The multiplicative inverse of zero

This is silly post, but was inspired while reading something which mentioned that zero has no multiplicative inverse. Since grade school I’ve always been puzzled by how math has weird corner cases like this, another: 1 divided by 3, never ends! Anyway, what if zero did have an inverse? Is there some Abstract Algebra thingy that has such a property?

So, lets call the inverse “j”, not ∞, it already has some meanings. Then we have: j0=1. Or, 0=1/j. This means that 0/0 = 1. I.e., (1/j)/(1/j) = 1*j/j*1 = 1.

Is there a physical meaning? What could 1/0 mean? Since multiplication can be interpreted as a scaling, then what happens when you scale something to zero? Well, using the analogy of a map or engineering drawing, a scaling doesn’t destroy a physical measurement, it allows a smaller or large value represent the other value. In a floor plan we may see something like 1/4 inch = 1 foot. In a full size scale, the multiplier is 1. As the multiplier becomes smaller we can represent larger measures. At zero scale, we represent infinity. Thus, scaling infinity to zero, still scales the whole thing, which is 1.

Yeah, doesn’t really work, unless your a mystic. Oh well, like I said silly. I won’t quit my day job.