"ROAST PIG MATH" (THE HARD WAY: 301-400 OF FAQS UNTAUGHT, PROVE OTHERWISE!)
"ROAST PIG MATH" (THE HARD WAY: 301-400 OF FAQS UNTAUGHT, PROVE OTHERWISE!)301. Students are not taught THAT THIS VECTOR OF REALS FORMULATION OF COMPLEX NUMBER CLEARLY SHOWS SUCH NUMBERS TO CONSTITUTE A BIMODUL -- REALS A SEPARATE MODUL FROM THE IMAGINARIES MODUL.302. Students are not taught THE VECTOR-HIDING OF THE INTEGERS, RATIONALS, AND REALS NO LONGER WORKS FOR COMPLEX NUMBERS AND SETS OFF A INFINITE HIERARCHY OF "CLIFFORD" VECTORS, WITH MANY CONSEQUENCES.
303. Students cannot imagine that such as vast and rich MATHEMATICAL DOMAIN as THE ARITHMETIC OF CLIFFORD NUMBERS (a.k.a. CLIFFORD ALGEBRA, a.k.a. MULTIVECTORS) WOULD BE ignored by MAINSTREAM MATHEMATICS.
304. Students are not taught that a vast and rich MATHEMATICAL DOMAIN such as THE ARITHMETIC OF CLIFFORD NUMBERS can be introduced on a LEVEL OF HIGH SCHOOL ALGEBRA.
305. Except in special college classes, students are not taught about the HYPERCOMPLEX EXTENSION OF COMPLEX NUMBERS.
306. Even in these classes, students are not taught that any n-hypercomplex system can be used to GENERATE the next (n+1)-hypercomplex system.
307. Students are not taught that THE ARITHMETIC OF CLIFFORD NUMBERS EMBRACES MORE THAN 25 REGULAR FIELDS OF MATHEMATICS.
308. Students are not taught that A SCALAR IS A STRUCTURE INVARIANT FOR ZERO-ANGLE OR ROTATION OPERATOR AS THE IDENTITY, whereas A VECTOR IS A STRUCTURE INVARIANT FOR ROTATION OPERATOR EQUAL TO POSITIVE ONE.
309. Students are not taught that one can DERIVE THE INNER PRODUCT FROM THE LAW OF COSINES.
310. Students are not taught the CAUCHY DERIVATION OF ALL TRIGONOMETRY FROM ANALYTIC GEOMETRY VIA THE UNIT CIRCLE CENTERED AT ORIGIN OF AXES.
311. Students are not taught about IDEMPOTENTS.
312. Students are not taught about THE ROTATION OPERATOR AS A SPECIAL CASE OF AN ORTHOGONAL TRANSFORMATION.
313. Students are not taught about TRANSFORMATION INVARIANTS, such as THE INNER PRODUCT.
314. Students are not taught about THE DEFINITIVE CHARACTERISTIC OF EUCLIDEAN SPACE.
315. Students are not taught that a PSEUDO-EUCLIDEAN METRIC IS NEEDED FOR SPACE-TIME, DEMANDING SOMETHING BEYOND INNER PRODUCT.
316. Students are not taught about ANTISYMMETRIC FORMS even though the rise in the (untaught) traditional "casting out elevens" algorithm and it homology in factoring polynomials.
317. Except in special classes, students are not taught about COMMUTATORS of MATRICES and OPERATORS and their BRACKET FORM.
318. Students are not taught that the GIBBS-HEAVYSIDE (GH) VECTOR ALGEBRA does not achieve CLOSURE ON VECTORS, but MUST BIFURCATE VECTORS BY DISITNGUISHING "POLAR" AND "AXIAL".
319. Students are not taught that the GIBBS-HEAVISIDE (GH) VECTOR ALGEBRA INTRODUCES MORE PATCHWORK IN SPEAKING OF "FREE" AND "FIXED" VECTORS SINCE ROTATION REQUIRES SPINOR AND ROTORS.
320. Students are not taught that the VECTOR PATCHWORK of THE GIBBS-HEAVISIDE (GH) VECTOR ALGEBRA is BYPASSED BY WORKING WITH BIVECTORS.
321. Students are not taught that THE FAMILIAR ELIMINATION ALGORITHM OF HIGH SCHOOL ALGEBRA MOTIVATES THE DETERMINANT AND MATRIX STRUCTURES.
322. Students are not taught that ARTHUR CAYLEY USED DETERMINANT MULTIPLICATION TO MOTIVATE MATRIX MULTIPLICATION.
323. Students are not taught that THE NONASSOCIATIVITY OF VECTOR CROSS PRODUCT LIMITS ITS EXTENSION AND PROHIBITS GROUP DEVELOPMENT.
324. Students are not taught that INVARIANCE OF CROSS PRODUCT UNDER INVERSION (UNLIKE ITS COMPONENTS) REQUIRES A NEW TYPE OF VECTOR, THE "AXIAL" VECTOR, CONSTRASTED WITH THE STANDARD "POLAR" VECTOR.
325. Students are not taught that GH must especially distinguish the "pole" of ROTATION by speaking of a "FIXED" VECTOR, CONTRASTED WITH THE STANDARD ("FREE") VECTORS.
326. Students are not taught that CROSS PRODUCT IS EFFECTIVELY 3-D, SINCE THE RESULTING VECTOR IS OUTSIDE THE PLANE OF THE TWO VECTORS COMPOSING IT.
327. Students are not taught that NON-NULL VECTORS HAVE NO INVERSES IN GH, forcing a clumsy RECEIPROCAL SYSTEM.
328. Students are not taught that THE ROTATION OPERATOR HAS IMAGINARY EIGENVALUES.
329. Students are not taught that FAILURE OF ASSOCIATIVITY AND INVERSITY UNDER PRODUCT FORCES VECTOR SPACES TO BE DEFINED OVER A RING INSTEAD OF A FIELD.
330. Students are not taught that GH has NOWHERE TO GO since CANNOT BE EMBEDDED IN A SUPERALGEBRA, as it can be if THESE RESULTS ARE OBTAINED IN MULTIVECTOR THEORY.
331. Students are not taught that we can DERIVE THE LAW OF SINES FROM VECTOR OUTERPRODUCT.
332. Students are not taught the distinction of Russian mathematicians between the 0-, 1-, 2-, 3-triangles.
333. Students are not taught that, passing from UNDIRECTED GEOMETRY TO DIRECTED GEOMETRY, THE LAW COSINES LEADS TO VECTOR INNERPRODUCT, THE LAW OF SINES LEADS TO VECTOR OUTERPRODUCT.
334. Students are not taught that THREE INDEPENDENT BASIS VECTORS AND OUTERPRODUCT CREATE 1-, 2-, 3-SPACE.
335. Students are not taught that AREA IS ORIENTED, AS SHOWN BY OUTERPRODUCT.
336. Students are not taught that MULTIVECTORS UNDER OUTERPRODUCT ARE EQUIVALENT TO ANTISYMMETRIC TENSORS.
337. Students are not taught about SYMMETRIC and ALTERNATING FORMS IN NUMALGEBRA.
338. Students are not taught that "casting out nines" and "casting out elevens" in Arithmetic have homologues in NUMALGEBRA.
339. Students are not taught that ALL OF COMPLEX ARITHMETIC & ALGEBRA & THEORY OF A COMPLEX VARIABLE IN ANALYSIS CAN BE DONE IN MULTIVECTORS WITHOUT ANY REFERENCE TO COMPLEX NUMBERS & IMAGINARIES.
340. Students are not taught about INVOLUTIONS.
341. Students are not taught that PROJECTIVE GEOMETRY (as well as METRICAL GEOMETRY) CAN BE EMBEDDED IN MULTIVECTOR THEORY.
342. Students are not taught that DIFFERENTIAL GEOMETRY CAN BE EMBEDDED IN MULTIVECTOR THEORY.
343. Students are not taught that ONLY IN MULTIVECTOR THEORY CAN WE DISPENSE WITH COORDINATES, DEALING EXCLUSIVELY WITH INVARIANTS.
344. Students are not taught that SPACETIME PHYSICAL THEORY CAN BE READILY EMBEDDED IN MULTIVECTOR THEORY.
345. Students are not taught that EVERY BIVECTOR DETERMINES A UNIQUE ROTATION.
346. Students are not taught about the SIGNIFICANT DIFFERENCE BETWEEN HOMOGENEOUS AND HETEROGENEOUS POLYNOMIALS IN NUMALGEBRA.
347. Students are not taught that what SEEMS "DIMENSION-NONSENSE" IN THE MULTIPRODUCT FORM (SCALAR PLUS BIVECTOR) IS BELIED BY ITS DERIVATION IN TERMS OF HOMOGENEOUS & HETEROGENEOUS POLYNOMIALS.
348. Students are not taught that GRASSMANN & HAMILTON DISCOVERED THE MULTIPRODUCT, BEFORE CLIFFORD, BUT DID NOT USE IT.
349. Students are not taught that MULTIPRODUCT IS THE GREAT HOMOGENIZER.
350. Students are not taught that VARIOUS METRICS CAN BE EASILY EMBEDDED IN MULTIVECTOR THEORY.
351. Students are not taught that THE PAULI ALGEBRA OF QUANTUM THEORY AND THE DIRAC ALGEBRA OF QUANTUM FIELD THEORY ARE EASILY EMBEDDED IN MULTIVECTOR THEORY.
352. Students are not taught that GREIDER SHOWED IN 1984 THAT ONLY MULTIVECTOR THEORY ALLOWS ALL RELATIVISTIC QUANTUM FIELDS OF SPINS 0, 1/2, CAN BE ENCOMPASSED IN THE SINGLE DIRAC FORMALISM.
353. Students are not taught that THE DIFFERENTIAL FORM OF MULTIPRODUCT UNITIES GRAD, DIV, CURL OPERATORS INTO A SINGLE OPERATOR.
354. Students are not taught MULTIPRODUCT MAKES POSSIBLE THE DIRECTED INTEGRAL.
355. Students are not taught that HESTENES SHOWS THE INVERSE OF GRADIENT INVOKES THE CAUCHY INTEGRAL FORMULA ALLOWING A DIRECTED THEORY OF THE INTEGRAL EMBRACING BOTH RIEMANNIAN & LEBESQUE FORMS.
356. Students are not taught that HESTENES' ARGUMENT THAT HOMOLOGY THEORY IS MORE READILY EXPRESSED IN TERMS OF DIRECTED INTGRALS THAN IN THE SCALAR-VALUED INTEGRALS OF COHOMOLOGY.
357. Students are not taught that MULTIVECTOR THEORY ALLOWS REAL AND COMPLEX ANALYSIS TO BECOME ONE THEORY, EXTENDING COMPLEX ANALYSIS BEYOND WHAT IS POSSIBLE IN THE STANDARD FORM.
358. Students are not taught that MULTIPRODUCT ALLOWS MAXWELL'S ELECTROMAGNETIC EQUATIONS TO BECOME ONE EQUATION (FOR ONE PROCESS).
359. Students are not taught that MULTIPRODUCT IS A SUPERFACTOR OPERATOR, ALLOWING FACTORING IMPOSSIBLE OTHERWISE.
360. Students are not taught that the 4 pages in the MATRIX DERIVATION OF DIRAC'S FACTORING OF THE KLEIN-GORDON EQUATION CAN BE REDUCED TO 4 LINES BY THE MULTIPRODUCT.
361. Students are not taught that we can CONSTRUCT THE "IMAGINARY" NUMBER as a BIVECTOR.
362. Students are not taught that the "imaginary" CONSTRUCT AN "IMAGINARY"-PLANE, implicit in the work of Wessel, Argand and Gauss.
363. Students are not taught that any MULTIVECTOR IS ALSO AN OPERATOR WITH OTHER MULTIVECTORS AS OPERANDS.
364. Students are not taught that the "imaginary" is a ROTATING OPERATOR.
365. Students are not taught the DERIVATION OF OPERATOR FROM RELATION AND FUNCTION, and that AN OPERATOR HAS DOUBLE CLOSURE.
366. Students are not taught that Norbert Weiner and Max Born used OPERATORS to move QUANTUM THEORY from PERIODIC to NONPERIODIC PROCESSES.
367. Students are not taught that "the imaginary" is PRESENT IN MAXWELL'S ELECTROMAGNETIC FIELD EQUATIONS in the guise of ROTATION.
368. Students are not taught that EULER'S EXPONENTIAL-TRIG EQUATION CAN BE DERIVED AS A MULTIVECTOR.
369. Students are not taught that HYPERBOIC FUNCTIONS ARE EMBEDDED IN MULTIVECTOR THEORY (ALONGG WITH BESSEL FUNCTIONS, and similar FUNCTIONS).
370. Students are not taught that COMPLEX EQATIONS WHICH ARE IMPLICIT IN MULTIVECTOR THEORY CAN BE DERIVED EXPLICITLY.
371. Students are not taught that MULTIPRODUCT OF TWO BASIS VECTORS CREATES A 2-D OPERATOR, AND ONE OF THREE CREATES A 3-D OPERATOR (WHOSE NEGATIVE IS THE FAMOUS HODGE STAR OPERATOR).
372. Students are not taught the story (legend?) as to how HAMILTON CARVED THE QUATERNION EQUATIONS INTO A BRIDGE OUTSIDE DUBLIN.
373. Students are not taught that COMPLEXES, QUATERNIONS, OCTONIONS, and their SUCCESSIVE "DOUBLES" (EVEN NUMBER BASIS VECTORS) are n-MODULS.
374. Students are not taught that it's easy to TRANSLATE MULTIVECTORS INTO QUATERNION LANGUAGE.
375. Students are not taught of William Clifford's discovery that the PRIMARY EXTENSION OF BASIS UNITS IS BINARY: 1, 2, 4, 8, ....
376. Students are not taught that ALL OF LINEAR ALGEBRA CAN BE DERIVED MERELY BY ADDITION AND MULTIPRODUCT, in fact, more.
377. Students are not taught that MULTIVECTOR INVOKES THE "CONJUGACY ALGORITHM" (Melzak's STRATEGY OF BYPASS).
378. Students are not taught that MULTIPRODUCT IS IMPLICIT IN EULER'S THEOREM, THE SIMPLEST FORM OF A SPINOR.
379. Students are not taught ANY MULTIVECTOR CAN BE USED AS A LEFT- OR RIGHT-OPERATOR UPON ANY MULTIVECTOR AS OPERAND.
380. Students are not taught that MULTIPRODUCT CAN BUILD REFLECTION OPERATORS, AS "SQUARE-ROOTS" OF ROTATION OPERATORS.
381. Students are not taught that REFLECTION OPERATORS ACTING CONJUGALLY UPON A VECTOR TRANSFORM IT INTO A SPINOR.
382. Students are not taught HAMILTON'S THEOREM: A ROTATION IS THE PRODUCT OF TWO REFLECTIONS.
383. Students are not taught CARTAN'S THEOREM: EVERY ISOMETRY CAN BE CONSTRUCTED FROM REFLECTIONS. Hence, MULTIVECTOR THEORY IS AN ARITHMETIC OF REFLECTIONS AND ROTATIONS.
384. Students are not taught that EVERY ROTATION OPERATOR CAN BE FACTORED INTO A MULTIPRODUCT OF SPINOR-VECTOR-SPINOR, leading to some interesting GROUPS (INCLUDING THE SPIN GROUP).
385. Students are not taught that EVERY MATRIX IS THE INNER PRODUCT OF VECTORS, AND ALL DETERMINANT PROPERTIES DERIVE FROM INNER AND OUTER PRODUCT.
386. Students are not taught that REPLACEMENT OF SINE, COSINE IN THE ROTATION OPERATOR YIELDS THE HYPERBOLIC "ROTATION OPERATOR" FOR SPECIAL RELATIVITY THEORY.
387. Students are not taught that MULTIVECTORS ALLOW COMPLEX VARIABLE THEORY TO BE EXTENDED FAR BEYOND ITS USUAL PRESENTATION.