Not "flaw", rather, "could be defined differently". Introducing new definitions for division by 0, including new numbers, follows a traditional path trod by others. For general background, see Patrick Suppes' Introduction to Logic, Chapter 8, Sections 5 and 7, titled respectively The Problem of Division by Zero and Five Approaches to Division by Zero.[a] Specific examples defining division by 0 include Meadows[b], Wheels (Carlström, 2004), Wheels (Setzer, 1997), as well as the extended complex plane and its relatives that use one or more point(s) at infinity (e.g., the (affinely) extended real numbers), or Möbius transformations (i.e., the exact real arithmetic of Edalat and Potts).[c] Meadows, following Suppes' approach 3, is alone in not using a new definition.
Introducing a new type of number as you suggest follows Suppes' fourth approach, the one he says is "most consonant with ordinary mathematical practice". Like you, I've thought about introducing unique quotients, but there are reasons it's not normally done with 0 - the arithmetic becomes very limited. To avoid at least some of these limitations, I've played around with going a step further. I've replaced 0 with a redefined zero from which unique quotients arise.
A couple of things that I've done so far to define a different number of nothing:
- Used a different notion of nothing.
Placeholder zeros provide a guide. They indicate the absence of some particular things, as, for example, 0 does the other nine digits. So a replacement zero could represent the absence of the real numbers or some other set of numbers. The importance of the idea of "absence" allows for "presence" and I'll try to show the importance of this for unique quotients.
- Used a different notation. The notation, while awkward, makes it simple to see how to divide according to the usual rules for division.
To accomplish 2, the new zero will be in two parts. One part will indicate "absence" and the other part will indicate, in this case, the reals. Division, and only division, will change the absent into the present. Otherwise, the new, "absent", zero functions just like 0. So
- Part A. introduce a bar to indicate absence
- Part B. use the reciprocal of the Roger Penrose / John A. Wheeler / John Wallis definition[d] for
$\infty$
So the notation $/1/\infty $ or $ \overline{\frac{1}{\infty}} $ would read "the absence of the reciprocal of the reals".
An example of obtaining a(n) unique quotient.$$ \frac{4}{\overline{\frac{1}{\infty}}} = \infty_4 $$
Division inactivates the absence bar, thus making the absent present. Now we can apply the usual rules for complex fractions. The "revealed" reals then extend orthogonally from 4. Geometrically, this would be a line that extends out from the point at 4 on the real number line.
This zero does not have a reciprocal because $$ \overline{\frac{1}{\infty}} \times \infty_4 $$ still equals zero. Yet, by definition, $\infty_4 $ has a reciprocal - the "free" or "revealed" part of the new zero. $$ \infty_4 \times \frac{1}{\infty} = 4 $$ With repeated division a plane can be constructed, as well as other objects. Basic arithmetic operations seem to make it possible to construct $n$-real-dimension space. Speaking visually, with a redefined number zero it becomes possible to carry out arithmetic operations, not only with points on the real number line, but also with lines, planes, and other higher dimensional constructs.
A less undeveloped version of the somewhat simplified arithmetic introduced here that I've been playing around with may be found in my paper Replacing 0.
Footnotes.
[a] Sections 8.7 and 8.5 begin on pages 181 and 184 of the linked PDF, not the pages listed in the table of contents.
[b] ArXiv has papers by a number of mathematicians using "meadows" in the title. An overview of their research program and links to papers is here http://staff.science.uva.nl/~janb/FAM/topFAM.html
[c] Links to referenced authors can be found in the bibliography to the paper "Replacing 0" linked at the end of my answer above and on my user profile.
[d] Penrose defines $\infty$ as an array of the real numbers. John Wallis, the inventor of the infinity symbol, did this implicitly.