At the end of last century, the concept of Wheel Theory was introduced to Algebra, which does something similar to what you propose. It extends the real numbers (or other number fields like the rationals or the complex numbers) with two new elements. One represents 1/0, basically infinity. The other new element represents 0/0. To be able to calculate with these new elements, the definitions of addition, subtraction, multiplication and division need to be amended somewhat.

The Wikipedia page on Wheels links to a very technical paper on the subject.

The idea of adding infinity to the set of reals has been around since the 19th century and has lead to many interesting developments in geometry, notably Projective Geometry. As far as I know, Wheel theory is still looking for applications. ]]>

Would be nice if WordPress has decent support for Latex.

]]>