed by their combination ("CM"). DEFINED LOGARITHM (for any OPERATION o):  S o S = S; usually of the form CM. ALL ARITHMETIC OPERATIONS, EXCEPT ROOT EXTRACTION AND LOGARITHM, WORK FOR DEFINED LOGARITHMS. The number, logba is IRRATIONAL whenever a,b are COPRIME (with no shared integral factor), prevailing for an UNCOUNTABLE number of cases. 
         CLOSURE ON A TOTAL NUMBER SYSTEM       
           ------------------------->
 SHIFT FROM|                        ^ALL OPERATIONS
    DEFINED|                        |EXCEPT LOGARITHM
LOGARITHMIC|                        |CLOSED IN
 ARITHMETIC|                        |REAL NUMBER
           |                        |SYSTEM
           V----------------------- >
  DEVELOP CAUCHY SEQUENCE VECTORS ARITHMETIC
         COMBINING
   A CAUCHY SEQUENCE (GENERALIZING ARITHMETIC PROGRESSION) IS: A SEQUENCE {an} SUCH THAT, FOR EVERY e > 0, THERE EXISTS AN INTEGER N FOR ALL |an - am| WITH m, n > N. (CAUCHY) SUMS Of INFINITE SEQUENCES CAN BE DEFINED. THIS IS ACCOMPLISHED BY ADJOINING (TO FINITE OPERATIONS OF ARITHMETIC) THE TRANSFINITARY OPERATION OF LIMIT (AN ANTITONIC PROCESS), PROVIDING THAT EVERY CAUCHY SEQUENCE EXISTS, WHEREIN EVERY FINITE SUBSEQUENCE IS RATIONAL. THIS ALLOWS FORMATION OF VECTORS OF THE FORM [ c, 0]; [0, m]; [c, m], MIXED TYPE. WHEREAS DEFINED LOGARITHMS ARE RESTRICTED, NO SUCH RESTRICTION EXISTS ON VECTOR COMPONENTS, PROVIDED NO VIOLATING OF RATIONAL ARITHMETIC. 
         "AWKWARD" VECTOR NOTATION       
         ------------------------->USE DECIMAL
    SHIFT|                        ^NUMBER NOTATION
     FROM|                        |FOR THE THREE
  VECTORS|                        |VECTOR CLASSES:
OF CAUCHY|                        |CLOSED FOR ALL
     SUMS|                        |OPERATIONS EXCEPT
         V----------------------- >ROOT EXTRACTION
  REDUCE: 3 EQUIVALENCE CLASSES OF VECTORS
         COMBINING
 SO REAL NUMBERS ARE VECTORS OF CAUCHY SUMS OF RATIONAL NUMBERS. BUT THIS VECTOR NOTATION IS AWKWARD & CAN BE BYPASSED. HOW? BY USING WRITING THE THREE VECTOR EQUIVALENCE CLASSES AS DECIMAL NUMBERS. THE VECTOR [c, 0] BECOMES AN INTEGRAL REAL NUMBER. THE VECTOR [0, m] BECOMES A DECIMAL NUMBER WITH CHARACTERISTICS ZERO, MANTISSA NONZERO. AN THE MIXED VECTOR, [c, m], BECOMES AN OBVIOUS FORM OF DECIMAL NUMBER. THIS GENERATES THE REAL NUMBER SYSTEM IN WHICH THE EXPONENTIATION INVERSE OF LOGARITHM ALWAYS EXISTS.