Does the definition of division by zero in Wheel theory actually make sense? I came across this question : Will Division by Zero be Defined Eventually? 
and was very surprised that there is a theory, called Wheel theory, which tries to make the division by zero meaningful.

Must this theory be taken serious ?

Googling Wheel theory was not very enlightening. If I understand right , $\frac{1}{0}$ is considered to be an additional element, like the point of infinity. But I cannot see how the theory deals with the usual issues occuring. In particular, multiplying the additional element with $0$ could be any number because $\frac{1}{0}$ and $\frac{2}{0}$ , for example lead to the same element, right ?
Can anyone shed some light on this strange theory ?
 A: There are two separate questions here: whether wheels are meaningful and whether they are significant/useful. The answer to the first question is clearly yes: the definition, regardless of whether it matches intuitions, makes perfect sense and gives a well-defined class of algebraic structures. So a certain amount of the negative tone of the question, in my opinion, is not justified.
But that's not the big question: more importantly, should we care - or better, what reasons are there for one to care? At a glance I can't find any real applications of the notion. That said, there could be subtle ones: for example, while the generalization from unital to non-unital rings may appear uninteresting, it in fact has some important consequences (in many ways the collection of non-unital rings is better behaved than the collection of unital rings). So I would tentatively say: based on my current knowledge, the best reason for you to care about wheel theory is the extent to which wheels are interesting in their own right to you. For me personally (at least, at the moment) this means that I"m not interested in wheel theory - although of course that could change, either by virtue of learning about some cool applications or by virtue of having a "oh wait that is cool!" moment.
