"ROAST PIG (THE HARD WAY)" MATH (201-300 OF FAQS UNTAUGHT, PROVE OTHERWISE!)

201. Students are not taught the STRATEGY of BYPASSING THE SUBTRACTION OF NATURALS PROBLEM BY FORMING ORDERED PAIRS OF NATURALS.

202. Students are not taught that AN EQUIVALENCE ROLE CAN COLLECT A SET (POSSIBLY INFINITE) INTO A FINITE SET OF EQUIVALENCE CLASSES, as in EQUIVALENT ALLOWED DIFFERENCES OF NATURALS.

203. Students are not taught that SUBTRACTION can be taught without the suspicion of CHEATING invoking WEIRD RULES.

204. Students are not taught that about CLOSURE, "the most sacred Rule in Mathematics".

205. Students are not taught that TWO-WAY STRATEGY FOR PROOF.

206. Students are not taught that a restricted part of the NATURAL NUMBER SYSTEM can be studied: ALLOWED DIFFERENCES.

207. Students are not taught the GENERAL OPERATIONAL FORM FOR ALLOWABLE DIFFERENCES CLOSURE.

208. Students are not taught how to DERIVE AN EQUIVALENCE RULE, such as for ALLOWED DIFFERENCES.

209. Students are not taught the FORM of ADDITION and SUBTRACTION OPERATIONS for ALLOWABLE DIFFERENCES.

210. Students are not taught that the ADDITION RULE for ALLOWABLE DIFFERENCES is ISOTONIC (MATCH).

211. Students are not taught that the SUBTRACTION RULE for ALLOWABLE DIFFERENCES is ANTITONIC (MIX).

212. Students are not taught what the CLOSURE FORM of a MULTIPLICATION RULE for ALLOWABLE DIFFERENCES.

213. Students are not taught that THE MULTIPLICATION RULE for ALLOWABLE DIFFERENCES is ISOTONIC-ANTITONIC (MATCH&MIX).

214. Students are not taught how a RESTRICTED MODEL of a NUMBER SYSTEM can be used.

215. Students are not taught that the PROBLEM OF MEANINGLESS SUBTRACTIONS OF NATURAL NUMBERS (SUBTRAHEND GREATER THAN MINUEND) CAN BE BYPASSED BY VECTORS OF NATURAL NUMBERS WITH NO CONDITIONS ON 1ST AND 2ND COMPONENTS.

216. Students are not taught that a VECTOR ADDITION RULE can be adapted from an ALLOWABLE DIFFERENCE ADDITION RULE.

217. Students are not taught that a VECTOR EQUIVALENCE RULE (adapted from an ALLOWABLE DIFFERENCE EQUIVALENCE RULE) allows SIMPLIFICATION OF NATURAL NUMBER VECTORS.

218. Students are not taught that an EQUIVALENCE RULE allows us to REDUCE AN INFINITY OF SUCH VECTORS A FINITE NUMBER OF EQUIVALENCE CLASSES.

219. Students are not taught that an ISOTONIC VECTOR ADDITION RULE can be adapted from the ALLOWABLE DIFFERENCE ADDITION RULE.

220. Students are not taught that an ANTITONIC NATURALS VECTOR SUBTRACTION RULE can be adapted from the ALLOWABLE DIFFERENCES SUBTRACTION RULE.

221. Students are not taught that, whereas one of these NATURALS DIFFERENCES, (a — b) — (c — d) — & (c — d) — (a — b) may be MEANINGLESS, nevertheless, both NATURALS VECTORS, [a,b] — [c,d] & [c,d] — [a,b] ARE ALWAYS MEANINGFUL.

222. Students are not taught that BOTH WAYS OF WRITING NATURAL VECTOR SUBTRACTIONS ARE MEANINGFUL BECAUSE THE RULE TURNS NATURAL NUMBER SUBTRACTION (ONLY CONDITIONALLY MEANINGFUL) INTO NATURALS VECTOR COMPONENT ADDITION: ALWAYS MEANINGFUL -- BYPASSING THE PROBLEM.

223. Students, even those of ABSTRACT ALGEBRA, are not taught that a NATURALS VECTOR EQUIVALENCE RULE ALLOWS REDUCTION OF AN INFINITY OF VECTOR DIFFERENCES TO JUST THREE EQUIVALENC CLASSES ("POSITVE, NEGATIVE,NULL").

224. Students are not taught that the 223 BYPASS allows VECTOR-VECTOR HIDING such as [2,0] which HIDES AN INFINITY OF VECTORS OF THE EQUIVALENT FORM [3,1], [4,2], [5,3], [6,4], [7,5], ETC.

225. Students are not taught that the VECTOR-VECTOR HIDING described in 224 allows HIDING VECTORS ENTIRELY BY INTEGRAL SIGNS -- WHICH IS NOT CHEATING!

226. Students are not clearly taught that THE INTEGER SYSTEM is A NUMBER SYSTEM in which ADDITION, SUBTRACTION, MULTIPLICATION, and EXPONENTIATION ARE TOTAL (ALLWAYS ALLOWED) BUT DIVISION AND EXPONENTIATION INVERSION ARE NOT.

227. Students are not taught that "The Integral Law of Signs" is not a WEIRD RULE, imposed by fiat, BUT LOGICALLY FOLLOWS FROM TWO NECESSITIES: CLOSURE OF DIFFERENCE OPERATIONS, and EQUIVALENCE OF SHORT AND EXPANDED CALCULATION CHAINS.

228. Students are not taught that, IN TAKING UP INTEGERS, WE DID NOT LEAVE THE NATURALS BEHIND: INTEGERS ARE ORDERED PAIRS OF NATURALS WITH CALCULATION RULES BASED UPON THE NATURALS CALCULATION RULES.

229. Students are not taught that THE TEDIUM OF REPEATED ADDING CAN BE SHORTCUT OR BYPASSED THROUGH RECURSIVE DEFINITION OF MULTIPLICATION IN TERMS OF ADDITION.

230. Students are not taught that TWO MULTIPLICATION INVERSES EXIST BECAUSE MULTIPLICATION IS WELL DEFINED -- but the TWO INVERSES MERGE INTO ONE because MULTIPLICATION IS COMMUTATIVE.

231. Students are not taught that DIVISION AS INVERSE TO MULTIPLICATION CAN BE DEFINED IN TERMS OF MULTIPLICATION.

232. Students are not taught that DIVISION IS NOT ALWAYS DEFINED OR ALLOWED IN THE INTEGER SYSTEM, hence, NEED TOTALING.

233. Students are not taught that the VECTORBYPASS&HIDING-SHORTCUT, which worked for THE SUBTRACION OF NATURALS, now also works for DIVISION OF INTEGRS (EXCEPT BY ZERO).

234. Students are not taught that -- IN ORDER TO BE AN INVERSE TO MULTIPLICATION -- THE OPERATION OF DIVISION IS DEFINED IN TERMS OF MULTIPLICATION.

235. Students are not taught that this DEFINITION, or any equivalent one, RESTRICTS THE DIVISION ALLOWED OPERATIONS, MAKING DIVISION A PARTIAL OPERATION.

236. Students are not taught that the STANDARD TEACHING OF ARITHMETIC MAKES IT OPEN TO THE CHARGE OF CHEATING ON DIVISION -- CAN DO IT ONLY WITH A FUNNY DIVIDING MARK (SLASH or SOLIDUS).

237. Students are not taught that ALL OF DIVISION ARITHETIC CAN BYPASS APPARENT CHEATING OR WEIRDNESS BY GENERATIVE METHODS.

238. Students are not taught that THE DIVISION CLOSURE FAILURE OF NATURALS AND INTEGERS CAN BE GENERATIVELY CORRECTED.

239. Students are not usually taught that that ALL DIVISORS MUST BE NONZERO, NOR WHAT HAPPENS TO A CALCULATOR WITHOUT THIS PRECAUTION.

240. Students are not reminded at appropriate educational stages (e.g., defining operations) of the GENERAL MEANING OF CLOSURE.

241. Students are not taught that the 2-WAY STRATEGY USED WITH INTEGERS CAN DISCOVER THE RULES FOR ALLOWABLE INTEGRAL QUOTIENTS.

242. Students are not taught that THE RESTRICTED PART OF INTEGERS CONSISTING OF ALLOWABLE QUOTIENTS CAN PROVIDE A MODEL FOR TOTALIZING DIVISION.

243. Students are not taught the SIGNIFICANCE of DEFINING DIVISION WITH AND WITHOUT REMAINDER.

244. Students are not taught the MEANING OF ALLOWABLE NATURAL AND INTEGRAL QUOTIENTS.

245. Students are not taught how to DERIVE AN EQUIVALENCE RULE FOR ALLOWABLE NATURAL AND INTEGRAL QUOTIENTS.

246. Students are not taught ADDITION and SUBTRACTION RULES for ALLOWABLE NATURAL and INTEGRAL QUOTIENTS.

247. Students are not taught a MULTIPLICATION RULE for ALLOWABLE NATURAL and INTEGRAL QUOTIENTS.

248. Students are not taught a DIVISION RULE for ALLOWABLE NATURAL and INTEGRAL QUOTENTS.

249. Students are not taught that ORDERED PAIRS (VECTORS) CAN BYPASS THE DIVISION PROBLEM FOR NATURALS AND INTEGERS.

250. Students are not taught that, from an INTEGRAL SYSTEM WITH PARTIAL DIVISION, we can GENERATE A SYSTEM WITH TOTAL (NONZERO) DIVISION.

251. Students are not taught that ALLOWABLE INTEGRAL QUOTIENTS can provide THE MODEL for TOTAL VECTOR QUOTIENTS OF INTEGERS.

252. Students are not taught that ADDITION AND SUBTRACTION RULES FOR ALLOWABLE INTEGRAL QUOTIENTS can provide THE MODEL for ADDITION AND SUBTRACTION RULES FOR TOTAL VECTOR QUOTIENTS OF INTEGERS.

253. Students are not taught that THE MULTIPLICATION RULE FOR ALLOWABLE QUOTIENTS OF INTEGERS CAN PROVIDE THE MODEL FOR THE MULTIPLCATION OF VECTORS OF INTEGERS.

253. Students, even those of ABSTRACT ALGBRA, are not taught that an EQUIVALENCE RULE FOR VECTORS OF INTEGERS ALLOWS REDUCTION OF AN INFINITY OF VECTORS TO JUST THREE EQUIVALENCE CLASSES (EFFECTIVELY TWO).

254. Students are not taught that VECTORS OF INTEGERS FALL INTO THE THREE CLASSES OF INTEGRAL, EGYPTIAN, AND FRACTIONAL.

255. Students are not taught that it is EQUIVALENCE REDUCTION TO THREE CLASSES THATALLOWS HIDING VECTORS BY SLASH-SOLIDUS SIGNS, WHICH IS NOT CHEATING.

256. Students are not taught that THE INTEGRAL DIVISION PROBLEM IS BYPASSED WITH QUOVECTORS BY PASSING FROM DIVISION TO MULLTIPLICATION, WHICH IS ALWAYS ALLOWABLE -- NO CHEATING!.

257. Students are not taught that -- IN THE RATIONAL NUMBER SYSTEM -- ADDITION, SUBTRACTION, MULTIPLICATION, (NONZERO) DIVISION, EXPONENTIATIOON ARE TOTAL OPERATIONS, BUT THAT ANY INVERSION OF EXPONENTIATION CAN ONLY BE PARTIAL.

258. Students are not taught that THE RULE FOR DIVIDING A FRACTION BY A FRACTION (INVERT DENOMINATOR FRACTION AS MULTIPLIER) IS NOT WEIRD, BUT IS REQUIRED IN THE INTEGRAL SYSTEM FOR CLOSURE OF ALLOWABLE QUOTIENTS.

259. Students are not taught that WE HAVEN'T LEFT THE NATURALS IN ATTAINING TOTAL (NONZERO) DIVISION, since RATIONAL NUMBERS ARE SIMPLY ORDERED PAIRS OF ORDERED PAIRS OF NATURAL NUMBERS!

260. Students are not taught that THE TEDIUM OF REPEATED MULTIPLICATION LEADS TO RECURSIVE DEFININITION OF EXPONENTIATION IN TERMS OF MULTIPLICATION.

261. Students are not taught that EXPONENTIATION IS WELL DEFINED BUT NO COMMUTATIVE, HENCE (UNLIKE ADDITION AND MULTIPLICATION) HAS TWO DISTINCT INVERSES.

262. Students are not taught that EXPONENTIATION "SPLITS" WITH TWO DISTINCT INVERSES: LOGARITHM AND ROOT EXTRACTION.

263. Students are not taught that LOGARITHM AND ROOT EXTRACTION ARE ONLY PARTIAL OPERATIONS, AND REQUIRE A "NEW NUMBER SYSTEM" TO BECOME TOTAL.

264. Students are not taught that the NATURALS, INTEGERS, RATIONALS ARE CLOSED UNDER EXPONENTIATION.

265. Students are not taught that EXPONENTIATION IS DOUBLY WELL-DEFINED.

266. Students are not taught that EXPONENTIATION IS NOT COMMUTATIVE, HENCE (UNLIKE ADDITION, MULTIPLICATION) HAS TWO DISTINCT INVERSES.

267. Students are not taught that THE TWO INVERSES OF EXPONENTIATION ARE LOGARITHM AND ROOT EXTRACTION.

268.Students are not taught that the TWO INVERSES OF EXPONENTIATION ARE BOTH PARTIAL, REQUIRING "NEW NUMBER SYSTEMS" TO BECOME TOTAL.

269. Students are not taught that TOTALIZING INVERSION OF EXPONENTIATION LEADS TO TWO "NEW NUMBER SYSTEM" -- REAL NUMBERS to TOTALIZE LOGARITHM, COMPLEX NUMBERS to TOTALIZE ROOT EXTRACTION.

270. Students are not regulalry taught that RATIONAL NUMBER (FRACTION) CAN BE REPRESENTED BY A PERIODIC DECIMAL.

271. Students are not taught that EVERY PERIODIC DECIMAL CAN BE REPRESENTED AS A GEOMETRIC SUM OF INFINITE LENGTH.

272. Students are not taught that EVERY GEOMETRIC SUM OF INFINITE LENGTH EQUALS, IN THE LIMIT, A RATIONAL NUMBEr.

273. Students are not taught that TOTALIZING LOGARITHM (AS INVERSE OF EXPONENTIATION) CANNOT BE ACCOMPLISHED FINITARILY, BUT ONLY BY ADJOINING TO ARITHMETIC'S FINITE OPERATION THE TRANSFINITE OPERATION OF LIMIT.

274. Students are not taught that REPEATING DECIMALS ARE CAUCHY SEQUENCES WHICH CAN BE GENERALIZED BY THE LIMIT OPERATION TO COMPREHEND NONREAPEATING DECIMALS.

275. Students are not taught that the device of METRIC can be used to obtain CAUCHY SEQUENCES TO TOTALIZE LOGARITHM.

276. Students are not taught that CAUCHY SEQUENCES ARE IMPLICIT IN EVERY NONENDING DECIMAL ENCOUNTERED.

277. Students are not taught that an IRRATIONAL NUMBERS, SUCH AS THE SQUARE ROOT OF TWO, IS PINCHED BETWEEN UPPER AND LOWER BOUNDS.

278. Students are not taught that THE PINCHING OF AN IRRATIONAL BETWEEN UPPER AND LOWER BOUNDS IS ANTITONIC (COMING DOWN FROM UPPER, GOING UP FROM LOWER).

279. Students are not taught that KNOWLEDGE GAINED FROM DECIMALS CAN BE USED TO CREATE INFINITE VECTORS WHICH TOTALIZE LOGARITHM BY A "NEW NUMBER SYSEM", THE REALS.

280. Students are not taught that THE STANDARD DEFINITION OF LIMIT IS DUE TO AUGUSTIN CAUCHY.

281. Except in a Calculus Class, students are not taught CAUCHY'S DEFINITION OF LIMIT.

282. With rare exception, students are not taught Zeno's Paradox of Achilles and The Tortoise.

283. Students are not taught that the CAUCHY LIMIT DEFINITION CAN BE REFORMULATED BY MEANS OF THE METRIC CONCEPT.

284. Students are not taught that STUDY OF PERIODIC AND NONPERIODIC DECIMAL NUMBERS ARE IMPORTANT TO UNDERSTANDING REAL NUMBERS AND LOGARITHM.

285. Students are not taught the meaning of "tone" intervals of a musical scale or intervals in a Process.

286. Students are not taught the meaning of "isotonic" and "antitonic".

287. Students are not taught that climbing a stairs or finding a sock in a drawer or file in a cabinet is an ANTITONIC PROCESS.

288. Students are not taught that the LIMIT PROCESS IS CANTATONIC -- a CONTINUOUS ANTITONIC PROCESS.

289. Students are not taught that THE DECIMAL NUMBER FORM is a SHORTCUT to the ARITHMETIC OF RATIONAL NUMBERS.

290. Students are not taught the DECIMAL NUMBER FORM as (1)DECIMAL POINT; (2)CHARACTERISTIC BEFORE DECIMAL POINT; (3)MANTISSA AFTER DECIMAL POINT.

291. Students are not taught the THEOREM WITH PROOF THAT EVERY RATIONAL NUMBER HAS A PERIODIC DECIMAL REPRESENTATION, AND VICE VERSA.

292. Students are not taught how to CONVERT ANY PERIODIC DECIMAL INTO FRACTIONAL FORM.

293. Students are not regularly taught that THE DECIMAL REPRESENTATION OF EACH RATIONAL NUMBER CAN BE SUMMED AS A GEOMETRIC PROGRESSION.

294. Students are not taught that 293 explains the failure of LOGARITHMIC TOTALIZING IN THE RATIONALS, YET ALSO HOW TO CORRECT THIS.

295. Students are not taught that LOGARITHMS ARE INFINITE VECTORS, but fall into THREE TYPES, ALLOWING US TO HIDE THE VECTOR-FORM BY THE DECIMAL-FORM.

296. Students are not taught that WILLIAM ROWAN HAMILTON INVENTED THE NAME "VECTOR", HENCE, STRICTLY SPEAKING, CREATED THE FIRST VECTOR, WRITING THE COMPLEX NUMBER AS A VECTOR OF REALS, WITH SPECIAL RULES.

297. Students are not taught that HAMILTON SAW A DIFFERENCE BETWEEN THE SUM  2 + 2 = 4  AND THE "SUM"  x + iy , for i = √ ¯1.

298. Students are not regularly taught the EQUIVALENCE RULE FOR COMPLEX NUMBERS, emphasizing its VECTOR-FORM (REAL PART INDEPENDENT OF IMAGINARY PART).

299. Students are not taught that the WEIRD MULTIPLICATION RULE i² = ¯1 DERIVES FROM THE PRODUCT RULE FOR INTEGERS.

300. Students are not taught about Ore's discussion of THE MODUL, specially that THE IMAGINARIES CONSTITUTE A MODUL.

301-400 OF FAQS UNTAUGHT.