Computus

a primer to the basic algebraic notion of computation
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[under construction]

"There exists no finitesmal value, however small finitely, separating SQRT 2, from all the numbers in the list"
"Therefore, there exist numbers in the list, in-finitesmally close to SQRT 2 (the Rationals near SQRT 2) however indeterminably"

Foreword

The concepts introduced in school mathematics are adaptable to college computus. Calculators and computers, though but simpleton machines, need not lack significant figures, nor clear to zero; function limits and sensitivities, units checking and conversions, coordinate transformations and correspondences, need not be left as repetitive exercise skills of the programmer; and infinitesmals and complexors, need not remain mere chalk-talk theories and canonical forms: Real numbers can be realized, without petty rationalizations; What is foremost resolves to identificable constructions controlling diversation and perpetuation; And our early waking concept develops and matures into a working concept ... Representation Theory, approximately what we'd been calling, Algorithmic or Automata Theory, makes a decent introduction to the subject of the computus.

Preface

In this course on the computus we'll examine in very basic depict, the means and methods of computation by numeric representation of measure, count, decision, and likelihood, and type-coding Because the technology of today supports rapid computation by electronic micro-circuit representation, we must also review some of our early arithmetic, and update it for use and iteration by machine. And because we are yet the refined mathematicians, we shall continue our preference for finding the fun solutions, answers, choices, and discussions. *

* [The author uses an Hewlett Packard HP21S STAT/MATH electronic programmable calculator for tabular excerptations]

Included as appendices are, prerequisite discussion: a consumus of arithmetic methods for the student-computor, a matriculus (though I am ever inclined to call this, gut arithmetic, for its iteratively simple linear systematic transformation processes and products), and a calculus done in the here-preferred style of infinitesmals - all useful predicates to the purposes, designs, and application-opportunities of the general purpose digital computer, as it was once known industrially.

And we begin by meeting a challenge intimated and forworded preceding, inasmuch as to extricate ourselves from a certain blythe tendency among early pupils of the mathematica to accede to proof-by-intimidation.

Introduction

You may recall from high school or earlier, the proof that there are more Reals (Irrationals plus Rationals) than Rationals (alone): It proceeded on thiswise: suppose one has a (countable) list of Real fractions on the unit interval, represented decimally, then construct a number comprised of digits each not equal to the corresponding digit in the numbers in the list - being sure to touch at least one digit of every number: this constructed number cannot equal any of the listed numbers, because it differs finitesmally in at least one digit, and therefore one's list must be shy of all Real fractions, even though it [the list] may include all Rationals, all i/j : i>j of N; plus zero, which can be enumerated in a spiral-like list) - whence there is at least one uncountable Real (Irrational if the list includes all rationals). However, if the list looks like 1.000..., 0.?72..., 0.5?3..., 0.46?2... etc. with "?" a traveling non-nine, the constructed number may be 0.999... - [this can be forced in binary, radix 2] - and, if you accept the supposed equality of 0.999... and 1.000..., already in the list somewhere: the method thus fails to construct a Real (Irrational) outside the list. * Obviously now, the equality of 0.999... nad 1.000... is untenable in some frame of mathematical discussion, being here decimal inequality. In fact, we may conside that there are or may be infinitely many arguments against the suspect equality; and these infinitely many arguments may converge (while none is the final argument) toward a proof: a meta-proof, meaning that, if 0.999... equals 1.000..., then the arguments taken as a whole altogether equal a disproof: one which contradicts this claimed equality. Since we learned mathematics by argument, the sequence of arguments is stronger than the numeric claim of one argument (being a subject of the school of argument). This seems extraneous but alarming.

* or conversely, if 1.000... were outside, then 0.999... is already excluded

Of course, in finite-smal measure, 0.999...=1.000..., because there is no measurably finite difference; But equality in general, no: Since adjacent numbers with equally many digits divide the interval equally [0.0,0.1,...0.9,1.0] divides the interval [0,1] equally, in equal intervals of 0.1, (then) when all digits are specific (hypothetically possible, but not within our finite canon), then, {0.000...,...,0.999...,1.000...} must be the endpoints of [0.1] plus all the equally spaced dividers of the same interval: And then, if 0.999... equals 1.000... so that its subinterval 0.000... is identically 0, zero, all would equal 1.000..., as all the subintervals were so identically zero, and (intuitively) the integral summation of identically zero, is zero. Here we see the final break in the generalization of equality: We tacitly assume that the sum (the infinite sum) of all the subintervals must equal the whole interval [0,1], but infinity times zero, is zero: only infinity times infinite-smal, may accumulate to some finite (or other arbitrary value), and thereby we have distinguished infinitesmals as real quantities - as irrational as that first seemed. In short, we must be especially aware of our meaning of equality: For this reason we include calculus by infinitesmals. *

* [space-time physicists aren't alone who integrate zero to nonzero]

Computus (on the other hand) is by definition a finite covering; And we begin with our notion of equality. The argument preceding is not lost in the computus, but gives it a special flavor: As we shall see immediately, adjacent numbers in precise measure are vestigially indistiguishable: a new tenor of equality. But, in the finite cleavage of computus it takes only a step across (an) adjacent number(s) to reach distinguishability. Then also we must develop the notions of inequality and inclusion: numbers with their own precision lie constrainted upon specifiable activity domains (or even within bounds). Taking one last overview so as to not escape our imminent entry in the computus, we foresee our first concern for significant digits, shall also be our last concern: Our new concept of number must carry with it a sense of precision adaptable to grand accumulations and iterations; We will revise all estimates and notions until we have a stable base concept; and our finished product must be a simple as when we began in the earliest grades of school to learn the arithmetic of decimal places and natural numbers, then operations advanced beyond counting.

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"The tack of the true mathematician is to go quickly to the best right answer integrating left and right."

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The theory of measurement propounded in this work is not to be cited (as) considering contraband or corpses; Nor are the intellectual appurtenances herein to be used for or in the commission of crimes against persons, peoples, properties, or powers (states). May your tabernacle measure true.