                        MATHEMATICS WORKS BY TURNING BYPASS INTO SYNTAX
NUMBER OPERATIONS BY RECURSIONS AS BYPASS DIAGRAMS
                                               STRATEGY                                                 TACTICS

BYPASS TEDIOUS CARDINATION ---------------------> SHIFT| ^LABEL FROM| |COUNTS "TALLYING"| |FOR NATURAL | |NUMBER | |CARDINATION V--------------------> RECURSION ON COUNTING
replace tedious TALLYING CARDINATION by
COUNTING - thus, |||| Þ 4 - GENERATED BY
RECURSION: INITIAL COUNT, 0 -- RECURSION
BY SUCCESSOR OPERATION, S( ) , s. t., FOR
ANY COUNT n , S(n) º n + 1 , where n + 1 is
SUCCESSOR of n - thus, S(0) = 1 -- S(1) = 2 --
S(2) = 3 - COUNTS ARE "NATURAL NUMBERS"
- EVERY NATURAL NUMBER HAS SUCCESSOR
so SUCCESSION IS A TOTAL OPERATION --
SUCCESSOR INDUCES INVERSE OPERATION,
PREDECESSOR, P( ), s. t. n is PREDECESSOR of
n + 1, i.e., P(n + 1) = n IF,AND ONLY IF, S(n) = n + 1
-- HOWEVER, 0 HAS NO PREDECESSOR, so P( )
IS A PARTIAL (NOT A TOTAL) OPERATION.

BYPASS TEDIOUS MULTIPLE COMBING OF COUNTS -------------------------> SHIFT| ^ADDITION FROM| |OPERATION FOR COUNTS| |1ST PRIMARY | |OPERATION OF | |NATURAL NUMBER V----------------------- >SYSTEM RECURSION ON COUNTS FOR COMBINING
A CHILD COUNT OFF 3 FINGERS, THEN 4
FINGERS, & COMBINES TO COUNT OFf THE
7 FINGERS - SUCH TEDIOUS COMBINING IS
REPLACED BY ADDITION CONSTRUCTED BY
RECURSION ON COUNTING OPERATION:
S(a) º a + 1, a + S(b) º S(a + b) - AS WITH
COUNTING, ADDITION ALWAYS YIELDS A
NATURAL NUMBER, SO ADDITION IS A TOTAL
OPERATION - PARTIAL LIMIT ON "COUNTING
BACKWARDS" PUTS LIMIT ON INVERSE OF
ADDITION: SUBTRACTION - BUT (NOTED
ELSEWHERE) MORE THAN INDUCTION ON
ADDING IS NEEDED TO DEFINE ITS INVERSE.

BYPASS TEDIOUS MULTIPLE COMBINING OF ADDENDS -------------------------> SHIFT| ^MULTIPLICATION FROM| |OPERATION FOR ADDENDS| |2ND PRIMARY | |OPERATION OF | |NATURAL NUMBER V----------------------- >SYSTEM RECURSION ON ADDENDS FOR COMBINING
REPEATED DDITION, SUCH AS 5 + 5 + 5 + 5 IS
TEDIOUS -- BYPASSED BY DEFINING
MULTIPLICATION AS RECURSION ON
ADDITION: a * 1 º a, a * S(b) º a * b + a - LIKE
COUNTING AND ADDITION, MULTIPLICATION
IS TOTAL, ALWAYS YIELDING A NATURAL
NUMBER - BUT PARTIAL LIMITS ON INVERSE
OF COUNTING AND INVERSE OF ADDITION
PUT SUCH A LIMIT ON MULTIPLICATION.

BYPASS TEDIOUS MULTIPLE COMBINING OF PRODUCTS -------------------------> SHIFT| ^EXPONENTIAL FROM| |OPERATION FOR PRODUCTS| |3RD PRIMARY | |OPERATION OF | |NATURAL NUMBER V----------------------- >SYSTEM RECURSION ON PRODUCTS FOR COMBINING
REPEATED MULTIPLICATION OF FACTORS,
SUCH AS 4*4*4*4 IS TEDIOUS - BYPASSED BY
DEFINING EXPONENTIATION, DENOTED
b^e = p, WHERE b IS BASE, e IS EXPONENT, p IS
POWER (eTH POWER OF b), DEFINED
RECURSIVELY BY MULTIPLICATION:
b⁰ º 1, b^S(e) º (b^e)*b -- LIKE COUNTING,
ADDITION, MULTIPLICATION,
EXPONENTIATION IS A TOTAL OPERATION -
BUT LIMIT ON INVERSE OF, RSP., COUNTING,
ADDITION, MULTIPLICATION, PUTS LIMIT ON
INVERSE OF EXPONENTIATION.

BYPASSING NATURALS FOR INTEGERS
                                                  STRATEGY TACTICS

CLOSURE ON NATURAL NUMBER SUBTRACTION ---------------------> SHIFT| ^CLOSURE ON FROM| |DEFINED NATURAL| |DIFFERENCE NUMBERS| |SUBTRACTION | |(RESTRICTED V-------------------->SYSTEM) DEVELOP DEFINED DIFFERENCE ARITHMETIC
MINUEND @ m; SUBTRAHEND @ s;
DFIND DIFFERENCE: [m - s] s. t. m < s;
DIFF. CLOSURE: for OPERATION o and DEF. DIFFERENCE d : [d₁] o [d] = [d₃] (COMBINED DEF. DIFFS IS DEF. DIFF):
[m₁ - s₁] > [m₂ - s₂] <--> m₁ + s₂ >m₂ + s₁;
[m₁ - s₁] < [m₂ - s₂] <--> m₁ + s₂ < m₂ + s₁;
[m₁ - s₁] = [m₂ - s₂] <--> m₁ + s₂ = m₂ + s₁;
[m₁ - s₁] + [m₂ - s₂] = [m₁ + m₂] - [s₁ + s₂];
[m₁ - s₁] - [m₂ - s₂] = [m₁ + s₂] - [m₂ + s₂];
[m₁ - s₁] _* [m₂ - s₂] = [m₁ _* m₂ + s₁ _* s₂ ] -
[m₁ _* s₂ + m₂ _* s₁]

CLOSURE ON A TOTAL NUMBER SYSTEM -------------------------> SHIFT FROM| ^ADDITION & DEFINED| |SUBTRACTION DIFFERENCE| |CLOSED IN ARITHMETIC| |TOTAL NUMBER | |SYSTEM V----------------------- > DEVELOP NATURAL VECTORS ARITHMETIC COMBINING
  1st VECTOR COMP. = u   2nd VEC. COMP. = v
VECTOR: [u, v] s.t. the COMPS are NATURALS;
ARITH. OF VECTORS OF NATURAL NUMBERS:
[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 + u₂], [v₁ + v₂];
[u₁, v₁] - [u₂, v₂] = [u₁ + v₂,u₂ + v₁];
[u₁,v₁] _* [u₂,v₂] = [u₁_*u₂ + v₁_*v₂, u₁_*v₂ + u₂_*v₁];
WHEREAS DEFINED DIFFENCES HAVE RESTRICTION ON COMPONENTS, NO SUCH RESTRICTION ON VECTOR COMPONENTS, PROVIDED NO VIOLATING OF NATURAL NO. ARITHMETIC, HOWEVER, NOTE THAT SUBTRACTION ON VECTORS BECOMES ADDITION ON COMPONENTS, HENCE ALWAYS ALLOWED! THUS, CLOSURE GOAL ACHIEVED FOR VECTOR SUBTRACTION!

"AWKWARD" VECTOR NOTATION ------------------------->USE POSITIVE, SHIFT| ^NEGATIVE SIGNS, FROM| |ZERO: 3 CLASSES VECTORS| |OF INTEGERS OF| |CLOSED FOR NATURALS| |ADDITION & V----------------------- >SUBTRACTION REDUCE: 3 EQUIVALENCE CLASSES OF VECTORS COMBINING
SO "INTEGERS" ARE VECTORS OF NATURAL NUMBERS, BUT THE VECTOR NOTATION IS AWKWARD & CAN BE BYPASSED. HOW? BY THE VECTOR EQUIVALENCE RULE: [u₁, v₁] = [u₂, v₂] <--> u₁ + v₂ = u₂ + v₁;
LET u > v IN A VECTOR, SO THAT u - v = w; THEN THIS NATURAL NUMBER SUBTRACTION IS ALLOWED: [u, v] = [u - v, v - v] = [w, 0]: CALL THIS A "POSITIVE VECTOR"; ALTERNATIVELY, LET u < v, SO THAT v - u = x, THEN [u - u, v - u] = [0, x]: CALL THIS A "NEGATIVE VECTOR"; OR, IF u = v, [u - u, v - v] = [0,0]: CALL THIS THE NULL VECTOR, SIMPLY DENOTED AS 0 INTEGER. THEN THE POSITVE VECTOR CAN BE HIDDEN BY A SIGN: ⁺w; THEN NEGATIVE VECTOR BY ANOTHER SIGN: ^-x. AND WE HAVE OUR FAMILIAR INTEGER NOTATION (HIDING VECTORS OF NATURALS!), PROVIDING CLOSURE FOR SUBTRACTION -- WITHOUT CHEATING!
SOMETHING OCCURRED ABOVE. WE INVOKE IT VIA THE VECTOR PRODUCT RULE -- [u₁,v₁] _* [u₂, v₂] = [u₁_*u₂ + v₁_* v₂, u₁_*v₂ + u₂_* v₁] -- WHEN APPLIED TO POSITIVE AND NEGATIVE UNIT VECTORS: [1,0] * [1,0] = [1 * 1 + 0 * 0, 1 * 0 + 0 * 1] = [1,0]: POSITIVE; AND [0,1] * [0,1] = [0 * 0 + 1 * 1, 0 * 1 + 1 * 0] = [1,0]: ALSO POSITIVE; WHEREAS [1,0] * [0,1] = [1 * 0 + 0 * 1, 0 * 0 + 1 * 1] = [0,1]: NEGATIVE! LOOKY! THAT "WEIRD" RULE OF SIGNS FOR VECTORS. WHY? IT ORIGINATED BACK IN NATURAL NUMBER ARITHMETIC BY REQUIRING CLOSURE ON OPERATING WITH DEFINED DIFFERENCES!
TO OBTAIN A MULTIPICATION RULE FOR DEFINED DIFFERENCES YIELDING A DEFINED DIFFERENCE ANSWER, WE ADOPT THE FORM S. T. DEFINED DIFFERENCE TIMES DEFINED DIFFERENCE EQUALS A DEFINED DIFFERENCE: [m₁ - s₁] _* [m₂ - s₂] = [m₁ _* m₂ + s₁ _* s₂] - [m₁ _* s₂ + m₂ _* s₁]. AND WHEN WE USE THIS TO MODEL VECTORS OF NATURALS, THOSE VECTORS PRODUCTS (POS. TIMES POS. IS POS.; NEG. TIMES NEG. IS NEG; POS. TIMES NEG. = NEG., AND VISY VERSY!) POP OUT!
NOBODY MADE UP A WEIRD RULE OF SIGNS. IT FOLLOWED FROM THE MOST "SACRED" RULE IN ARITHMETIC -- PERHAPS IN ALL MATH! -- CLOSURE. DIG?

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?
BYPASSING RATIONALS FOR REALS
                                                  STRATEGY TACTICS

CLOSURE ON RATIONAL NUMBER LOGARITHM ---------------------> SHIFT| ^CLOSURE ON FROM| |DEFINED RATIONAL| |RATIONAL NUMBERS| |LOGARITHM | |(RESTRICTED V-------------------->SYSTEM) DEVELOP DEFINED RATIONAL ARITHMETIC
("INFINITE") RATIONAL CHARSUM @ C; ("INFINITE") RATIONAL MANSUM @ M; ("INFINITE") RATIONAL MIXEDSUM @ X; ("INFINITE:) SUM @ C/M/X; DEFINED LOGARITHM (for any OPERATION o): S o S = S; (COMBINED DEFINED LOGARITHM IS DEFINED LOGARITHM). ALL ARITHMETIC OPERATIONS, EXCEPT ROOT EXTRACTION AND LOGARITHM, WORK FOR DEFINED LOGARITHMS. IN PARTICULAR, SOME LOGARITHMS ARE NOT DEFINED, SUCH AS log₁₀(2)^1/2.
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 {a_n} SUCH THAT, FOR EVERY e > 0, THERE EXISTS AN INTEGER N FOR ALL |a_n - a_m| 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.
W. R. Hamilton (1805-65) created the vector concept and name to create complex numbers as vectors of real number components to clarify their nature. The natural number system has a single UNIT: 1; the integral, rational, and real number systems have two UNITS: ⁺1, ^-1. The complex number system has four UNITS: ⁺1, ^-1, i = -1, ^-i = --1. Hamilton formulated a vector product that allowed a negative square to represent the square of -1 to clarify tne nature of i.
BYPASSING REALS FOR COMPLEX NUMBERS
                                                  STRATEGY TACTICS

CLOSURE ON ROOT EXTRACTION ---------------------> SHIFT| ^CLOSURE ON FROM| |ROOT EXTRACTION REALS| |BY ALLOWING THE | |SQUARE OF AN | |IMAGINARY TO BE V-------------------->HEGATIVE DEVELOP VECTORS OF REALS
FOR REAL NUMBER COMPONENTS, [a,b] = [c,d] iff a = c, b - d; [a,b] < [c,d] iff a < c, b < d; [a,b] > [c,d] iff a > c, b > d. [a,b] + [c, d] = [a + c,b + d]; [a,b] - [c, d] = [a - c,b - d]. [a,b] * [c, d] = [a + c,b + d]. DIVISION IS THE INVERSE OF MULTIPLICATION; its complicated nature is irrelevant to our present concerns.
The above PRODUCT RULE ACCOMPLISHES THE NEEDED RESULT THAT THE SQUARE OF A NEGATIVE VECTORS BE NEGATIVE. But whence the RULE? Hamilton doesn't tell us, nor does the literature. It seems to "descend from Heaven", as seems so much of MATHEMATICS. But it is easily explained. Consider the PRODUCT RULE FOR VECTORS OF NATURALS TO MODEL INTEGERS: [a,b] * [c,d] = [a*c + b*d, a*d + b*c]. Let's apply this RULE to NEGATIVE UNITS: [0,1] * [0,1] = [0*0 + 1*1, 0*1 + 1*0] = [1,0], which is POSITIVE. But let's CHANGE THAT FIRST COMPONENT SUM TO A DIFFERENCE: [a,b] * [c,d] = [a*c - b*d, a*d + b*c]. Applied to a NEGATIVE UNIT: [0,1] * [0,1] = [0*0 - 1*1, 0*1 + 1*0] = [=1,0] , NEGATIVE. Thus, whereas the quasi-AXIOMATIC PRESENTATION OF ARITHMETIC is mysterious, the GENERATIVE PRESENTATION explains!
I knew for over forty years that THIS NUMBER SYSTEM HAS A NEW UNIT that induced Gauss, Argand, and Wessel to REPRESENT A COMPLEX NUMBER BY THE ARROWED RAY that GEOMETRICALLY REPRESENTS A VECTOR, even though that name had not yet been invented by Hamilton. For this has long been hand-waved in the literature. a while back -- because I was on the alert for it -- I found the EXPLANATION, in what I thought an unlikely format: Introduction to Number Theory, by Oystein Ore, p. 159 (Dover 1988).
Ore Ore introduces the MODUL (not to be confused with a module over a ring), perhaps guided by the notion of CLOSURE UNDER ADDITION. Thus, for COMPLEXES, [a, b] + [c, d] = [a + c, b + d]. Note that the REALS remain SEPARATE from the IMAGINARIES under ADDITION.
In Ore's definition, A MODUL IS A STRUCTURE CLOSED UNDER SUBTRACTION. And Ore notes

The NATURALS are not CLOSED UNDER SUBTRACTIONS, BUT THE INTEGERS ARE -- HENCE, A MODUL.

The RATIONALS form a MODUL.

The REALS form a MODUL.

COMPLEX NUMBERS form a MODUL.

THE IMAGINARIES form a MODUL.

Ore apparently does not see that THE COMPLEX NUMBERS FORM A BIMODUL (my definition!): TWO INDEPENDENT SUBTRUCTURES EACH FORMING A MODUL -- CLOSED UNDER SUBTRACTION. We can easily see this from our work with VECTORS OF NATURALS (for INTEGERS) and VECTORS OF INTEGERS (for RATIONALS). I color code with respect to the vector components: RED for 1st COMPONENTS OF OPERANDS OF INPUT, BLUE for 2nd COMPONENTS OF OPERANDS OF INPUT, in order to show what happens in the OUTPUT:

SUBTRACTION FOR VECTORS OF NATURALS: [a, b] - [c, d] = [a + d, c + b].

SUBTRACTION FOR VECTORS OF INTEGERS: [a, b] - [c, d] = [a � d - b � c, b � d].

SUBTRACTION FOR VECTORS OF REAL NUMBERS: [a, b] - [c, d] = [a - c, b - d].

See the difference? In the first two cases, THE VECTOR SECOND COMPONENT (coded BLUE) "GETS INTO THE FIRST COMPONENT" OF THE RESULT (mixing RED and BLUE). In the third case, THIS DOES NOT HAPPEN (no COLOR MIXING within COMPONENT scope).
We can highlight by considering "THE REALS" SEPARATELY FROM "THE IMAGINARIES":

THE "REALS": [a, 0] - [c, 0] = [a - c, 0] -- "THE REALS MODUL": set, M_R.
THE "IMAGINARIES": [0, b] - [0, d] = [0, b - d] -- "THE IMAGINARIES MODUL": set, M_I.
M_RM_I = �. (Their SET INTERSECTION IS NULL -- THEY FORM INDEPENDENT MODULS. This is not the case for previous NUMBER SYSTEMS!)

What does that mean? It means THAT VECTORS OF REAL NUMBERS (COMPLEX NUMBERS) HAVE AN INDEPENDENT UNIT WHICH CANNOT BE HIDDEN BY ANY SIGN-TRICK! VECTORS ARE HERE TO STAYYYYYYYY!
So we have A NUMBER SYSTEM SUSTAINING TOTALLY THE OTHER INVERSE OF EXPONENTIATION: ROOT EXTRACTION. (Mathematicians speak of "solution by radicals).) But, again, we have not left the NATURAL NUMBERS TOTALLY BEHIND.
A COMPLEX NUMBER IS

AN ORDERED PAIR OF REAL HUMBERS;
WHICH IS AN ORDERED PAIR OF INFINITE VECTORS, EACH A CAUCHY SEQUENCE OF RATIONAL NUMBERS;
AND EACH OF THOSE COMPONENT RATIONALS IS AN ORDERED PAIR OF ORDERED PAIRS OF NATURAL NUMBERS;
SO A COMPLEX NUMBER IS AN ORDERED PAIR OF CAUCHY SEQUENCES OF ORDERED PAIRS OF ORDERED PAIRS OF NATURAL NUMBERS -- SORT OF TRANSFINITE 2-D ABOVE THE NATURALS.
CONSEQUENCES: VECTORS OF REAL NUMBERS (COMPLEX NUMBERS):

MODEL ROTATION IN 2-D;

MODEL ALTERNATING CURRENT ELECTRICITY (making possible its spread in a few years).

Made possible RADIO TRANSMISSION.

Provide THE SIMPLEST FORM OF A SPINOR ("the square root of a vector"),

Needed to explain critical SPECTRA.

Correcting Heisenberg-Schr�dinger formulation of QUANTUM THEORY.

Provide the "take-off" for THE ARITHMETIC OF CLIFFORD NUMBERS (a.k.a. GEOMETRIC ALGEBRA, MULTIVECTORS).
All this has been explicated via BYPASS TURNED INTO SYNTAX.

STRATEGY	TACTICS
BYPASS TEDIOUS CARDINATION ---------------------> SHIFT\| ^LABEL FROM\| \|COUNTS "TALLYING"\| \|FOR NATURAL \| \|NUMBER \| \|CARDINATION V--------------------> RECURSION ON COUNTING	replace tedious TALLYING CARDINATION by COUNTING - thus, \|\|\|\| Þ 4 - GENERATED BY RECURSION: INITIAL COUNT, 0 -- RECURSION BY SUCCESSOR OPERATION, S( ) , s. t., FOR ANY COUNT n , S(n) º n + 1 , where n + 1 is SUCCESSOR of n - thus, S(0) = 1 -- S(1) = 2 -- S(2) = 3 - COUNTS ARE "NATURAL NUMBERS" - EVERY NATURAL NUMBER HAS SUCCESSOR so SUCCESSION IS A TOTAL OPERATION -- SUCCESSOR INDUCES INVERSE OPERATION, PREDECESSOR, P( ), s. t. n is PREDECESSOR of n + 1, i.e., P(n + 1) = n IF,AND ONLY IF, S(n) = n + 1 -- HOWEVER, 0 HAS NO PREDECESSOR, so P( ) IS A PARTIAL (NOT A TOTAL) OPERATION.
BYPASS TEDIOUS MULTIPLE COMBING OF COUNTS -------------------------> SHIFT\| ^ADDITION FROM\| \|OPERATION FOR COUNTS\| \|1ST PRIMARY \| \|OPERATION OF \| \|NATURAL NUMBER V----------------------- >SYSTEM RECURSION ON COUNTS FOR COMBINING	A CHILD COUNT OFF 3 FINGERS, THEN 4 FINGERS, & COMBINES TO COUNT OFf THE 7 FINGERS - SUCH TEDIOUS COMBINING IS REPLACED BY ADDITION CONSTRUCTED BY RECURSION ON COUNTING OPERATION: S(a) º a + 1, a + S(b) º S(a + b) - AS WITH COUNTING, ADDITION ALWAYS YIELDS A NATURAL NUMBER, SO ADDITION IS A TOTAL OPERATION - PARTIAL LIMIT ON "COUNTING BACKWARDS" PUTS LIMIT ON INVERSE OF ADDITION: SUBTRACTION - BUT (NOTED ELSEWHERE) MORE THAN INDUCTION ON ADDING IS NEEDED TO DEFINE ITS INVERSE.
BYPASS TEDIOUS MULTIPLE COMBINING OF ADDENDS -------------------------> SHIFT\| ^MULTIPLICATION FROM\| \|OPERATION FOR ADDENDS\| \|2ND PRIMARY \| \|OPERATION OF \| \|NATURAL NUMBER V----------------------- >SYSTEM RECURSION ON ADDENDS FOR COMBINING	REPEATED DDITION, SUCH AS 5 + 5 + 5 + 5 IS TEDIOUS -- BYPASSED BY DEFINING MULTIPLICATION AS RECURSION ON ADDITION: a * 1 º a, a * S(b) º a * b + a - LIKE COUNTING AND ADDITION, MULTIPLICATION IS TOTAL, ALWAYS YIELDING A NATURAL NUMBER - BUT PARTIAL LIMITS ON INVERSE OF COUNTING AND INVERSE OF ADDITION PUT SUCH A LIMIT ON MULTIPLICATION.
BYPASS TEDIOUS MULTIPLE COMBINING OF PRODUCTS -------------------------> SHIFT\| ^EXPONENTIAL FROM\| \|OPERATION FOR PRODUCTS\| \|3RD PRIMARY \| \|OPERATION OF \| \|NATURAL NUMBER V----------------------- >SYSTEM RECURSION ON PRODUCTS FOR COMBINING	REPEATED MULTIPLICATION OF FACTORS, SUCH AS 4444 IS TEDIOUS - BYPASSED BY DEFINING EXPONENTIATION, DENOTED b^e = p, WHERE b IS BASE, e IS EXPONENT, p IS POWER (eTH POWER OF b), DEFINED RECURSIVELY BY MULTIPLICATION: b⁰ º 1, b^S(e) º (b^e)b -- LIKE COUNTING, ADDITION, MULTIPLICATION, EXPONENTIATION IS A TOTAL OPERATION - BUT LIMIT ON INVERSE OF, RSP., COUNTING, ADDITION, MULTIPLICATION, PUTS LIMIT ON INVERSE OF EXPONENTIATION.

STRATEGY	TACTICS
CLOSURE ON NATURAL NUMBER SUBTRACTION ---------------------> SHIFT\| ^CLOSURE ON FROM\| \|DEFINED NATURAL\| \|DIFFERENCE NUMBERS\| \|SUBTRACTION \| \|(RESTRICTED V-------------------->SYSTEM) DEVELOP DEFINED DIFFERENCE ARITHMETIC	MINUEND @ m; SUBTRAHEND @ s; DFIND DIFFERENCE: [m - s] s. t. m < s; DIFF. CLOSURE: for OPERATION o and DEF. DIFFERENCE d : [d₁] o [d] = [d₃] (COMBINED DEF. DIFFS IS DEF. DIFF): [m₁ - s₁] > [m₂ - s₂] <--> m₁ + s₂ >m₂ + s₁; [m₁ - s₁] < [m₂ - s₂] <--> m₁ + s₂ < m₂ + s₁; [m₁ - s₁] = [m₂ - s₂] <--> m₁ + s₂ = m₂ + s₁; [m₁ - s₁] + [m₂ - s₂] = [m₁ + m₂] - [s₁ + s₂]; [m₁ - s₁] - [m₂ - s₂] = [m₁ + s₂] - [m₂ + s₂]; [m₁ - s₁] _* [m₂ - s₂] = [m₁ _* m₂ + s₁ _* s₂ ] - [m₁ _* s₂ + m₂ _* s₁]
CLOSURE ON A TOTAL NUMBER SYSTEM -------------------------> SHIFT FROM\| ^ADDITION & DEFINED\| \|SUBTRACTION DIFFERENCE\| \|CLOSED IN ARITHMETIC\| \|TOTAL NUMBER \| \|SYSTEM V----------------------- > DEVELOP NATURAL VECTORS ARITHMETIC COMBINING	1st VECTOR COMP. = u 2nd VEC. COMP. = v VECTOR: [u, v] s.t. the COMPS are NATURALS; ARITH. OF VECTORS OF NATURAL NUMBERS: [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 + u₂], [v₁ + v₂]; [u₁, v₁] - [u₂, v₂] = [u₁ + v₂,u₂ + v₁]; [u₁,v₁] _* [u₂,v₂] = [u₁_u₂ + v₁_v₂, u₁_v₂ + u₂_v₁]; WHEREAS DEFINED DIFFENCES HAVE RESTRICTION ON COMPONENTS, NO SUCH RESTRICTION ON VECTOR COMPONENTS, PROVIDED NO VIOLATING OF NATURAL NO. ARITHMETIC, HOWEVER, NOTE THAT SUBTRACTION ON VECTORS BECOMES ADDITION ON COMPONENTS, HENCE ALWAYS ALLOWED! THUS, CLOSURE GOAL ACHIEVED FOR VECTOR SUBTRACTION!
"AWKWARD" VECTOR NOTATION ------------------------->USE POSITIVE, SHIFT\| ^NEGATIVE SIGNS, FROM\| \|ZERO: 3 CLASSES VECTORS\| \|OF INTEGERS OF\| \|CLOSED FOR NATURALS\| \|ADDITION & V----------------------- >SUBTRACTION REDUCE: 3 EQUIVALENCE CLASSES OF VECTORS COMBINING	SO "INTEGERS" ARE VECTORS OF NATURAL NUMBERS, BUT THE VECTOR NOTATION IS AWKWARD & CAN BE BYPASSED. HOW? BY THE VECTOR EQUIVALENCE RULE: [u₁, v₁] = [u₂, v₂] <--> u₁ + v₂ = u₂ + v₁; LET u > v IN A VECTOR, SO THAT u - v = w; THEN THIS NATURAL NUMBER SUBTRACTION IS ALLOWED: [u, v] = [u - v, v - v] = [w, 0]: CALL THIS A "POSITIVE VECTOR"; ALTERNATIVELY, LET u < v, SO THAT v - u = x, THEN [u - u, v - u] = [0, x]: CALL THIS A "NEGATIVE VECTOR"; OR, IF u = v, [u - u, v - v] = [0,0]: CALL THIS THE NULL VECTOR, SIMPLY DENOTED AS 0 INTEGER. THEN THE POSITVE VECTOR CAN BE HIDDEN BY A SIGN: ⁺w; THEN NEGATIVE VECTOR BY ANOTHER SIGN: ^-x. AND WE HAVE OUR FAMILIAR INTEGER NOTATION (HIDING VECTORS OF NATURALS!), PROVIDING CLOSURE FOR SUBTRACTION -- WITHOUT CHEATING!

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.

STRATEGY	TACTICS
CLOSURE ON RATIONAL NUMBER LOGARITHM ---------------------> SHIFT\| ^CLOSURE ON FROM\| \|DEFINED RATIONAL\| \|RATIONAL NUMBERS\| \|LOGARITHM \| \|(RESTRICTED V-------------------->SYSTEM) DEVELOP DEFINED RATIONAL ARITHMETIC	("INFINITE") RATIONAL CHARSUM @ C; ("INFINITE") RATIONAL MANSUM @ M; ("INFINITE") RATIONAL MIXEDSUM @ X; ("INFINITE:) SUM @ C/M/X; DEFINED LOGARITHM (for any OPERATION o): S o S = S; (COMBINED DEFINED LOGARITHM IS DEFINED LOGARITHM). ALL ARITHMETIC OPERATIONS, EXCEPT ROOT EXTRACTION AND LOGARITHM, WORK FOR DEFINED LOGARITHMS. IN PARTICULAR, SOME LOGARITHMS ARE NOT DEFINED, SUCH AS log₁₀(2)^1/2.
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 {a_n} SUCH THAT, FOR EVERY e > 0, THERE EXISTS AN INTEGER N FOR ALL \|a_n - a_m\| 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.

STRATEGY	TACTICS
CLOSURE ON ROOT EXTRACTION ---------------------> SHIFT\| ^CLOSURE ON FROM\| \|ROOT EXTRACTION REALS\| \|BY ALLOWING THE \| \|SQUARE OF AN \| \|IMAGINARY TO BE V-------------------->HEGATIVE DEVELOP VECTORS OF REALS	FOR REAL NUMBER COMPONENTS, [a,b] = [c,d] iff a = c, b - d; [a,b] < [c,d] iff a < c, b < d; [a,b] > [c,d] iff a > c, b > d. [a,b] + [c, d] = [a + c,b + d]; [a,b] - [c, d] = [a - c,b - d]. [a,b] * [c, d] = [a + c,b + d]. DIVISION IS THE INVERSE OF MULTIPLICATION; its complicated nature is irrelevant to our present concerns.