Irrationals - Once Again

This account was temporarily lost. The password was forgotten, and the account could not be recovered via the traditional means. This was saddening; because, the readership had reach nearly 25K and it was gaining momentum. And, with the commotion that teaching college courses be, a concerted effort at recovering the account could not be achieved. By chance, Google sent a e-mail message which asked if the account could be validated and it used a smartphone number for recovery. So, the account was available for usage again.

Hopefully, that readership will resume. Although, it has been nearly a decade since the account was lost.

This forum was used for sharing some ideas which seemed valuable; however, they were not yet commonplace knowledge or were controversial. All of the work is "original" emanating from this sole author.

One topic and lesson that shall be taught again is that concerning "irrational" numbers. Which after this short mathematical development will seem quite possibly like a "irrational" concept.

During fourth grade, our class learnt the art of long division along with other much more advanced concepts often met in secondary school and college. By chance, one of the more advanced topics taught by our instructor, Mrs. Kaiser, was the development of the number hierarchy with its various sets: natural numbers, integers, rational numbers, irrational numbers, reals, and etcetera. It is believed that she also mentioned those who be complex, transcendental, and such; however, she said that we would learn more about those later. She was just introducing a few of the fundamentals.

Seeing that, we had just learnt about long division. We had discovered the power of the "vinculum". It is that bar who rest above the repeating part of a decimal number when one is determining the quotient of the division problem with its decimal expansion.

It dawned upon the author that the usage of the vinculum might yield a means for writing irrational numbers as ratios of integers. Which would make them rational and not irrational.

How might this be done? One asks!

The observation was made when the instructor described "pi" as a number with an infinite and never-ending decimal expansion. And, a comment was shared by the author. At which, the instructor exclaimed, "Do you want a Field Medal?"

That comment was this. [If, one places that bar used for repeating decimals above the zero in a ten. He would produce a "big honking ten" (it was fourth-grade), and he could multiply pi by a special form of one, the "big honking ten" divided by the "big honking ten" and produce a ratio of integers who each have an infinite length.]

So, seeing that, this editor will not produce a vinculum, we will describe our ten with a infinite number of zeros after it as 1(0)' . The ' signifies that the number or numbers in the parentheses repeats indefinitely. 

And,

[pi * 1(0)'] / 1(0)' = [pi * 1000...] / [1000...]

Which represents the ratio of an integer with a infinite number of digits divided by the same. So, although it might never be rendered by hand or mechanically, it is a ratio of integers. Unless, the number system does not allow for integers of infinite length. Which is does. 

At the age of nine, the author knew very little about Field Medals. And, life never permitted pursuing mathematics as a career field. Such that, a thesis, a dissertation, a research publication could be prepared that merited a Field Medal. If, such an idea is potentially sufficient for one.

So, this notion is shared once more; because, it seems that the concept of the irrational number is simply that, irrational. From, a simple-minded perspective that uses discrete mathematics concepts. An unending number of integers exist. When, these are composed with themselves once. The count of the total possible compositions is the square of infinity. Which is also infinity. So, the number of ratios must cover and map with all possible numbers. This is by the pigeon-hole principle.

Albeit, one might "prove" otherwise using a "diagonalization" argument which fixes the locus of infinity. Which by simple reasoning is unreachable and not fixable.

This is why Poincare said that Cantor [should cease with his insanity and arguing that infinity was reachable via an argument of diagonalization].

Arguments of diagonalization versus a very large fixed, finite value are great for comparing and contrasting the growth rate of mathematical objects like functions and sets; however, they place a hard-fixed and artificial cap on mathematical reasoning about the infinite. When, one claims that it is a fixed point with nothing beyond it. Such permits the "irrational" among mathematical reasoning.