"JEOPARDY" MATHEMATICS

On the typical TV "Quiz Show", the panelist is asked a question to be answered. Part of the long popularity of the "Jeopardy" Show is that (in reverse) the panelist is told an answer and is supposed to formulate a question matching the Interrogator's answer. The relative difficulty of this intrigues TV audiences.

ARITHMETIC RESEMBLES TYPICAL QUIZZING: QUESTIONS TO BE ANSWERED.

ALGEBRA (in any form) RESEMBLES "JEOPARDY" QUIZZING: ANSWERS TO BE MATCHED TO RELEVANT QUESTIONS.

In other words, ALGEBRA (in any form) IS "BACKWARDS" of some "FORWARD MATH". NUMALGEBRA IS "BACKWARDS" ARITHMETIC.

Before defining NUMALGEBRA, I'll show you that NUMALGEBRA is implicit in what is done in First or Second Grade Arithmetic -- whenever you meet that "creepy" stranger, SUBTRACTION.

The most common method for teaching SUBTRACTION is the "take-away method": "Five take away three: what's left?" Mathematicians call this "rhetorical math" -- in WORDS, not SYMBOLS. Symbolically, 5 — 3 = ?

The other commonly taught method sometimes goes by the label, "The Austrian Method", for some historical reason. Actually, it appeals to THE DEFINITION OF SUBTRACTION IN TERMS OF ADDITION: a — b = c if, and only if, a = b + c. So, for example, it transforms that SUBTRACTIVE form, 5 — 3 = ?, into an ADDITIVE form: 3 + ? = 5. Well, just change that "?" into "x", for 3 + x = 5, and you have the familiar NUMALGEBRAIC FORM. (So, they sneaked "algebra" onto your desk when you were too innocent to defend yourself!)

I'll define NUMALGEBRA in extended terms to be explicated in files that follow. As a start (whereas each ARITHMETIC OPERATION yields a SINGLE NUMBER), NUMALGEBRA IS THE ARITHMETIC OF SETS OF NUMBERS (or) THE ARITHMETIC OF FUNCTIONS OF SETS OF NUMBERS.

How do SETS get into this? The reason relates to "backwards math":

• the primary operations of addition, multiplication, and exponentiation are binary operatioms, which can be modeled as an ordered pair or ordered pairs: [[input₁, input₂], output]. (Examples, respectively for addition, multiplication, exponentiation: [[2,3], 5], [[2,3], 6], [[2,3], 8].
• In the natural (counting) number system, these (apart from binary commutativity) are one-one, that is, for addition, [[a,b], 5] if, and only if, [a,b] = [2,3], or [a,b] = [3,2] -- no two other natural number inputs yield the sum 5. (Comparable cases arise for multiplication and exponentiation.)
• But this situation changes for multiplication in the integers, due to the rule of signs, wherein "positive times positive is positive, and negative times negative is positive". Thus, we cna have the following multiplication forms: [[2,3], 6] = [-2,-3], 6].
• Because of this, and similar extensions, in going back from a product or other extension to the input, we must allow for multiple inputs.
• A single input case need not be represenbted by a set, but a multiple case must be.
• This is why we use "x" in numalgebra (backwards math).

You know this! Even if you think you don't.

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Remember? In Elementary School, you studied "The Three R's" -- in slang, "Reading, Riting, Rithmetic". But behind "The Three R's", or implicit within, is a "4th R": REWRITE.

I'll explain REWRITE; its CONNECTIONS to "The Three R's"; its CONNECTIONS to Arithmetic; its CONNECTIONS to NUMALGEBRA. To do so, I introduce some (syntactic) SIGNS:Let "«_»" denote (be syntactic sign of) terms in the METALANGUAGE, for talking about the LANGUAGE. In other words, given "«_»", the blank sign ("_") REPRESENTS WHAT IS TO BE REWRITTEN. Let denote (be the syntactic sign of) the REWRITE command. It says "what is on the LEFT SIDE of the ARROW is to be REWRITTEN ON ITS RIGHT".

For general purposes, let "..." denote (be the syntactic sign of) "et cetera".

Given these (syntactic) signs, we can REWRITE:

• "greeting» Hello.
• In Grammar, we have: «sentence» «subject» «predicate»
• In Statement Logic,«disjunctive 'or'» |. -- Example: «True or False» «True»|«False»
• For Counting, «digit» 0|1|2|3|4|5|6|7|8|9
• In Arithmetic, we begin with «tallies» «natural or counting numbers». -- Example: ( ||| and || ||||| ) 3 + 2 = 5.

It's almost the same with NUMERICAL ALGEBRA, after some UNRAVELING occurs. (In another file, I refer to UNRAVELING as "THE KIERKEGAARD KIKBAK".)

Example: x + 2 = 5 x + 2 - 2 = 5 - 2 x + 0 = 3 x = 3. That is, x + 2 = 5 x = 3. That is, x + 2 = 5 (x {3}). (See! see! Spot runs with the SET CONTAINING THE NUMBER 3. SETWISE, WE'VE RETURNED TO ARITHMETIC ( ), because THE CHECKING yields 3 + 2 = 5.)

Did you notice the "unraveling"? Starting out, "x" is ADDED TO 2. To ISOLATE x (UNRAVEL x from 2), we use THE KIKBAK (INVERSE OPERATION) OF ADDITION, namely, SUBTRACTION. BY THE GOLDEN RULE OF EQUATIONS ("DO YE TO THE RIGHT-SIDE OF THE EQUATIONS WHAT YE DO TO THE LEFT"), 2 MUST BE SUBTRACTED FROM BOTH SIDES OF THE EQUATION. The x is UNRAVELED (or KIKEDBAK) to 3. A slightly more complicated unravelling (KIKBAK): x² = 4 (x²)^1/2 = x = ^±2, because (¯2)² = 4 and (⁺2) ² = 4. That is, x² = 4 (x → {¯2,⁺2}).

In the first example, "x" is REWRITTEN AS A SINGULAR (or 1-tple) SET (set with a single member). In the second example, "x" is REWRITTEN AS A DOUBLETON (or 2-tple) SET (set with two members). In either case, we were RESTORED TO ARITHMETiC. And one of the meanings of "algebra" is "restore".

Repeating, NUMERICAL ALGEBRA IS THE ARITHMETIC OF SETS OF NUMBERS, or, THE ARITHMETIC OF FUNCTIONS OF SETS OF NUMBERS.

But that is entirely in keeping with my saying that ALGEBRA, in any form, IS "JEOPARDY" MATHEMATICS OR "ARITHMETIC BACKWARDS". THAT'S WHY IT SEEMS LIKE "LOOKING-GLASS ARITHMETIC".

Yet WHYSO? AH! ALL SHALL BE REVEALED! ANON! ANON!

				Jeopardy Math hath
				Coloquery downderry
				For Answer formtransfer
                                Dig or redig

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