ALLOWABLE DIFFERENCE MODEL

SUBTRACTION for NATURAL NUMBERS is DEFINED in terms of ADDITION, to be its INVERSE: For NATURALS a, b, c: a — b = c iff ("if and only if") a = b + c -- that is, MINUEND (a) MINUS SUBTRAHEND (b) EQUALS DIFFERENCE (c) iff MINUEND (a) EQUALS SUBTRAHEND (b) PLUS DIFFERENCE (c). Thus, 7 — 3 = 4, because 3 + 4 = 7.

This RESTRICTS NATURAL NUMBER SUBTRACTION TO THOSE ALLOWABLE DIFFERENCES such that SUBTRAHEND IS NOT GREATER THAN MINUEND! "allowable" because IT ALLOWS A NATURAL NUMBER RESULT; deviation YIELDS NO NATURAL NUMBER. But "allowability" severely RESTRICTS SUBTRACTION AS A PARTIAL INVERSE FOR ADDITION IN THE NATURALS, AND WE WISH IT TO BE TOTAL (AS ARE ADDITION AND MULTIPLICATION IN NATURALS).

Typical teaching looks like CHEATING! "You can't subtract 5 from 2 -- except by putting a funny dash sign in front of a 2: -2." Saying this is a new and different NUMBER SYSTEM -- THE INTEGERS -- does little for CREDIBILITY, since it invokes what seems to be a WEIRD RULE, "The Law of Signs": "Minus times minus is plus? Whahhhh?"

However, STUDENTS AREN'T SHOWN HOW SUBTRACTION ARITHMETIC CAN BE GENERATED WITHIN THE NATURAL NUMBER SYSTEM -- WITHOUT CHEATING or WEIRDNESS!

We seek CLOSURE FOR SUBTRACTION -- BUT CLOSURE FAILS in the NATURAL NUMBER SYSTEM.

CLOSURE ("All in The Family") is the "most sacred rule of all MATHEMATICS". (I CLAIM THAT EVERY MATHEMATICAL SYSTEM INVOLVES CLOSURE UNDER RELATIONS OR FUNCTIONS OR OPERATIONS. Find an EXCEPTION to this claim, or forever hold your peace!) But what is, say, OPERATIONAL CLOSURE?

Let o denote ANY OPERATION (ADDITION, MULTIPLICATION, SUBTRACTION, etc.) Then, let p, q belong to the SAME SYSTEM ("family") OF ELEMENTS (say, both are NATURAL NUMBERS); then, given p o q = r, CLOSURE requires r to be of the same type ("family") as p,q.

(Assignment: Find that THE EVEN NATURAL NUMBERS are CLOSED -- retain EVENNESS -- under ARITHMETIC OPERATIONS. But ODDS are not.)

What mathematicians call the 2-WAY STRATEGY is used in PROOFS, going back to Euclid, and can be used to CHECK CALCULATIONS. (Arthur Cayley used it.) Its LOGIC is NECESSARY, but NOT SUFFICIENT -- meaning that A NEGATIVE RESULT IS UNQUESTIONABLE.

  • You "prove" (or calculate) in one legitimate way.

  • You do so in another legitimate way.

  • If the two results (answers) agree, maybe you're "OK". But if they disagree, you definitely know "something's awry".

The 2-WAY STRATEGY can also be used when

  • Teaching a new solution-method that handles problems the old solution-method founders on.
  • Solving a problem never attempted before.
It's too difficult for most students or other feathermerchants to do both of these in a single mood-swing. So you "distribute the labor".

  • You teach the new solution-method by applying it to a problem easily solvable by the old method, using the latter for checking use of the new solution-method.
  • When new solution-method is "mastered", apply to a problem you wouldn't attempt with the old solution-method.

Many students will whimper about this. "Why do I have to do it the hard way, when I can do it the easy way? WAHHHHHH!" But tough-love must prevail.

So, we'll use the 2-WAY STRATEGY in DISCOVERING THE RULES LEADING TO INTEGER ARITHMETIC, wherein SUBTRACTION "ALWAYS WORKS". That is -- without CHEATING and encountering WEIRD RULES -- we'll GENERATE THE INTEGER NUMBER SYSTEM WITH ITS ARITHMETIC!!!

We start with the RESTRICTED PART of THE NATURAL NUMBER SYSTEM and treat that as a SPECIAL NUMBER SYSTEM: THE ALLOWABLE (NATURAL) DIFFERENCE SYSTEM. And proceed by FULFILLING TWO CONDITIONS:

  1. CLOSURE ("keep it in the Family");
  2. 2-WAY STRATEGY (the "longer CLOSURE way" of solving a problem yields the same answer as the familiar way).

By Definition, DIFFERENCE: <<minuend>> — <<subtrahend>> = <<difference>>.

ALLOWABLE DIFFERENCE: (am — bs) = rd, where a, b, r are ALL NATURAL NUMBERS. (Note subscripts, "m", "s", "d", respectively for "minuend", "subtrahend", "difference".)

Let o DENOTE ANY OPERATION (ADDITION, SUBTRACTION, ETC.). Then we consider the CLOSURE form: (a — b) o (c — d) = (e — f) Þ "an allowable difference combined with an allowable difference yields an allowable difference".

Now we need ADDITION and MULTIPLICATION OPERATIONS for ALLOWABLE DIFFERENCES.

CLOSURE: (cm — ds) o (em — fs) = (gm — hs), where o represents ADDITION or SUBTRACTION.

ADDITION: (am — bs) + (cm — ds) = (? — ?). You can find that (am — bs) + (cm — ds) = ((a + c)m — (b + d)s). ("ADD MINUENDS OF BOTH ALLOWABLE DIFFERENCES TO FORM THE MINUED OF THE RESULT; ADD SUBTRAHENDS OF BOTH ALLOWABLE DIFFERENCES TO FORM THE SUBTRAHEND OF THE RESULT." That is, "Add 1st to other 1st; add 2nd to other 2nd".)

Here's where THE 2-WAY STRATEGY comes in. Check by an example: (7 — 3) + (12 — 5) = ?. By the familiar way, 7 —: 3 = 4; and 12 — 5 = 7; so 4 + 7 = 11. And by the long CLOSURE way: (7 — 3) + (12 — 5) = ((7 + 12) — (3 + 5)) = (19 — 8), which, of course, is 11 -- CHECKS! THE SUM OF DIFFERENCES IS A DIFFERENCE (CLOSURE!).

ATTENTION!!! The ADDITION RULE FOR ALLOWABLE DIFFERENCES INVOKES AN ISOTONIC CORRESPONDENCE: IT MATCHES CORRESPONDING ORDINANDS OF TWO ORDERINGS -- FIRST ORDINAND OF ONE ORDERING ADDED TO FIRST ORDINAND OF THE OTHER ORDERING, SECOND ORDINAND OF ONE ORDERING ADDED TO SECOND ORDINAND OF THE OTHER ORDERING. (THIS CAN ALSO BE LABELED A "MATCHING" RULE.)

Now, REVIEW THE DEFINITION OF SUBTRACTION OF NATURALS: For NATURALS a, b, c: a — b = c iff ("if and only if") a = b + c. That REVERSAL IN THE SUBTRACTION RULE FOR NATURALS INDUCES A REVERSAL IN THE SUBTRACTION RULE FOR ALLOWABLE DIFFERENCES OF NATURALS. REVERSING FROM AN ISOTONIC CORRESPONDENCE TO AN ANTITONIC CORRESPONDENCE: FIRST ORDINAND OF ONE ORDERING ADDED TO SECOND ORDINAND OF THE OTHER ORDERING, SECOND ORDINAND OF ONE ORDERING ADDED TO FIRST ORDINAND OF THE OTHER ORDERING: (am — bs) — (cm — ds) = ((a + d)m — (b + c)s). (THIS CAN ALSO BE CALLED A "MIXING" RULE.)

We invoke 2-WAY STRATEGY (checking by an example): given (9 — 3) — (7 — 4), since 9 — 3 = 6, and 7 — 4 = 3, then (9 — 3) — (7 — 4) = 6 — 3 = 3. And the long CLOSURE way yields: (9 — 3) — (7 — 4) = ((9 + 4) — (3 + 7)). THUS, SUBTRACTION ALSO PASSES CLOSURE BECAUSE NOTICE, IN THE SUBTRACTION RULE, SUBTRACTION IS BYPASSED FOR ADDITION, WHICH ALWAYS WORKS!

We need a MULTIPLICATION RULE such that THE PRODUCT OF TWO ALLOWABLE DIFFERENCES IS A ALLOWABLE DIFFERENCE: (a — b) · (c — d) = (? — ?). How do we multiply on paper or on blackboard?

		45
	        17
               315
               45
               765
For a better model, let's redo that this way:
                40  +  5
                10  +  7
                40 · 10 + 10 · 5 + 7 · 40 + 5 · 7 = 400 + 50 + 280 + 35 = 765
Now, our ALLOWABLE DIFFERENCE PRODUCT:
               am  —  bs
               cm  —  ds
                _________________________________
               (cm · am) + (cm · ¯bs) + (¯ds · cm) + (¯ds · ¯bs).

This is a case of MATCH-&-MIX. MATCH: minuend times minued, subtrahend times subtrahend; MIX: minuend times subtrahend, vice versa. Or, in previous language: ISOTONIC-&-ANTITONIC.

Assignment: Use the 2-WAY STRATEGY with an example (say, (7 — 3) · (9 — 5) -- or one of your own) to find that YOU CAN SATISFY CLOSURE and "short cut" arithmetic only by a MULTIPLICATION RULE of the form: (am — bs) · (cm — ds) = ((a · c + b · d)m — ((a · d + b · c)s).

And note that this is MATCH-&-MIX (ISOTONIC-&-ANTITONIC): The MATCHES (ISOTONES) FORM MINUED OF THE DIFFERENCE -- that is, the POSITIVE PART OF THE DIFFERENCE; while the MIXES (ANTITONES) FORM THE SUBTRAHEND OF THE DIFFERENCE -- THE NEGATIVE PART OF THE DIFFERENCE.

Assignment: Find in this the pattern of THE LAW OF SIGNS FOR INTEGERS. But it was forced on us IN THE NATURAL NUMBER SYSTEM by CLOSURE and THE 2-WAY STRATEGY!

  • We could obtain the DIVISION RULE FROM THIS; then go on to obtain an EXPONENTIATION RULE and EXTRACTION OF ROOT RULE. But our purpose is the get some idea how this works out. And to understand that "weird" LAW OF SIGNS -- implicit in in the PRODUCT RULE just found.

    However, this MODEL OF ALLOWABLE DIFFERENCES forms a small part of the NATURAL NUMBER SYSTEM. We wish to have AN ENTIRE NUMBER SYSTEM CLOSED UNDER SUBTRACTION. What do we do?

    1. We notice that SUBTRACTION IS A BINARY ORDERED OPERATION, RELATING TWO NATURALS, WHERIN ORDERING MAKES A DIFFERENCE.

    2. This suggests shifting to another BINARY ORDERED STRUCTURE: THE 2-TPLE or 2-COMPONENT VECTOR. That is, write a NEW TYPE OF NUMBER (INTEGER) in the format: [p, q], with NATURAL NUMBER p as "party of the first part" (lst component) in the VECTOR (2-TUPLE); and with NATURAL NUMBER q as "party of the second part" (2nd component) in the VECTOR (2-TUPLE).

    3. But what are the OPERATIONAL RULES (ADDITION, ETC.) FOR THESE VECTORS? We already have them! THE OPERATIONAL RULES FOUND FOR CLOSURE OF ALLOWABLE DIFFERENCES. So, from NATURAL NUMBER ARITHMETIC, we GENERATE SUBTRACTION OF 2-TUPLES or VECTORS or INTEGERS without playing Moses, without cheating! For we discover a BYPASS: TURNING SUBTRACTION INTO ADDITION, WHICH IS TOTAL FOR NATURAL NUMBERS.

      Let's see.