what a quagmire...
the question: "what is 1 divided by 3?", and the question: "what is the size of 1/3?" are two different questions.
1/3 (as defined to be a certain equivalence class in the field of fractions of the integral domain of the grothendieck groupification of the free monoid formed by creating a minimal inductive set from the one set postulated existentially by the zermelo fraenkel axioms...isn't THAT a mouthful? and did i miss anything there?) is a perfectly good answer to the first.
the answer to the second is a bit more complicated. 1/3 is a rational expression (literally, a ratio of 1 to 3), and the greeks measured such expressions through comparison. so, it's easy to say, given a/b and c/d, which is "bigger", but relating these quantities to some common scale depends on finding lcm(b,d), and then using that as a unit of measurement. and perhaps here you can see some practical dificulty, because thre is no "universal" scale that will work for every rational number (or for every finite set of measurements).
the decimal system is a compromise of sorts, in that it allows us to establish a "scalable" scale of measurement, good to whatever precision we're satisfied with. it is rather troubling that we cannot represent what we consider an "exact" quantity (such as 1/3) exactly in this system (designed mainly for ease of computation).
but changing our arbitrarily chosen base of 10 will not help matters, because NO natural number is divisible by every smaller natural number, except two. the natural expression of fractions in base 2 is depicted very succintly in the subdivisons of an inch-ruler, and carpenters (for example) have been know to decide on a "scale of resolution" and call out their measurements (say 1/8 of an inch) as 4-8-4 (meaning 4 feet 8 inches and 1/2 inch). in the truss industry i used to work in, feet-inches-sixteenths was the standard (a peculiar system to do arithmetic in, i assure you).
but while base 2 may represent some kind of "ideal" system, for fractions with odd denominators, it does spectacularly poorly. for small numbers, one might be satisfied with something like the base 60 the babylonians used (and they were pretty handy with fractions), but the number of prime integers is infinite, so no "greatest common denominator" for all integers can be found, to use as a common base.
and these are just inherent difficulties with doing arithmetic operations with rational expressions, the situation gets very out of hand quite quickly in dealing with solutions of even fairly simple polynomial equations (such as x^2 - 2 = 0).
even considering the sides of triangles based on "even" divisions of a 360 degree circle, lead us in short order to consideration of various irrational quantities, and when we extend the ratio of the sides of such triangles to a continuous function, we encounter numbers that aren't even solutions of polynomial equations. and yet a circle is such a clean, "whole" thing, so it is very counter-intuitive that it should imply we need numbers that are "unmeasurable".
to go even further, i don't think we have ever really agreed amongst ourselves, as to what should properly qualify as a "number". are matrices numbers? (if you think not, then what about this one:
[a -b]
[b a] ?)
are polynomials numbers (don't be too hasty answering this one, either)? what about power series? how about functions themselves, surely they have "sizes" we can measure at least some of them do)? where do we draw the line, and say: "these are proper numbers, these are not?" my point is, this is actually a hard question to get a handle on, especially when we take for granted a number system the vast majority of whose members are totally unknowable to us (so how can we say we know what real numbers are? I've only personally made the acquaintance of a handful).
in the end, the more sophisticated among us exclaim: the things we shall consider as numbers, are those things which have the properties we want! and such things are unique up to isomorphism! which kind of works, conceptually, but is somewhat at odds with how we actually use numbers to calculate stuff. so we wind up with a situation where we say: "in a perfect world, i would have this, but i am finite, so here is my best approximation". there is something deeply unsatisfactory about that, but smarter people than i have attempted various reconciliations between our abstractions and our abilities, and failed.
@OP: it doesn't sound like you have something truly novel or useful, but that's no reason not to give us the details, so at the very least we can mock you :)
