In the file, "ALLOWABLE QUOTIENT MODEL", we display a limited ARITHMETIC OF ALLOWABLE QUOTIENTS, with EQUALITY-INEQUALITY RELATIONS and OPERATIONAL RULES, wherein THE OPERATIONS (ADDITION, SUBTRACTION, MULTIPLICATION, EXPONENTIATION) ARE TOTAL (ALWAYS DEFINED), but the other INVERSE OPERATIONS OF THIS ARITHMETIC ARE ONLY PARTIAL WITH DIVISION EXISTING ONLY FOR ALLOWABLE QUOTIENTS (DIVIDEND AN INTEGRAL MULTIPLE OF DIVISOR).

We now wish to GENERATE AN ENTIRE NUMBER SYSTEM with these properties. Since DIVISION IS A BINARY ORDERED OPERATION, we MODEL with a BINARY ORDERED STRUCTURE: THE 2-VECTOR or 2-TUPLE or ORDERED PAIR, with INTEGERS as its COMPONENTS. Thus, [p, q], where ALLOWABLE QUOTIENTS.

1. EQUALITY:
   • For ALLOWABLE QUOTIENTS: a ÷ b = c ÷ d → a · d = b · c.

   • For VECTORS OF INTEGERS: [a, b] = [c, d] → a · d = b · c.    Thus, [8, 5] = [12, 9] because 8 + 9 = 5 + 12 = 17.    Please note: [8, 5] = [8 - 5, 5 - 5] = [3, 0], that is, [8, 5] = [3, 0] because 8 + 0 = 5 + 3.    Similarly, [12, 9] = [12 - 9, 9 - 9] = [3, 0].    And I said above that we can write [2, 3], for which we now find [2, 3] = [2 - 2, 3 - 2] = [0, 1]. That is, [2, 3] = [0, 1] BECAUSE 2 + 1 = 3 + 0. Dig?

   Assignment: Find INEQUALITY RULES for VECTORS from THE EQUALITY RULE. We need only the latter rule below.

   • ADDITION and SUBTRACTION:

     • For ALLOWABLE QUOTIENTS: (g_D÷ h_d) ± (j_D ÷ k_d) = (g_D · k_d ± j_D · h_d ÷ (h_d · k_d)..

     • For VECTORS OF INTEGERS: [g, h] ± [j, k] = [g · k ± j · h, h · k]. Example. [8, 2] + [6, 3] = [8 · 3 + 2 · 6, 2 · 3] = [24 + 12, 6] = [36, 6] = [36 ÷ 6, 6 ÷ 6] = [6, 1]. (Please note this reduction via the EQUIVALENCE RULE, for reference below.)

   • MULTIPLICATION:
     • For ALLOWABLE QUOTIENTS: (g ÷ k) · (j ÷ k) = (g · j) ÷ h · k).

     • For VECTORS OF INTEGERS: [g, h] · [j, k] = [g · j, h · k].

I'll now show you some GENERAL RESULTS, which lead to "the hiding".

And, again, students of abstract algebra will realize that our VECTOR EQUIVALENCE RULE (MODELED ON OUR ALLOWABLE QUOTIENT EQUIVALENCE RULE) can do "what an Equivalence Rule does best": REDUCE ITS SYSTEM INTO A NUMBER OF EQUIVALENCE CLASSES (often a considerable reduction). Here, we'll see that THE VECTOR EQUIVALENCE RULE YIELDS THREE EQUIVALENCE CLASSES -- effectively, just TWO CLASSES.

PROOF: Given INTEGERS a, b, c for VECTORS of the form, [a, b]:

1. If a = b · c , then [a, b] = [a ÷ b, b ÷ b] = [c, 1], -- which I label "AN INTEGRAL PAIR".

2. But if b = a · c, then [a, b] = [a ÷ a, b ÷ a] = [1, c], -- which (for historical reasons) I label "AN EGYPTIAN PAIR". Example, [3,6] = [1,2].

3. If a, b are the same multiple, including 1, of a COPRIME pair of integers (no factor in common), then EQUIVALENCE can reduce them only, say, to [p, q], an irreducible pair -- which I shall label "A FRACTIONAL PAIR".

The VECTOR EQUIVALENCE RULE (MODELED ON OUR ALLOWABLE QUOTIENT EQUIVALENCE RULE) REDUCES the infinity of INTEGRAL VECTORS TO JUST THREE EQUIVALENCE CLASSES, which I choose to label CANNONICAL TYPES: INTEGRAL, EGYPTIAN, FRACTIONAL!

REDUCTION TO JUST 3 CLASSES ALLOWS (as with NATURAL NUMBER VECTORS) TO HIDE INTEGRAL VECTORS BY SIGNS:

1. For any INTEGER, n, we have [n, 1] → INTEGER n.    Example: [2, 1] → 2.
2. [1, n] for NONERO INTEGER n can be written as "1" over "n", with solidus (fractional bar) between vector components.    Example: [1,2] → ½.    In Computer Science, the SOLIDUS is replaced by the SLASH ("/") MARK: 1/n.
3. Given INTEGERS a, b, if [a, b] reduces to COPRIME INTEGERS [c, d], then [c, d] can be written as "c SOLIDUS d", or c/d.

Note: We don't need any special sign for INTEGRAL VECTORS. And a single sign (SOLIDUS or SLASH) can be used for EGYPTIAN and FRACTIONAL VECTORS.

SUMMARY:

• We didn't CHEAT. For INTEGERS 2, 3, the operation, 2 ÷ 3, is still MEANINGLESS. But we can write THE RATIONAL NUMBER, 2/3.

• Remember the DIVISION RULE (either for DEFINED QUOTIENTS or VECTOR OF INTEGERS). It YIELDS THE QUOTIENT AS A PRODUCT! WHICH IS ALWAYS POSSIBLE. Hence, the DIVISION PROBLEM is BYPASSED.

• So, now, we have an RATIONAL NUMBER SYSTEM in which ADDITION, MULTIPLICATION, EXPONENTIATION, AND ALSO SUBTRACTION AND NONZERO DIVISION ARE TOTAL. (LOGARITHM and ROOT EXTRACTION ARE STILL PARTIAL!)

• And DIVISION RULES don't turn WEIRD.

  We need a MULTIPLICATION RULE FOR ALLOWABLE QUOTIENTS SATISFYING TWO CONDITIONS: CLOSURE (PRODUCT OF QUOTIENTS EQUALS A QUOTIENT ALREADY IN THE SYSTEM), and DISTRIBUTIVITY (SHORT AND LONG WAYS OF CALCULATING WHICH SHOULD BE EQUIVALENT ARE INDEED EQUIVALENT).

  Satisfying these two conditions FORCES ON US A "FLIP-&-MULTIPLY" DIVISION RULE AMID THE INETEGERS, WHICH IMMEDIATELY ADAPTS AS OUR RATIONAL NUMBER (FRACTIONAL) DIVISION RULE.

  But IT IS INTEGRAL ARITHMETIC WHICH REQUIRES IT! NOT SOME KIND OF "NEW MATH"!

• Yet RATIONAL NUMBERS ARE SIMPLY ORDERED PAIRS Of INTEGERS, which (in turn) ARE SIMPLY ORDERED PAIRS OF NATURAL NUMBERS: [[a,b], [c,d]], for NATURALS, a, b, c, d. So, we never left the NATURAL NUMBER SYSTEM. RATIONAL NUMBERS are just sort of "NATURALS in 3D".

  And this explains the development of THE RATIONAL NUMBERS SYSTEM to render TOTAL THE DIVISION OPERATION OF NATURAL NUMBERS AND INTEGERS AS INVERSE OF THE MULTIPLICATION OPERATION GENERATED TO AVOID REPEATED ADDING.

  And we find that REPEATED MULTIPLICATION OF NATURALS, INTEGERS, RATIONALS -- as in 4 · 4 · 4 -- is TEEEDJUS!

  SHORTCUT: DEFINE EXPONENTIATION OPERATIONS RECURSIVELY ("in the Fourth Degree") IN TERMS OF MULTIPLICATION: b¹ = b; b^{(p + 1)} = b^p · b. THIS PERFORMS ANY EXPONENTIATION, SO NATURALS, INTEGERS, RATIONALS ARE CLOSED UNDER EXPONENTIATION.

  Kikbak? Yes ("in The Fourth Degree"), because of WELL DEFINEDNESS: (b^p)^q = (b^p)^r Þ q = r.

  BUTTTTTTTTTTTTTTTTTTTTT! EXPONENTIATION IS NOT COMMUTATIVE: 2³ = 8, while 3² = 9. Hence, a DOUBLEKIKBAK.

  EXPONENTIATION SPLITS! TWO INVERSES! LOGARITHM ("Fourth Degree Kikbak Passive"): b^p = e Þ log_e = b; and ROOT EXTRACTION ("Fourth Degree Kikbak Active"): b^p = e Þ (e)^1/p = b (example: 2³ = 8 Þ (8)^1/3 = 2 -- the third root of 8 is 2).

  Both INVERSES are only PARTIAL in the NATURAL, INTEGRAL, and RATIONAL NUMBERS SYSTEMS. This requires TWO MORE NUMBER SYSTEMS (REAL and COMPLEX) to RENDER THE INVERSES TOTAL.

  Briefly, MAKING LOGARITHM TOTAL REQUIRES A NEW OPERATION, called LIMIT, which LEADS TO "THE CALCULUS".